Three Essays on Communication
in Signalling Games
vorgelegt von
MSc Lilo Wagner
von der Fakultät VII Wirtschaft und Management
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktorin der Wirtschaftswissenschaften
Dr. rer. oec
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. Georg Meran
Gutachter: Prof. Dr. Pio Baake
Gutachterin: Prof. Dr. Dorothea Kübler
Tag der wissenschaftlichen Aussprache: 15. Dezember 2014
Berlin 2015
Summary
This dissertation is concerned with various economic applications of signalling games. In
this type of games, a sender transmits information to a receiver, who interprets it.
Educational institutions senders evaluate students based on a xed grading scale.
Full disclosure of information is the best response by schools to a student body that
consists of expected utility maximizers. The impact of a deviation from this behaviour
is investigated: the human mind typically focuses on salient states the very best or
worst grades here. It is demonstrated how this behaviour brings educational institutions
to employ coarse, rather than ne, grading schemes.
While educational institutions presumably disclose information truthfully, this is not
necessarily the case for privately organized certiers. If information manipulation is not
possible, a certier should be interested in fully revealing information; at least if producer
investment correlates with product quality. It is shown that if this so called threat of
capture exists, information disclosure may be coarse. A reduced transparency may also
increase social welfare because it prevents the market from breaking down.
In the market for life insurances, rms which are asymmetrically informed about the
risk of applicants, might benet hereof. Information is implicitly transmitted to appli-
cants and competitors via contract oers. It is shown how the non-ability of an informed
rm to persuasively transmit information serves all insurers. Industry prots increase at
the expense of consumers. This structure is created by the exchange of information on
the bargaining behaviour of applicants.
Keywords: Signalling games, Information disclosure, Grading, Education, Behavioural
theory, Certication, Bribery, Insurance markets, Asymmetric information
i
Zusammenfassung
Diese Dissertation befasst sich mit verschiedenen ökonomischen Anwendungen von Si-
gnalspielen. In dieser Art von Spielen übermittelt ein Sender einem Empfänger Informa-
tion, die dieser interpretiert.
Bildungsinstitutionen Sender bewerten Studenten nach einem festgelegten No-
tensystem. Die vollständige Oenlegung von Information ist die beste Strategie von In-
stitutionen, wenn Studenten ihren erwarteten Nutzen maximieren. Untersucht wird eine
Abweichung hiervon: menschliche Wahrnehmung konzentriert sich oftmals auf besonders
auällige Zustände hier sehr gute oder sehr schlechte Noten. Es wird gezeigt, wie dieses
Verhalten Bildungsinstitutionen dazu bringt, Notensysteme gröber zu gestalten.
Während davon ausgegangen werden kann, dass Bildungsinstitutionen Informationen,
wenn schon nicht vollständig, so doch wahrheitsgemäÿ oenlegen, ist dies bei privatwirt-
schaftlich organisierten Zertizierern nicht notwendigerweise der Fall. Ist die Manipula-
tion von Information nicht möglich, sollte ein Zertizierer Interesse an der vollständigen
Oenlegung von Informationen haben, wenn Produzenten Einuss auf die Qualität ihrer
Güter nehmen. Hier wird gezeigt, dass, besteht die Möglichkeit zur Falschzertizierung,
dies nicht mehr der Fall ist. Eine reduzierte Transparenz ist nicht nur im Sinne des
Zeritizierers wünschenswert, sondern sichert auch die Funktionsfähigkeit von Märkten.
In dem Markt für Lebensversicherungen können Firmen, welche asymmetrisch über
das Risiko von Bewerbern informiert sind, von diesem Zustand protieren. Information
wird implizit über Vertragsangebote an Bewerber und Wettbewerber übermittelt. Es
wird gezeigt, wie durch die Tatsache, dass eine informierte Firma nicht in der Lage ist,
Information überzeugend zu übermitteln, Industriegewinne zulasten der Konsumenten
gesteigert werden können. Diese Marktstruktur wird geschaen durch den Austausch von
Informationen über das Verhandlungsverhalten der Bewerber.
Schlüsselwörter: Signalspiele, Informationsoenlegung, Notengebung, Bildung, Verhal-
tensökonomie, Zeritzierung, Bestechung, Versicherungsmärkte, Asymmetrische Infor-
mation
iii
Acknowledgements
I am indebted to many people for accompanying me on the path of completing this
dissertation. First and foremost, I would like thank my supervisor Pio Baake. He
continuously guided and supported my research and shared his profound knowledge on
the topic of microeconomic theory with me.
I would also like to express my sincere gratitude to Dorothea Kübler for very kindly
agreeing to be my supervisor and lending her expertise from the eld of microeconomics.
All chapters of this thesis have immensely beneted from the comments of many
people many thanks especially to Georg Weizsäcker, Ives Breitmoser, Paul Viefers,
Daniel Huppmann, Andreas Harasser, Laura Wagner and Alfonso Caiazzo for this. I
also owe a huge debt of gratitude to Tobias Schmidt for sharing many insights and
his enthusiasm on new developments in decision theory, to Vanessa von Schlippenbach
and Balkan Akbaba for introducing me to the DIW, to my coauthors for productive
and delightful cooperation and to Alex Gilbert, Kathy Sartori and Adam Lederer for
proof-reading.
I wrote this dissertation while being a doctoral student at DIW Berlin. This pro-
gram's research environment helped tremendously in completing this thesis. Financial
support from DIW Berlin is gratefully acknowledged.
Last but certainly not least, I would like to express many thanks to my family and
my dear friends without whose support writing this thesis would have been a much more
arduous path.
v
Contents
1 Introduction 1
1.1 Sender-receiver games . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Methodological approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Objectives and contributions . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Pass/Fail, A-F or 0-100? Optimal grading of eager students 11
2.1 Introduction.................................. 12
2.2 Themodel................................... 16
2.3 Optimal grading with expected utility . . . . . . . . . . . . . . . . . . . . 17
2.4 Salience of extreme grades . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Optimal grading with salience . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6 Discussion................................... 27
2.6.1 Dierentiated students . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6.2 Concluding comments . . . . . . . . . . . . . . . . . . . . . . . . 28
2.7 Appendix ................................... 29
2.7.1 Proofs ................................. 29
2.7.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Informational opacity and honest certication 35
3.1 Introduction.................................. 36
3.2 Themodel................................... 38
3.3 Optimal honest certication . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Thecaptureproblem............................. 43
3.4.1 Capture under full disclosure . . . . . . . . . . . . . . . . . . . . . 44
3.4.2 Capture under partial disclosure . . . . . . . . . . . . . . . . . . . 47
3.4.3 Sub-optimality of full disclosure . . . . . . . . . . . . . . . . . . . 52
3.4.4 Welfare properties of partial disclosure . . . . . . . . . . . . . . . 54
3.5 Discussion................................... 55
3.6 Appendix ................................... 57
vii
CONTENTS viii
3.6.1 Proofs ................................. 57
3.6.2 Extensions............................... 62
4 Non-persuasiveness mitigates competition in the market for life insur-
ances 65
4.1 Introduction.................................. 66
4.2 Themodel................................... 70
4.3 Preliminary analysis: persuasiveness . . . . . . . . . . . . . . . . . . . . . 73
4.4 List price competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.5 Discussion................................... 82
4.6 Appendix ................................... 84
4.6.1 Proofs ................................. 84
4.6.2 Information sharing systems . . . . . . . . . . . . . . . . . . . . . 90
Bibliography ix
List of Figures xiv
1 Introduction
1.1 Sender-receiver games
Sender-receiver relationships are prevalent in everyday life. By transmitting signals, a
sender often reveals valuable information to a receiver. For instance, good grades in
college may give information about a high educational quality, but also about the good
productivity of a potential employee (as suggested in the canonical example due to Spence
1973).
Signals are subject to interpretation by the receiver. If a product is labelled bio, its
quality may meet a dened minimum standard, but is it possibly even better? Also, what
can be inferred from the absence of this label?
These relationships show dierent properties depending on the natural settings in
which information is transmitted.
One important way to classify sender-receiver games also: signalling games is
sender manipulation. For instance, a job applicant will typically not be suspected to
have forged degrees since doing so may entail high nes. By contrast, an insurance
broker is not made responsible for having sold inappropriate products to his clients.
In case signals are subject to manipulation, a further distinction is whether signals
are costly, or more specically, whether dierent signals incur dierent costs. In the job
market signalling example due to Spence (1973), persons incur costs for high degrees,
their utilities are hence directly connected to the signal choice. An insurance broker on
the other hand receives commission for selling products, but talking is costless (he is
`talking cheap'). His payo therefore solely depends on the receiver's interpretation.
This dissertation treats dierent topics and settings of sender-receiver relationships.
Its chapters integrate into signalling games as follows.
1
1
A more detailed description of the objectives and results of the single chapters is given further below.
Also, note that structures are richer than in the basic sender-receiver game. In Chapter 3, this inverts
the motivation: while cheap talk games are concerned with how persuasive communication can possibly
be, the work is motivated by the question why information is not exactly revealed.
1
Introduction 2
Signals are not subject to manipulation
: Chapter 2.
This chapter explores how educational institutions (the senders) should optimally choose
their grading scales through which they reveal information about the educational level
or quality of their students to potential employers (the receivers). Although universities
may have an incentive to make their students appear as brilliant as possible, they re-
peatedly interact on the market. Attempts to overstate the real market value of students
quickly become common knowledge.
Signals are subject to manipulation and signalling is non-costly
: Chapter 3.
It is analysed in which way privately organized certiers (the senders) choose to reveal
information at hand. Most of the economic literature presumes honest information dis-
closure. Albeit rather rare, intentional mis-certication is discovered from time to time
the ADAC and ZDF-scandals are two recent examples. The chapter explicitly focuses on
the possibility of information manipulation and in particular on how it aects informa-
tion revelation.
Signals are subject to manipulation and signalling is costly
: Chapter 4.
Here, an insurer signals the personal default risk of an applicant by oering him a con-
tract. Because these oers are binding, information transmission is costly. The chapter's
focus is on demonstrating how senders may benet from not being persuasive (to be
dened).
1.2 Methodological approach
A sender
S
sends a message
m∈M
to a receiver
R
. The message is chosen from a
message space
M
.
R
observes
m
and responds by choosing an action
a
from some action
space, which aects both players' payos.
S
possesses some private information
θ∈Θ
(his type), where
Θ
is the type space.
Its cardinality is not larger than that of
M
, meaning that messages are allowed to be
dierent for each type.
R
's type space consists of a single element, that is,
R
is not
privately informed. He holds, however, beliefs about
θ
. Prior to information being
sent, these beliefs correspond to the true probability distribution over
Θ
with probability
function
p(θ)
.
Perfect Bayesian Equilibrium.
After
m
has been transmitted and observed,
R
updates his
beliefs according to Bayes' Rule when possible, meaning that beliefs have to be consistent
in equilibrium. More specically, in equilibrium,
R
knows
S
's strategy
σ∗
, which is a
Introduction 3
specication of a message
m
for each type in
Θ
. Therefore,
R
is able to infer information
from the observed message. Upon observation of an on-path message
m
,
R
's believed
probability that
S
's type is
θ
,
˜p(θ|m)
, is given by
˜p(θ|m) = p(θ)σ∗(m|θ)
Σˆ
θ∈Θp(ˆ
θ)σ∗(m|ˆ
θ)).
Equilibria in which
R
updates his beliefs accordingly are called
Perfect Bayesian Equi-
libria
.
Signal manipulation.
If signalling is not subject to manipulation by
S
, he commits to
some previously announced signalling strategy
σ
. A game without manipulation has the
following basic structure.
(1)
S
announces and commits to some strategy
σ
,
(2)
θ
realizes or is observed by
S
,
(3)
S
reveals according to
σ
,
(4)
R
takes an action
a
.
If
S
does not commit to some strategy prior to learning
θ
, stage (1) is omitted. In Chapter
3, however, the game formulation still incorporates this stage. Instead,
S
announces
σ
,
from which he can deviate in stage (3). Basically, the formulation does not aect the set
of equilibria. Pre-announcements however can be understood as a reference point, for
which deviation incentives have to be checked. That is,
S
`chooses' an equilibrium. By
contrast, if (1) is omitted, it is rather that equilibria arise.
Costly signalling.
In both settings costly or non-costly signalling equilibria usually
contain unused messages in
M
. The concept of Perfect Bayesian Equilibrium makes a
statement only about
R
's signal interpretation on the path. It says nothing about the
interpretation of o path messages, however. Multiple equilibria arise as a result.
If messages do not aect
S
's payo, their value is generated only through their use
in equilibrium. In that case, every o-path message can just be interpreted like some
on-path message, and equilibria persist (Sobel 2009). Stated dierently, o-path beliefs
are of concern only when signalling is costly, as in Chapter 4.
Several restrictions to o-path beliefs have been proposed, the most famous being the
Intuitive Criterion
by Cho and Kreps (1987). It works as follows.
Consider some candidate equilibrium strategy
σ∗(m|θ)
, the corresponding sender equi-
librium payo
U∗
S(θ)
and some message
m0∈M
which is not sent for any
θ
according
Introduction 4
to
σ∗
. When observing this unexpected message
m0
,
R
best responds to
m0
by choosing
action
a∗
. This choice depends on
R
's o-path belief about the probability that the type
is
θ
,
˜pd(θ|m0)
and possibly also on
m0
.
The
Intuitive Criterion
then states that if for some
θ
, the sender's deviation prot
US(θ, m0, a∗(˜pd(θ|m0), m0))
is smaller than
U∗
S(θ)
for all possible receiver beliefs, then
R
must put zero probability on the event
θ
. A deviation to
m0
is said not to be
admissible
on
θ
. All candidate equilibria which required the o-path belief
˜pd(θ|m0)
to be strictly
greater than zero are eliminated.
In words, upon seeing an unexpected message that is not admissible on
θ
, it is as if
the receiver is implicitly making the following speech.
`If
S
's type is
θ
, he would not have sent me that message (
m0
). Because if he had
done so, he would have been worse o as compared to sending the equilibrium message
no matter what my belief about his type is.`
1.3 Objectives and contributions
Chapter 2: optimal grading in schools.
In this chapter, I seek to provide an explanation for the prevalence of coarse grading
schemes in schools. While students' examination results are recorded on a very precise
scale, like 1,2,..,99,100, nal reporting is much less informative. American colleges for
instance often adopt letter grading systems or report on a pass/fail basis. Also, at
universities with letter schemes, not all grades are used especially top schools like
Harvard are known to generously award
A
's, whereas grades below
B
are virtually non-
existent. This phenomenon known as grade ination makes grades even less informative.
This coarse grading is puzzling, as it generates informational asymmetries between
the school/the students and employers. This asymmetry however should not be in the
interest of the sender (the school) it reduces the welfare of all market participants.
To see why, consider a game with the commitment structure as outlined above: the
school which is assumed to act in the best interest of its students announces a system (a
signalling strategy); students employ eort (and thereby inuence the realization of the
type, or educational quality); grades are awarded according to the previously announced
strategy and students go on the job market, where employers (the receivers) pay higher
salaries for higher educational quality.
2
If dierent qualities are pooled together and if studying creates costs of some sort to
2
Note also that this is not a signalling game in the sense of Spence (1973): employers are not awarded
talent or costs for studying, but the outcome.
Introduction 5
students, their best response to coarse schemes will be to study less. In the extreme case
where no information is revealed at all, that is, the same message is sent for all types,
students will not study at all. Therefore, the probability for high educational quality is
low.
3
Employers update according to Bayes' rule, and pay only low salaries to graduates
from the respective school. If by contrast information is fully revealed, students will be
awarded for their studying eorts.
It is therefore surprising that most top tier schools employ particularly coarse schemes
whereas institutions of lower reputation tend to reveal more information. The explanation
I oer is well captured by the following conjecture.
Ivy League educational institutions attract a disproportionate share of grade obsessed
overachievers. [...] Their compulsion to succeed as others dene it and their sheepish
failure to prioritize higher-order benets with their time at college perhaps makes a grading
system that is based on obvious grade ination the best option available.
4
It is long known in social science that people tend to draw special attention to the
most salient events. In terms of choice under uncertainty, lottery payos that stand out
are overweighted at the expense of rather average outcomes. As shown by Bordalo et al.
(2012), this phenomenon accounts for several puzzles of decision theory, like the Allais
(1953) paradox. Allais has demonstrated how actual observed behaviour contradicts the
independence axiom of expected utility. For illustration, consider the famous version of
Kahnemann and Tversky (1979). Individuals are asked to make a choice between two
lotteries,
L1(z)
and
L2(z)
:
L1(z) =
$2500
with prob.
0.33
$0
xxxxxxxxxxIn
0.01
$z
xxxxxxxxxxIn
0.66
L2(z) =
$2400
with prob.
0.34
$z
xxxxxxxxxxIn
0.66
By the independence axiom, expected utility theory predicts that an individual's prefer-
ence ranking should be stable in variations of the common consequence
z
. However, when
z
is
2400
, most people prefer lottery
L2(2400)
whereas if
z= 0
, people more frequently
opt for
L1(0)
.
As opposed to Kahnemann and Tversky's prospect theory, Bordalo, Gennaioli, and
3
Ineciencies from informational asymmetries may arise in the absence of moral hazard (i.e. in
adverse selection settings) if higher qualities imply higher reservation utilities. This is Akerlof's famous
lemon problem (Akerlof 1970). In this educational setting, this would be the case if good students
preferred to not work at all rather than work at a salary which is considered too low.
4
In Defense of Grade Ination at Harvard,
The Atlantic
, December 6, 2013.
Introduction 6
Shleifer (2012)'s salience theory suggests that distortions are based on the value of
z
rather than on the value of the underlying probabilities.
5
If
z= 2400
, the payo
$0
is
eye-catching or salient in
L1(2400)
, and therefore overweighted. On the other hand,
when
z= 0
, the payo of
$2500
in
L1(0)
is more salient than the payo of
$2400
in
L2(0)
,
which lets
L1(0)
appear relatively more attractive.
In particular, Bordalo, Gennaioli, and Shleifer (2012) propose that the decision maker
considers dierent payo states in the state space
Λ
. For instance, consider
z= 2400
.
Then, the state space
Λ
is given by
{(2500,2400),(0,2400),(2400,2400)}
. The salience
of each of these states is then described by a salience function which satises what is
denoted as the ordering property: consider two states
s
and
s0
. If the lowest payo in
s
is lower than the lowest payo in
s0
, and the highest payo in
s
is higher than the
highest payo in
s0
, then
s
is more salient than
s0
. In the example above, this means
that
(2500,2400)
and
(0,2400)
are both more salient than state
(2400,2400)
, whereas the
property says nothing about the salience relation of states
(2500,2400)
and
(0,2400)
.
6
The individual then weights the states following a weighting function. The degree to
which salience distorts perceptions is measured by some variable
δ∈(0,1]
. If it is
1
, the
decision maker is an expected utility maximizer, if it is
0
, he considers only the most
salient state(s).
Applied to the grading setting, a student's eort choice can be interpreted as a lottery
choice. His attention is drawn to the most salient grades or, more precisely, to the most
salient payos associated with grades. This aects incentives to study. I assume that
educational institutions seek to maximize its students' expected utilities.
Clearly, if students are expected utility maximizers themselves, a fully revealing rule
is optimal. It is shown that extreme distortions bring students to employ too much
eort from the school's perspective. Roughly speaking, this is the case because the true
probability for the state
(worst grade when choosing no eort, best grade when choosing
eort)
is higher than that of the state
(best grade when choosing no eort, worst grade
when choosing eort)
. Therefore, when multiplied by a constant larger than
1
, the eect
is more important for the rst state.
Subsequent to this result, it is then shown that every coarseness category (that is,
the number of factually assigned grades in a system) is optimal for some bias measure
δ
. Moreover, optimal rules exist for all
δ
for which students employ too much eort.
5
Prospect theory suggests that very small probabilities are overweighted. Therefore, if
z= 2400
, the
probability
0.01
of getting
$0
is overweighted while when
z= 0
, the total probability of getting
$0
is
0.67
.
6
Bordalo, Gennaioli, and Shleifer (2012) also suggest that the function should satisfy diminishing
sensitivity, but I do not need this for my results.
Introduction 7
Numerical analysis suggests that this is often the case, and that lower
δ
require coarser
rules.
The bias measure
δ
can be interpreted as students' inherent competitiveness, or school
selectivity. Put dierently, grade obsession in the above citation is translated into a focus
on both the very good and very bad results. In that sense, the results account for both the
occurrence of coarse schemes and the dierent degrees to which information is suppressed.
Chapter 3: information disclosure for product certication.
This article, which is co-authored by Martin Pollrich, treats a related topic. Here, we seek
to give an explanation to why privately organized certiers do not fully reveal information.
Like in educational institutions, information is usually recorded very precisely, whereas
revelation is not. For instance, biofood-labelling is often based on a pass/fail decision,
where only `passes' receive certication. The German organization Peta dierentiates
two passing animal-rights categories (one or two stars), the cosmetic label Natrue oers
three passing certicates.
The basic market structure diers from the grading setting in Chapter 2: privately
organized certiers are prot-maximizers, their incomes are generated through fees which
typically have to be paid upon application for certication. Producers of goods typically
have an idea of its quality, which also determines their decision for application. In that
sense, producers are also senders here. If some qualities remain uncertied, consumers
(the receivers) also hold beliefs about these types and they have to be consistent in
equilibrium (see above).
The following is the basic game structure: in a rst step, the certier (the sender)
announces a disclosure rule (a signalling strategy) and a fee which has to be paid by
producers who apply for certication. Sellers make an investment which relates to the
realization of the quality (the type). Quality is observed by sellers who then decide
whether to apply or not. If so, quality is revealed according to the rule. On the market,
consumers are willing to pay more for higher qualities.
Despite the dierent setting, as in the previous chapter, full disclosure can be shown
to maximize certier utility her prots and should therefore be observed more often.
7
As an explanation for why this is not the case, we propose the following: certiers may
be tempted to accept bribes for releasing favourable certicates. This behaviour, which
we call
capture
, enables the certier to extract payments other than the certication fee.
Since consumers are aware of this threat of capture, the certier must nd a way to
credibly commit to honesty. The best way to do so is to employ a coarse disclosure rule.
7
Lizzeri (1999) has shown in a seminal paper that coarseness is, in a similar setting, optimal when
quality is exogenously given. This should however not often be the case.
Introduction 8
The basic intuition for this is that coarseness reduces a seller's willingness to pay for
bribery, since it lowers dierences in the market values of certicates.
Methodically, as mentioned above, o-path behaviour is not much of a concern here.
It does not make much sense to consider messages outside the previously announced rule
a certier whose rule it is to rate on a one to ve star scale and who then certies a
product `12.7' is assumed to have rated a worthless good. Instead, deviations are possible
on the path, i.e. a two-star product may receive ve stars instead.
Certier incentives are shaped by consumer beliefs about his actual behaviour. We
analyse whether equilibria exist in which consumers are condent, meaning that they
faithfully trust certicates. In order for non-babbling equilibria to exist at all then, the
certier must be punished for deviations.
8
This is done by a reputation mechanism in
which consumers adopt a trigger strategy: they trust certicates as long as no deviation
is detected (after purchase), and never buy again otherwise.
9
This is the hardest pos-
sible punishment. If such an equilibrium does not exist, then neither does it for other
punishment strategies.
Our conclusion that coarseness reduces the threat of capture then implies that equi-
libria can be sustained for lower discount factors than if information is fully revealed.
Therefore, informational asymmetries are socially desirable in our setting they might
help to prevent market breakdowns, and should therefore not be objected per se.
Chapter 4: non-persuasive insurers.
This paper is a joint work with Julian Baumann. Like the previous chapter and unlike
Chapter 2, it treats a rational choice model in which the sender cannot commit to a
signalling strategy.
Providers of life or disability insurances oer contracts on the basis of the outcome of
a screening. More specically, a person who wishes to apply for insurance is required to
provide detailed information on his medical record. Because rms dispose of advanced
technological means to estimate the statistical risk of a loss, they may gain an informa-
tional advantage versus the applicant. In this chapter, we consider this possibility. By
oering an individual contract to some person, the insurer (the sender) may reveal infor-
mation about the person's default risk. Clearly, an insurer would always like to signal
8
Babbling equilibria are equilibria in which no information is transferred. In non-costly signalling
games, these equilibria always exist. The qualitative dierence to classic cheap talk models à la Crawford
and Sobel (1982) in which non-babbling equilibria may exist without reputation mechanisms is sender
utility: here, the certier would always like to signal high quality (against the bribery payment).
9
More precisely, we require that deviations be detected with certainty. This restricts the set of feasible
rules because if a rule is announced for which deviations are not unambiguously detected, equilibria cease
to exist. In that sense, the paper does not make general statements on desirability of coarse rules, but
provides a possible explanation.
Introduction 9
high risk because an applicant's willingness to pay increases in his default probability.
In this market, insurance premiums are paid constantly but only until default occurs.
This implies that the insuree has no possibility to punish sender misbehaviour. This
forces, unlike in the previous chapter, receivers to be suspicious against signals.
More precisely, all equilibria require the following receiver belief.
`Whenever accepted,
I believe that my risk is low. Only when rejected, I am willing to accept that my risk is
high.'
This has been shown by Villeneuve (2005). The reason is that if a contract oer
existed that could make the applicant believe that he is a high risk, an insurer would
always oer him this contract even if he is a low risk.
10
The market for life or disability insurances has another special feature: applicants
are not only required to provide information on their medical data, but also on their
application history. In particular, they are asked whether they have applied for insurance
before and whether they have been rejected or accepted.
From this observation, we conclude that at least some applicants should be bargainers,
on the search for the best oer they can possibly get. Price takers on the other hand are
assumed to consider only one oer.
We analyse a game in which insurers compete in two stages. In particular, the game
has three steps: rst, two insurers compete in list prices. Second, the rm that makes
the best oer wins all applicants, i.e. price takers and bargainers. This rm can either
accept the applicant and oer the previously announced list price or reject him. Only
bargainers continue their search, and rms compete for them in a third step.
We allow rms to decide whether to collect, upon receiving an application, information
on the default risk of each single applicant. This decision is not observable by other players
(applicants and competitors). Every equilibrium proves that rms are asymmetrically
informed on each single applicant (the winner of the list price competition is informed).
This game formulation generates a special eect, which we call
persuasiveness
, or
non-persuasiveness
respectively. What is understood by this is the following. Consider
some candidate equilibrium where applicants with low default probability are accepted
in the rst place. The informed rm may consider to deviate from this equilibrium by
rejecting an applicant of this type. By doing so, she trades o loosing low risk price
takers against being able to ask high risk premiums from the low risk bargaining type
who consequently believes to be a high risk.
If the informed insurer is tempted to deviate in this way from an equilibrium where
10
As opposed to Spence (1973), signalling is not costly, and therefore not possible. However, pooling
equilibria may not exist if the low risk applicant is not willing to pay at least the fair price for the high
risk type.
Introduction 10
all bargainers are oered their fair prices, she is said to be non-persuasive. Otherwise,
she is said to be persuasive.
Whether an informed rm is persuasive determines market outcomes: if the rm is
persuasive, all bargainers are accepted in the competition stage and receive fair oers. If
not, by contrast, high risk bargainers accept the oer from the uninformed rm even if
it exceeds the fair price.
The following belief of an applicant who has been rejected previously forms an equi-
librium when the rm is not persuasive.
`Whenever in the second round of competition, the best oer comes from the unin-
formed insurer, believe that the risk is high. If the best oer comes from the informed
rm, believe that the risk is low.`
An equilibrium with this belief structure survives the Intuitive Criterion: the informed
rm would like to deviate, upon having observed a high risk type, in the second round of
competition, to a price which exceeds the fair price for the high risk type but undercuts
her competitor's oer. Suppose he does. If the applicant believes that his risk is high,
he accepts the deviation oer. But if the applicant is in reality a low risk, the insurers'
prots are large if the person can be convinced to be a high risk. On the other hand, the
regular equilibrium prots on low risks are lower by denition of non-persuasiveness.
Following the logic of the Intuitive Criterion, the deviation is then admissible on the low
risk type, and the equilibrium is not eliminated.
As a consequence, when the informed rm is not persuasive, the uninformed rm
benets. This impacts list price competition: knowing that not winning this competition
earns this rm positive prots mitigates competition in the rst stage. Put dierently,
all rms benet from the eect. In fact, it can be shown that rms manage to always
coordinate on this non-persuasiveness outcome. Perhaps surprisingly, industry prots
increase in the share of bargainers in the market.
Finally, if rms were prohibited from collecting information on the bargaining be-
haviour of applicants, they would be symmetrically informed, and eective competition
would be restored.
2 Pass/Fail, A-F or 0-100? Optimal grading
of eager students
Chapter Abstract
Most schools grade their students using a very coarse grading scheme, making grades
essentially less informative. Especially highly selective schools suppress more infor-
mation than schools with lower reputation. We formalize an explanation which is
intuitively described as the `excessive preoccupation with grades' in top schools.
Coarsening is the best response by schools to a student body that consists of in-
dividuals who overweight grades that stand out on their school's scale. Such over-
weighting of salient grades leads students to exert more eort than expected utility
maximizers. Schools which attempt to maximize an EU-welfare criterion counter-
balance this bias through the use of broad grading schemes.
11
Pass/Fail, A-F or 0-100? Optimal grading of eager students 12
2.1 Introduction
Students' examination results are typically recorded on a very accurate scale, like
1,2,
..., 99,100
. Yet, educational institutions pool dierent scores together in broad categories,
for instance by adopting a letter grading system. In fact, perfectly accurate nal reporting
can rarely be found. This prevalence of coarse grading systems is puzzling as it reduces
the informativeness of diplomas to employers. Even more surprisingly, selectivity seems
to be a good indicator for the degree to which information is suppressed. Yale Law
school for instance employs a broad system on the basis of fail/low pass/pass/honours.
Other top law schools like Harvard and Stanford have adapted similar systems while less
selective schools tend to stick to the ner traditional (and usually curved) letter grading
system.
A more subtle method to install a coarse system is to factually abolish lower marks
by just never or very rarely assigning them, a phenomenon recently discussed in the
news under the headline of grade ination. Stuart Rojstaczer, one of the initiators of
the debate states in a
Washington Post
article:
We recently handed in our grades for an
undergraduate course we teach at Duke University. They were a very limited assortment:
A, A-minus, B-plus, B and B-minus. There were no C's of any avor and certainly no
D's or F's. It was a good class, but even when classes aren't very good, we just drop down
slightly, to grades that range from A-minus to B-minus.
1
At Harvard, the most frequently given grade is now an
A
, the median an
A−
.
2
When
the
Business Insider
lists
13 schools where it's almost impossible to fail
3
, the list com-
prises almost exclusively elite private schools. More selective institutions assign higher
GPA's to their students than do their peers for the same education level (Healy and
Rojstaczer, 2012), presumably corresponding to a lower support in the scale.
4
The purpose of the present paper is to give an explanation for the prevalence of such
coarse grading and for the dierent degrees to which information is pooled.
The factual non-existence of lower grades at the elite schools resulting from grade
ination is commonly considered a problem, the main arguments against it being both
low informativeness and reduced eort incentives. This view provokes calls on school
administrations to take action. Their behaviour however mostly suggests that this phe-
1
Where All Grades Are Above Average",
Washington Post
, January 28, 2003
2
Substantiating Fears of Grade Ination, Dean Says Median Grade at Harvard College Is A-, Most
Common Grade Is A,
The Harvard Crimson
, December 3, 2013
3
13 schools where it's almost impossible to fail,
The Business Insider
, May 29, 2013
4
Albeit not perfectly correlated, the fact that the share of D's at private colleges has now dropped
to less than
3%
(from the same report) leads us to conclude that. Also, many reports indicate that this
phenomenon is prevalent most and forall at the elite schools.
Pass/Fail, A-F or 0-100? Optimal grading of eager students 13
nomenon is wilfully neglected, if not actively initiated.
5
One might be led to believe that
this coarsening is the equilibrium outcome of a game in which employers can be mislead
about educational quality (as for instance investigated in Chan et al. 2007), it is how-
ever widely accepted that grading standards are common knowledge among institutions,
students and prospective employers.
We call attention to a dierent explanation. Information suppression may be the best
response of schools to a student body who consists of individuals who place too much
weight on salient grades.
Conjectures pointing to the same direction are well captured in the following state-
ments:
Ivy League educational institutions attract a disproportionate share of grade-
obsessed overachievers. [...] Their compulsion to succeed as others dene it and their
sheepish failure to prioritize higher-order benets with their time at college perhaps makes
a grading system based on obvious ination the best option available.
6
Similarly, when Yale Law School changed its system from numerical grading to the
broader system in 1967, it was reported that
it is believed by those members of the faculty
who voted in favor the plan that it will oer some relief from what one educator described
as `the excessive preoccupation with number or letter grades' .
7
To formalize what is understood as `grade obsession' or `excessive preoccupation'
in the above citations, we posit that students' attentions are drawn to salient payos
according to Bordalo, Gennaioli, and Shleifer (2012) henceforth BGS. We further claim
that the degree to which this is done is determined by students' inherent competitiveness,
or school selectivity.
Thus, grade obsession is translated into a focus on both the very good and the very
bad outcomes at the expense of average results.
8
Since students can never be certain
about the success of their eorts, they are faced with the choice between independent
lotteries, with payos being determined on the job market. Due to the salience of extreme
outcomes, students replace the true outcome probabilities by salience-distorted weights.
More specically, the salience of a payo is context-dependent, meaning that outcomes
are compared to those of other lotteries which realize in the same state of the world.
We take it for granted that all grades of all lotteries are taken into account, however
small their occurrence probability may be. With states being evaluated according to a
salience function that exhibits ordering, meaning that for a consideration set with two
5
In fact, when the issue came up in 2002, the only institution that would revise its grading policies
was Princeton University.
6
In Defense of Grade Ination at Harvard,
The Atlantic
, December 6, 2013
7
Yale College inaugurates pass-fail marking system,
The Heights
, November 13, 1967
8
The hypothesis that focusing equates to focussing on large dierences is also explored by Bordalo
et al. (2013), K®szegi and Szeidl (2013) and Tversky (1969).
Pass/Fail, A-F or 0-100? Optimal grading of eager students 14
lotteries, if the payos associated with state
s
are nested within the payos associated
to some other state
s0
, then
s0
is more salient, the most salient states are
{
lowest payo
in lottery
1
, highest payo in lottery
2}
and
{
highest payo in lottery
1
, lowest payo in
lottery
2}
.
Consider the following setting: an institution which seeks to maximize the expected
utility of its students commits to and publicly announces a grading system. Based on
this, students of dierent cost types
9
make an unobserved decision between employing
costly high eort, and costless low eort. The associated probability mass functions are
standardly assumed to satisfy the monotone likelihood ratio property. Scores then realize,
grades are assigned, and students go on the job market where risk neutral employers are
willing to pay higher salaries for better expected scores.
Clearly, when students maximize expected utility, a grading system which fully reveals
scores is socially ecient, and even uniquely optimal.
A focus on the most salient states however brings students to exert ineciently high
eort, from the school's perspective: roughly speaking, there are two most salient states,
but a student who considers employing eort overweights the good grades relatively
more than the bad grades. Similarly, when considering to exert low eort, the student
overweights bad grades relatively more than good grades. This eect arises due to the
monotone likelihood ratio property.
The share of hard working students is a continuous function of the degree of salience,
implying that coarse rules which mimic full disclosure at the extreme have an intersection
with the expected utility maximizing cut-o and are therefore optimal for some distortion
level. It can further be shown that optimal coarse rules exist whenever full disclosure
induces too many students to choose the eort lottery, from the school's perspective.
Numerical analysis suggests that this is typically the case for all levels of distortion.
Further: (a) given a coarseness category (e.g. Pass/Fail or A,B,C), optimal rules exist
for a closed interval of the bias measure, (b) stricter biases require coarser rules, (c)
within one coarseness category, extreme grading (e.g. almost all pass when the grading
system is Pass/Fail or almost all receive an
A
when grading category has multiple cuts)
is optimal for high distortions. Therefore, our results account for both the occurrence of
coarse grading schemes and for the dierent degrees to which information is suppressed.
9
This may the opportunity costs for what is named higher order benet in the above citation: it is a
well established thesis among psychologists that while extrinsic motivation responds well to incentives,
it comes at the cost of a reduced intrinsic motivation (for an overview and critical assessment: Cameron
and Eisenberger (1996)). Similarly, institutions may be concerned about the health of their students: at
Harvard Law School, the system change reportedly served to reduce stress and anxiety levels.
Pass/Fail, A-F or 0-100? Optimal grading of eager students 15
Related literature.
Coarse grading may be advantageous in contest games. In particular,
Dubey and Geanakoplos (2010) show that if students care about their rank in class and
grading schemes are chosen such that students are best motivated, coarse schemes may
be the best choice and are always so when students are disparate. The basic idea is that
when disparate, students of lower ability should be given the chance to outperform their
high ability classmates. Similarly, these should not feel too comfortable. Their conclusion
is qualitatively dierent from ours as coarse schemes motivate students to work more,
not less.
10
Further, Ostrovsky and Schwarz (2010) analyse best disclosure policies when
employers rank students according to their exogenously given abilities. Partial disclosure
can be a market equilibrium if student bodies exhibit dierent exogenously given talent
levels. In particular, an average school, if faced with low quality competitors, may take
into account non-revelation of highly talented students in order to attain a good placement
for less able persons. A similar eect is present in Boleslavsky and Cotton (2014).
11
In a non-contest setting, Costrell (1994) analyses optimal Pass/Fail schemes under
dierent welfare conceptions. Rayo and Segal (2010) show that pooling can be optimal
if grading is based not solely on the abilities or educational achievement of students but
also on their protability to the sender (the paid tuition fees).
In industrial organization, a stream of literature starting with Lizzeri (1999) shows
why prot maximizing certiers may nd it optimal to employ broad disclosure policies.
In Lizzeri (1999), a certier who is bound to oer his service at a xed non-discriminatory
fee to sellers with dierent quality goods considers disclosure on a Pass/Fail basis optimal.
In comparison to an educational setting, participation is voluntary with the decision being
based on a previously observed quality level. Similarly, intermediaries who seek maximize
not their own prots but the information being provided to the public, partial disclosure
may outrule fully revealing rules if a selection process is at work and certication is costly
(Harbaugh and Rasmusen, 2013).
The article proceeds as follows. Section 3.2 describes the model. In Section 2.3,
we derive two general insights, which are independent of the focus on salient payos
but relate only to the game structure. The salience bias is characterized in Section 2.4,
and optimal rules are analysed in Section 2.5. The last section discusses extensions and
concludes. All proofs are relegated to the appendix.
10
Unfortunately, empirical studies exploring this relationship in educational settings do not exist.
Some works investigate the eect of raising a Pass/Fail standard (e.g. Figlio and Lucas (2004), Betts
and Grogger (2003)) and others nd that eort is chosen strategically, i.e. such that scores closely above
a threshold become probable (Oettinger, 2002), but none systematically explores the eect of coarsening
grades in relation to ner schemes.
11
Many more examples of how information suppression takes place is presented in both Ostrovsky and
Schwarz (2010) and Boleslavsky and Cotton (2014).
Pass/Fail, A-F or 0-100? Optimal grading of eager students 16
2.2 The model
Students are assumed to have a cost for studying of
θ∈[0,1]
.
θ
is uniformly distributed
across students. A student's choice set is
{L0,Le}
where the choice of lottery
Le
costs
θ
whereas that of
L0
is free. Without loss of generality, students are assumed to be risk
neutral. An expected utility maximizing student of type
θ
then chooses lottery
Le
if
and only if the expected value from doing so, minus the costs is at least as high as the
expected value of choosing
L0
.
The lottery choice determines the probability distribution of exam scores. In particu-
lar, an exam score is an element of the ordered set
Ω = {q1, q2, ..., qN}
with
N≥3
. The
probability that a score in
Ω
is
qn
is given by
pn
e
if the student chooses lottery
Le
and it is
pn
0
otherwise.
pe
and
p0
are the respective probability vectors. All elements are assumed
to be strictly positive.
The values of scores to students are determined by the salaries paid on the job market.
There, high scores are rewarded. In particular, it is assumed that
qn=n/N
for every
qn∈Ω
. Hence, when the market exhibits symmetric information, the expected valuations
for both lotteries are
E[qn|Le] = PN
n=1 qnpn
e
or
E[qn|L0] = PN
n=1 qnpn
0
respectively.
Teachers convert scores into grades using a grading rule that maps every possible
score
qn
into a grade. Denote a grading rule by
ΓH={γ1, γ2, ..., γH}
with
γH=qN
and
the associated grades by
C={C1, C2, ..., CH}
. For instance,
γ1
could be
30
out of
100
points and
C1
would be grade
D
. Further, let
Ψ
be the set of all possible grading rules.
In particular, a grading rule in
Ψ
is a partition of
Ω
into consecutive intervals. It can be
expressed as a subset of
Ω
:
ΓH⊆Ω
.
The elements of a rule
ΓH
are dened such that all scores
qn
that are equal to or worse
than
γ1
are granted grade
C1
, all scores that are strictly better than
γ1
but equal to or
worse than
γ2
are assigned grade
C2
and so forth. The best grade
CH
is awarded to scores
greater than
γH−1
. An example for a rule
Γ3
is depicted in gure 2.1: if
γ1=qm, γ2=qn
,
then the lowest grade
C1
is awarded to all scores lower or equal to
qm
,
C2
is awarded
to scores
qm+1
to
qn
and the best grade to all scores strictly greater than
qn
. From this
follows that
γH≡qN= 1
and
1≤H≤N
.
If
H= 1
, no information is disclosed, if
H=N
, information is fully revealed. We use
Γ
to describe a general rule and
ΓH
to describe some rule with
H−1
cuts. The same
holds for
Ψ
and
ΨH
respectively:
ΨH⊂Ψ
is the set of disclosure rules that are described
by some
ΓH
. For instance, all possible Pass/Fail schemes are contained in the set
Ψ2
,
where
γ1
is an element of
Ω/{qN}
.
Note that describing grading rules accordingly has two implications: rst, probabilis-
Pass/Fail, A-F or 0-100? Optimal grading of eager students 17
qm−2qm−1qm
γ1
qn
γ2
qn+1 qn+2
C1C2C3
qN
γ3
Figure 2.1: A grading rule
Γ3⊂Ψ3
.
tic rules are not accounted for (i.e. every score is deterministically assigned a grade)
12
and second, every grade occurs with strictly positive probability.
13
Employers are informed about the grading rule, on the basis of which they form
expectations about the value of a grade
Ch
, denoted by
vh
. Also, the distribution of
costs and the probability distributions are assumed to be common knowledge while the
students' lottery choices cannot be observed. The game then proceeds as follows:
(1) The grading rule is publicly announced,
(2) students make a choice between
Le
and
L0
,
(3) scores are disclosed according to the grading rule,
(4) students sell their education on the job market.
2.3 Optimal grading with expected utility
An expected utility maximizing student
θ
chooses lottery
Le
if and only if
VΓ
e−θ≥VΓ
0
,
where
VΓ
e
and
VΓ
0
are the expected valuations from choosing the respective lottery for a
given disclosure rule. Dene
γ0= 0
. Then,
VΓ
e
and
VΓ
0
are given by
VΓ
e=E[vh|Le] =
H
X
h=1
vh
γhN
X
i=γh−1N+1
pi
e,
VΓ
0=E[vh|L0] =
H
X
h=1
vh
γhN
X
i=γh−1N+1
pi
0.
(2.1)
Note that the second sum in both expressions of equation (2.1) is the probability for
grade
h
to occur given the lottery choice and the grading rule: if for instance
γh−1=qm
12
That is, for the moment. Simple probabilistic rules are used to derive an existence condition of
optimality for given bias measures, see below.
13
Similarly, this assumption is not crucial as shortly discussed in the last section.
Pass/Fail, A-F or 0-100? Optimal grading of eager students 18
and
γh=qn
, then the probability that a student choosing lottery
Le
is assigned grade
Ch
is given by
pm+1
e+pm+2
e+... +pn
e
.
In order to ensure inner solutions, it is standardly assumed that probability functions
satisfy the monotone likelihood ratio property. Dene
τn≡pn
e/pn
0
.
Assumption 2.1.
(
Monotone likelihood ratio property
)
τn> τm∀n > m
The condition guarantees that students who choose the costly lottery
Le
will attain higher
expected scores than the ones whose choice is
L0
. Therefore, it holds that
θ≤VΓ
e−VΓ
0
for at least some
θ
if information is revealed at all, that is, if
H≥2
.
The principal/the school seeks to maximize his conception of student welfare, which
is overall expected utility. Since it is never optimal that high types choose
Le
when lower
types do not, it is given by
WΓ(ˆ
θ) = Zˆ
θ
0
(VΓ
e−θ)dθ +Z1
ˆ
θ
VΓ
0dθ
(2.2)
for a given rule
Γ
. Transforming equation (2.2) and maximizing with respect to
ˆ
θ
gives
the rst best critical value of
θ
such that all students with lower costs should choose
Le
whereas higher types should choose
L0
. It is given by
ˆ
θr=VΓ
e−VΓ
0
. Therefore,
expected utility maximizing students share the principal's conception of welfare. The
market exhibits symmetric information if the education level is fully revealed to the
public. Then,
ˆ
θr
is the dierence in expected quality for a given lottery choice
Le
or
L0
.
ˆ
θF B =E[qn|Le]−E[qn|L0] =
N
X
n=1
qn(pn
e−pn
0).
Denote
WF D(ˆ
θ)
the
WΓ(ˆ
θ)
when information is fully revealed. In words,
WF D(ˆ
θ)
is
the expected utility with full disclosure as a function of the student behaviour
ˆ
θ
. The rst
best expected utility level is then given at
WF D(ˆ
θF B)
, that is, if students are expected
utility maximizers and scores are fully revealed.
When information is not fully revealed, employers form, for a given disclosure rule, a
belief
φ
about the students' lottery choices,
ˆ
θ
(i.e. about the critical type). The market
values are then a function of this belief. To see why, consider some rule such that grade
CH
is coarse: if many students are believed to have chosen
Le
, that is, if
φ
is large,
then the value of this grade is high as compared to a lower
φ
because the likelihood of a
student with an awarded grade
CH
having attained a top score rather than a still good
score is high whereas it is lower for lower
φ
. If information is fully revealed however, a
Pass/Fail, A-F or 0-100? Optimal grading of eager students 19
grade is simply worth the score. The value of a grade
Ch
is given by
vh(Γ|φ) = PγhN
i=1+γh−1Nqi(φpi
e+ (1 −φ)pi
0)
PγhN
i=1+γh−1N(φpj
e+ (1 −φ)pj
0).
(2.3)
This given, the true expected valuations from choosing lotteries
Le
and
L0
can be
derived by inserting equations (2.3) into equations (2.1),
VΓ
e(φ)
and
VΓ
0(φ)
. Students'
actions are a (not necessarily rational) reaction to the market values and therefore to
φ
,
hence
ˆ
θ(Γ|φ)
. The used equilibrium concept is the Perfect Bayesian Equilibrium. This
implies that on the end market, students' lottery choices can be foreseen and beliefs are
correct. Denote
˚
θ(Γ)
the
φ
that solves
φ=ˆ
θ(Γ|φ)
.
For a given disclosure rule
Γ
, the school maximizes
14
WΓ(˚
θ(Γ)) = Z˚
θ(Γ)
0
(VΓ
e(˚
θ(Γ)) −θ)dθ +Z1
˚
θ(Γ)
VΓ
0(˚
θ(Γ))dθ.
This is to say, even if students are not expected utility maximizers, what counts for
the school is still the expected valuation of the lotteries, albeit their values depend on the
equilibrium behaviour. If information is fully revealed,
VΓ
e
and
VΓ
0
equal the expected
scores for a given lottery choice, and thereby do not depend on student behaviour. Hence,
WF D(˚
θ(Γ))
is a function of the bounds of integration. The following then holds:
Lemma 2.1.
For a given disclosure rule
Γ⊂Ψ
, for any
˚
θ(Γ) ∈[0,1]
, it holds that
WΓ(˚
θ(Γ)) = WF D(˚
θ(Γ))
.
Lemma 2.1 states that expected utility is exclusively determined by the student be-
haviour, the lottery choice. Put dierently, with rational actors on the end market,
coarseness itself does not aect the expected market value, only the eect on the eort
choice does. This holds although
VΓ
e(˚
θ(Γ))
and
VΓ
0(˚
θ(Γ))
are typically dierent from
E[qn|Le]
and
E[qn|L0]
respectively. As a result, the rst-best outcome is restored when-
ever
˚
θ(Γ) = ˆ
θF B
, in which case we call student behaviour 'optimal'. Similarly, a rule that
implements
ˆ
θF B
is referred to as 'rst-best' or 'optimal'. It can easily be veried that
WF D(ˆ
θ)
is concave and symmetric around
ˆ
θF B
. Hence, welfare is improved as
|˚
θ(Γ)−ˆ
θF B|
becomes smaller.
It is this study's main purpose to demonstrate why schools may nd it optimal to em-
ploy a coarse grading scheme rather than fully revealing examination results. If students
14
We implicitly assume a minimum degree of rationality which is that there exists a critical type such
that all higher types choose
L0
and all lower types choose
L0
. When students focus on salient payos,
this is given as will be shown below.
Pass/Fail, A-F or 0-100? Optimal grading of eager students 20
share the school's conception of welfare, a rule is optimal if and only if
ˆ
θr(Γ|φ) = φ=ˆ
θF B
.
The following can be shown:
Proposition 2.1.
For all rules in
Ψ/ΨN
, it holds that for any belief
φ
,
ˆ
θr(Γ|φ)<ˆ
θF B
.
Therefore, when students are expected utility maximizers, no other than the full dis-
closure rule is optimal. More precisely, any rule that reveals less than full information
induces also too less eort from the school's point of view, a result which does not depend
on the belief
φ
. Here, the intuition that incentives to work are lowered if better scores
are not fully rewarded, applies due to the simple structure of the game.
2.4 Salience of extreme grades
Following BGS, students evaluate
H2
dierent payo states. Each state
sgh ∈S
occurs
with a commonly known probability, given by
πsgh =
γhN
X
i=γh−1N+1
pi
0
γgN
X
i=γg−1N+1
pi
e∀g, h ∈ {1,2, ..., H}.
Accordingly, it holds that
Psgh∈Sπsgh = 1
.
The salience of a state
sgh ∈S
is described by a continuous, non-negative and sym-
metric (states
sgh
and
shg
are equally salient) salience function
σsgh ≡σ(vg, vh)
. As
proposed by BGS,
σsgh
satises the ordering property.
15
Formally:
Assumption 2.2.
(Ordering)
If
min{vg0, vh0}<min{vg00 , vh00 }
and
max{vg0, vh0}>max{vg00 , vh00 }
, then
σsg0h0> σsg00 h00
.
The ordering property says that whenever one state's payos are nested within the payos
of some other state, the latter is more salient. Here, it implies that
s1H
and
sH1
are the
most salient states, whereas some state
sgg
,
g∈ {1,2, ..., H}
, exhibits the least salience.
To obtain the decision weights attached to some state
sgh
, dene the weighting func-
tion as
ωsgh =δ−σsgh
Psst∈Sδ−σsst πsst
.
(2.4)
15
BGS also propose that the salience function should exhibit diminishing sensitivity, that is, for any
two states
sg0h0, sg00 h00
, where
vg00 =vg0+ε
and
vh00 =vh0+ε
, for any
ε > 0
,
σsg0h0≥σsg00 h00
. We do
not need this property to derive the general result, we will however, use a function that satises it when
conducting the numerical analysis.
Pass/Fail, A-F or 0-100? Optimal grading of eager students 21
δ∈(0,1]
measures the degree of distortion, where
δ= 1
means that students are expected
utility maximizers. The perceived probability for a state
sgh
is given by
˜πsgh =ωsgh πsgh
where
Psgh∈S˜πsgh = 1
. Then, what a student perceives to be the value of lottery
Le
for some rule in
ΨH
is given by
PH
h=1 vhPH
i=1 ˜πih
. Accordingly, lottery
L0
is valued
PH
h=1 vhPH
i=1 ˜πhi
and the critical cost type is given by:
ˆ
θ(ΓH|φ, δ) =
H
X
g=1
H
X
h=g+1
(vh(ΓH|φ)−vg(ΓH|φ))ωsgh (ΓH|φ, δ)(πsgh (ΓH|φ)−πshg (ΓH|φ)),
(2.5)
where
ˆ
θ(ΓH|φ, 1) = ˆ
θr(ΓH|φ)
.
Biased students prefer lottery
Le
over
L0
if their cost type is smaller than what they
perceive to be the value dierence for a given belief
φ
. Whatever this belief is,
ˆ
θ(Γ|φ, δ)
is bound from above by
qN−q1<1
and from below by
q1>0
. Further, the function
is continuous in
φ
, and dened on the interval
[0,1]
, which completes Lemma 2.1: a
˚
θ(Γ|δ)∈[0,1]
exists for any
Γ
and
δ
. In other words, a Perfect Bayesian Equilibrium
always exists.
2.5 Optimal grading with salience
Our focus lies on rules that induce optimal student behaviour for a given bias
δ
. Therefore,
we restrict our analysis to exploring whether
ˆ
θ(Γ|ˆ
θF B, δ) = ˆ
θF B
has a solution. In words,
given that employers believe students to behave optimally, is there a scheme which induces
students to indeed do so?
First, consider the case where students are extremely biased. For
δ→0
, students
take into account only the most salient states,
s1H
and
sH1
, all payos not contained in
these states are ignored. To see this, write
ωsgh =δσs1H−σsgh
Psst∈Sδσs1H−σsst πst
∀g, h ∈ {1,2, ..., H}.
By the ordering property,
ωs1H=ωsH1= (πs1H+πsH1)−1
and
ωsgh = 0
for all other
states, and the critical type under a fully transparent grading scheme is given by
lim
δ→0
ˆ
θ(ΓN|φ, δ) = (qN−q1)τN−τ1
τN+τ1
.
Pass/Fail, A-F or 0-100? Optimal grading of eager students 22
The following can be shown:
Lemma 2.2.
lim
δ→0
ˆ
θ(ΓN|φ, δ)>ˆ
θF B
.
Lemma 2.2 states that if results are fully revealed, and students are extremely biased,
eort levels are higher than what is considered desirable by the school.
To give an intuition for the result, consider the overweighting of
Le
relative to
L0
.
The choice of
Le
attaches too much weight to the lowest grade in state
sN1
and to the
highest grade in state
s1N
. The opposite holds true for
L0
. The reason why the eect
that is created by
s1N
overweights, is that although the degree by which initial weights
are distorted is equal, the initial weights themselves are not, that is,
π1N> πN1
by
Assumption 2.1.
The lemma has an important implication: Since students focus only on the most
salient states
s1H
and
sH1
, the only grades that are taken into account are
v1
and
vH
. In
the case of full disclosure,
v1=q1
and
vH=qN
. Therefore, any other rule with
γ1=q1
and
γH−1=qN−1
yields the same limit result. From Proposition 2.1, we know that for
δ= 1
, every rule dierent from the full disclosure rule induces too few students to choose
Le
for any belief. As a result, since
ˆ
θ(Γ|φ, δ)
is continuous in
δ
, there exists some
δ < 1
for which a rule of this form
γ1=q1
and
γH−1=qN−1
is optimal. This insight
provides an explanation for the occurrence of coarse grades and shall therefore be stated
separately:
Proposition 2.2.
For every
H≥3
,
ΨH
contains at least one rule
˜
ΓH
for which
lim
δ→0
ˆ
θ(˜
ΓH|ˆ
θF B, δ)>ˆ
θF B
. Therefore, for every
H≥3
,
ΨH
contains at least one rule
that is optimal for some
δ∈(0,1)
.
A natural question to ask next is whether optimal rules exist for all
δ
. However,
ΓH
is
a subset of
Ω
, its entries thereby being rational numbers. This implies that even for large
N
, the rst best outcome may not exactly be restored by a deterministic rule, although
some optimal probabilistic counterpart exists. Therefore, in order to be able to answer
this question, it is necessary to extend the set of rules to probabilistic schemes. As will
be shown, to derive a general result, it suces to consider rules which are probabilistic
only to a small degree.
Extended set of rules: probabilistic schemes.
We strive to generate continuity of
ˆ
θF B(ΓH|ˆ
θF B, δ)
both within and across coarseness categories
H
. In particular, dene
G
as the set of probabilistic rules with the following properties: every score
qn
which is
not placed on a cut, that is,
qn6=γh
for any
h
, is deterministically awarded a grade,
Pass/Fail, A-F or 0-100? Optimal grading of eager students 23
as before. All other scores may be assigned grades stochastically. In particular, denote
rh
a probability vector, with its elements summing up to one. In particular, an element
rg
h
denotes the probability with which score
qn=γh
is assigned grade
g
. As before, we
require that every grade
g
is awarded with strictly positive probability. A rule in
G
is then
characterized by the set
GH={γ1, γ2, ..., γh, ...γH, r1, ..., rH}
, with
γH=qN
, as before. If
rh
is such that
rh
h= 1
for all
h
, the rule equals the respective deterministic rule. As turns
out, it suces to consider rules which have that (a) at most one
rh
contains elements
dierent from
1
and
0
, and (b) at most two elements of
rh
are dierent from
0
, and
these are neighbouring elements. In words, these rules are probabilistic only to a small
degree: at most one score is assigned to at most two grades with positive probability.
The following can be shown:
Proposition 2.3.
For any
δ
for which
ˆ
θ(ΓN|δ)≥ˆ
θF B
, an optimal rule in
G
exists.
The proposition states that optimal rules exist for each
δ
which induces too many students
to choose
Le
when information is fully revealed. The idea of the proof is as follows: the
rule which does not disclose any information to the market veries that
ˆ
θ(Γ1|φ, δ) = 0
for
any tuple
(φ, δ)
. That is, whatever the belief and however strong the bias, all students
choose lottery
L0
. Probabilistic rules generate continuity between student reactions to
a no-disclosure policy and a fully revealing rule. This is done as follows: Consider the
following class of deterministic rules, from which probabilistic rules are derived: for any
coarseness category
H
,
˜
ΓH
is such that
γh=qh
up to
H−1
. Also,
γH=qN
(by
denition). That is,
˜
Γ2={q1, qN}
,
˜
Γ3={q1, q2, qN}
,
˜
Γ4={q1, q2, q3, qN}
and so forth.
Then, continuity between
˜
ΓH
and
˜
ΓH+1
is generated by implementing a rule in which
score
qH−1
is awarded grade
CH−1
with probability
r∈(0,1]
and
CH
with probability
1−r
.
Consider for instance
˜
Γ2
and
˜
Γ3
.
q2
is then awarded either grade
C2
(probability
r
)
or
C3
(probability
1−r
). If it is assigned
C2
for sure, the behaviour induced is the same
as under
˜
Γ3
. On the other hand, when
r→0
, it limits
ˆ
θ(˜
Γ2|φ, δ)
for some given
φ
and
δ
. Respective probabilistic rules generate continuity of the behaviour function for all
combinations of
H
and
H+ 1
. As a result, whenever
ˆ
θ(ΓN|δ)≥ˆ
θF B
, some rule of this
kind is optimal.
For a given salience function
σ(·)
and given probability vectors
pe
,
p0
, an example
is depicted in gure 2.2 for
N= 5
.
16
In particular, the picture shows the probability
r
, denoted by
r∗(δ, H)
, which, for the described class of rules then yields optimality for
given
δ
and
H
. For instance, for the approximate interval
(0.2,0.54]
, a rule with three
16
In particular, salience function and probability vectors are the same as used in the numerical example
further below. The probability distributions are linear with
β0= 0
and
βe= 1
Pass/Fail, A-F or 0-100? Optimal grading of eager students 24
distinct grades, where
q1
is awarded grade
C1
,
q2
is awarded grade
C2
with probability
r
and
C3
with probability
1−r
while all other scores are assigned
C3
, is optimal.
Figure 2.2: An example for optimal rules for
N= 5
Also, note that probabilistic rules in
G
can be constructed such that deterministic rules
within a coarseness category are linked. To see how, consider the following example: let
there be two rules of the same coarseness category
H
, and assume they are identical
with respect to
H−1
elements, and dier in one single element where the dierence
between these two elements is one score, i.e.
1/N
. For instance, let
H= 2
and consider
Γ20={qn, qN}
and
Γ200 ={qn+1, qN}
Now consider the following probabilistic rule: the
score
qn+1
is assigned grade
C1
with probability
r
and
C2
with probability
1−r
, while
all other scores are graded according to the initial deterministic rules, that is,
q1
to
qn
are assigned
C1
and
qn+2
to
qN
are assigned
C2
. Denote this new continuous rule
G2(r)
.
Then, student behaviour can be written as a function hereof, i.e.
ˆ
θ(G2(r)|φ, δ)
with
ˆ
θ(G2(0)|φ, δ) = ˆ
θ(Γ20|φ, δ)
and
ˆ
θ(G2(1)|φ, δ) = ˆ
θ(Γ200 |φ, δ)
for each tuple
(φ, δ)
.
So far, we have found that the focus on salient payos can account for the occurrence of
coarse schemes and we have derived conditions under which optimal rules exist. However,
nothing has been said about how the bias measure relates to optimal standard setting,
that is, how strict should the passing standard be for instance when Pass/Fail schemes
turn out to be optimal? Also, how does the optimal number of cuts
H
behave in
δ
?
In the following, we compute solutions for a salience function proposed by BGS,
found to well predict choice behaviour in experimental settings. It is given by
σshg =
(|vh−vg|)/(|vh|+|vg|)
. For the probability mass functions, both linear and binomial
distributions are considered.
When scores are distributed according to binomial distributions
B(N−1, α0)
or
B(N−
1, αe)
respectively, with
αe, α0∈(0,1)
,
pn
0
and
pn
e
describe the probability that the number
Pass/Fail, A-F or 0-100? Optimal grading of eager students 25
of successes is
n−1
. Thereby, an exam is understood to be separable into
N
subproblems
with the probability of one such problem being given by some absolute term which again
is determined by the lottery choice.
By contrast, when we speak of linear distributions, we mean that probability mass
functions can be represented by a linear function, i.e.
pn
0(β0) = 1
N(1 + β0
2n−(N+ 1)
(N−1) )
pn
e(βe) = 1
N(1 + βe
2n−(N+ 1)
(N−1) ),
with
βe, β0∈[−1,1]
. The monotone likelihood ratio property is then fullled if and
only if
αe> α0
or
βe> β0
respectively. Since, as has been demonstrated above, simple
probabilistic rules can be constructed to link rules within each coarseness category
H
,
results are computed for deterministic rules only.
Results.
Figure 2.3 displays
ˆ
θ(ΓN|δ)
for three linear and three binomial distributions
and
N= 50
, showing that a fully revealing grading system is optimal only for
δ= 1
.
Therefore, from Proposition 2.3, optimal rules exist for all
δ∈(0,1]
. Moreover, the share
of students choosing lottery
Le
is strictly decreasing in
δ
.
δ0(·)
denotes a value of
δ
for
which a rule is optimal, that is,
˚
θ(ΓH|δ0) = ˆ
θF B
. For Pass/Fail schemes, i.e.
H= 2
,
a rule
Γ2⊂Ψ2
is fully described by some
γ1∈Ω
. For such schemes and the same
distributions,
δ0(Γ2)
is depicted in Figure 2.4. For all these distributions,
ˆ
θ(Γ2|ˆ
θF B, δ)
is
strictly decreasing in
δ
,
δ0(Γ2)
thereby being unique. All distributions have in common
that Pass/Fail schemes are optimal for
δ
very close to zero, but never for high
δ
. In
particular, with linear distributions, extreme biases require rules to be extreme, that is,
the passing standard should either be very high or very low. Albeit intuitive, this is not
an obvious result: with lenient or strict standards, incentives to invest are low, but the
decision weight
ω12
is large. The rst eect however turns out to outweigh the second.
On the other hand, when scores are distributed according to binomial distributions,
most Pass/Fail schemes induce too few students to employ eort even for extreme biases.
Moreover,
δ0(Γ2)
is larger for rules which let fail the expected score for lottery
L0
,
α0N
,
but let pass the expected score for lottery
Le
,
αeN
.
Next, consider rules with two cuts. Figure 2.5 shows graphs of
δ0(Γ3)
for a xed second
cut
γ2
. Uniqueness of
δ0
is given, as before. The pictures reveal that for all distributions,
these rules outrule both a fully revealing system as well as simple Pass/Fail schemes
for intermediate values of
δ
. The contour plots for
δ0
are presented in the appendix.
The range very much depends on the distribution: while for binomial distributions with
Pass/Fail, A-F or 0-100? Optimal grading of eager students 26
Figure 2.3: Full disclosure:
ˆ
θ(ΓN|δ)
for
N= 50
and dierent distributions
Figure 2.4: Pass/Fail schemes:
δ0(γ1)
for
N= 50
and dierent distributions
α0= 0.3
and
αe= 0.8
,
A, B, C
rules are optimal for almost all
δ
, they are roughly for
δ
between
0.002
and
0.085
for
(α0, αe) = (0.4,0.55)
. Also, while
δ0
maximizes where
each grade contains about the same number of scores when distributions are linear, it
does where the number of scores is relatively low in the
B
grade when distributions are
binomial.
17
Rules where the most common grade is an
A
(
γ1
and
γ2
low) are chosen for
rather low values of
δ
, a result which holds for all distributions.
17
This is basically the strategy to induce high eort levels given the constraint that there be no more
than two cuts. The whole calculation is however biased by the weighting function which exhibits polyno-
mials. This becomes also obvious in the contour plots which show that for some binomial distributions,
δ0
exhibits two local maxima.
Pass/Fail, A-F or 0-100? Optimal grading of eager students 27
Figure 2.5: A,B,C schemes:
δ0(γ1, γ2)
for given
γ2
,
N= 50
and dierent distributions
2.6 Discussion
2.6.1 Dierentiated students
So far, we have assumed that students of the same institutions are equal with respect to
(a) the probabilities distributions and (b) their inherent competitiveness (their
δ
). In the
following, we shortly discuss deviations from these assumptions.
By contrast, consider students of dierent ability levels. In particular, let there be
M
student groups
t1, t2, ..., tM
, whose probability distributions have dierent supports.
That is, students of group
tm
score between
qm
and
qm
,
qm< qm
and
qm, qm∈Ω
.
Denote
φm
the belief employers hold about the critical cost type in groups
m
,
ˆ
θm
, and
denote
φt= (φ1, φ2, ..., φM)
. The market value of a certicate
vh
is given by
vh(Γ|φt, δ) = PM
m=1 PγhN
i=1+γh−1Nqi(φmpim
e+ (1 −φm)pim
0)
PM
m=1 PγhN
i=1+γh−1N(φmpjm
e+ (1 −φm)pjm
0),
where
pm
e
and
pm
0
are probability vectors for group
m
with
zero
elements outside the
support interval. If students are disparate in the sense that their supports do not overlap,
our results from the previous analysis apply accordingly: every group can be analysed
separately, and cuts are chosen such that optimality is restored. There is no benet from
further distorting information. As an example, when all groups are optimally incentivized
by employing a Pass/Fail scheme, the now optimal rule is one with
M
cuts.
Similarly, if supports overlap and the optimal cuts are placed outside the overlapping
area, the optimal solution can easily be restored. If not so, the decision maker has to
trade o the eect of merging the dierent cuts against that of moving cuts outside
Pass/Fail, A-F or 0-100? Optimal grading of eager students 28
the support intervals of students of dierent abilities. Optimality is then typically not
restored. We leave the in-depth analysis of these eects for further research.
A related question is, in how far does the behaviour of dierently biased students
aect the behaviour their classmates? Assume the school has chosen a coarse rule
˜
Γ
which in equilibrium implemented just the right eort levels given the average
δ
, denoted
by
˜
δ
. More specically,
ˆ
θ(˜
Γ|ˆ
θF B,˜
δ) = ˆ
θF B
. Consider a single (atomistic) student whose
salience bias is lower, namely
ˆ
δ
, therefore
ˆ
θ(˜
Γ|ˆ
θF B,ˆ
δ)<ˆ
θF B
.
Consider by contrast an institution where the average
δ
is
ˆ
δ
but which has employed
the same, then non-optimal, rule
˜
Γ
. It can be veried that market values for each grade
are lower, the reason being that employers foresee the reduced willingness to work. As a
result, our analysis predicts that for a given grading scheme, less focused students employ
more eort in the presence of biased classmates.
2.6.2 Concluding comments
By presenting a model in which a decision maker overweights salient payos, BGS account
for a range of identied basic violations of expected utility theory. This study applies their
approach to a simple educational model. In particular, an institution seeking to maximize
its students' expected utility designs a grading system that does so when students are
drawn to salient payos associated with grades. Our results suggest that this focus
induces too much eort in students, from the institutions' perspective, and the more so
the stronger the bias. The common intuition that coarsening reduces incentives applies.
By assuming that this `focus on grades' is present rst and foremost in student bodies
of very selective colleges, the model provides an explanation for the occurrence of infor-
mation suppression in these institutions. Further, the numerical analysis suggests that
within one coarseness category, choosing standards such that most students receive
A0s
is a best response to large distortions.
Our results also account for dierent conceptions of desired states: for instance, con-
sider schools that wish to maximize tuition fees. If students are aware of their bias,
and institutions compete for students, employing the grading rule that maximizes overall
student welfare is an equilibrium outcome.
Pass/Fail, A-F or 0-100? Optimal grading of eager students 29
2.7 Appendix
2.7.1 Proofs
Proof of Lemma 2.1.
It is to show that
WF D(˚
θ(Γ)) = WΓ(˚
θ(Γ))
. Denote
E[qn|Le] =
VF D
e
and
E[qn|L0] = VF D
0
. Then
WF D(ˆ
θ) = WΓ(ˆ
θ)
is given if and only if
ˆ
θV F D
e+ (1 −ˆ
θ)VF D
0=ˆ
θV Γ
e(ˆ
θ) + (1 −ˆ
θ)VΓ
0(ˆ
θ).
(2.6)
Using
VΓ
e
and
VΓ
0
as given in equation (2.1) and
vh
as in equation (2.3) transforms
equation (2.6) to
N
X
i=1
qi(ˆ
θpi
e+ (1 −ˆ
θ)pi
0) =
H
X
h=1 PγN
i=γh−1N+1 qi(φpi
e+ (1 −φ)pi
0)
PγN
i=γh−1N+1 φpi
e+ (1 −φ)pi
0
γN
X
i=γh−1N+1
ˆ
θpi
e+ (1 −ˆ
θ)pi
0.
Since
˚
θ(Γ) = φ
, the condition becomes
N
X
i=1
qi(φpi
e+ (1 −φ)pi
0) =
H
X
h=1
γhN
X
i=1+γh−1N
qi(φpi
e+ (1 −φ)pi
0),
which is clearly given.
Proof of Proposition 2.1.
Dene
∆ΓH:=ˆ
θr(ΓH|φ)−ˆ
θF B
=
H
X
h=1
vh(ΓH|φ)
γH
X
i=1+γh−1N
(pi
e−pi
0)−
N
X
n=1
qn(pn
e−pn
0)
It is to show that
∆ΓH<0
for all
H < N
and
ΓH⊂ΨH
. The proof is by induction.
First step: consider some rule
˜
ΓN−1⊂ΨN−1
such that grade
Ct
pools scores
qt
and
qt+1
.
∆˜
ΓN−1= (vt−qt)(pt
e−pt
0)−(qt+1 −vt(pt+1
e−pt+1
0))
Dening
zn= (φpn
e+ (1 −φ)pn
0)
and using
vt
as given in equation (2.3) gives
∆˜
ΓN−1zt+zt+1
qt+1 −qt=(zt+1(pt
e−pt
0)−zt(pt+1
e−pt+1
0))
=pt+t
0pt
e−pt
0pt+1
e=pt
0pt+1
0(τt−τt+1)<0.
Pass/Fail, A-F or 0-100? Optimal grading of eager students 30
Second step: consider some
∆˜
ΓH⊂ΨH
and assume
∆˜
ΓH<0
. Now consider some
∆˜
ΓH−1⊂ΨH−1
which pools two neighbouring grades
Ct0
and
Ct0+1
into a new grade
Cs
,
all other grades remain unchanged.
∆˜
ΓH−1−∆˜
ΓH=ˆ
θr(Γ ˜
H−1|ˆ
θF B)−ˆ
θr(Γ ˜
H|ˆ
θF B)
=(vs−vt0)
γt0N
X
i=1+γt0−1N
(pi
e−pi
0)−(vt0+1 −vs)
γt0+1N
X
i=1+γt0N
(pi
e−pi
0)
with
vs−vt0=Pγt0N
i=γt0−1N+1 Pγt0N
j=γt0−1N+1 zizj(qi−qj) + Pγt0N
i=γt0N+1 Pγt0N
j=γt0−1N+1 zizj(qi−qj)
Pγt0N
i=γt0−1N+1 ziPγt0+1N
i=γt0−1N+1 zi
=Pγt0+1N
i=γt0N+1 Pγt0N
j=γt0−1N+1 zizj(qi−qj)
Pγt0N
i=γt0−1N+1 ziPγt0+1N
i=γt0−1N+1 zi
vt0+1 −vs=Pγt0+1N
i=γt0N+1 Pγt0+1N
j=γt0N+1 zizj(qi−qj) + Pγt0+1N
i=γt0N+1 Pγt0N
j=γt0−1N+1 zizj(qi−qj)
Pγt0+1N
i=γt0N+1 ziPγt0+1N
i=γt0−1N+1 zi
=Pγt0+1N
i=γt0N+1 Pγt0N
j=γt0−1N+1 zizj(qi−qj)
Pγt0+1N
i=γt0N+1 ziPγt0+1N
i=γt0−1N+1 zi
Therefore,
∆˜
ΓH−1−∆˜
ΓH<0
if and only if
Pγt0+1N
i=γt0N+1 Pγt0N
j=γt0−1N+1 zizj(qi−qj)
Pγt0+1N
i=γt0−1N+1 ziPγt0N
i=1+γt0−1N(pi
e−pi
0)
Pγt0N
i=γt0−1N+1 zi−Pγt0+1N
i=1+γt0N(pi
e−pi
0)
Pγt0+1N
i=γt0N+1 zi<0
⇔Pγt0N
i=1+γt0−1N(pi
e−pi
0)
Pγt0N
i=γt0−1N+1 zi−Pγt0+1N
i=1+γt0N(pi
e−pi
0)
Pγt0+1N
i=γt0N+1 zi<0
⇔
γt0N
X
i=1+γt0−1N
γt0+1N
X
j=1+γt0N
pj
0pi
e−pi
0pj
e=
γt0N
X
i=1+γt0−1N
γt0+1N
X
j=1+γt0N
pi
0pj
0(τi−τj)<0,
which is given due to the monotone likelihood ratio property.
Proof of Lemma 2.2.
The problem can be written as a maximization problem where
p0
is being xed. It is to show that there the maximium is not strictly positive. For
technical convenience, we consider
τn= 0
, i.e.
pn
0= 0
for some small
n
and
τn=τn+1
for
Pass/Fail, A-F or 0-100? Optimal grading of eager students 31
some
n
(in contradicion to Assumption (2.1) where we assumed strictly increasing
τn
).
max
τf(τ) := −(N−1)τN−τ1
τN+τ1
+
N
X
n=1
npn
0(τn−1)
s.t. τn≥τn−1≥0
for all
n,
N
X
n=1
pn
0τn= 1
τN
can be written as a function of
τ−N
. Then, for every
2≤n≤N−1
,
∂f(τ−N)
∂τn
=pn
0(n−N+2(N−1)τ1
(τN+τ1)2pN
0
)
Possible solutions to the program are:
(a)
τ1→0
. Dene
En
e:= PN
n=1 npn
e=PN
n=1 npn
0τn
and
En
0:= PN
n=1 npn
0
. Then:
lim
τ1→0=−(N−1) + (En
e−En
0)<0
.
(b)
1> τ1>0
. Since
∂f(τ−N)/∂τn
is monotonically increasing in
n
, a candidate solution
has the property that there exists some critical
n
, denoted by
ˆn
such that
τ1
ˆn:= τ1=
τ2=... =τˆn< τˆn+1 =... =τN:= τN
ˆn
. Denote
pˆn:= PN
i=ˆn+1 pi
0
.
Then
τN
ˆn= (1 −(1 −pˆn)τ1
ˆn)/pˆn
and
f(τ) = −(N−1)(1 −τ1
ˆn)
pˆn(τN
ˆn+τ1
ˆn)−(1 −τ1
ˆn)
ˆn
X
i=1
ipi
0+ (1 −τ1
ˆn)1−pˆn
pˆn
N
X
i=ˆn+1
ipi
0.
f(τ)<0
i
−(N−1) + (τN
ˆn+τ1
ˆn)(1 −pˆn)
N
X
i=ˆn+1
ipi
0−(τN
ˆn+τ1
ˆn)pˆn
ˆn
X
i=1
ipi
0<0.
It holds that
Pˆn
i=1 ipi
0>(1 −pˆn)
and
PN
i=ˆn+1 ipi
0< Npˆn
. Therefore,
f(τ)<0
if
1−(τN
ˆn+τ1
ˆn)(1 −pˆn)pˆn>0.
⇔1>(1 −pˆn)(1 + τ1
ˆn(2pˆn−1))
which can be checked to hold for all
τ1
ˆn∈(0,1)
and
pˆn∈(0,1)
.
(c)
τ1→1
yields
lim
τ1→1f(τ) = 0
.
Proof of Proposition 2.2.
From the analysis in the text, it follows that for any
ΓH
,
H≥3
with
γ1=q1
and
γH−1=qN−1
, it holds that
ωs1H=ωsH1= (p1
0pN
e+pN
0pN
e)−1
and
Pass/Fail, A-F or 0-100? Optimal grading of eager students 32
ωshg = 0
for all other states, thereby:
lim
δ→0
ˆ
θ(ΓH|φ, δ) = lim
δ→0
ˆ
θ(ΓN|φ, δ),
which is strictly greater than
ˆ
θF B
by Lemma 2.2. From Proposition 2.1, these rules are
strictly smaller than
ˆ
θF B
at
δ= 1
. By continuity of
ˆ
θ(ΓH|φ, δ)
in
δ
, there exists some
δ
for which
ˆ
θ(ΓH|φ, δ) = ˆ
θF B
.
Proof of Proposition 2.3.
First, if no information is revealed, i.e.
H= 1
, then
ˆ
θ(Γ1|φ, δ) = 0
for all tuples
(φ, δ)
.
Some probabilistic rule in
G
establishes continuity between any
ˆ
θ(Γ1|φ, δ)
and
ˆ
θ(Γ2|φ, δ)
for any
Γ2⊂Ψ2
. Consider the following rule in
Ψ2
:
˜
Γ2=q1
and the probabilistic rule in
G
,
˜
G2={q1, qN,(r, 1−r),(0,1)}
with
r∈(0,1]
. That is, score
q1
is awarded grade
C1
with probability
r
and
C2
with probability
1−r
. All other scores are awarded grade
C2
.
The function
ˆ
θ(˜
G2|φ, δ)
is clearly continuous in
r
. When
r= 1
,
˜
G2=˜
Γ2
.
Also, when
r→0
,
˜
G2→Γ1
, and
ˆ
θ(˜
G2|φ, δ)→ˆ
θ(Γ1|φ, δ)=0
, as can be veried by
inspection of equation 2.5. Then, all sum elements of equation 2.5 which contain
g= 1
or
h= 1
approach
0
, while the value of grade
C2
limits the value of
C2
of
Γ0
. Further,
the probability that grade
C2
occurs if
L0
is chosen, is given by
PqNN
i=q2Npi
0+ (1 −r)p1
0
(analog presentation for
Le
), while the probability that grade
C1
occurs is
rp1
0
, which is
clearly zero for
r→0
. As a result, the weighting functions as given in 2.4 also limit those
of
Γ0
, i.e.
1
.
The same procedure links all higher coarseness categories up to the full disclosure rule
ΓN
. More specically, the new probabilistic rules are all derived from the determinis-
tic rules
˜
ΓH={γ1=q1, γ2=q2, ...γH−1=qH−1, γH=qN}
. Then, the rules
˜
GH=
{q1, ..., qH−1, qN,(1,0, ..., 0),(0,1,0, ..., 0), ...(0, ..., 1,0,0),(0, ..., 0, r, 1−r),(0, ..., 1)}
,
r∈
(0,1]
establish the link between
˜
ΓH−1
and
˜
ΓH
for all
H
. The last step is the transition
from
˜
ΓN−1
to the full disclosure rule
ΓN
.
By continuity of
ˆ
θ(˜
GH|φ, δ)
in
r
for any tuple
(φ, δ)
and any rule
˜
GH
, continuity is es-
tablished between
ˆ
θ(Γ0|ˆ
θF B, δ)
- student behaviour when no information disclosure - and
ˆ
θ(ΓN|ˆ
θF B, δ)
- student behaviour for full disclosure. As a result, optimal rules exist for
all
δ
for which
ˆ
θ(ΓN|ˆ
θF B, δ)≥ˆ
θF B
.
Pass/Fail, A-F or 0-100? Optimal grading of eager students 33
2.7.2 Numerical results
Figure 2.6: A,B,C schemes:
δ0(γ1, γ2)
for
N= 50
and dierent
(p0, pe)
3 Informational opacity and honest certication
Chapter Abstract
This paper studies the interaction of information disclosure and reputational con-
cerns in certication markets. We argue that by revealing less precise information a
certier reduces the threat of capture. Opaque disclosure rules may reduce prots
but also constrain feasible bribes. For large discount factors a certier is uncon-
strained in the choice of a disclosure rule and full disclosure maximizes prots. For
intermediate discount factors, only less precise, such as noisy, disclosure rules are
implementable. Our results suggest that contrary to the common view, coarse dis-
closure may be socially desirable. A ban may provoke market failure especially in
industries where certier reputational rents are low.
35
Informational opacity and honest certication 36
3.1 Introduction
In markets that exhibit informational asymmetries, product quality is typically reduced.
This in turn may provoke a breakdown of trade. The lack of credible communication
between informed and uninformed parties may result in the emergence of certication
intermediaries. Certiers inspect products whose characteristics are private information
to agents, and publicly reveal this information. Examples abound: rating agencies, eco-
labels, wine certicates or technical inspections. Often however, results are revealed on a
coarse scale, although the information at hand would allow for a more precise disclosure.
A stream of literature starting with Lizzeri (1999) has identied prot concerns as
the reason for imprecise information disclosure. We provide an alternative explanation
for such opacity. We show that partially revealing rules can serve as a safeguard against
fraud: certiers may be tempted to accept bribes for releasing favourable certicates.
This behaviour, which we call capture, enables the certier to extract payments other
than the certication fee. If consumers are aware of this threat of capture, then the
certier must nd ways to credibly commit to honesty. We show that one way to do
so is to employ an opaque disclosure rule. opacity reduces the producer's willingness to
pay for bribery, because a more opaque disclosure rule lowers dierences in the values of
certicates. Hence, opacity can be welfare enhancing since it may prevent market failure.
This result is surprising because it contradicts the commonly held view that reducing
informational asymmetries is socially desirable per se.
We show our result in a model with moral hazard where, in each period, short-
lived producers rst have to make an investment choice, which in turn determines the
probability distribution of their products' qualities. Thus, the payos assigned to each
quality outcome determine the incentives to invest. The long-lived certier has two
instruments at his disposal: a at certication fee and the disclosure rule. Consumers
experience the true quality of a product only after consumption. If it does not match
the awarded certicate, capture is detected. This makes the certier face a classical
reputation dilemma because he trades o short-run gains from capture against regular
future prots.
We characterize feasible disclosure rules in this setting. Our major nding is that for
suciently low discount factors, honest certication requires partial disclosure of quality
information, which in our model implies noisy disclosure. In the short run, a certier
may gain from making a capture oer that is acceptable for at least some producers. The
maximum producer willingness to pay for bribes is directly aected by the publicly an-
nounced disclosure rule. It is greatest for full disclosure and can be substantially reduced
Informational opacity and honest certication 37
by revealing less precise information. But if consumers detect a bribe and therefore lose
trust, a certier gives up his future prots. Static certier prots are maximal for full
disclosure and any deviation will typically reduce the long-run loss from losing credibility.
As will be shown, the rst eect exceeds the latter.
We moreover obtain the counterintuitive result that a threat of capture increases so-
cial welfare.
1
Whenever information is fully revealed, sharing prots necessarily reduces
producer investments as compared to the rst-best level, obtained under complete in-
formation. We show that whenever capture oers are made before a certier observes
the true quality level, these are such that they are accepted by either all producers or
only by low quality producers. If the highest threat of capture stems from oers that are
accepted by all producers and the disclosure rule is noisy, credibility can be maintained
by making honest certication more attractive to high quality producers. This in turn
increases equilibrium investment levels as compared to full information disclosure.
Related literature.
A stream of literature seeks to explain why certiers often choose
to only partially reveal quality information. Lizzeri (1999) nds that it is optimal for
a monopolistic certier in a static adverse selection environment to reveal almost no
information. In this setting, this result is robust to introducing capture because a no rev-
elation policy simply annihilates producer incentives to bribe. In the presence of moral
hazard however, information revelation is necessary to create incentives for the provision
of quality. Albano and Lizzeri (2001) study optimal disclosure rules in a static model of
both moral hazard and adverse selection. In their setting, a certier chooses to employ
noisy disclosure if his set of actions is restricted to at fees. According to Farhi et al.
(2014), opacity in certication markets is caused by information averse sellers. In Dubey
and Geanakoplos (2010), it is shown that coarse grading schemes can help to induce all
students to employ eort if they are disparate and care about their status in class. Kar-
tasheva and Yilmaz (2013) explain imprecise ratings in a model with partially informed
investors and heterogeneous liquidity needs of issuers. A static adverse selection model
where quality is not fully observable by the seller is analysed by Faure-Grimaud et al.
(2009). They identify conditions under which the ownership of certication results is left
to rms and under which rms reveal their ratings.
The threat of capture in certication markets has been analysed by Strausz (2005).
In a pure adverse selection setting with full disclosure, he analyses the eects of a threat
of capture on certication prices. He nds that in order to maintain credibility, for low
1
We analyse a belief system that substantially restricts the set of feasible disclosure rules. For dierent
belief systems and suciently low discount factors, other (opaque) rules may be chosen by the certier.
The eect on social welfare is therefore not a general result.
Informational opacity and honest certication 38
discount factors, a certier raises fees above the static monopoly price. This result is
consistent with our nding in that as less information is disclosed, the certication fee
generates a cut-o value that species a minimal certied producer quality. A larger
fee increases this cut-o but this implies that less information is revealed in equilibrium.
Although this eect is also present in Strausz (2005), he however does not explicitly
point it out. As in the present paper, credibility is maintained by reducing the maximal
willingness to bribe. In Strausz (2005), this is aected by the value of not being certied,
which, in turn, is an increasing function of the certication fee.
There is a rich literature on reputation building in markets with informational asym-
metries. For example, Shapiro (1983) analyses the forces at work when sellers build
reputation. Biglaiser (1993) investigates the role of market intermediaries when sellers
are unable to build their own reputation.
Examples of works that treat reputational concerns of rating agencies are Mathis et al.
(2009) and Bolton et al. (2012). In contrast to the present paper, these works follow the
asymmetric information approach to reputations, where certiers are assumed to always
be committed (i.e. honest) with positive probability.
2
This, however, restricts the set of
allowed certication fees and disclosure rules for non-committed certiers. The reason
is that a departure from the equilibrium strategy immediately reveals the certier type.
Instead of assuming that testing by the certier is imperfect as is done in those works,
we show how imperfect testing may endogenously arise in equilibrium.
Levin (2003) extends the standard moral hazard setting to situations where contrac-
tual agreements are enforceable only to a certain degree and where reciprocal relations are
long-term. The optimal contract derived by Levin has a coarse structure, which parallels
our nding of coarse disclosure being optimal.
The remainder of the paper is organized as follows. Section 2 presents the model.
Section 3 analyses the static game in the absence of bribery. In section 4, we treat the
general case of certication under the threat of capture. Section 5 concludes. All proofs
are presented in the appendix.for.
3.2 The model
We consider a dynamic framework in discrete time. In each period
t= 1,2, . . . , ∞
,
a short-lived monopolistic producer is born. He produces a single unit of quality
qt∈
{ql, qh}
, where
0≤ql< qh
. In the following, we refer to a
high type
producer if his product
quality is
qh
and to a
low type
producer otherwise. Prior to production, a producer chooses
2
See Mailath and Samuelson (2012, Chapter IV) for a discussion of this approach.
Informational opacity and honest certication 39
begin of period
t
producer chooses
et
producer learns
qt
good sold in auction
consumers learn
qt
begin of period
t+ 1
Figure 3.1: Timing in one period without certication
some investment level
et∈[0,1]
. Quality is stochastic and the probability of the produced
good being of high quality
qh
is given by
Prob(qt=qh|et) = et
. This probability function
is independent of
t
, i.e. quality levels are independent across time. Investment costs
are given by the function
k(·)
. We assume
k(·)
to be increasing and strictly convex. For
technical reasons we assume a non-negative third derivative, such that the certier's prot
function is concave. To guarantee interior solutions we additionally assume
k0(0) ≤ql
and
k0(1) ≥qh
.
Consumers' reservation prices equal (expected) qualities. Both investment and quality
level are private information to the producer. Consumers observe the product quality
only after consumption. All other components of the model are common knowledge. The
equilibrium concept we use throughout is the perfect Bayesian equilibrium.
Each producer is short-lived and leaves the market at the end of a period. Goods are
sold in a second-price auction.
3
Figure 3.1 summarizes the timing in period
t
.
To simplify notation, we set
ql≡0
and dene
v:= qh−ql
. In the benchmark case with
complete information, high quality goods are sold in the second-price auction at price
v
and low quality goods are sold at price
0
. The producer then chooses
e
to maximize his
expected prots
ev −k(e)
. The rst-best investment level
e∗
is thus given by
k0(e∗) = v
.
It lies in the interval
[0,1]
, because
k0(1) ≥v
by assumption. In particular,
e∗>0
.
When the market exhibits asymmetric information, a producer cannot persuade con-
sumers that he oers a high quality good. As a result, the market price cannot be made
contingent on a good's quality. It is standard to show that the Perfect Bayesian market
outcome involves a market breakdown. Consumers form a belief
qe
t
about the oered
quality reecting their willingness to pay. In equilibrium, this belief has to be consis-
tent with the actual expected quality,
E(qt|et)
. Given any belief, a producer's optimal
investment choice will be
et= 0
as he maximizes
qe
t−k(et)
. But since
E(qt|0) = 0,
in
the unique equilibrium, producers choose
et= 0
in every period. Thus, quality is zero in
each period. The result is a market failure: high quality is never oered in equilibrium.
3
The second price auction results in a standard monopoly price that equals consumers' valuations. It
circumvents signaling issues, e.g. letting the informed party take a publicly observed action that might
be interpreted as a signal.
Informational opacity and honest certication 40
We summarize this nding in the following lemma.
Lemma 3.1.
Without certication, producers choose
et= 0
in each period. In equilib-
rium, quality is given by
qt= 0
and the price is
0
in each period.
This ineciency calls for the emergence of alternative market institutions to facilitate
supply of high quality. The focus of this paper lies on certication as one such institution.
Assume that an innitely long-lived certier enters the market. She screens quality and
oers to disclose the result of some potentially imperfect test, prior to it being sold. More
precisely, at the beginning of the game, in period
t= 0
, the certier announces a fee
f≥0
and a disclosure rule
D= (C, αl, αh)
.
4 5
The fee has to be paid by any producer who
wishes to have his product tested. The disclosure rule consists of a set
C={C1, . . . , Cm}
of potential certicates and probability vectors
αl
and
αh
, where the
k
-th entry of vector
αi
indicates the probability that a product of quality
qi
is awarded certicate
Ck
if tested.
We do not assume that those probabilities add up to one, i.e. we allow for
Pm
k=1 αi
k<1
.
Hence, a product may remain uncertied with the conforming probability and will be
sold as such. We assume that consumers cannot observe whether a product was tested,
unless it is oered with a certicate.
6
Possible disclosure rules encompass for example
full
disclosure
, where
C={C1, C2}
and
α2= (0,1)
as well as
α1= (1,0)
, or
no disclosure
,
where
C={C}
and
αi= (1)
.
7
For a given certicate
Ck
, consumers form a belief
eqCk
about the true quality of a product. The belief for uncertied products is denoted
eq∅
.
For notational convenience we henceforth add
∅
to the set of certicates
C
, which refers
to uncertied products. Hence,
C={C1, . . . , Cm,∅}
.
An interpretation of the disclosure rule, which we shall use throughout the paper, is
the following: the certier can create any test that leads to a grading scheme with grades
from the set
C
and results in the respective grades with conforming probabilities. This
may be done by a computer program or a statistical test. In particular, after the test
result is obtained, both certier consumers share the same beliefs on product quality.
4
Assuming a single fee
f
, that does not depend on the certicate, is without loss in the setting with
only two quality levels. The best a certier could do is, following the revelation principle, to oer a menu
of `contracts' for the two potential producer types. Eventually, there is one payment referring to the high
type and one referring to the low type. It can be easily shown that the optimal contract corresponds to
the full disclosure rule, where high types pay
f
and low types pay
0
and true quality is revealed.
5
The fee
f
creates a distortion as will become clear later on. The certier could implement the rst-
best outcome, but only when moving rst, i.e. when demanding an upfront payment
before
producers
choose their investment. This timing however seems unreasonable in many certication markets.
6
Hence products which failed the test are sold under the same label as products that didn't even
take the test. This assumption is not crucial, since the certier can replicate any outcome of a game
where consumers are able to observe whether a product applied for certication.
7
Note that certicates do not carry an intrinsic value. In the case that quality is fully revealed,
whether
C1
or
C2
is the valuable certicate depends on the choice of
α
.
Informational opacity and honest certication 41
prod. chooses
et
prod. learns
qt
prod. certify: y/n
Ct
disclosed if y
∅
disclosed if n
in auction
good sold
consumers
learn
qt
Figure 3.2: Timing of a period
t
with certication
Finally we assume that the certier's inspection costs are zero
8
and that she discounts
future prots at rate
δ∈(0,1)
. Figure 3.2 illustrates the timing of the game with
certication.
3.3 Optimal honest certication
In this section, we analyse certier equilibrium strategies when the certier is honest. By
the stationary structure of the model, we can restrict our analysis to the certier decision
(D, f)
plus a single period of production. Let
πD(f)
denote the equilibrium prot of the
certier, when adopting disclosure rule
D
with certication fee
f
.
We rst study a full disclosure rule in some detail. It will turn out that this disclosure
rule maximizes certier prots. Consider the case where quality is fully revealed such
that
αh= (1,0)
and
αl= (0,1)
. Any product that is awarded
C1
is sold at a price
v
, whereas
C2
is worth nothing. The only plausible equilibrium is one where only high
types apply for certication.
9
A producer chooses his investment according to
e=arg max
eeee·(v−f)−k(ee).
(3.1)
This implies
k0(e) = v−f
in equilibrium and certier expected equilibrium prots can
be expressed as
bπF D(e) = e·(v−k0(e)).
(3.2)
Denote
eF D
the equilibrium eort level under a full disclosure rule and
fF D
the
corresponding prot maximizing fee. The following lemma proves that these values exist
and are unique.
8
This assumption simplies the analysis without substantially aecting the results, which continue
to hold as presented here for small but strictly positive inspection costs. High inspection costs do not
invalidate most of our results, but create cumbersome case distinctions.
9
Trivially, low quality producers do not demand certication when
f > 0
since their revenues are zero
at most.
Informational opacity and honest certication 42
Lemma 3.2.
Under full disclosure, there exists a unique fee
fF D
that maximizes certier
prots. The uniquely dened equilibrium investment level
eF D
is implicitly given by
k00(eF D)·eF D =v−k0(eF D).
(3.3)
The fee is
fF D =v−k0(eF D)
and the (subgame-) equilibrium prot is
πF D =eF D ·fF D
.
We continue with an analysis of general disclosure rules. The entire set of disclosure
rules is complex and dicult to handle analytically. A closer look at equation (3.2), which
allows us to express the certier prot as a function of the implemented investment level
e
, points to the advantages of using an indirect approach. We can express the attained
prot of any certier policy
(D, f)
solely in terms of the induced investment level
e
. This
allows for a straightforward comparison of attained prots and leads us to the following
proposition.
Proposition 3.1.
For any disclosure rule
D=C, α1, α0
and any fee
f≥0
, it holds
that
πD(f)≤πF D
in equilibrium.
Proposition 3.1 states that the certier will always nd it optimal to employ a full dis-
closure rule. The reason is that investment incentives depend on the dierence between
payos from selling high and low quality products. Given full disclosure, the certication
fee is sucient to fully control this dierence.
We complete this section by pointing to the fact that full disclosure is not the unique
optimal (i.e. prot maximizing) disclosure rule. First of all, the same outcome can
be implemented by adding redundant certicates to various rules either additional
certicates for high types, which then all have the same value in equilibrium, or by
adding certicates for low types that will not be issued in equilibrium.
Because certication is assumed to be costless for the certier, other rules also maxi-
mize certier per-period prots: issue two dierent certicates
C1
and
C2
. Low quality
products are only eligible for certicate
C2
, hence
αl= (0,1)
. High quality products re-
ceive certicate
C1
with probability
α∈(0,1)
and
C2
otherwise, therefore
αh= (α, 1−α)
.
With rules of this structure, it is possible to sustain equilibria in which all producer types
apply for certication.
10
The optimal certier prot
πF D
can be obtained by choosing
f
and
α
accordingly.
11
This class of disclosure rules plays a crucial rule in the remainder
of this paper. We henceforth refer to these as
partial disclosure
rules.
10
For this, we have to set the o-equilibrium belief
eq∅= 0
and all other beliefs underly Bayesian
updating.
11
We formally show this in the proof of Proposition 3.6.
Informational opacity and honest certication 43
prod. chooses
et
prod. learns
qt
capture oer
(C, b)
certier makes
prod. a/r
prod. certify: y/n
only if r:
disclosure:
Ct
if y,
∅
if n,
C
if a
in auction
good sold
consumers
learn
qt
Figure 3.3: Timing of a period
t
with certication and capture
3.4 The capture problem
So far we have assumed that the certier sticks to the previously announced disclo-
sure rule. This implies that she conducts the lottery honestly and grants the respective
certicate. However, there is pressure from producers who wish to be awarded better
certicates. For instance, if disclosure is meant to be noisy, a certier might be willing
to guarantee a producer a high value certicate in exchange for a bribery payment. In
this section we address this issue by formally introducing the possibility of capture.
We follow Strausz (2005) in modelling the threat of capture, using the framework of
enforceable capture as initiated by Tirole (1986). Hence we assume that the certier and
the producer can write an enforceable side-contract with transfers. Consumers are fully
aware of the possibility of these side-contracts, but cannot observe them. Specically, we
model capture as follows: after a producer has learned his type
qt
, but before deciding
whether to apply for certication, the certier, without having observed
qt
, may make an
oer
(C, b)
to the producer.
The oer consists of a certicate
C
, issued in case of acceptance, and a nancial trans-
fer
b
to be paid by the producer. The certier thus oers to sell the sure certicate
C
at the price
b
, circumventing the customary certication procedure. Hence,
(C, b)
are
the terms at which she is willing to become captured. A producer however can reject
this oer and insist on being honestly certied by paying the fee
f
. This assumption
is motivated following Kofman and Lawarrée (1993) who argue that the certier cannot
forge certication without the help of the producer. Figure 3.3 displays the timing in a
representative period
t
, allowing for the possibility of capture.
Note that the choice of disclosure rule puts some limits on the set of
feasible
capture
oers. For a general disclosure rule
D={C, α}
only oers of the form
(C, b)
with
C∈ C
Informational opacity and honest certication 44
are feasible.
12
Within the framework presented here, capture may subvert honest certication for
two reasons.
13
First, producers with low quality products are willing to side-contract
with the certier in order to obtain better certication. Second, high types may want to
avoid uncertainty if disclosure is noisy.
In this section, we are interested in the existence and characterization of equilibria
where the certier resists the temptation of making a capture oer. Throughout, we will
work with dierent specications of trigger beliefs. This becomes necessary as the ability
of consumers to detect capture varies across disclosure rules. We assume consumers to
be able to perfectly observe quality after consumption. Therefore, if
D
is full disclosure
or if certain certicates are awarded exclusively to high types, capture detection is also
perfect.
In particular, when speaking of consumer trigger beliefs we mean the following: they
stop trusting the certier immediately as soon as a false testimony about a product's
quality is detected. Knowing this, producers are not willing to pay for certication
anymore. Consequently, the certier will lose future demand and makes zero prots
henceforth. This prevents the certier from becoming captured in the rst place. We
shall make this more precise in the following subsections.
3.4.1 Capture under full disclosure
Because, by Proposition 3.1, a certier wishes to employ full disclosure whenever possible,
we start by investigating capture under a full disclosure rule. We assume that consumers
trust certicates as long as no deviation has been detected. A certier who anticipates this
behaviour may be prevented from succumbing to the temptation of becoming captured
by the fact that losing credibility will leave her without demand in future periods.
Denote
ht= (Ct, qt)
the certication outcome in period
t
, where
Ct
is the issued
certicate in period
t
and
qt
is the true quality observed after consumption. If certication
in period
t
did not take place, then
Ct=∅
. Now let
Ht= (h1, . . . , ht−1)
summarize the
history of certication at the beginning of period
t
. Finally, we denote
eqt(Ct, Ht)
a
consumer's belief in period
t
when faced with a product carrying certicate
Ct
and when
having observed history
Ht
. The following assumption on consumer beliefs formalizes the
12
This will be made more precise when formally introducing consumer beliefs. Granting a certicate
which is not contained in
D
is certainly perceived as cheating by consumers. Consequently consumers
believe to be faced with a worthless product and they will lose trust in the certier's credibility.
13
When certication is costly for the certier, there is a third reason: saving certication cost! As
already mentioned in Footnote 8 our analysis can be extended to
c > 0
, but this involves some troubling
case-by-case distinctions.
Informational opacity and honest certication 45
described behaviour.
14
Assumption 3.1.
Consumer beliefs
eqt(Ct, Ht)
satisfy
eqt(Ct, Ht) = eqCt
whenever
{τ <
t|qCτ6=qτ∨Cτ/∈ C ∪ {∅}} =∅
. Moreover
eqt(Ct, Ht) = 0
whenever
{τ < t|eqCτ6=
qτ∨Cτ/∈ C ∪ {∅}} 6=∅
and
eqt(Ct, Ht) = 0
whenever
Ct/∈ C
.
The assumption states that consumers trust the certier if capture was not observed in
the past. They however lose trust forever, once cheating is detected. Losing trust implies
that consumers, whatever the certier's claim may be, believe in being faced with a low
quality product.
With full disclosure, there are (at most) two types of bribing oers that can be made:
(C1, b)
and
(C2, b)
. Obviously, an oer
(C2, b)
is turned down by all types of producers, as
it is worth nothing. Hence, in the following we focus on oers
(C1, b)
and talk of a bribe
b
rather than
(C1, b)
. An oer
b
is accepted by high producer types whenever
b < f
. Low
quality producers accept any bribe
b < v
because acceptance will yield positive prots as
compared to zero prots for a rejection. In equilibrium, the certier assigns probability
e(f)
to the event that a producer is of high type, where
e(f)
is the producer's optimal
investment under full disclosure, derived from (3.1). We are interested in equilibria where
capture does not occur. In all these equilibria, a producer chooses his optimal investment
level while being aware of the fact that he will not receive an acceptable capture oer.
The acceptance probability
p(b|f)
of a bribing oer
b
given the a certication fee
f
is
given by
p(b|f) =
1,
if
b < f,
1−e(f),
if
f≤b < v,
0,
if
b≥v.
(3.4)
We denote by
ΠD(f) = P∞
t=1 δt−1πD(f) = πD(f)/(1−δ)
the certier's expected prot
from honest certication under disclosure rule
D
and fee
f
. The certier's expected
prot from oering bribe
b
is denoted by
b
ΠD(b|f)
and depends on whether the consumer
detected capture as follows: whenever
b < f
, all producer types will accept the bribe,
but only for low quality producers this is detected. Hence,
b
ΠF D(b|f) = b+e(f)δΠF D(f)
.
For
f≤b < v
, only low quality producers accept the bribe and
b
ΠF D(b|f) = (1−e(f))b+
e(f)(f+δΠF D(f))
. Whenever
b≥v
, all producers reject the bribe and the certier
obtains
b
ΠF D(b|f) = ΠF D(f)
.
If
b
ΠF D(b|f)
exceeds
ΠF D(f)
, the certier is actually better o becoming captured
14
Note that consumers do not lose trust in the certier when a product is awarded certicate
C2
,
although this should not happen in equilibrium. It is not necessary to include this case into consumer
beliefs, because any such event can only follow a non-protable deviation by the certier.
Informational opacity and honest certication 46
with the associated probability
p(b|f)
. We say that certication at a fee
f
is
capture
proof
if and only if
ΠF D(f)≥b
ΠF D(b|f)
(3.5)
for all
b
. Note that
b
ΠF D(b|f)
is increasing in
b
, both on
[0, f)
and
[f, v)
and it is constant
for
b≥v
. Furthermore
b
ΠF D(·|f)
is continuous at
b=f
.
15
Therefore, certier prots from
bribery are largest when
b
approaches
v
. Evaluating this yields the following proposition:
Proposition 3.2.
Under a full disclosure rule, an equilibrium satisfying Assumption 3.1
is capture proof. It exists if and only if
δ≥δF D(f)≡v
v+πF D(f).
(3.6)
The proposition highlights the crucial role the discount factor plays for the existence of
honest, i.e. capture proof, equilibria: the critical discount factor determines the relative
weights of the short run gain the bribe
b
and the long run loss of capture foregone
future prots from certication. To see this, note that all bribes
b < v
are accepted
with some positive probability and therefore, the largest possible short-run gain equals
v
. In the long run, a certier risks her per-period prots
πF D(f)
. As the certication fee
only enters via the per-period prot,
δF D(f)
depends on
f
only through
πF D(f)
, which
is concave in
f
. Therefore
δF D(f)
must be convex in
f
and minimized at the prot
maximizing fee
fF D
. The following corollary summarizes.
Corollary 3.1.
For any discount factor
δ≥δF D
there exists an interval of fees
[fl(δ), fh(δ)]
,
which sustains capture-proof certication under full disclosure, where
δF D ≡v
v+πF D .
(3.7)
In the right part of Figure 3.4 the set of feasible
(δ, f)
-combinations for full disclosure is
depicted.
An immediate consequence from this is that the static monopoly fee
fF D
can sustain
honest certication for all discount factors
δ≥δF D
. Alternatively one might ask the
question, what level of producer investment can be implemented via capture-proof certi-
cation with a full disclosure rule? The analysis follows the same arguments as above,
only that certier prots in the inequality of Proposition 3.2 are expressed in terms of
e
.
15
To see this compare the left and right limit:
limb%fb
ΠF D(b|f) = f+e(f)δΠF D(f) = (1 −e(f))f+
e(f)f+δΠF D(f)= limb&fb
ΠF D(b|f)
.
Informational opacity and honest certication 47
e
δ
eF D
δF D
e∗
1
1
δF D(e)
CP
f
δ
fF D
δF D
v
1
δF D(f)
CP
Figure 3.4: Capture proof combinations of
(e, δ)
resp.
(f, δ)
under full disclosure.
Proposition 3.3.
For any
δ≥δF D
there exists an interval of investment levels
[eF D
l(δ), eF D
h(δ)]
that can be implemented in a capture-proof equilibrium. A particular investment level
e∈[0, e∗]
can be implemented in a capture-proof equilibrium with full disclosure if and
only if
δ≥δF D(e)≡v
v+e·(v−k0(e))
(3.8)
.
The set of feasible
(e, δ)
-combinations is depicted in the left part of Figure 3.4.
Note that the rst-best investment level
e∗
can only be (virtually) implemented for
δ= 1
.
Whenever
δ < 1
, fees must be strictly positive in order to induce the certier to remain
honest. But then, the producer does not obtain the entire return on his investment.
Hence, it must be that
e < e∗
.
3.4.2 Capture under partial disclosure
We next argue that alternative noisy disclosure rules can improve certier credibility in
the sense that they increase the range of discount factors that allow for capture-proof
equilibria.
To gain an intuition for this consider condition (3.6). This condition summarizes the
trade-o between short-run gains and long-run losses. A larger prot
πD(f)
reduces the
critical discount factor and full disclosure guarantees maximal per-period prots. On
the other hand,
δF D(f)
is decreasing in
v
, which represents the the maximal bribe still
accepted by low-type producers and therefore the largest possible short-run gain from
capture. Using noisy disclosure the certier can aect the maximal short-run gain in
various dimensions. First of all, lowering the value of the best certicate or increasing
the value of the worst certicate (resp. the value of uncertied products) decreases the
Informational opacity and honest certication 48
gap between particular certication outcomes. This eect can be used to reduce the
maximal bribe which producers are willing to pay. Second, with noisy disclosure the
certier can sustain an outcome where both producer types demand certication. Upon
colluding with a producer type the certier foregoes the regular certication fee, which
reduces the eective gain from becoming captured.
Before analysing noisy disclosure rules, we have to reconsider the detection possibili-
ties by consumers. An implication of noisy rules is that consumers may hold probabilistic
beliefs about a product's quality. In order to simplify matters and because it suces to
make our point clear, we focus on partial disclosure rules as introduced in section 3.3.
Other noisy disclosure rules are discussed in section 3.5 and in the appendix. Under par-
tial disclosure, there are again two certicates
C1
and
C2
, where certicate
C1
is awarded
exclusively to high quality products and
C2
is awarded to high quality sellers with prob-
ability
1−α
and to low quality sellers with probability
1
. With an appropriately chosen
fee
f
, all producer types demand certication, hence there are no uncertied products
in equilibrium. The corresponding o-equilibrium belief is
q∅= 0
. The fact that
C1
is
awarded exclusively to high quality products makes eective trigger punishment possible.
In particular, it then suces that the certier is punished only if probability zero events
(a low quality product was awarded certicate
C1
) are observed.
The fact that capture detection is not possible if bribes are being paid in exchange
for the low value certicate
C2
, which is assigned to both high and low types, turns out
not to be crucial. This relies on the fact that in the equilibria under consideration all
producer types demand certication, hence receiving certicate
C2
is the worst possible
outcome. Certicate
C2
can therefore not be part of a protable bribing oer, as we will
argue later.
To specify consumer beliefs, let
ht= (Ct, qt)
denote the certication outcome in period
t
and, as before,
Ht= (h1, . . . , ht−1)
describes the history of certication before period
t
.
Consumer beliefs are specied as follows
Assumption 3.2.
Consumer beliefs
eqt(Ct, Ht)
satisfy
eqt(Ct, Ht) = eqCt
whenever
{τ <
t|
Prob
(C=Cτ|q=qτ) = 0 ∨Cτ/∈ C ∪ {∅}} =∅
. Moreover
eqt(Ct, Ht) = 0
when either
Ct/∈ C
or
{τ < t|
Prob
(C=Ct|q=qt) = 0 ∨Cτ/∈ C ∪ {∅}} 6=∅
.
Note that in contrast to Assumption 3.1, consumers trust the certier unless probability
zero events occurred in the past. Because the crucial bribe entails certicate
C1
, which is
exclusively awarded to high quality producers, this essentially says that consumers stop
trusting the certier whenever they nd a low quality product carrying certicate
C1
.
Bribing oers can now be of two kinds:
(C1, b)
and
(C2, b)
. Oer
(C2, b)
is never
benecial. It would only be accepted for
b < f
since any producer receives at least the
Informational opacity and honest certication 49
certicate
C2
when applying for (honest) certication and the certier gets
f
from any
producer who is honestly tested. Thus, we can focus on bribing oers of the form
(C1, b)
,
which we will simply refer to as
b
. Recall that certicate
C1
can only be awarded to
high quality products. Hence,
qC1=v
. To simplify notation, denote
V2
the value of a
C2
-certied product, i.e.
V2=qC2
. Furthermore, recall that
α
is the probability with
which a high type is awarded
C1
.
A bribe
b
is accepted by low types whenever
V2−f < v −b
. High quality producers
accept
b
if
αv + (1 −α)V2−f < v −b
. Denote
e(α)
the equilibrium investment.
16
Then
bribery acceptance probabilities are
p(b|α, f) =
1
if
b < f + (1 −α)(v−V2),
1−e(α)
if
f+ (1 −α)(v−V2)≤b < f + (v−V2),
0
if
b≥f+ (v−V2).
Let
ΠP D(α, f)
denote the expected prot from employing a partial disclosure rule
and honestly disclosing information in each period. The corresponding expected certier
prots from bribing oer
b
are
b
Π(b|α, f) =
b+e(α)δΠP D(α, f)
if
b < f + (1 −α)(v−V2),
(1 −e(α))b+e(α)f+δΠP D(α, f)
if
f+ (1 −α)(v−V2)≤b < f + (v−V2),
ΠP D(α, f)
if
b≥f+ (v−V2).
Note that whenever high types accept the bribery oer, this is not perceived as cheat-
ing because the certicate then matches the observed quality level. Again,
b
Π(b|α, f)
is
increasing in the respective subintervals. But the function now exhibits a downward-
jump at
b=f+ (1 −α)(v−V2)
. The reason is that high types are willing to accept
bribes strictly larger than the certication fee
f
to avoid the lottery between the good
and the bad certicate. Therefore, at least locally, the certier is better o bribing all
producers instead of only the low types as it was the case with full disclosure. Further-
more, the maximal bribe that is accepted by at least some types is now
f+v−V2
, which
is weakly lower than under full disclosure, where the maximal bribe is
v
.
17
The analysis
of condition (3.5) yields the following proposition.
16
The investment decision does not depend on the fee because in equilibrium, all types apply for
certication and therefore pay
f
anyway. The expected producer prot is
e(αV1+ (1 −α)V2) + (1 −
e)V2−f−k(e)
and consequently the optimal investment level depends on
α
but not on
f
.
17
In order to have all producer types demand certication it has to hold that
f≤V2
. Consequently
f+v−V2≤v
.
Informational opacity and honest certication 50
Proposition 3.4.
With partial disclosure, an equilibrium satisfying Assumption 3.2 is
capture-proof. It exists if and only if
δ≥δP D(α, f)≡max δl(α, f), δl,h(α, f),
(3.9)
where
δl(α, f) = v−V2
v−V2+f
and
δl,h(α, f) = (1−α)(v−V2)
(1−α)(v−V2)+(1−e(α))f
.
The result gives a lower bound on the discount factor
δ
to guarantee existence of a
capture-proof equilibrium with partial disclosure. The critical discount factor
δP D(α, f)
depends on the parameters in the way how they aect short-run gain and long-run loss
from capture and on which producer types accept the bribing oer that yields largest
deviation prots. The term
δl(α, f)
refers to the case where the largest threat stems from
bribes accepted only by low types. The numerator
v−V2
is the eective bribe, dened
as the bribery payment minus foregone payments. In the denominator we nd again the
eective bribe and the per-period prot
f
, reecting the long-run loss from capture. The
term
δl,h(α, f)
refers to the case where the largest threat stems from bribes accepted by
all types. Here the eective bribe is
(1 −α)(v−V2)
. Since the long-run prot is only at
stake if quality is low, long-run prots are lost with probability
(1 −e(α))
. Although the
classical trade-o between short-run gain and long-run loss, that we already identied
for full disclosure, prevails, the derivation of the maximal short-run gain is more involved
for partial disclosure.
From Proposition 3.4 we identify a third notable dierence between capture under full
and noisy disclosure. Short-run gains from capture can be reduced due to the dierent
equilibrium structure: all producers certify in equilibrium which implies that the certier
always loses fee payments if he is captured. Therefore, a larger fee
f
not only increases
the long-run losses but at the same time reduces the short-run gains from capture.
It is now straightforward to see that
δP D(α, f)
is decreasing in the certication fee
f
. This implies that for any partial disclosure rule (i.e. any
α
) the threat of capture
is lowest when
f
is maximal. To ensure that all producer apply for certication,
f
is
not allowed to exceed
V2
. It is therefore optimal to set
f=V2
, which leaves low quality
producers with an expected prot of zero. The following corollary summarizes.
Corollary 3.2.
With partial disclosure a capture-proof equilibrium satisfying Assumption
3.2 exists if and only if
δ≥δP D(α)≡max δl(α), δl,h(α),
(3.10)
where
δl(α) = v−e(α)(v−k0(e(α)))
v
and
δl,h(α) = 1
1+e(α)
.
Informational opacity and honest certication 51
Corollary 3.2 allows us to reduce the problem of nding the critical discount factor
for partial disclosure to the one-dimensional problem of nding the optimal level of
α
,
the probability that high quality is revealed. In fact,
δP D(α)
depends on
α
only through
the equilibrium value for producer investment
e(α)
. The set of investment levels that
can be implemented by partial disclosure is
(0, e∗)
, the same set as for full disclosure.
Dening
δP D ≡minαδP D(α)
allows us to formulate the analogue of Proposition 3.3 for
partial disclosure.
Proposition 3.5.
For any
δ≥δP D
exists an interval of investment levels
[eP D
l(δ), eP D
h(δ)]
that can be implemented in a capture-proof equilibrium. A particular investment level
e∈[0, e∗]
can be implemented in a capture-proof equilibrium with noisy disclosure if and
only if
δ≥δP D(e) = max δP D,l(e), δP D,l,h(e)
(3.11)
where
δP D,l(e) = v−e(v−k0(e))
v
and
δP D,l,h(e) = 1
1+e
.
Proposition 3.5 makes implementation of capture-proof equilibrium under full and
partial disclosure directly comparable. Before investigating this in the next section we
want to highlight some properties of the function
δP D(e)
. Writing
e(v−k0(e)) = πP D(e) =
f
the term
δP D,l(e)
can be expressed as
(v−f)/(v−f+πP D(e))
. This resembles the
trade-o between short-run gain and long-run loss, already identied above. Only the
maximal short-run gain with partial disclosure is the maximal bribe minus foregone
regular payments. The same trade-o leads to
δP D,l,h(e)
, which is however independent
of the producer's cost function
k(e)
. The maximal bribe that is accepted from both
producer types in particular must be accepted from high quality producers. For them, the
dierence between the sure certicate
C1
and the lottery faced when certifying honestly
matters. This dierence is closely related to a producer's investment incentives. In fact
one can show that the maximal bribe equals
v−k0(e)
. Now both short-run gain and
long-run loss depend in a similar way on the investment incentives
18
and consequently
the fraction
δP D,l,h(e)
does not depend on the producer's cost function anymore.
Which of the two terms,
δP D,l(e)
and
δP D,l,h(e)
, is now larger?
δP D,l,h(e)
is decreasing
in
e
, starting at
1
for
e= 0
towards
1/2
for
e= 1
. On the other hand
δP D,l(e)
is convex in
e
with a unique minimum at
e=eF D
. Furthermore
δP D,l(0) = δP D,l(1) = 1
. Therefore,
δP D
is either
δP D,l(eF D)
, that is the minimum of
δP D,l
, or it is the intersection of both
fractions lying to the right of
e=eF D
. Figure 3.5 illustrates the two cases, the latter in
its left part.
18
As discussed, the short-run gain equals
v−k0(e)
. The long-run loss is the per-period prot, which
was already shown to be
e(v−k0(e))
.
Informational opacity and honest certication 52
e
δ
δP D,l,h
δP D,l
ePD
δP D
eFDe∗
CP
1
e
δ
δP D,l,h
δP D,l
ePD=eF D
δP D
e∗
CP
1
Figure 3.5: Capture-proof
(e, δ)
-combinations for low (left) and high (right) marginal
costs
k0
at
e=eF D
.
3.4.3 Sub-optimality of full disclosure
In the previous sections, we have identied the conditions under which capture-proof
equilibria exist for full disclosure and a special class of noisy disclosure rules. These
conditions are expressed in terms of the critical discount factors
δF D
and
δP D
. It is the aim
of this study to show that opaque disclosure rules can be used by the certier to improve
her credibility. Comparing the critical discount factors
δF D
and
δP D
is short-hand for
comparing the entire sets of
(e, δ)
-combinations, for which a capture-proof equilibrium
exists with the respective disclosure rule. We are going to prove in this section that the
two sets are dierent and, more importantly, that the respective set for full disclosure
is contained in the respective set for partial disclosure. Consequently there exists an
intermediate range of discount factors for which a capture-proof equilibrium with full
disclosure does not exist. Yet, it is still possible to sustain capture-proof equilibria with
partial disclosure.
As discussed earlier, the key trade-o for implementing a capture-proof equilibrium
is that of short-run gain versus long-run loss. Either disclosure rule leads to a per-period
prot of
π(e) = e(v−k0(e))
when implementing eort level
e
, the potential long-run
loss is therefore the same. However, with partial disclosure, the short-run gain from
being bribed only by low quality producers is
v−f
, as compared to
v
for full disclosure.
Therefore, partial disclosure accounts for lower discount factors. This presumes that
Informational opacity and honest certication 53
the largest threat of capture indeed stems from low quality producers. This is generally
true for full disclosure, but ceases to hold for partial disclosure. When the maximum
threat stems from a bribe accepted by all producer types, the long-run loss is reduced.
Obviously, this is perceived as fraud only when the product quality was low. As a result,
per-period prots are lost with probability
1−e
only. On the other hand, the bribe
need to be lower in order to be accepted by high quality producers, which again reduces
the short-run gain. The following proposition proves that the latter eect outweighs the
former.
Proposition 3.6.
It holds that
δP D < δF D
. For any
δ∈[δP D, δF D]
, a capture-proof
equilibrium can only be sustained applying a noisy disclosure rule. Furthermore, for any
δ≥δF D
, we have that
[eF D
l(δ), eF D
h(δ)] ([eP D
l(δ), eP D
h(δ)]
.
Proposition 3.6 shows our main result, namely that opacity can be used as a tool
to improve certier credibility. For any level of producer investment
e
, the range of
discount factors that allow for capture-proof implementation of
e
is strictly larger for
partial disclosure as compared to full disclosure. Similarly, for any discount factor
δ
,
the set of investment levels that are implementable in a capture-proof equilibrium with
partial disclosure is strictly larger than the corresponding set for full disclosure. There-
fore, the superiority of partial disclosure takes place along two dimensions. Figure 3.6
displays these dierences. The dark-grey area corresponds to
(e, δ)
-combinations that
can be implemented as a capture-proof equilibrium under full disclosure. The light-grey
area shows the
additional
(e, δ)
-pairs that allow for implementation in capture-proof
equilibrium under partial disclosure.
In Section 3.3, we show that a certier would always want to implement
eF D
as this
maximizes her per-period prots. With full disclosure, this is only possible when
δ≥δF D
.
Partial disclosure allows for capture-proof equilibria also for lower discount factors. It is
remarkable that, at least for a range of discount factors, this can be achieved without
waiving any prots. To see this, denote
˜
δ(πF D)
the smallest discount factor, such that
a capture-proof equilibrium is sustained and achieves per-period prots of
πF D
. The
following corollary is an immediate consequence of Proposition 3.6.
Corollary 3.3.
It holds that
˜
δ(πF D) = max v−πF D
v,1
1 + eF D < δF D.
Informational opacity and honest certication 54
e
δ
eP D
δP D
˜
δ(πF D)
eF D
δF D
e∗
CP
1
Figure 3.6: Dark-grey: capture-proof certication with full disclosure. Light-grey: (ad-
ditional) capture-proof certication with noisy disclosure.
3.4.4 Welfare properties of partial disclosure
In this subsection, we study welfare properties of capture-proof equilibria with partial
disclosure. When
˜
δ(πF D) = δP D
we also have
δP D = (v−πF D)/v
. In this case, the largest
threat of capture stems from low quality producers, i.e. the largest deviation prot for the
certier is achieved for
b=v
. Then the certier can still achieve the maximal per-period
prots
πF D
in a capture-proof equilibrium for any
δ≥δP D
, which implies implementing
e=eF D
.
This is however not true when
˜
δ(πF D)> δP D
. As can be seen from Figure 3.6, for
discount factors below
˜
δ(πF D)
the prot maximizing level of investment
eF D
is no longer
capture-proof implementable. Instead only larger values of producer investment can
be implemented when
δ∈[δP D,˜
δ(πF D))
. To provide an intuition for this, note the
following: Bribing oers
b
that are accepted by all producer types pose the largest threat.
Now, implementing a larger
e
leads to a reduction in
V2
, as otherwise prots would
increase beyond
πF D
. To incentivize producers to make larger investments, the certier
therefore has to increase
α
. As now shown, for high quality producers the dierence in
expected prots between the lottery of the certication process and the sure certicate
v
is reduced.
19
This in turn lowers the maximum bribe they are willing to pay for capture
and therefore reduces the short-run gain for the certier from any such oer. From a
19
Honest certication yields an expected payo
αv +(1−α)V2
. This value is reduced when
α
increases
and
V2
decreases at the same time.
Informational opacity and honest certication 55
welfare perspective this increase in investment is benecial. Social welfare is given by
e·v−k(e)
in each period. The rst-best investment level
e∗
was shown to be strictly
larger than
eF D
and welfare is strictly increasing on
[0, e∗]
. Implementing certication
with partial disclosure for discount factors
δ∈[δP D,˜
δ(πF D)]
therefore increases social
welfare as compared to doing so for larger levels of the discount factor. Put dierently, a
severe threat of capture increases welfare. We summarize this in the following proposition.
Proposition 3.7.
Assume
˜
δ(πF D)> δP D
. For intermediate levels of the discount factor,
i.e.
δ∈[δP D,˜
δ(πF D))
, only investment levels that are strictly larger than
eF D
can be
capture-proof implemented with partial disclosure. This leads to increased social welfare.
3.5 Discussion
We have analysed the eects of reputational concerns on optimal disclosure rules from the
point of view of a monopolistic certier. Our main nding is that if capture is an issue,
a certier benets from resorting to coarser certication in order to reduce the threat
of capture. In particular, for medium discount factors, sustaining honest certication
is impossible if information is fully disclosed whereas it is still possible if information
disclosure is noisy.
Implications of our analysis are manifold. First of all we provide a novel explanation
for the occurrence of imperfect testing. In many papers on e.g. rating agencies (examples
include Mathis et al. (2009) and Bolton et al. (2012)) imperfect testing is exogenously
given, whereas here it arises in equilibrium. An empirical implication is that for low
discount factors we expect disclosure to be coarser. This is consistent with the casual
observation that certication in markets with low volume, such as wine, technical inspec-
tions or eco-labels often involves only a few dierent certicates. On the other hand,
the high volume rating market exhibits a rather wide variety of dierent but still coarse
certicates per rating agency.
Our ndings also have important policy implications. Politics tend to push certiers
to precisely reveal information. Our results suggest that doing so may have unforeseen
consequences for the functioning of those markets. In particular, reputation building and
the resistance to capture is impeded. Similarly, regarding the current nancial crisis,
forcing rating agencies to issue more precise information might even exacerbate capture
problems.
We have demonstrated our results in a highly stylized model, but the intuition be-
hind our results is general. In particular, they carry over to more than only two quality
specications. The appendix provides examples for honest equilibria for the case of more
Informational opacity and honest certication 56
than two quality specications.
Our restriction to a particular class of noisy disclosure rules is without loss of gen-
erality. First, oering various coarse certicates generates incentives for the certier to
always oer the best among the noisy certicates in a bribing oer. This will be accepted
(at least by low quality producers) in order to avoid a lottery that includes the worst
certicates. As deviations of this kind remain undetected they will occur with certainty,
and this destroys the equilibrium. Second, disclosure rules that do not allow for unam-
biguous detection of deviations call for a dierent type of trigger beliefs. Consumers lose
trust in the certier whenever they rst detect low quality sold with the best certicate.
This leads to punishments even if collusion did not take place. The harsher punishments
makes it impossible to sustain capture proof equilibria for low discount factors. Proposi-
tion 3.8 in the appendix makes this statement precise.
Finally, we have used a specic extensive form to model capture. More sophisticated
forms to study imply non-uniform bribing oers, e.g. menus, to elicit the producers'
private information. A possible extension is the analysis of posterior bribing, i.e. after
the certier learned
q
. Also, producers might be equipped with tools to to signal their
private information. The exact extensive form may well aect parts of the analysis albeit
not the main nding.
Informational opacity and honest certication 57
3.6 Appendix
3.6.1 Proofs
Proof of Lemma 3.1.
Follows immediately from the arguments given in the text.
Proof of Lemma 3.2.
Following the arguments given in the text the certier maximizes
(3.2). Recall that we assume
k000(·)≥0
, which ensures that this prot function is concave
in
e
, thus the rst-order condition is sucient for an optimum. This rst-order condition
is
0 = v−k0(e)−ek00(e)
. Dene
Ψ(e) = v−k0(e)−ek00(e)
. We have
Ψ(0) = v > 0
and
Ψ(1) = v−k0(1) −k00(1) ≤0
by our assumptions on
k(·)
. Furthermore
Ψ
is strictly
decreasing due to strict concavity of
k(·)
. Hence there exists a unique
eF D
such that
Ψ(eF D) = 0
, which consequently is the unique maximizer of the certier prot. The
formulas for
eF D
and
fF D
follow easily from the formulas above.
Proof of Proposition 3.1.
First of all a disclosure rule can potentially lead to four dif-
ferent subgames: (1) no producer demands certication, (2) only low quality producers
demand certication, (3) only high quality producers demand certication, and (4) all
producers demand certication. Note that we do not explicitly consider mixed strategies
by producers. The reason is that any outcome where some producers randomize their
certication decision can be replicated by a disclosure rule that adds the respective prob-
abilities for not certifying to the probabilities of remaining uncertied though paying for
certication. To see this, assume type
i
chooses to certify with probability
γ∈(0,1)
.
Now multiply every
αi
by
γ
and increase the probability of remaining uncertied ap-
propriately. After changing the fee from
f
to
γf
, it is easy to see that this adjusted
disclosure with the reduced fee leads to the same investment incentives and also to the
same equilibrium prices for (un-)certied products and the certiers prot is unchanged.
Case (1) trivially leads to zero prots and the claim is proven.
Case (2) leads to consumers paying zero in equilibrium for certied products.
20
To make
low quality producers pay for certication we consequently must have
f= 0
which
leads to zero prots and proves our claim also in this case.
Case (3) can be analysed as follows: If only high types certify, rational behaviour by
consumers dictates that a certied product is sold at a price
v
. Uncertied products
however can be of either high or low quality and have some price
q∅∈[0, v)
.
20
A disclosure leading to this particular subgame is given by
C={C}, αl= 1
and
αh= 0
.
Informational opacity and honest certication 58
A producer's investment decision is given by the solution of
max
eeX
k
α1
kv+ (1 −X
k
α1
k)q∅−f+ (1 −e)q∅−k(e),
which yields the following rst-order condition for producer investment:
X
k
α1
k(v−q∅)−f=k0(e).
Rewriting this constraint in terms of induced investment yields
f=v−k0(e)−(1 −
Pkα1
k)(v−q∅)−q∅
. Now we have for the certier prot
πD(f) = e(f, D)·f=e·v−k0(e)−(1 −X
k
α1
k)(v−q∅)−q∅≤e·(v−k0(e)) ≤πF D.
This proves the claim for case (3).
Finally consider case (4): When both producer types demand certication, the resulting
certier prot in the subgame is
πD(f) = f
. The price at which a product holding
certicate
Ci
can be sold is
qCi=v·eαh
i
eαh
i+ (1 −e)αl
i
.
Uncertied products are sold at price
q∅=v·e1−Piαh
i
e1−Piαh
i+(1−e)1−Piαl
i
. A producer's
investment decision follows from maximizing his expected payo from certication, given
by
e· X
i
αh
iqCi+ 1−X
i
αh
i!q∅!+ (1 −e)· X
i
αl
iqCi+ 1−X
i
αl
i!q∅!−f−k(e).
The resulting investment constraint is
k0(e) = X
i
(αh
i−αl
i)(qCi−q∅).
(3.12)
On the other hand, from the formula given for
qCi
we have
eαh
iqCi+(1−e)αl
iqCi=evαh
i
.
Similarly
e(1−Piαh
i)q∅+(1−e)(1−Piαl
i)q∅=ev(1−Piαh
i)
. Summing those expressions
Informational opacity and honest certication 59
yields
X
ieαh
iqCi+ (1 −e)αl
iqCi+e(1 −X
i
αh
i)q∅+ (1 −e)(1 −X
i
αl
i)q∅=ev.
(3.13)
Rewriting the left hand side of equation (3.13) yields
eX
i
(αh
i−αl
i)(qCi−q∅) + X
i
αl
iqCi+ 1−X
i
αl
i!q∅=ev.
(3.14)
Finally, to make all producer types demand certication we must have in particular
f≤X
i
αl
iqCi+ 1−X
i
αl
i!q∅
(3.15)
i.e. low quality producers ecpected payo from certication must be non-negative.
21
From this we can derive an upper bound on certier prots:
πD(f) = f
(3.15)
≤X
i
αl
iqCi+ 1−X
i
αl
i!q∅
(3.14)
=ev −eX
i
(αh
i−αl
i)(qCi−q∅)
(3.12)
=ev −ek0(e) = e(v−k0(e)).
But
e(v−k0(e))
is the prot from implementing eort level
e
optimally with a full dis-
closure rule, therefore we have proven
πD(f)≤πF D
.
Proof of Proposition 3.2.
In any equilibrium in which Assumption 3.1 holds capture
may not take place, since otherwise the beliefs of consumers are not consistent with the
behaviour of the certier. Hence, condition (3.5) must be satised for all b. As mentioned
in the text, certier prots from deviating
b
ΠF D(b|f)
are largest for
b
approaching
v
.
Taking this limit yields
lim
b%vb
ΠF D(b|f) = (1 −e(f))v+e(f)·f+δΠF D(f)
= (1 −e(f))v+πF D(f) + δ
1−δe(f)πF D(f)
= (1 −e(f))v−δ
1−δ(1 −e(f))πF D(f) + ΠF D(f).
21
More conditions are required in subgame where all producer types demand certication, but the one
presented her is the only required for our proof.
Informational opacity and honest certication 60
Condition (3.5) is thus equivalent to
(1 −e(f))v≤δ
1−δ(1 −e(f))πF D(f).
Rearranging this expression yields that condition (3.5) is satised if and only if
δ≥δF D(f)≡v
v+πF D(f).
Proof of Proposition 3.3.
We rst argue how condition (3.6) can be translated to-
wards (3.8). Recall
πF D(f) = e(f)·f
and optimal investment by producers requires
k0(e) = v−f
. Replacing
f
by
v−k0(e)
yields (3.8). All other statements are straight-
forward reformulations of Proposition 3.2 and Corollary 3.1.
Proof of Proposition 3.4.
In any equilibrium in which Assumption 3.2 holds capture
may not take place, since otherwise the beliefs of consumers are not consistent with the
behaviour of the certier. Hence, condition (3.5) must be satised for all b. We compute
the respective critical discount factors. Taking the limit of
b
ΠD(b|f)
as
b
approaches
f+ (1 −α)(v−V2)
we get
lim
b%f+(1−α)(v−V2)b
ΠD(b|f) = f+ (1 −α)(v−V2) + e(α)δΠP D(f)
=f+ (1 −α)(v−V2) + e(α)δ
1−δf
= (1 −α)(v−V2)−δ
1−δ(1 −e(α))f+ ΠP D(f).
Consequently this limit lies below
ΠP D(f)
if and only if
(1 −α)(v−V2)≤δ
1−δ(1 −e(α))f,
respectively whenever
δ≥δl,h(α, f) = (1 −α)(v−V2)
(1 −α)(v−V2) + (1 −e(α))f.
Informational opacity and honest certication 61
Similarly the limit of
b
ΠD(b|f)
as
b
approaches
f+ (v−V2)
can be rewritten as follows
lim
b%f+(v−V2)b
ΠD(b|f) = (1 −e(α)) ·(f+ (v−V2)) + e(α)f+δΠP D(f)
= (1 −e(α))(v−V2)−δ
1−δ(1 −e(α))f+ ΠP D(f).
Therefore
limb%f+(v−V2)b
ΠD(b|f)≤ΠP D(f)
if and only if
δ≥δl(α, f) = v−V2
f+v−V2
.
Because capture-proofness requires
b
ΠD(b|f)≤ΠP D(f)
for all
b
, (3.9) follows.
Proof of Corollary 3.2.
As discussed in the text, the certier may set
f=V2
to
minimize the threat of capture. We consider
δl(α, f)
rst. Making use of
f=V2
allows
us to simplify it to
(v−V2)/v
. From the proof of Proposition 3.1 we get
V2=e(v−k0(e))
and therefore
δl(α) = v−V2
v=v−e(α)(v−k0(e(α)))
v.
Now consider
δl,h(α, f)
. With
f=V2
we may rewrite
δl,h(α, f) = (1 −α)(v−V2)
(1 −α)(v−V2) + (1 −e(α))V2
By Bayesian updating we have
V2=v·(1 −α)e(α)/1−αe(α)
in equilibrium, which
implies
v−V2=v·1−e(α)(1−αe(α)
. Replacing
V2
and
v−V2
accordingly yields
(1 −α)(v−V2)
(1 −α)(v−V2) + (1 −e(α))V2
=1
1 + e(α).
Proof of Proposition 3.6.
Recall, that with full disclosure the critical discount factor
is
δF D(e) = v
v+πF D(e)=v
v+e(v−k0(e))
and this term is minimized for the prot maximizing
eort
e
, yielding
mineδF D(e) = v
v+πF D
. For all
e∈(0, e∗)
we have
v−e(v−k0(e))
v< δF D(e)
.
To see this:
v−e(v−k0(e))
v< δF D(e) = v
v+e(v−k0(e)) ⇔(e(v−k0(e)))2>0.
Also
1
1 + e< δF D(e) = v
v+e(v−k0(e)) ⇔ek0(e)>0
Informational opacity and honest certication 62
Therefore also
max{1
1+e,v−e(v−k0(e))
v}< δF D(e)
for all
e∈(0, e∗)
and hence we can dene
δP D := min
emax 1
1 + e,v−e(v−k0(e))
v
and it follows that
δF D > δP D
. Since both
δP D,l(e)< δF D(e)
and
δP D,l,h(e)< δF D(e)
the
last statement follows immediately.
Proof of Proposition 3.7.
When
δP D <˜
δ(πF D)
we must have
˜
δ(πF D) = δP D(eF D) =
1
1+eF D
. Since
1/1 + e
decreases in
e
we have
δP D(e)>˜
δ(πF D)
for any
e < eF D
. Con-
sequently we must have
δP D(e)<˜
δ(πF D)
on some interval
[eF D,ˆe]
. This proves our
result.
3.6.2 Extensions
Examples for more than two levels of quality
Let quality levels be
{0,0.5,1}
and
P(q= 0.5|e) = P(q= 1|e) = e/2
. Consequently
P(q= 0|e) = 1 −e
. The cost of eort is
k(e) = e2/2
. If we restrict the analysis
to deterministic disclosure rules, it is straightforward to show that full disclosure with
a fee
f= 3/8
maximizes certier prots. With this fee both quality levels
0.5
and
1
get certied in equilibrium. Using the same line of argument as in the main text, this
disclosure rule can be sustained as a capture-proof equilibrium whenever
δ≥16
19
.
A cut-o disclosure rule that certies any product whose quality exceeds
0
, but does
not distinguish any further, achieves the same static prot as the mentioned full disclosure
rule. However, the largest possible bribe is then not equal to
1
since no certicate which
yields a price of one is available. Instead, the best certicate yields
3/4
, the value of
a certied product. Consequently, a capture-proof equilibrium with this disclosure rule
exists whenever
δ≥16
20
. While prots remain the same, the largest acceptable bribe is
lowered.
Alternative disclosure rules for the two-quality case
Proposition 3.8.
For any
δ < δF D
and any disclosure rule which is such that the highest
certicate's value is dierent from
v
, no capture-proof equilibrium exists.
Proof.
We restrict the proof to the following simple disclosure rule
22
: there are two
22
For any other rule, the argument is the same for selling the best certicate in a capture oer to the
low quality producer. However, there are even more feasible bribing oers, which make it even harder
to resist the threat of capture.
Informational opacity and honest certication 63
certicates,
C1
and
C2
, where high quality always receives
C1
and low quality receives
C1
with probability
α∈(0,1)
. Denote
V
the value of
C1
, certicate
C2
is always worth
zero (in equilibrium). The rst-order condition for producer investment reads as
k0(e) = (1 −α)V
and from Bayes' rule we have
V=ve
e+α(1 −e).
Thus, to implement a particular
e
, the certier has to set
23
α=e(v−k0(e))
e(v−k0(e)) + k0(e)
The fee must be such that low quality producers are willing to get their product certied,
i.e.
f≤αV
.
When a purchased product with certicate
C1
turns out to be of low quality, con-
sumers cannot be sure whether this was due to bad luck or to a captured certier.
Appropriate trigger beliefs have to be such that the certier is punished whenever low
quality is sold with certicate
C1
. This can well happen without any deviation by the
certier. The probability of entering punishment, absent any deviation, is
p= (1 −e)α
and expected prots from honest play are given by
Πh(α, f) = f+ (1 −p)δf + (1 −p)2δ2f+. . . =f
1−(1 −p)δ.
The maximal bribe is given by
b≈(1−α)V+f
, where only low quality producers accept
it. The prot from making such an oer is
Π(b|f, α) = (1 −e)b+e(f+δΠh(α, f))
We have
Π(b|f, α)≤Πh(α, f)
for
b→(1 −α)V+f
whenever
δ≥(1 −e)b−(1 −e)f
(1 −e)(1 −p)b−epf =b−f
1−(1 −e)αb−eαf
This is both increasing in
b
and in
f
, such that the largest threat is exercised for
f=αV
23
Note that
lime→0α
equals
1
whenever
k00(0) = 0
and otherwise equals
v
v+k00 (0)∈(0,1)
, that is in the
latter case not all
α
are implementable.
Informational opacity and honest certication 64
and
b=V
, which results in the condition
δ≥1
1 + eα.
We have
1
1+eα ≥v
v+e(v−k0(e))
if and only if
v−k0(e)≥vα ⇔1≥e.
Hence, for all
e
to be implemented, this is only possible with a noisy rule without sure
high quality certicate, when this is also possible using a full disclosure rule.
4 Non-persuasiveness mitigates competition in
the market for life insurances
Chapter Abstract
In this article, we analyse a duopolistic insurance market with bargainers and price
takers. Firms compete in two stages. We set up a model in which rms, but not
applicants themselves, have the option to screen an individual's risk of incurring a
loss. In equilibrium, insurers are asymmetrically informed on individual applicants.
When an informed insurer is able to signal high risk, she is said to be
persuasive
.
If not, competition is invalidated on high risk bargaining applicants, resulting in
strictly positive expected prots for the uninformed rm. It is shown that all equi-
libria are non-persuasive, list prices may be asymmetric and both rms expect to
earn strictly positive prots.
65
Non-persuasiveness mitigates competition in the market for life insurances 66
4.1 Introduction
In insurance markets that provide economic coverage for a biometric risk (e.g. life, health
or disability insurances), contract oers are based on the outcome of a screening, which
is conducted by an insurance rm. More specically, if a person wishes to buy insurance,
he is required to provide inter alia detailed information on his medical record. Based on
the screening result, the person is either accepted and oered a pre-specied contract, or
rejected.
The medical record is not the only basis for the decision: consumers are furthermore
required to provide information on their application history. In particular, insurers ask
whether a consumer has applied for this kind of insurance before and if so, what the
outcome was. The information provided is twofold: rst, if the answer is `yes', the
respective insurer will be aware of being faced with a person in search for the best oer,
a so called bargainer. Second, information about previously received oers may help the
rm to engage in competition without having to incur costs for screening.
1
Another common method to collect the same kind of information is by means of
an information exchange system. In these systems acceptance/rejection decisions are
made public to competitors. Examples for these systems are the German Hinweis- and
Informationssystem (H.I.S.) and the U.S. Medical Information Bureau (MIB).
In the present study, we analyse this specic market with two competing insurers. In
particular, we are interested in the role of bargainers versus price takers (persons that
apply only once) in determining industry prots. Surprisingly, we nd that the presence
of bargainers helps both rms to positive expected prots. By contrast, as the share
of price takers converges to one, competition works eectively and rms do not make
positive prots.
These results also help to explain the occurrence of information sharing systems: with
bargainers being present, both rms earn positive prots. By contrast, if information
about acceptance/rejection decisions is not shared, both rms opt to screen applicants,
producing a perfectly competitive market environment. This outcome is counterintuitive
only on rst sight because an insurer shares information also about the bargaining
behaviour of applicants, the competitor understands he will not earn positive prots,
and consequently forgo costly screening.
2
It is not obvious why rejections happen at all. With informed consumers, higher risk
1
Note that there are no incentives for misreporting, as far as the application history and the medical
record are concerned. If misreporting is detected by the time a claim is submitted, or earlier, an insurer
may rescind the contract without having to pay back any paid premiums.
2
This result is immediate and its analysis considered in the appendix.
Non-persuasiveness mitigates competition in the market for life insurances 67
types would not remain uninsured, since they would be willing to pay more for their
insurance. On the other hand, rms dispose of advanced technological means to estimate
the statistical risk of a loss. Although a consumer is aware of his medical condition, he
usually does not have the necessary experience to evaluate his own risk. A relatively
young stream of literature considers the possibility that insurers are better informed
than consumers. In particular, Villeneuve (2005) studies a model in which applicants
can infer their default probabilities only from the contract oers they receive. This
signalling structure provides an explanation for the actual occurrence of rejections: since
an applicant's willingness to pay for insurance directly depends on the beliefs he holds
about his risk type, an informed rm may not be able to persuade consumers that their
risk is high by setting high prices. In that case, the only way for a rm to convincingly
signal high risk may be to reject the applicant.
In this study, we analyse a model in which rms, but not applicants themselves, have
the option to screen an individual's risk of incurring a loss. More precisely, the two-
stage competition game proceeds as follows. Firms compete in list prices. Persons who
wish to purchase insurance apply to the rm that oers the best list price. There, the
person's medical prole is screened if the rm spends a small amount (this decision is not
observable). Depending on the screening outcome (or the belief about the personal risk
of the applicant), an applicant is considered eligible for insurance at the list price, or not,
in which case he is rejected. Price takers consider only this oer, while bargainers induce,
in a next stage, competition between both insurers. Before entering the competition for
bargainers, the competitor chooses whether to screen the applicant, or not, and observes
the applicant's previously received oer.
In equilibrium, the rm to which applicants apply rst (the winner of the list price
competition), screens his risk prole, while her competitor does not. This implies that
the acceptance/rejection decision transmits information to both the competitor and the
applicant.
One eect, which we shall call
persuasiveness
of the informed rm, is central to our
analysis: consider the rm that has won the list price competition and assume she faces
a low risk applicant. By accepting this applicant, she reveals the true risk type. This
earns him the list price minus potential compensations in case of damage, but only if the
applicant is a price taker. If he is a bargainer, the true risk is revealed and competition
works eectively, leaving the insurer without positive prots.
He could also reject the applicant. If he is a price taker, the person remains uninsured
and the rm earns nothing. But in case he is a bargainer, both the competitor and the
applicant believe that the person's risk is high. In the second round, the competitor oers
Non-persuasiveness mitigates competition in the market for life insurances 68
the fair price for the high risk type. That implies that the informed rm can benet, and
earns strictly positive prots.
We say that the insurer is
persuasive
whenever she weakly prefers to accept the
low risk applicant to rejecting him. If not, then in equilibrium, a high risk bargaining
applicant is rejected twice by the informed rm
3
for the benet of her competitor. The
insurer would like to undercut her competitor's oer, but she can't: doing so induces the
applicant to believe to have a low default risk. Note that the term persuasiveness refers
only to bargaining applicants. High risk price takers by contrast are always rejected, in
the sense of Villeneuve (2005).
Our rst main nding is that signalling is never persuasive in equilibrium. The reason
is that, in order to give persuasion, list prices would need to be relatively high. In
particular, the lowest list price would have to exceed some critical threshold. When
persuasion is given, the standard Bertrand logic applies and competition works. Once
competition drives price oers into an interval in which persuasion is not given, being
the ignorant rm becomes relatively more attractive however.
This impacts list prices: on account of the incentives of not winning the price com-
petition and thereby incurring the ignorant rm's position list price competition remains
imperfect. Under list price competition, the 'natural' symmetric equilibrium in which
both rms earn the same prots may not exist, either as a result of the non-persuasion
requirement or of the maximum willingness to pay for a low risk individual. In this case
our second important nding dierent list prices are oered in equilibrium: one rm
will oer a high price in the persuasion interval. Thereby, the winner of the list price
competition, although earning less, has no incentives to switch roles. The observation
that list prices dier between rms is in line with the real market situation.
Another nding is that overall industry prots increase in the share of bargainers in
the market. If this share is low, on the contrary, list prices converge to the competitive
solution: since the uninformed rm does not, in the absence of bargainers, expect to earn
positive prots, winning the list price competition becomes relatively attractive again.
A high share of bargainers creates the opposite eect: the future ignorant rm expects
high prots.
Related literature
A small stream of literature analyses the eects of consumer bargaining
behaviour on prices and rm prots. In Gill and Thanassoulis (2013), rms rst compete
for price takers in list prices and only in a second step for bargainers. They nd that a
higher share of bargainers dampens competition both in the price listing stage and in the
3
More precisely, the informed rm's oer is never accepted.
Non-persuasiveness mitigates competition in the market for life insurances 69
bargaining stage. Their result is in line with our ndings, although the eect is somewhat
dierent: there, list prices serve as an outside option for second stage prices and thereby
reduce incentives to undercut the rival. In Gill and Thanassoulis (2009), the second stage
eect is also present, while the unique market list price is determined through Cournot
competition. In the present study, by contrast, list prices rise due to the inability of an
informed rm to convincingly signal high risk. Desai and Purohit (2004) analyse a model
in which the bargaining strategy is an endogenous decision variable of rms. Raskovich
(2007) demonstrates that list prices can jump from the fully competitive outcome to the
monopoly price if the share of bargainers exceeds some critical threshold. The logic is
that in this juncture rms nd it more worthwhile to negotiate prices with bargainers
individually.
In our paper, meaningful signalling is possible since the beliefs restrict the maximum
willingness to pay for insurance. Beliefs may be such that an insurer and a high risk
applicant cannot agree on a price, forcing rms to reject these individuals. As a result,
equilibria with xed and commonly known sender preferences need not necessarily be
pooling but can be separating. Put dierently, the only tool to signal high risk may be
to reject an applicant.
To the best of our knowledge, we are the rst to point out how customer bargaining
behaviour aects such persuasive signalling. We do not allow for rm tools to generate
persuasiveness, like reputation building (e.g. Sobel 1985, Gentzkow and Shapiro 2006),
advertising (e.g. Milgrom and Roberts 1986) or multiple senders (e.g. Battaglini 2002,
Villeneuve 2005). As to what the latter is concerned, with endogenous information ac-
quisition rms will be asymmetrically informed in equilibrium.
A large stream of literature analyses quality signalling through prices (e.g. Bagwell
and Riordan 1991, Ellingsen 1997, Adriani and Deidda 2011, Janssen and Roy 2010).
There, high quality sellers manage to signal their type by distorting prices upwards and
thereby reducing sold quantities. Adriani and Deidda (2011)) show that this might not
be possible, i.e. high quality sellers drop out of the market if competition is strong,
while Janssen and Roy (2010) demonstrate that equilibria exist even in a setting without
additional product dierentiation. This literature diers from the present study mainly in
that we consider a market with inelastic demand. There, in order for separating equilibria
to arise, additional conditions are required, such as buyers diering in their willingness
to pay (Ellingsen 1997).
Our model adds to the literature of asymmetrically informed rms. As far as we know,
a structure where one rm has private information over the customer and its competitor
has only been considered in the credit lending literature, namely by Inderst and Mueller
Non-persuasiveness mitigates competition in the market for life insurances 70
(2006) and Inderst (2008). Asymmetrically informed rms with informed customers have
been studied by de Garidel-Thoron (2005) in an insurance setting and by Sharpe (1990)
and Hauswald and Marquez (2006) in the bank lending market. Asymmetrically informed
rms are also considered in the literature on information sharing in competitive markets
(see for instance Vives 1984, Gal-Or 1985, Spulber 1995, Kühn and Vives 1995 and Vives
2008). These works trade o the ex-ante eects of decreasing uncertainty against those
of a changing competitive environment. There, the symmetrically imperfectly informed
rms agree ex-ante on a certain way of sharing information and are subsequently bound
to report honestly.
This study is organized as follows. Section 4.2 introduces the model. In Section 4.3, a
modied version of the game is considered in order to grasp the role of persuasiveness. The
game is then analysed in Section 4.4. Section 4.5 concludes. The eect of a prohibition
of information sharing systems is discussed in the appendix. Then, both rms opt to
screen applicants, resulting in fully revealing and perfectly competitive oers. All proofs
are relegated to the appendix.
4.2 The model
Consumers with wealth level
W
incur the risk of running a loss
d
, which is normalized
to
1
. Their utility is given by a von Neumann-Morgenstern-function
u(·)
where
u0(·)>
0, u00 (·)<0
. A loss occurs with some probability
θ∈ {L, H}
where
0< L < H < 1
.
We say
θ
is a consumer's risk type. Let
qθ
with
Pθqθ= 1
be the commonly held prior
on the distribution of risk types in the population. At the beginning of the game,
θ
is
unobservable for all players. A share
λ
of consumers does not bargain. In the following,
we refer to them as price takers. Correspondingly, bargaining consumers are referred to
as bargainers. Furthermore, we consider two competing risk-neutral insurance companies
i∈ {A, B}
.
4
The rms can perfectly observe an individual's risk type (although not his
bargaining behaviour) after having invested some arbitrarily small but strictly positive
amount
γ
. We assume that this decision is not observable by other players. If a rm
does not acquire information, prior beliefs are preserved unless the rm receives some
meaningful signal from an informed rm. Consumers remain uninformed.
Oers.
If a person applies for insurance, insurers make oers. In particular, oers may be
considered acceptable or non-acceptable by consumers, contingent on their beliefs about
4
In this setting we analyse, the results are supported only in a duopolistic structure. We discuss this
in the last section.
Non-persuasiveness mitigates competition in the market for life insurances 71
their own risk type. If informed rms are not able to signal high risk because they have
no tool to do so, these persons must be rejected, as shown in Villeneuve (2005).
Let
Ω
be the set of possible price oers. Formally, a rejection is identical to a price
that is never acceptable to any consumer, whatever his beliefs. To make rejections more
explicit, we denote by
∆
the set of all oers in
Ω
that are not acceptable to a consumer
who believes that he is an
H
-type. A consumer will accept an oer if his expected utility
from it is at least as high as his expected utility from not buying insurance (his reservation
utility).
Beliefs in every period can be expressed as the expected risk which is a mapping of the
set of vectors of oers in
t
,
Pt
into the interval
[L, H]
. It is denoted as
ˆ
θt=ˆ
θ(Pt)∈[L, H]
.
Ex-ante, it equals prior risk expectations, i.e.
ˆ
θ0=qLL+qHH
. Then, a person's expected
utility
U(·)
from accepting an oer
p
is given by
U(p) = u(W−p),
while his expected utility from not buying any insurance is
U0(ˆ
θt) = (1 −ˆ
θt)u(W) + ˆ
θtu(W−1).
A consumer judges an oer
p
acceptable if and only if
U(p)≥U0(ˆ
θt)
. In the follow-
ing, the maximum willingness to pay for given beliefs will play an important role. In
particular, three prices will be of relevance:
˜pL
and
˜pH
are the prices that correspond to
the certainty equivalents for
L
and
H
-types, i.e.
U(˜pθ) = U0(θ)
. A consumer with the
prior risk expectation
θ0
is indierent between insurance at price
˜pu
and no insurance.
To circumvent existence problems (see below), we assume the following.
Assumption 4.1.
˜pu< H
.
From the denition of
∆
follows that all prices in
∆
are strictly greater than
˜pH
. A
rm's posterior prot is
πi(p) = p−θ
if her oer
p
is selected by a consumer and zero
otherwise. The lowest price a rm asks if she is informed about
θ
is the fair price, i.e.
θ
.
The game proceeds as follows:
t=0 Firms set list prices
pA
and
pB
. Consumers apply to the rm that oers the lowest
list price if this price does not exceed
˜pL
. If it does, the game ends. Applicants
randomize with probability
1/2
if
pA=pB≤˜pL
.
Non-persuasiveness mitigates competition in the market for life insurances 72
t=1 Price taking stage: the rm where the consumer applies, denoted rm
i
, makes
an investment choice
γi∈ {0, γ}
and contingently observes
θ
. Contingent on this
observation, the rm publicly and individually accepts the applicants and oers the
list price
pi
or rejects him.
t=2 Bargaining stage: bargainers ask both rms to make better oers: rms choose
whether to acquire information on consumers that have not been assigned to them.
Firms simultaneously oer
pi,2, pj,2
for
i, j ∈ {A, B}
and
j6=i
. Bargainers choose
among all oers, i.e.
pi, pi,2, pj,2
and no insurance.
Therefore, rms compete for price taking applicants. More specically, competition takes
place on low risk consumers only, while high risks are being rejected. By modelling list
price competition accordingly, we follow Inderst (2008) and Inderst and Mueller (2006).
In the following section, it is demonstrated in a modied version of this game why rms
have no tool to signal high risk. The basic idea is that if an applicant was willing to
believe that he has a high risk when oered a high price, rms would always want to
induce this belief to make him pay higher prices. By Assumption 4.1, it must also hold
that
˜pL< H
, and all equilibria must be such that high risk applicants end up rejected.
It is assumed that all applicants rst apply to the rm that oers the best list price.
That is, we abstract from the possibility that bargaining consumers may wish to employ
a search strategy that diers from that of price taking consumers.
In this game, the decision whether to acquire information about risk types is made
on an individual basis (every applicant is an `individual market'). That is to say, a rm
may have an informational advantage on some consumers while not on others vis-à-vis
her competitor. Also note that the insurer who does not win the list price competition
in
t= 0
has observed the other rm's acceptance/rejection decision. This game speci-
cation is motivated by the fact that insurers indeed share information about oers via an
information sharing system.
5
As a result, in equilibria in which the insurer who has not
won the price competition in
t= 0
chooses to remain uninformed while her competitor
does not, the price oer
pi
serves both uninformed players this insurer and the applicant
to update beliefs, see further below. A further note refers to consumer preferences: for
technical convenience, we assume that whenever a consumer is indierent between oers
in
t= 2
, he chooses the oer from the rm to which he has applied in the rst place.
Equilibrium selection.
Consumers hold beliefs
ˆ
θ0
about their own risk type and the rms'
informational state. Updating takes place after the price taking stage, i.e. after the rst
5
The eects of a prohibition are being discussed in the appendix.
Non-persuasiveness mitigates competition in the market for life insurances 73
oer has been made, and after the bargaining stage. Since bargainers do not have to
make a decision after the price taking stage, we are not concerned with their beliefs after
the price taking stage. We assume that whenever consumers observe o-path behaviour,
they make inferences only about their risk type. Put dierently, beliefs on investment
behaviour remain xed and are never updated. We require that o-path beliefs satisfy
the Intuitive Criterion by Cho and Kreps (1987).
The equilibrium concept is the Perfect Bayesian Nash Equilibrium, meaning that
beliefs have to be consistent in equilibrium. We restrict our analysis to pure strategy
equilibria and full insurance contracts.
6
Comments on game structure.
Bertrand games may not have equilibria if utility functions
exhibit discontinuities (see e.g. Dasgupta and Maskin 1986). This problem arises here
because consumers update their beliefs based on the obtained contract oers which may
render oers in some intervals unacceptable.
7
Equilibria in the bargaining game are
aected whenever rms are asymmetrically informed and the market situation is such
that a consumer who believes to be an
L
type is not willing to pay at least
H
while, if
priors are preserved, the willingness to pay is at least
H
. Since we are more concerned
with the equilibria in which an informed rm reveals information in the price taking
stage, we circumvent this problem by considering only situations where the willingness
to pay of a consumer who holds prior beliefs is less than
H
(Assumption 4.1).
4.3 Preliminary analysis: persuasiveness
In this section, in order to grasp the role of persuasiveness here, we analyse the subgame
starting in
t= 1
. This is to say, we abstract from list price competition for a moment
and assume that in
t= 0
, applicants are assigned randomly (with probability
1/2
) to one
of the two rms. The rm to which a consumer has been assigned makes him an oer,
and bargainers then proceed to the second stage of the game. More precisely, the game
we are considering here is the following:
t=0 With probability
1/2
, nature assigns consumers to rm
i
for
i∈ {A, B}
.
6
This may require further assumptions on the market situation. We discuss this topic in the last
section.
7
In particular, consider the following situation: let rm
A
be informed about
θ
, while
B
only knows
that
θ∈ {L, H}
. If the consumer, whose belief is
L
, is not willing to accept at least the fair price for
H
,
i.e.
H
, there is an interval of prices that would never be accepted by the consumer if the oer is made
by
A
.
Non-persuasiveness mitigates competition in the market for life insurances 74
t=1 Price taking stage: the rm to which the applicant has been assigned, denoted rm
i
, makes an investment choice
γi∈ {0, γ}
and contingently observes
θ
. Contingent
on this observation, the rm publicly announces an individual price oer
pm
i
. Price
takers accept the oer or choose no insurance.
t=2 Bargaining stage: bargainers ask both rms to make better oers: rms choose
whether to acquire information on consumers that have not been assigned to them.
Firms simultaneously oer
pm
i,2, pm
j,2
for
i, j ∈ {A, B}
and
j6=i
(and where
i
denotes
the rm to which an applicant has been assigned in
t= 0
). Bargainers choose
among all oers, i.e.
pm
i, pm
i,2, pm
j,2
and no insurance.
Note also that in this game formulation, we allow period-one rm
i
to freely choose
oers from the contract space
Ω
on each individual applicant in
t= 1
, instead of being
restricted to the formerly set list price. Here, we demonstrate that no distinct price oers
are made in
t= 1
to applicants of dierent risk types. Put dierently, the restriction to
a single price in the general formulation of the game is without loss of generality.
This section is organized as follows: it is rst shown that only certain informational
states may arise in equilibrium. In a next step, we analyse properties of such potential
equilibria before investigating which of these equilibria exist.
In particular, consider some equilibrium in which each rm chooses to gather infor-
mation on each single applicant, that is, no matter whether being his period-one rm or
not.
Let
pm,θ
i,t
for
θ∈ {L, H}
be denoted the equilibrium oers made by an informed rm
i
in
t
. From Villeneuve (2005), Proposition
6
, we know that whenever both rms are informed,
in any equilibrium it holds that
(pm,L
A,2, pm,L
B,2)6= (pm,H
A,2, pm,H
B,2)
and that
pm,L
A,2=pm,L
B,2=L
.
8
In
words, any equilibrium in the bargaining game is separating and the competitive outcome
arises on the
L
-type. But because prots are zero on the
L
-type, the second rm may as
well not acquire information and always oers the equilibrium contract for the
H
-type
in order to save costs. As a result, these equilibria never exist.
On the other hand, consider an equilibrium in which no rm gathers information on an
applicant's risk type, that is, whether being his period-one rm or not. But in this case,
rms will always benet from clandestinely acquiring information. We can therefore
focus on equilibria in which either the period-one rm acquires information while her
competitor does not, or the inverse. Consider an equilibrium in which
j
is informed while
the period-one insurer
i
is not. By Assumption 4.1, it holds that if
i
is not informed,
8
The result also applies to the general game.
Non-persuasiveness mitigates competition in the market for life insurances 75
pm,u
i= ˜pu
and
pm,u
i∈∆
are equilibrium candidates, where
pm,u
i
denotes the price oer
made by an uninformed rm
i
in
t= 1
. Further, because of the rm's ignorance, any
deviation does not shape consumer beliefs. Therefore, a deviation is always protable.
Proposition 4.1 proves this intuition.
Proposition 4.1.
Any equilibrium veries that the period-one rm
i
acquires information
while her competitor
j
does not.
Therefore, we can focus on this class of equilibria.
By Assumption 4.1, the consistency condition requires, rst, that price taking con-
sumers must rule out the possibility that they are a high risk type whenever accepted and
second, that high risk applicants are indeed rejected in the price taking stage. The rough
intuition for the rst point is that
i
never wins on
H
-type bargainers in
t= 2
, since her
competitor's oer is not restricted by beliefs. As a result, it must be that either
j
's oer
is accepted in equilibrium or that
i
's oer is accepted but the outcome is competitive.
Put dierently,
i
never wins on
H
-types in the bargaining stage (i.e. she earns no positive
prots on these applicants), and she is therefore not willing to accept losses on the same
type in the price taking stage. Further,
i
's equilibrium prots in
t= 2
are at least as
high on
H
-types as they are on
L
-types on which they are zero in separating equilibria
(dierent prices in
t= 1
) by the Bertrand logic resulting in the lack of signalling op-
portunities for the
L
-type. Following the logic of Villeneuve (2005), price taking
H
-types
must therefore end up rejected. As a result, equilibria are either separating or pooling
with both types being rejected in the rst place.
At the beginning of the bargaining stage, the uninformed rm shares the price taker's
(consistent) belief. Consequently, in separating equilibria, the uninformed rm's infor-
mation sets on the equilibrium path are singletons:
IA={L}
is the information set when
a person has been accepted and was oered the equilibrium price for an
L
-type in
t= 1
.
The corresponding information set for a rejection is
I∆={H}
. In pooling equilibria, the
rejection decision does not reveal any information, i.e.
I∆
then contains both types. We
denote by
pm,IA
j,2
and
pm,I∆
j,2
equilibrium oers made by the uninformed rm.
The following proposition characterizes the set of possible equilibria given that period-
one rm
i
is informed while her competitor remains ignorant.
Proposition 4.2.
Any equilibrium where rm
i
has acquired information, veries that
either
(A) (POOLING) in the price taking stage
pm,L
i,1=pm,H
i,1∈∆
, and in the bargaining
Non-persuasiveness mitigates competition in the market for life insurances 76
stage,
pm,L
i,2≤pm,IA
j,2
and
pm,L
i,2≥L+λ
1−λ(˜pL−L).
Or,
(B) (SEPARATING) in the price taking stage,
pm,L
i,1= ˜pL
,
pm,H
i,1∈∆
, and in the bar-
gaining stage,
pm,L
i,2=L
and
pm,IA
j,2=L
.
Furthermore, there exists a
˚
λ∈(1/2,1)
such that
(p)
pm,H
i,2=pm,I∆
j,2=H
i
λ≥˚
λ
, or
(np)
pm,I∆
j,2∈(H, ˜pH]
and
pm,H
i,2=pm,I∆
j,2+ε
or
pm,H
i,2> pm,H
j,2
if
pm,H
j,2= ˜pH
,
where
ε
is arbitrarily small.
˚
λ
is given by
˚
λ=H−L
H−L+ ˜pL−L.
Part (A) of Proposition 4.2 states that for a pooling equilibrium to arise, period-one
rm
i
must earn strictly positive prots on the
L
-type in the bargaining stage. The
reason is that otherwise, a deviation to an oer which is considered acceptable by some
applicant whatever his beliefs is clearly a protable deviation.
Part (B) of Proposition 4.2 characterizes candidates for separating equilibria, i.e. equi-
libria where only the
H
-type is rejected in the rst place. Low risk applicants are oered
the monopoly price
˜pL
in the price taking stage. It cannot be lower since any deviation
from a lower price oer will be protable and is not restricted by beliefs. Bargaining
L
-type applicants receive the actuarially fair price. Competition works here because in-
formation is revealed already in
t= 1
, which means that both rms are informed in
t= 2
and price oers are never restricted by beliefs on these types.
By contrast, consider
θ=H
and an equilibrium in which the bargaining
H
-type
applicant is oered the actuarially fair price by both rms. This can be an equilibrium
only if a deviation on low risk consumers is not protable in the rst round, that is to
say, if the rm will not earn higher prots on bargainers by rejecting all applicants in
t= 1
in order to make them believe that they are high risks. This in turn depends on
the share of price takers in the market. A higher share implies that opportunity costs
from rejecting low risk applicants are high.
As remarked in the introduction, the informed insurer is said to be persuasive if the
period-one rm
i
would not want to deviate and reject the
L
-type applicant in
t= 1
given that the competitive equilibrium arises on
H
-type bargainers. This implies that
Non-persuasiveness mitigates competition in the market for life insurances 77
if period-one rm
i
is persuasive, a bargainer that has been rejected in the rst period
accepts the informed rm's oer in
t= 2
. Here, persuasiveness is given whenever
λ≥˚
λ
(Part (B), (p) of Proposition 4.2).
The competitive outcome is unique whenever persuasiveness is given. Consider instead
a consumer who is rejected in the rst stage but oered, by the same rm, a high price in
the bargaining stage. If the rm is persuasive, this person must believe that he is a high
risk. This is inherent in the denition of persuasiveness and the logic of the Intuitive
Criterion: since by employing this reject/accept strategy, the informed party would, in
case of a deviation to rejecting the
L
-type, earn less than equilibrium prots on the same
person, the consumer must not put positive probability on being a low risk (the deviation
is not admissible on the
L
-type).
On the other hand, if the rm is not persuasive, all equilibria must be such that the
informed rm earns nothing on high risk bargainers. This is because the rm would oth-
erwise deviate and reject the
L
-type in the rst place. Proposition 4.2, Part (B), (np) (for
`non-persuasiveness') states that if the informed rm is not persuasive, in all equilibria,
the uninformed rm earns strictly positive prots while her informed competitor earns
nothing. All these equilibria are such that the consumer, in
t= 2
, upon observation of
an oer that he would accept from the informed rm, puts strictly positive probability
on
θ=L
. These beliefs are always allowed by the denition of non-persuasiveness: any
deviation
p0
i,2
that would be protable on the
H
-type, i.e.
pm,I∆
j,2≥p0
i,2≥H
, is also
protable on the
L
-type.
Proposition 4.2 characterizes equilibria for a given informational state. To see whether
these indeed exist, it is necessary to consider incentives to deviate. Consider a pooling
equilibrium as presented in Part (A) of the proposition. Clearly, rm
j
will benet
from clandestinely collect information because in order for this equilibrium to exist, the
winning oer implies strictly positive prots on the
L
-type in the second stage. Note
that we have assumed costs for information acquisition to be arbitrarily small.
On the other hand, equilibria as presented in Part (B) of the proposition do not induce
rms to deviate from the informational state. However, in order to guarantee existence,
λ
has to be suciently large.
Assumption 4.2.
λ≥1
2
.
Separating equilibria can exist only if this assumption is met. The reason is that other-
wise, rm
i
could still protably deviate to rejecting the
L
-type applicant in
t= 1
and
oering him
˜pL
in
t= 2
. Proposition 4.3 summarizes:
Non-persuasiveness mitigates competition in the market for life insurances 78
Proposition 4.3.
In any equilibrium, the rm to which the consumer has been initially
assigned acquires information while the second rm does not. Any equilibrium is fully
characterized by Proposition 4.2, Part (B). Such an equilibrium always exists.
Therefore, although the rms' expected prots are the same at the beginning of the
game, (initial assignment in
t= 0
to each rm takes place with probability
1/2
), they are
not once applicants have been assigned. This provides incentives to compete for initial
assignment, a topic we will treat in the following section.
4.4 List price competition
In this section, we analyse the game as specied in section 4.2. In the previous section,
we have given an intuition for the notion of persuasiveness and we have shown that it
may or may not be given if applicants are randomly assigned, that is, if competition in
list prices does not take place.
Further, we have found that in all equilibria of the modied game, the period-one rm
i
is informed while her competitor is not. In order to analyse the game with list price
competition, we now assume that the rm which wins the list price competition in
t= 0
(to which the applicant applies rst) acquires information. Afterwards, we check whether
this indeed is an equilibrium. Investigating equilibria accordingly essentially simplies
the analysis.
Whether or not the informed rm is persuasive in equilibrium now depends on the
lowest list price which is oered in equilibrium. It turns out that with list price com-
petition, persuasiveness is never given in equilibrium and that, although list prices are
sometimes asymmetric, all rms expect to earn strictly positive prots.
The informed rm can only be persuasive if she makes strictly positive prots on
price taking
L
-type consumers. Otherwise, rejecting low risks in the price taking stage
and thereby hoping to convince consumers and the other rm that the type is
H
would
always be protable.
Lemma 4.1.
There exists some
ˆp(λ)> L
such that persuasiveness is given if and only
if
pi≥ˆp(λ)
. It is given by
ˆp(λ) = 1−λ
λ(H−L) + L
This is to say, if the lowest price does not exceed some threshold
ˆp(·)
, persuasiveness is
not given. In that case, as before, the competitor makes strictly positive prots,
(pH−H)
Non-persuasiveness mitigates competition in the market for life insurances 79
on
H
bargaining type consumers. Here,
pH
denotes the equilibrium price of the modied
game,
pm,I∆
j,2
if the informed rm is not persuasive, see Proposition 4.2, Part (B), (np).
Now consider some equilibrium price
pi≥ˆp(λ)
, such the rm is persuasive. Because
the uninformed rm earns nothing on bargainers, both rms strive to win the list price
competition. Hence, competition works eectively until
ˆpL(λ)
is reached. Now consider
a possible equilibrium with
pA=pB= ˆpL(·)
. Firms have an incentive to deviate to
a lower price. However, once one rm does so, the competitor may not wish to keep
up. The reason is that as soon as the lowest oer is smaller than
ˆp(·)
the insurer which
does not win the list price competition expects to earn strictly positive prots. As a
result, persuasive equilibria cease to exist under list price competition, independently of
the share of price taking consumers in the market. Therefore, all equilibria must exhibit
non-persuasiveness and
pi<ˆp(λ)
in equilibrium.
A 'natural', symmetric equilibrium price would be one where both rms earn the same
in expectation. Dene the corresponding list price as
˚p(·)
with
˚p(λ) = 1−λ
λ
qH
qL
(pH−H) + L.
This list price depends on the distribution of risk types in the population and on the
share of price takers. More high risks drive the price up, because positive prots are less
likely to be made, the same holds true for the share of bargainers.
It turns out that the list price is bound from above by
˚p(λ)
. The reason is that a
deviation is always protable for rm
j
if her competitor's prots exceeded her owns. It
should also not be lower, since otherwise, both rms will nd it preferable to embrace the
position of the uninformed rm. However, two restrictions apply: rst,
L
-type consumers
may not be willing to pay the corresponding price; second, as demonstrated above, the
equilibrium list price is necessarily lower than the persuasion threshold
ˆp(λ)
.
These equilibria exist if an informed rm will not nd it protable to reject
L
-types in
the rst place in order oer them
˜pL
in the bargaining stage. To secure this, the following
assumption is made.
Assumption 4.3.
Let
qH(˜pH−H)≥qL(˜pL−L)
.
This is to say, expected prots of a monopolistic insurer in a market that exhibits sym-
metric information should be higher on high risk applicants than on low risks. Let
pθ
i,2
be denoted the informed rm's equilibrium oer to an applicant of risk type
θ
in the bar-
gaining stage, and
pIA
j,2
and
pI∆
j,2
her competitor's equilibrium oer to applicants that she
believes to be
L
or
H
-types, respectively. The following proposition species candidate
equilibria.
Non-persuasiveness mitigates competition in the market for life insurances 80
Proposition 4.4.
Assume that rm
i
is informed. Then an equilibrium has the following
properties:
pL
i,2=pIA
j,2=L
,
pI∆
j,2∈(qL/qH(˜pL−L)+H, ˜pH]
and
pH
i,2=pI∆
j,2+ε
or
pH
i,2> pH
j,2
if
pH
j,2= ˜pH
, with
ε
being arbitrarily small.
In the list price competition,
pi= min{˜pL; ˆp(λ)−ε;˚p(λ)}
, and
pj
≥ˆp(λ)
if
pi= ˜pL
= ˆp(λ)
if
pi= ˆp(λ)−ε
= ˚p(λ)
if
pi= ˚p(λ).
Therefore, as argued above, the lowest list price is the minimum of (a) the maximum
willingness to pay for insurance for a low risk,
˜pL
, (b) the persuasion threshold,
ˆp(λ)
, and
(c) the price at which both rms' expected prots are equal,
˚p(λ)
.
It is most notable that asymmetric equilibrium list prices arise if the lowest list price is
˜pL
or
ˆp(·)
. In the case where
˜pL
is the equilibrium price, one rm will oer some list price
that equals or exceeds the persuasion threshold
ˆp(λ)
, ensuring thereby that the other
rm has no incentives to seek to switch roles by oering a higher price and the ignorant
rm earns strictly more than her competitor. Also,
j
has no interest in undercutting
the best oer since in expectation, these rm's prots are higher than her competitor's,
since
pi<˚p(·)
. Asymmetry is also given in (b), that is, when the persuasion threshold
is just met. This has been seen above: while one rm has an incentive to undercut the
persuasion threshold, the competitor has not if her expected prots are higher. Again,
in this case, this is given because
pi<˚p(·)
.
As a result, the rm that does not win the list price competition expects to earn at
least as much as the winning rm. Figure 4.1 shows in an example how the list price
develops in the share of price takers for two dierent values of
qH
.
qH= 1/3
is chosen
such that Assumption 4.3 is met with equality. It then holds that
˜pL= ˚p(0.5)
. For larger
shares of high risk applicants,
˜pL
is the equilibrium list price for small values of
λ
. As the
share of price takers increases, however, competition works ne and the list price limits
the competitive outcome.
So far, it was assumed that
i
acquires information. Therefore, it remains to show that
the oers as given in Proposition 4.4 indeed form an equilibrium.
Proposition 4.5.
An equilibrium where rm
i
acquires information and price oers are
such as given in Proposition 4.4 exists.
This is a straightforward result: assume
i
chooses to remain uninformed. Then she
can either accept or reject all applicants. Accepting
H
-types is not protable because
rms play the competitive outcome on bargaining
L
-types. Therefore, doing so would
imply negative expected prots. On the other hand, rejecting all applicants is also not
Non-persuasiveness mitigates competition in the market for life insurances 81
Figure 4.1:
pi(λ)
for
L= 0.1,˜pL= 0.15, H = 0.3, pH= 0.4
and dierent
qH
protable, since the rm's oer on
H
-types is never accepted.
Industry prots.
Now consider overall industry prots. They are given by
Π(λ) = λqL(pi(λ)−L) + (1 −λ)qH(pH−H).
Deriving with respect to
λ
gives
dΠ(λ)
dλ =qL(pi(λ)−L)−qH(pH−H) + λqL
dpi
dλ .
(4.1)
The last term of equation (4.1) is non-positive. In particular, as shown in gure 4.1,
it is zero for small
λ
and negative for large
λ
. Some transformations then reveal that
industry prots are decreasing for all
λ≥1/2
. This is shown in the proof of the following
proposition:
Proposition 4.6.
dΠ(λ)
dλ ≤0
for all
λ≥1/2
.
This means that the more bargainers in the market, the higher are industry prots. This
result is independent of the distribution of risk types in the population. The reason for
it is that the expected prots of the insurer that has not won the list price competition
weakly exceed that of her competitor. This is the case even if the share of high risk
applicants is low. This then results in almost competitive list prices, yielding both rms
lower prots.
Non-persuasiveness mitigates competition in the market for life insurances 82
4.5 Discussion
We have analysed the role of bargainers in the market for life insurances. This market
exhibits special features: on the one hand, applicants are required to provide information
on their bargaining history. On the other hand, since rms dispose of advanced techno-
logical means to estimate the statistical risk of applicants they may be better informed
than the applicants themselves. This produces a signalling environment in which prices
possibly reveal information about risk types.
We have shown that an informed rm may not have the tool to signal high risk to no
applicant of that type (neither price taker nor bargainer), in which case we say she is not
persuasive. In this case, her oer in the bargaining stage is never accepted, resulting in
strictly positive prots for her competitor.
In a modied setting where applicants randomly choose to which rm to apply rst,
an informed rm can convincingly signal high risk in the bargaining stage if the share
of bargaining consumers does not exceed some critical threshold. List price competition
however undermines persuasiveness because competition works eectively as long as it is
given. It stops to do so only when the lowest list price is such that persuasiveness is not
given. As a result, the informed rm is never persuasive and the uninformed rm always
earns strictly positive prots. This aects the rst competition stage: list prices are not
competitive. As a result, and perhaps surprisingly, both rms also the informed insurer
benet from this ability to signal high risk only through a non-acceptable oer.
Moreover, the overall industry prot increases in the share of bargainers as their
absence fosters competition on the list price level. This is because the rm that does
not oer the best list contract will earn strictly positive prots on bargaining applicants
exclusively.
We have mentioned in the introduction that insurers usually exchange information
about customer data through the means of an information sharing system. Alternatively,
or additionally, they require consumers to reveal their application history. Our model
helps to understand the use of such procedures: they help rms to gather information
on the bargaining behaviour of applicants. In particular, a rm that is not aware of the
actual stage of the game may nd it useful to always acquire information that leaves
rms symmetrically informed in equilibrium. Implications hereof are discussed in the
appendix. In particular, symmetric information produces the fully competitive outcome.
In the model studied here, the value of non-persuasiveness is owed to the presence of
only two rms in the market. If more rms compete on the bargaining stage, the eect
vanishes due to the Bertrand structure. It can be restored, however, if competition is
Non-persuasiveness mitigates competition in the market for life insurances 83
imperfect. In particular, it is visible if the informed rm cannot nd any price at which
her expected equilibrium prot on a low risk type is higher than the deviation prot on
the same type. Otherwise, rejections are not expressed twice, although the persuasion
issue may still aect price setting behaviour. We leave this issue for further research.
In order to rationalize rejections in the way we have done, it is necessary to consider
a discrete risk type distribution. Alternatively, our model can be interpreted as a contin-
uous type space with imperfect screening technology: consumer willingness to pay will
change, but the logic remains the same.
Finally, rms may also oer partial insurance contracts while we have only taken into
account full insurance. We believe that this is not so much of a concern here, as applica-
bility seems limited in life or private occupational disability insurances. Furthermore, it
is of no relevance to our analysis, as long as risk types are suciently distinct.
Non-persuasiveness mitigates competition in the market for life insurances 84
4.6 Appendix
4.6.1 Proofs
Proof of Proposition 4.1.
.
First, there is no equilibrium in which both rms are informed. Assume by contrast
that there is a such equilibrium. From Villeneuve (2005), Proposition
6
, in equilibrium
with two informed rms, it must hold that
pm,L
i,2=pm,L
j,2=L
and prots are zero on a
bargaining
L
type. A rm that is not informed can always secure at least zero prots by
deviating to
pm,H
j,2
. This will be protable since information acquisition is costly.
Second, there is no equilibrium in which no rm is informed. Consider such an
equilibrium. Then, by the standard Bertrand logic,
pm
i,2=pm
j,2=E[θ]
for all applicants
in equilibrium. In that case, information acquisition by rm
j
and her deviating to
pL
j,2< E[θ]
on the
L
-type is a protable deviation.
Third, there is no equilibrium in which
i
is not informed while
j
is. Consider a such
equilibrium. In such an equilibrium,
i
oers, in the price taking stage some price
pm
i
to all
types. The price is acceptable if and only if
pm
i≤˜pu
. Consider by contrast
pm
i>˜pu
. In
that case, no consumer accepts. To acquire information is then always protable because
i
can make an oer to a
L
-type that is considered acceptable, yields strictly positive
prots, and does not inuence consumer beliefs. By contrast, consider
pm
i≤˜pu
. All
types accept this oer, but prots are strictly negative on
H
-type consumers. Therefore,
acquiring information and rejecting the
H
-type is a protable deviation.
Proof of Proposition 4.2.
.
(A)
First, consider a pooling equilibrium, i.e.
pm,L
i,1=pm,H
i,1
. There are two possibilities:
(i) Consider an equilibrium
pm,L
i,1=pm,H
i,1< H
. But rm
i
never strictly gains on bar-
gaining
H
-type consumers:
j
is not informed. Therefore,
j
's oer is not restricted
by beliefs and
i
cannot make positive prots on the bargaining
H
-type. A devia-
tion to the rst period oer
p0
i≥H
is then always protable, whatever the hereby
induced beliefs.
(ii)
pm,L
i,1=pm,H
i,1≥H
can only be an equilibrium if
pm,L
i,1, pm,H
i,1∈∆
since
˜pu< H
. For
this case, consider some equilibrium price
pm,L
i,2
in the bargaining game. Note that
(1) as in i),
j
's oer is not restricted by beliefs and
i
cannot make positive prots
on the bargaining
H
-type (see also argument below), and (2), if the belief following
a deviation in
t= 1
is
L
, the uninformed rm
j
oers
L
, according to standard
Bertrand logic.
Non-persuasiveness mitigates competition in the market for life insurances 85
According to the logic of the Intuitive Criterion, every deviation to a price greater
than
H
is admissible on
H
(which means that uninformed parties are allowed to
then believe
H
). If the belief is
H
, the deviation is protable. Therefore, the
equilibrium is eliminated if the deviation is not admissible on the
L
-type. This is
the case whenever the deviation is not protable on the
L
-type given that the belief
is
H
, that is if
(1 −λ)(pm,L
i,2−L)> λ(p0
i,1−L) + (1 −λ)(H−L)
for some deviation
oer
p0
i,1
. This is never the case, implying that any deviation is admissible on the
L
-type.
As a result, deviation incentives must be checked. For some
t= 1
deviation price
greater in than
H
, the oer will not be accepted by some applicant who believes
L
.
Therefore, a deviation in
t= 1
to some price
p0
i,1
greater than
H
is never protable.
The equilibrium is not destroyed for deviations to oers of at least
H
.
Further, a deviation to some price smaller than
H
is also admissible on the
L
-type,
while not on the
H
-type. Therefore, the belief on such a deviation must be
L
. It
can be protable if and only if it is accepted by the
L
-type. The deviation in
t= 1
is protable on the
L
-type if and only if for some deviation price
p0
i,1
:
λ(p0
i,1−L)>(1 −λ)(pm,L
i,2−L)
⇔pm,L
i,2<λ
1−λ(p0
i,1−L) + L
The equilibrium is not destroyed if the inequality is not fullled for any deviation
oer lower or equal to
˜pL
. Therefore if (part
(A)
of the Proposition):
pm,L
i,2≥λ
1−λ(˜pL−L) + L.
(B)
Separating equilibria.
All other equilibrium candidates must be separating. In that
case, for given updated beliefs of
j
, it must be that
pm,L
i,2=pm,IA
j,2=L
. This follows
immediately from the standard Bertrand logic.
The informed party makes no positive prots on a bargaining
H
-type either. Consider
by contrast an equilibrium in which a price
˜pH≥pm,H
i,2> H
is accepted by the bargainer
with strictly positive probability. Then, it must hold that
pm,I∆
j,2≥pm,H
i,2
. In any case,
a deviation for
j
to a lower price is always protable since
j
's oer does not inuence
consumer beliefs.
Consider rst possible outcomes in the price taking stage: we show that
pm,H
i,1∈∆
.
By contrast, consider
H≤pm,H
i,1≤˜pH
. This can be an equilibrium only if the oer is
considered acceptable in equilibrium (i.e. the consumer believes
H
) and if rm
i
does not
Non-persuasiveness mitigates competition in the market for life insurances 86
want to deviate to this oer upon observation of an
L
-type. But in any equilibrium with
pm,L
i,16=pm,H
i,1
, the informed rm cannot make positive prots on the bargaining
L
-type.
A deviation to a higher price is therefore always protable.
Also, the oer
pm,H
i,1< H
is not an equilibrium since rm
i
makes no positive prots
on bargaining
H
-type consumers. A deviation to a higher price is therefore protable
whatever the beliefs of the other players.
Then, we show that
pm,L
i,1= ˜pL
. Assume not and consider
pm,L
i,1>˜pL
. A deviation in
t= 1
to
p0
i,1= ˜pL
is clearly always protable no matter how this deviation aects beliefs.
Similarly, if
pm,L
i,1<˜pL
, a deviation to
p0
i,1= ˜pL
is protable on a
L
-type, no matter how
beliefs are aected.
In the following, we show that either in the bargaining stage, (a)
pm,H
i,2=H
and
pm,I∆∗
j,2=
H
(EQa) or (b)
pm,H
i,2> H
and
pm,I∆
j,2= min{pm,H
i,2−; ˜pH}
with
ε
being arbitrarily small
(EQb).
We also show that EQa is the unique separating equilibrium when
λ≥˚
λ
while an
equilibrium of the form EQb arises if
λ < ˚
λ
and no other separating equilibrium exists
in this case.
˚
λ
is given by
˚
λ=H−L
H−L+ ˜pL−L>1
2
(EQa) EQa is an equilibrium if
λ≥˚
λ
:
i
does not want to deviate on the
L
-type by
denition of
˚
λ
.
if
λ≥˚
λ
, EQa is the only equilibrium: consider some dierent candidate
and
λ≥˚
λ
. First, it is not possible that
i
oers the best price and makes
strictly positive prots on the
H
-type, since rm
j
's oer is not restricted by
beliefs. Second, consider the case where
j
wins, i.e.
j
's
H
-type oer is strictly
smaller than
i
's. This oer is part of an equilibrium if there exists a protable
deviation that is admissible on the
L
-type. It is admissible on the
L
-type
whenever
(1 −λ)(pm,I∆
j,2−L)> λ(˜pL−L)
, where
pm,I∆
j,2> H
. But
i
can then
always nd a protable deviation to some price oer which is admissible on
the
L
-type and suciently close to
H
.
(EQb) EQb is an equilibrium if
λ < ˚
λ
: if
λ < ˚
λ
, a deviation to reject
L
-type, oer
winning price is admissible on the
L
-type. Therefore, belief
L
is allowed on
such a deviation. Further, a deviation to reject
L
-type, oer
˜pL
or lower is
not protable whenever
λ≥1/2
.
if
λ < ˚
λ
, EQb is the only equilibrium: obviously,
i
cannot make a the winning
oer in
t= 2
on the
H
-type which is strictly greater than
H
. Also,
i
oers
Non-persuasiveness mitigates competition in the market for life insurances 87
H
cannot be an equilibrium since
i
would then deviate to reject
L
-type, oer
H
'.
Proof of Proposition 4.3.
.
Pooling equilibrium, Part (A) of Proposition 4.2
. Firm
j
wants deviate because the
actual prot on an
L
-type is zero while her deviation prot from acquiring information
is
pm,L
i,2−ε−L
on the same type while the prot on the
H
-type remains the same.
Separating equilibrium, Part (B) of Proposition 4.2
. The ex-ante prot for rm
i
from
acquiring information is
πm,e
i=λ(˜pL−L)>0
whereas the deviation prot is
πd,e
i= 0
if the
rm rejects the consumer in the rst place, and
πd,e
i=λ(qL(˜pL−L)−qH(H−˜pL)) < πm,e
i
if she oers
˜pL
. Any other oer is either not considered acceptable and yields no prots,
or yields even lower prots than the oer
˜pL
.
Firm
j
has no interest in acquiring information because the equilibrium is separating.
Proof of Lemma 4.1.
.
From Proposition 4.2, Part (B), if actions are separating on the price taking stage,
pm,L
i,2=
pm,IA
j,2=L
. The same holds for the bargaining stage in the general game. In analogy
to the proof of Proposition 4.2, it then follows that the
H
-type is rejected in the price
taking stage. The persuasion logic does not change, therefore, persuasion is given if and
only if
λ(pi−L)≥(1 −λ)(H−L)
ˆp(λ)
is the price at which the equation is strict, i.e.
ˆp(λ) = 1−λ
λ(H−L) + L
.
Proof of Proposition 4.4.
.
pL
i,2, pIA
j,2
are in analogy to the modied game.
Now consider
pi
and
pj
. We rst show that there is no equilibrium in which
pi>
min{˜pL,ˆp(λ)−ε,˚p(λ)}
.
(a) Assume
pj≥pi>˜pL
. Then, the game ends and all rms earn nothing. A deviation
in
t= 0
to
p0
i= ˜pL
is protable.
(b) Assume
pj≥pi≥ˆp(·)
. Persuasiveness is not given and
j
earns nothing if
pj> pi
and earns
1
20 + 1
2qL(pj−L)
if
pj=pi
. A deviation to an arbitrarily smaller price
pj
yields
qL(pj−L)
which is greater.
(c) Assume
pj≥pi>˚p(·)
. In that case,
i
expects to earns more than
j
. Therefore, a
deviation by
j
to a lower list price
p0
j< pi
is strictly protable.
Non-persuasiveness mitigates competition in the market for life insurances 88
Second, we show that there is no equilibrium in which
pi<min{˜pL,ˆp(λ),˚p(λ)}
.
(a) Assume
pj≥pi<˜pL
and
˜pL≤min{ˆp(·),˚p(·)}
. Then, a deviation to a higher price
is always protable: if
pj> pi
, a deviation by
i
to
p0
i= ˜pL
is protable. If
pj=pi
,
a deviation to some higher price is protable because persuasiveness is not given
and the uninformed rm earns more than the informed party.
(b) Assume
pj≥pi<ˆp(·)−ε
and
ˆp(·)−ε≤min{˜pL,˚p(·)}
. Then, persuasiveness is
given and the uninformed rm earns more than the informed rm. Therefore, a
deviation to a higher price
pi= ˆp(·)
is always protable.
(c)
pj≥pi<˚p(·)
and
˚p(·)≤min{˜pL,ˆp(·)}
. Then, the informed rm earns less than
the uninformed party and persuasiveness is given. A deviation by
i
to
p0
i= ˚p(·)
is
always protable.
Next, we investigate deviation incentives for
pi= min{˜pL,ˆp(λ)−ε,˚p(λ)}
.
(a)
min{˜pL,ˆp(·)−ε,˚p(·)}= ˜pL
.
Consider some
p∗
j≥ˆp(·)
.
j
does not want to deviate since
πe
j> πe
i
.
i
's best response
is
pi(p∗
j) = ˜pL
. Therefore,
p∗
i= ˜pL
and any price
p∗
j≥ˆp(·)
is an equilibrium.
By contrast, consider some equilibrium price
p∗
j<ˆp(·)
. But
i
's best response is
pi(p∗
j)> p∗
j
.
(b)
min{˜pL,ˆp(·)−ε,˚p(·)}= ˆp(·)−ε
.
Consider
pj= ˆp(·)
.
j
does not want to deviate since her expected prot is higher
than
i
's.
i
's best response is
pi(pj) = ˆp(·)−ε
. Therefore,
pi= ˆp(·)−ε
,
pj= ˆp(·)
is
an equilibrium.
By contrast, consider
pj= ˆp(·)−
.
i
's best response is
pi(pj)> pj
.
Further, consider
pj>ˆp(·)
. This cannot be an equilibrium.
i
's best response to
pj
is
pi(pj) = min{˜pL;pj−ε}
.
(c)
min{˜pL,ˆp(·)−ε,˚p(·)}= ˚p(·)
.
Consider
pj= ˚p(λ)
.
j
does not want to deviate because her prots are higher than
i
's.
i
's best response is
pi(pj) = pj
. Therefore,
pi=pj= ˚pL
is an equilibrium.
By contrast, consider
pj>˚p(λ)
.
i
's best response is
pi(pj) = min{pj−ε; ˜pL}
.
Last, for given
pi
, we have to check deviation incentives in
t= 1
. Persuasiveness is never
given, which implies that the deviation prots
i
can attain by rejecting the applicant in
Non-persuasiveness mitigates competition in the market for life insurances 89
the rst place are at most
(1 −λ)(˜pL−L)
. Then, it must hold that
λ(pi−L≥(1 −λ)(˜pL−L)
(4.1)
(a)
pi= ˜pL
.
λ(pi−L)≥(1 −λ)(˜pL−L)
since
λ≥1/2
.
(b)
pi= ˆp(λ)
. Inserting this into equation (4.1) gives
H≥˜pL
which is given.
(c)
pi= ˚p(λ)
. Inserting this gives
qH(pH−H)≥qL(˜pL−L)
. This is given if and only
if
qL/qH(˜pL−L) + H
.
Proof of Proposition 4.5.
.
In the text. Further, rejecting all types and oering them
˜pL
in
t= 2
is not a protable
deviation since acquiring information and rejecting
L
-types and oering them
˜pL
in
t= 2
is not a protable deviation: not being informed would imply losses on the
H
-type
bargaining applicants.
Proof of Proposition 4.6.
.
•pi= ˜pL
:
∂Π/∂λ =qL(˜pL−L)−qH(pH−H)
. If
pi= ˜pL
, it must hold that
˜pL≤˚p(λ)
.
Inserting
˚p(λ)
for
˜pL
gives that
∂Π
∂λ ≤qL(1−λ
λ
qH
qL
(pH−H) + L−L)−qH(pH−H)
=1−2λ
λqH(pH−H)≤0.
•pi= ˆp(λ)
: inserting
ˆp(·)
and
∂pi/∂λ =∂ˆp(λ)/∂λ =−1/λ2(H−L)
into equation
(4.1) in the main text gives
∂Π
∂λ =−qL(H−L)−qH(pH−H)<0.
•pi= ˚p(λ)
. Similarly, inserting
˚p(·)
and
∂˚p(λ)/∂λ =−1/λ2(pH−H)
gives
∂Π
∂λ =−2qH(pH−H)<0.
Non-persuasiveness mitigates competition in the market for life insurances 90
4.6.2 Information sharing systems
When information is shared, one would naturally assume asymmetries of information to
disappear. This depends however on the type of information that is being shared. In
insurance markets, only little information is transferred on the risk type of a prospective
customer,
9
leading us to conclude that these systems serve to inform competitors about
recent applications, thereby preventing them from screening clients. This in turn implies
that rms are symmetrically informed in equilibrium with customers being aware hereof.
To see this, consider the following modied game setting (modications in bold text):
(t=0) With probability
1/2
, nature assigns consumers to rm
i
for
i∈ {A, B}
.
This is
private information to applicants
.
t=1 Price taking stage:
Both rms
make an investment choice
γA, γB∈ {0, γ}
and
contingently observe
θ
. The rms publicly announce price oers
pS
A, pS
B
. Price
takers only observe the oer from the rm to which they have been assigned in
t= 0
, and accept the oer or choose no insurance.
t=2 Bargaining stage: Bargainers ask both rms to make better oers: rms choose
whether to acquire information on consumers that have not been assigned to them.
Firms simultaneously oer
pS
A,2
,
pS
B,2
. Bargainers choose among all oers, i.e.
pS
i
,
pS
i,2
,
pS
j
,
pS
j,2
and no insurance.
This game specication is a translation from the following idea: applicants choose one
rm where they apply in the rst place. Bargainers will then approach the competing
insurer to ask for a new oer. This second rm, since not aware of being faced with a
bargainer, will acquire information and make an oer for the price taker. It can easily
be shown that with informational costs being suciently small, acquiring information is
the only equilibrium outcome here. As before, rms then compete for bargainers in the
last stage.
A similar eect is generated by assuming commitment power: although information
is fully revealed in equilibrium, the lack of commitment power towards competitors is a
driver of the results. In particular, consider the case where the competitor is truthfully
informed about the true risk type. Then, an informed rm is always persuasive and
insurers are symmetrically informed in equilibrium (even if the second rm did not invest
in information). As a result, information sharing without commitment power serves to
increase asymmetries of information instead of reducing them.
9
In particular, only information of pass/fail decisions are being shared
Non-persuasiveness mitigates competition in the market for life insurances 91
This section sketches implications of a market in which rms are symmetrically in-
formed. As mentioned above already, Villeneuve (2005) has demonstrated that two rms
may, in a subset of the set of existing equilibria that have survived elimination accord-
ing to the Intuitive Criterion, make positive prots on bargaining consumers if they are
symmetrically informed. These equilibria however appear unintuitive.
By contrast, Bagwell and Ramey (1991) propose a two-step equilibrium renement
that turns out to uniquely select the ecient separating equilibrium. They propose the
following: Whenever the uninformed party (parties) observe(s) an o-path behaviour, he
(they) then count the number of deviating (informed) rms that are necessary to reach
this (these) particular disequilibrium oer(s). The belief will then be according to the
lower number of deviations. If this lowest number is equal for more than one equilibrium,
the Intuitive Criterion applies,
10
the criterion therefore being consistent with previous
analysis.
A Perfect Bayesian Equilibrium then requires the following:
(1) Beliefs are consistent with Bayes' rule: whenever both rms are informed, and
equilibrium pricing is fully separating, then the consumer must, if he observes some
price vector that is only played for given
θ
, believe
ˆ
θt(·) = θ
. If equilibrium pricing
is pooling for some
θ0, θ00
, then the consumer believes
ˆ
θt(·) = Σθ0,θ00 qθθ
Σθ0,θ00 qθ
.
(2) Consider a disequilibrium pricing tuple
(P0
i, P0
j)6= (Pi(θ), Pj(θ))
(the equilibrium
price vectors of both rms), and denote
N(θ)∈ {1,2}
the number of informed
deviating rms required to generate
(P0
i, P0
j)
. Then,
ˆ
θ2(P0
i, P0
j) = θ0
if
N(θ0)<
N(θ00)
.
(3) Consider a disequilibrium pricing strategy
(P0
i, P0
j)
for which
N(θ0) = N(θ00)
. Then
the Intuitive Criterion applies.
From this equilibrium selection rule, it follows that bargaining consumers never end
up uninsured. By contrast, consider the case where the
H
-type is rejected by both rms
and denote
pS,θ
i,t
an equilibrium candidate oer made by rm
i
to an applicant of type
θ
in
period
t
. Then,
(pS,H
A,1, pS,H
A,2, pS,H
B,1, pS,H
B,2)∈∆
. A rm would like to deviate to an oer that
is protable on
H
and that is considered acceptable by the consumer given his beliefs.
Because rejections are played only on
θ=H
, the consumer belief
ˆ
θ2(Pi∈∆, Pj∈∆)
must be
H
by the Bayes-consistency requirement. Therefore, if rm
i
deviates to a price
˜pH≥p0
i,2> H
,
N(H) = 1
, whereas
N(L) = 2
and the o-bath belief must be
H
too.
But then,
(pS,H
A,1, pS,H
A,2, pS,H
B,1, pS,H
B,2)∈∆
is not an equilibrium. The following are unique
10
Bagwell and Ramey (1991) dene a very similar concept which they call the
-Intuitive Criterion
Non-persuasiveness mitigates competition in the market for life insurances 92
equilibrium allocations, independent of the rm to which the applicant has been assigned
in the rst place (superscripts are being dropped again).
Proposition 4.7.
. The following contracts are oered in equilibrium:
pS,H
i,1=pS,H
j,1∈∆
,
pS,L
i,1=pS,L
j,1= ˜pL
for
i∈ {A, B}
and
pS,H
i,2=pS,H
j,2=H
,
pS,L
i,2=pS,L
j,2=L
.
Proof.
The unique equilibrium in
t= 2
is the perfect competition outcome. Suppose
not, then there exists a
θ
such that
pS,θ
A,2, pS,θ
B,2> θ
for some
θ
. Let
A
be the period-one
rm. Both rms wish to deviate to a lower price. If this price is not played for any other
θ
, denoted
θ0
, in equilibrium, then
N(θ)=1
and
N(θ0)=2
. Therefore, beliefs do not
change and the deviation is protable. On the other hand, there is no pooling equilibrium
in
t= 2
: suppose the type has not been revealed in the rst stage, then the rm to which
the applicant has not been assigned in the rst place has always an incentive to deviate to
˜pL< H
(it was assumed that with symmetric oers, an applicant prefers the initial rm)
if the equilibrium price is at least
H
. On the other hand, prices lower than
H
cannot be
oered in equilibrium, since rms would always want to deviate on the
H
-type. Further,
if the type has been revealed in the rst place, oers cannot be pooling since
˜pu< H
.
The logic for the equilibrium in
t= 1
follows the proof of Proposition 4.2.
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.
List of Figures
2.1 A grading rule
Γ3⊂Ψ3
. ........................... 17
2.2 An example for optimal rules for
N= 5
................... 24
2.3 Full disclosure:
ˆ
θ(ΓN|δ)
for
N= 50
and dierent distributions . . . . . . 26
2.4 Pass/Fail schemes:
δ0(γ1)
for
N= 50
and dierent distributions . . . . . 26
2.5 A,B,C schemes:
δ0(γ1, γ2)
for given
γ2
,
N= 50
and dierent distributions 27
2.6 A,B,C schemes:
δ0(γ1, γ2)
for
N= 50
and dierent
(p0, pe)
........ 33
3.1 Timing in one period without certication . . . . . . . . . . . . . . . . . 39
3.2 Timing of a period
t
with certication . . . . . . . . . . . . . . . . . . . 41
3.3 Timing of a period
t
with certication and capture . . . . . . . . . . . . 43
3.4 Capture proof combinations of
(e, δ)
resp.
(f, δ)
under full disclosure. . . 47
3.5 Capture-proof
(e, δ)
-combinations for low (left) and high (right) marginal
costs
k0
at
e=eF D
............................... 52
3.6 Dark-grey: capture-proof certication with full disclosure. Light-grey:
(additional) capture-proof certication with noisy disclosure. . . . . . . . 54
4.1
pi(λ)
for
L= 0.1,˜pL= 0.15, H = 0.3, pH= 0.4
and dierent
qH
..... 81
xiv
