Mülle , La s; Ka os, Dominik
Wo king Pape
Wel a e e ec s o a concealed in o ma ion exchange
Cen e o Ma hema ical Economics Wo king Pape s, No. 703
P o ided in Coope a ion wi h:
Cen e o Ma hema ical Economics (IMW), Biele eld Uni e si y
Sugges ed Ci a ion: Mülle , La s; Ka os, Dominik (2025) : Wel a e e ec s o a concealed in o ma ion
exchange, Cen e o Ma hema ical Economics Wo king Pape s, No. 703, Biele eld Uni e si y, Cen e
o Ma hema ical Economics (IMW), Biele eld,
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703
Ap il 2025
Wel a e E ec s o a Concealed In o ma ion
Exchange
La s M¨ulle and Dominik Ka os
Cen e o Ma hema ical Economics (IMW)
Biele eld Uni e si y
Uni e si ¨a ss aße 25
D-33615 Biele eld ·Ge many
e-mail: [email p o ec ed]
uni-biele eld.de/zwe/imw/ esea ch/wo king-pape s
ISSN: 0931-6558
Unless o he wise no ed, his wo k is licensed unde a C ea i e Commons
A ibu ion 4.0 In e na ional (CC BY) license. Fu he in o ma ion:
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Wel a e E ec s o a Concealed In o ma ion Exchange∗
La s M¨ulle †Dominik Ka os‡
Ap il 16, 2025
Abs ac
This pape analyzes he wel a e e ec s o p i a e and unila e al disclosu e o sensi-
i e in o ma ion in a sequen ial ba gaining con ex . We conside a model whe e wo
selle s each p opose a ake-i -o -lea e-i p ice o a homogeneous good o a single
buye . The buye accep s o ejec s he i s selle ’s o e be o e he second selle
p oposes he p ice. C ucially, he second selle migh lea n he i s selle ’s p ice
and whe he i was accep ed, allowing he o upda e he belie abou he buye ’s
willingness o pay and op imize he p icing s a egy. The wel a e e ec s caused by
his in o ma ion exchange a e e alua ed unde gene al condi ions. We show ha i
bene i s he buye i a ejec ion is e ealed bu migh ha m him i an accep ance is
e ealed. Addi ionally, he in o ma ion exchange imp o es he socie al wel a e by
educing ine iciencies and p omo ing addi ional ade. This pape s eng hens he
heo e ical amewo k o assessing he wel a e e ec s o in o ma ion exchanges by
o e ing new insigh s and p o iding ools o assess causali y o alleged damages.
Keywo ds: In o ma ion Exchange; Collusion; Unawa eness
JEL Classi ica ion: D82; D83; L41
∗The au ho s hank Ge i Bauch, Ju ek P eke , Ina Tane a, and Nikhil Vellodi o hei help ul
commen s and sugges ions.
†Cen e o Ma hema ical Economics, Biele eld Uni e si y, Pos ach 100131, 33501 Biele eld, Ge many.
lm[email p o ec ed], co esponding au ho . La s M¨ulle g a e ully acknowledges inancial suppo
by he Ge man Resea ch Founda ion (DFG) [RTG 2865/1 – 492988838].
‡Cen e o Ma hema ical Economics, Biele eld Uni e si y, Pos ach 100131, 33501 Biele eld, Ge many.
dominik.k[email p o ec ed]
1
1 In oduc ion and Mo i a ion
The exchange o in o ma ion among compe ing i ms plays a c ucial ole in shaping ma ke
ou comes. Addi ional in o ma ion can con ibu e o ma ke e iciencies, bu can also in-
luence p icing beha io o he consume s’ de imen . Because o he la e , an i us law
is qui e es ic i e wi h espec o in o ma ion exchange, and he las ew yea s ha e seen
se e al cases whe e i ms ha e aced subs an ial ines o exchanging sensi i e in o ma-
ion. Fo ins ance, he Eu opean Commission imposed ines o abou 32 million eu os on
wo me al packaging companies (Eu opean Commission, 2022), and he Ge man Fede al
Ca el O ice ined s eel o ging companies app oxima ely 35 million eu os o exchanging
in o ma ion in iola ion o an i us law (Ge man Fede al Ca el O ice, 2021).1
I seems o be an es ablished p inciple o assume ha p i a e announcemen s among
compe ing i ms can only be mo i a ed by assis ing p ice-collusion and do no enhance
e iciency, as a gued by he OECD (2012). This pape demons a es ha such a p inciple
mus be ques ioned. We show ha e en p i a e and unila e al disclosu e o in o ma ion
can enhance e iciency and enable wel a e-imp o ing ade, which migh e en bene i hi d
pa y consume s. Such an exchange among compe ing i ms is no hing ex ao dina y: o
ins ance, ade associa ions a e pla o ms whe e in o ma ion is easily and o en sha ed.2
Wi h hei shee ubiqui y o almos 60,000 in he U.S. alone,3 he impo ance o co ec ly
assessing he posi i e and nega i e e ec s o in o ma ion exchange becomes ob ious.
This pape analyzes hese e ec s in a sequen ial ading game and assesses he po en-
ial ha m o a consume unawa e o any in o ma ion exchange. This p o ides a na u al
wo s -case analysis om hei pe spec i e. In his model one buye sequen ially ba gains
wi h wo di e en selle s who o e a homogeneous good. Bo h selle s know he dis i-
bu ion unde lying he buye ’s willingness o pay bu no i s ealiza ion when o e ing a
ake-i -o -lea e-i p ice o hei good. The buye accep s o ejec s he i s o e be o e
he second selle p oposes he p ice. The e a e no sa ia ion e ec s and he buye migh
buy ze o, one, o wo goods. The impo an ea u e o he model is ha he subsequen
selle migh addi ionally ecei e in o ma ion abou he cou se o he p eceding nego ia-
1In nume ous o he cases, se e e ines we e imposed on i ms accused o ha ing exchanged comme cially
sensi i e in o ma ion. See Eu opean Commission (2020), whe e e hylene pu chase s we e ined 260 million
eu os, and Eu opean Commission (2021), whe e in es men banks we e ined 371 million eu os.
2See, o example, ines imposed on he Associa ion o he Ge man Con ec ione y Indus y o encou -
aging an an i-compe i i e exchange o in o ma ion (Ge man Fede al Ca el O ice, 2019).
3See IRS (2024), he IRS e e s o p o essional and ade associa ions as business leagues.
2
ion. Two scena ios a e compa ed: a benchma k scena io in which he second selle does
no ecei e any addi ional in o ma ion and an in o ma ion exchange scena io in which he
i s selle e eals o he second selle bo h he o e ed p ice and whe he o no i was
accep ed. In bo h scena ios, he buye is unawa e o he possibili y o an in o ma ion
exchange be ween he wo selle s.
The wel a e e ec s caused by his in o ma ion exchange a e e alua ed unde gene al
condi ions. Mos no ably, i is shown ha he buye bene i s i a ejec ion is e ealed
bu migh be ha med i an accep ance is e ealed. F om an ex-an e poin o iew he
in o ma ion exchange can be posi i e o nega i e o he consume . Socie al wel a e is
always inc eased by he in o ma ion exchange as i educes ine iciencies and p omo es
addi ional ade.
The emainde o his pape is s uc u ed as ollows. In Sec ion 2some ela ed li e a-
u e is p esen ed. Sec ion 3in oduces he bila e al ading s age game be o e Sec ion 4
se s up he sequen ial ading model and analyses he equilib ia ollowing an accep ance
o ejec ion. In Sec ion 5 he wel a e e ec s caused by he in o ma ion exchange a e anal-
ysed om di e en iming pe spec i es. Sec ion 6concludes. All p oo s a e elega ed o
he appendix.
2 Rela ed Li e a u e
While compe i ion law in he Uni ed S a es and he Eu opean Union clea ly p ohibi s
i ms om colluding on p ices, he e is no uni ied posi ion on he sole exchange o p ices
among i ms. Ha ing on (2022) c i icizes his lack o a common ea men and a ibu es
i , a leas in pa , o he “absence o a well-es ablished heo y o ha m”. He es ablishes
a heo e ical ounda ion ha shows how a p i a e exchange o p ices be ween compe i-
o s esul s in highe consume p ices in a s anda d duopoly se ing. Ou pape di e s
as i assesses he economic e ec s o in o ma ion exchanges be ween i ms, who ac as
in o ma ion p o ide s and ecei e s.4Albei om a di e en pe spec i e, ou pape hence
con ibu es o he es ablishmen o a solid heo y o ha m o in o ma ion exchanges by
dis inguishing gene al condi ions unde which hi d pa ies a e ha med by o bene i om
a p i a e exchange o sensi i e in o ma ion.
4See Ki by (1988) o a discussion o he incen i es o oligopolis ic i ms o use ade associa ions as
an in o ma ion exchange mechanism.
3
In o ma ion exchanges and hei impac on compe i ion ha e been ho oughly ex-
amined wi hin oligopolis ic models in which i ms ex-an e olun a ily disclose p i a e
in o ma ion. K¨uhn and Vi es (1995) su ey his s and o li e a u e. Ou pape di e s
om his pe spec i e as he analyzed in o ma ion exchange can no al e he wel a e o
he in o ma ion p o ide . This also dis inguishes he cu en pape om he in o ma ion
design li e a u e as su eyed by Be gemann and Mo is (2019).
The bila e al ading s age game in his pape esembles ha o Mye son and Sa -
e hwai e (1983) wi h independen ly dis ibu ed alua ions and cos s and asymme ic
in o ma ion. They show ha no incen i e-compa ible and indi idually a ional ading
mechanism can be ex-pos e icien . The minimal e iciency loss in his se ing when join ly
using a mechanism and he op imal in o ma ion s uc u e is quan i ied by Scho m¨ulle
(2023). In ou pape , howe e , he ading mechanism is ixed, and we compa e he
wel a e e ec s ha esul om he in o ma ion exchange.
Be gemann, B ooks, and Mo is (2015) e alua e he wel a e consequences o addi ional
in o ma ion ha is p o ided o a monopolis . The monopolis uses his in o ma ion
abou he alua ion o consume s and engages in hi d deg ee p ice disc imina ion. They
cha ac e ize he possible consume and p oduce su plus ha can be implemen ed in his
se ing. The limi s o hese cha ac e iza ions a e also he limi s o he wel a e o he
buye and he second selle in his model. Ali, Kleine , and K. Zhang (2024) p o ide
an equi alence esul be ween he achie able payo p o iles gi en by hese limi s and he
payo s suppo ed by an equilib ium in a game o olun a y disclosu e.
Glode and Opp (2016) is he closes pape in e ms o he model se ing. They also
analyze sequen ial bila e al ading games wi h selle s making ake-i -o -lea e-i o e s bu
ocus on he e ec o in e media ion chains be ween a selle and a buye . They show ha
he addi ion o mode a ely in o med in e media ies imp o es ade e iciency by educing
he incen i e o he selle o ine icien ly sc een he buye o his alua ion. This coun e -
incen i e o sc een o he buye ’s alua ion is also a he co e o he wel a e-enhancing
e ec s o he in o ma ion exchange in ou pape . Howe e , hei analysis ocuses on
implemen ing i s -bes ade and assumes ha ade is always mu ually bene icial. In a
e y simila se ing, Glode, Opp, and X. Zhang (2018) analyze he incen i es o a buye
o olun a ily disclose p i a e in o ma ion bo h in an ex-an e and in an in e im s age o
he game. Ye again, hei ocus is on he implemen a ion o i s -bes ade. Ou pape
hence di e s by sc u inizing o e s ha migh be ejec ed wi h posi i e p obabili y.
4
Wi hin his pape , playe s conside in o ma ion abou hei nego ia ions by de aul
as p i a e o a oid dis o ing he analysis by e ec s ha may a ise om he an icipa ion
o disclosed in o ma ion. On he one hand side, his allows us o achie e an unbiased
assessmen o he buye ’s wo s -case wel a e e ec s. On he o he hand side, he buye ’s
unawa eness ensu es ha we do no need o be conce ned abou epu a ion building as
in Milg om and Robe s (1982) o K eps and Wilson (1982).
3 The Bila e al T ading Game
To analyze he wel a e e ec s o an in o ma ion exchange in a sequen ial ba gaining con-
ex , i is necessa y o i s conside he indi idual s age game, which is played epea edly.
This sec ion de ines his s age game and analyses i s equilib ium s a egies and payo s.
3.1 No a ion and P elimina ies
In a bila e al ading game wi h incomple e in o ma ion, a isk-neu al selle (she) o e s
a single indi isible good a a p ice p≥0 o a isk-neu al buye (he) who ei he accep s
o ejec s ha o e . The buye ’s p i a e alua ion o he good is deno ed by ≥0, and
he selle ’s (oppo uni y) cos o p oducing i is gi en by c≥0. I he buye accep s, he
good is exchanged, and payo s a e −p o he buye and p−c o he selle . In case
o a ejec ion, playe s ob ain ze o payo s. Bo h and ca e independen ly dis ibu ed
acco ding o commonly known p obabili y dis ibu ions Fand Gwi h densi ies and
g. Thus, any bila e al ading game is uniquely de e mined by he pai ( , g), and, wi h
sligh abuse o no a ion, we iden i y he game by his pai .
We deno e he (se - heo e ic) suppo s o and gby
V:= { ∈R≥0: ( )>0}and C:= {c∈R≥0:g(c)>0},
espec i ely. To ease no a ion, le := in (V), := sup(V), c:= in (C), and c:= sup(C).
Th oughou his pape , all densi ies a e assumed o be con inuous on hei suppo s.5
A pu e s a egy o he selle is a map σ:C → R ha maps any ealized cos c o
a ake-i -o -lea e-i p ice p. A pu e s a egy o he buye assigns o any pai ( , p) o a
alua ion and an o e ed p ice a ejec ion o accep ance.
5Fo echnical easons, is w.l.o.g. assumed o be igh -con inuous a , i.e., ( ) = lim & ( ).
5
The ollowing s anda d assump ion is imposed on he buye ’s dis ibu ion.6
Assump ion 1. The haza d a e h:V { } → R≥0, de ined by h( ) := ( )
1−F( ), is
non-dec easing o all ∈ V { }.
No e ha Assump ion 1ensu es he con exi y o Vand, he e o e, gua an ees ha F( )
is s ic ly inc easing o all ∈[ , ).
3.2 Equilib ium S a egies
In a pe ec Bayesian equilib ium (hence o h, jus equilib ium), he buye will accep any
o e p < and ejec any o e p > .7Gi en his s a egy, he selle ’s expec ed payo
om o e ing p, gi en he own cos c, is
Π(p;c) = (1 −F(p)) (p−c).(1)
Hence, i c > , he e is no scope o ade in equilib ium; and i cis su icien ly low such
ha i is no wo hwhile o he selle o isk a ejec ion, she will simply o e and make
a p o i o −cwi h p obabili y 1.
In he middle g ound, he selle aces he adeo ha highe p ices a e ejec ed wi h
highe p obabili y. One can easily check ha an in e io poin p ha maximizes (1) mus
sa is y he i s o de condi ion
h(p)(p−c) = 1.(2)
We deno e he solu ion o (2) by ˆp(c). Then, he e exis s a unique cwi h p= ˆp(c) o
all p∈( , ) by Assump ion 1. Indeed, sol ing (2) o cquickly gi es ˆp−1( ) = −1
h( ).8
6The assump ion is, o ins ance, sa is ied by a log-conca e densi y (Bagnoli and Be gs om, 2005,
Co olla y 2) and he e o e by a wide ange o p obabili y dis ibu ions, e.g., he uni o m dis ibu ion on
any con ex se , he exponen ial dis ibu ion, and he gamma dis ibu ion o shape pa ame e s g ea e
o equal o one. I is also sa is ied by he Kuma aswamy dis ibu ion K(a, b) wi h pa ame e s a≥1
and b > 0 ha allows o a wide ange o densi ies and, in con as o he ela ed Be a-dis ibu ion,
can be exp essed in closed o m (Kuma aswamy, 1980). Assump ion 1is also igh ly connec ed o he
omnip esen egula i y assump ion in auc ion heo y, which ensu es s ic ly inc easing i ual alua ions
(Mye son, 1981) as well as o Assump ion 1 in Glode and Opp (2016) and equi alen o he second haza d
a e condi ion in McA ee (2002).
7In o de o a oid unin e es ing case dis inc ions, we assume ha o p= he o e will be accep ed.
8I migh be wo h men ioning he e ha he in e se ˆp−1 akes he o m o a i ual alua ion in
6
Thus, de ine
τ=τ( )≡lim
& ˆp−1( ) =
−1
h( ),i ( )>0
−∞,i ( )=0.
(3)
This p o ides us wi h a lowe bound o he domain o ˆp. The ollowing Lemma summa izes
all p ope ies o ˆpand is used o la e e e ence.
Lemma 3.1. The map ˆp: (τ( ), )→( , )is a con inuous and s ic ly inc easing
bijec ion wi h in e se ˆp−1( ) = −1
h( ) o all ∈( , ). In pa icula , ˆp(c)> c o all
c∈(τ( ), ).
Wi h sligh abuse o no a ion, we shall w i e ˆp−1( ) = τ( ) and ˆp−1( ) = , keeping in
mind ha τ( ) and migh be −∞ and +∞, espec i ely. A e ha ing de ined ˆp, we
can now desc ibe he selle ’s equilib ium s a egies in he a ea whe e hey a e unique.
Lemma 3.2. E e y bila e al ading game ( , g)has an equilib ium. In any equilib ium,
he selle ’s s a egy ˆσsa is ies o all c∈ C
ˆσ(c) = ˆσ(c; , g) =
, i c≤τ( )
ˆp(c),i τ( )< c < .
Mo eo e , ˆσis con inuous in c o all c∈ C wi h c < .
The mono onici y o ˆpimplies, unsu p isingly, ha highe cos lead o highe p oposed
p ices. So a , no assump ions abou he ela ion o and gwe e made. The ollowing
ema k cap u es he i ial cases.
Rema k 3.3. In a bila e al ading game ( , g) he ollowing hold:
i) I c < τ( ), hen ˆσ≡ and ade happens wi h ce ain y a p ice .
ii) I c > , hen mu ually bene icial ade is ex-an e impossible. In equilib ium, any
p ice p≥ccan be chosen, and any o e will be ejec ed.
classical auc ion heo y (see Mye son, 1981). As in his s and o li e a u e, he s ic mono onici y o ˆp
and ˆp−1is ensu ed by Assump ion 1.
7
The esul s o P oposi ion 4.6 and Co olla y 4.7 a e no ha su p ising: i he second
selle has obse ed a ejec ion, she will o e a lowe p ice; and he lowe he ejec ed p ice
she obse ed, he lowe he p ice will be. This has wo e ec s: Fi s , he e is a highe
chance ha he o e will be accep ed. Second, in case he o e is accep ed, he buye
will pay a lowe p ice han in he benchma k. As he i s e ec only appea s o buye
alua ions ha a e su icien ly small, he buye ’s gain depends on .
5 Wel a e E ec s
We shall now in es iga e he gains and losses o he buye and he second selle om
he unila e al in o ma ion disclosu e by he i s selle . We conside ou s ages: ex pos ,
i.e., a e all pa ame e s ha e been e ealed; in e im, i.e., a e all playe s know hei own
pa ame e s, bo h be o e and a e he in o ma ion o he i s selle has been disclosed;
and ex an e, i.e., be o e playe s e en know hei own pa ame e . As he e is no e ec on
he i s selle , we only in es iga e he e ec s on he buye ’s and he second selle ’s payo .
5.1 Ex-pos Wel a e E ec s
In his subsec ion, we in es iga e wel a e e ec s a e he buye ’s alua ion and he selle s’
cos s ha e ealized. P oposi ion 4.5 and P oposi ion 4.6 show ha he buye can be
ha med by he in o ma ion disclosu e only in case o an accep ance, and ha he can
p o i om he in o ma ion disclosu e only in case o a ejec ion in he i s game. The
ollowing p oposi ion o malizes hese insigh s.
P oposi ion 5.1. Le ( , g1, g2)be a sequen ial ading game, and le ∈ V,c1∈ C1, and
c2∈ C2.
1. I ≥ˆp(c1), hen ˆu , c2; A
c1, g2≤ˆu( , c2; , g2). The inequali y is s ic i and
only i c2< c1. Mo eo e , ˆu , c2; A
c1, g2is dec easing in c1and s ic ly dec easing
in c1i c2< c1.
2. I < ˆp(c1), hen ˆu , c2; R
c1, g2≥ˆu( , c2; , g2). The inequali y is s ic i and
only i τ( )< c2<ˆpR
c1−1( ). Mo eo e , ˆu , c2; R
c1, g2is dec easing in c1and
s ic ly dec easing in c1i τ R
c1< c2<ˆp(c1).
14
The o e all wel a e ha is gene a ed by ade is −c2and independen o he in o ma-
ion disclosu e. Thus, he in o ma ion exchange only a ec s o e all wel a e i i changes
whe he o no a ade occu s. By P oposi ion 4.5, an accep ed o e in he i s ound
does no in luence whe he o no he buye and he selle in he second ound ade. By
P oposi ion 4.6, a ejec ed o e in he i s ound ei he has no e ec o inc eases he
amoun o ade. Thus, we ob ain he ollowing esul s and omi he p oo s.
Theo em 5.2. Le ( , g1, g2)be a sequen ial ading game, and le ∈ V,c1∈ C1, and
c2∈ C2.
1. I ≥ˆp(c1), hen ˆu , c2; A
c1, g2+ ˆπ , c2; A
c1, g2= ˆu( , c2; , g2) + ˆπ( , c2; , g2).
2. I < ˆp(c1), hen ˆu , c2; R
c1, g2+ ˆπ , c2; R
c1, g2≥ˆu( , c2; , g2) + ˆπ( , c2; , g2).
This inequali y is s ic i and only i τ( )< c2<ˆpR
c1−1( ).
In e es ingly, he in o ma ion disclosu e by he i s selle migh also ha m he second
selle om an ex-pos pe spec i e. This happens i she adap s and lowe s he p ice a e
obse ing a ejec ion, al hough he benchma k o e would ha e been accep ed.
Co olla y 5.3. Le ( , g1, g2)be a sequen ial ading game, and le ∈ V,c1∈ C1, and
c2∈ C2. I τ( )< c2<ˆp−1( )< c1, hen ˆπ , c2; R
c1, g2<ˆπ( , c2; , g2).
5.2 In e im Wel a e E ec s A e In o ma ion Disclosu e
We nex conside in e im e ec s a e he in o ma ion disclosu e, i.e., e ec s a e bo h
he buye and he second selle know hei own ype and he selle has upda ed he belie
abou he buye ’s ype. As he ou come o he i s game is no a ec ed by he disclosu e
o in o ma ion, he e ec on he buye ’s payo a e an accep ed o ejec ed o e a e
∆ˆ
UA( ;c1)≡
ˆ
U ; A
c1, g2−ˆ
U( ; , g2),i ≥ˆp(c1)
0,i < ˆp(c1),
∆ˆ
UR( ;c1)≡
0,i ≥ˆp(c1)
ˆ
U ; R
c1, g2−ˆ
U( ; , g2),i < ˆp(c1),
espec i ely. (Recall ha ˆ
U( ; , g) deno es he buye ’s in e im expec ed payo in he
bila e al ading game ( , g), de ined in Equa ion (6).)
15
P oposi ion 5.4. Le ( , g1, g2)be a sequen ial ading game, le ∈ V and c1∈ C1.
1. Le ≥ˆp(c1). Then ∆ˆ
UA( ;c1)≤0. The inequali y is s ic i and only i c1> c2.
Mo eo e , ∆ˆ
UA( ;c1)does no depend on .
2. Le < ˆp(c1). Then ∆ˆ
UR( ;c1)≥0. The inequali y is s ic i and only i c2> τ( )
and G2ˆpR
c1−1( )> G2(τ( )). Mo eo e , ∆ˆ
UR( ;c1)is weakly inc easing in .
A he in e im s age a e he in o ma ion disclosu e e ealed an accep ance in he i s
game, he i s pa o P oposi ion 5.4 shows ha he buye can only su e om he in-
o ma ion disclosu e. Howe e , maybe su p isingly, he size o he damage is independen
o his own alua ion . The easoning is ha an accep ance o he i s o e has no e ec
on he likelihood ha he second o e will be accep ed as well. Indeed:
(i) I c2≤τ( ), hen in he benchma k he selle would ha e o e ed and his would
ha e been accep ed wi h ce ain y. Wi h he addi ional in o ma ion she o e s ˆp(c1),
which is accep ed wi h ce ain y as well;
(ii) I τ( )< c2< c1, hen in he benchma k she would ha e o e ed ˆp(c2)<ˆp(c1) and
he o e would ha e been accep ed. A e ob aining he addi ional in o ma ion she
o e s ˆp(c1), which is s ill accep ed wi h ce ain y.
(iii) I c2≥c1, he beha io o he second selle is no a ec ed a all.
In con as , he ealized cos o he i s selle c1has, condi ional on an accep ance in he
i s ound, wo e ec s: Fi s , i c1inc eases, he p obabili y ha c1exceeds c2inc eases
as well, which is exac ly he p obabili y ha he second selle inc eases he p ice due
o he ob ained in o ma ion. Second, i inc eases he (condi ional) expec ed di e ence
ˆp(c1)−ˆp(c2), which, i he second o e is accep ed as well, inc eases he buye ’s loss due
o he in o ma ion disclosu e.
A he in e im s age a e he in o ma ion disclosu e e ealed a ejec ion, he second
pa o P oposi ion 5.4 shows ha he buye can only p o i :
(i) I c2≤τ( ), he second selle will o e , ei he way, so he in o ma ion disclosu e
makes no di e ence.
(ii) I τ( )< c2≤τ( R
c1), he selle would ha e o e ed ˆp(c2) in he benchma k compa ed
o he lowe and ce ainly accep ed a e obse ing a ejec ion. Since ˆp(c2) is
16
ejec ed wi h s ic ly posi i e p obabili y, he buye p o i s in wo ways: a highe
p obabili y o ade and lowe p ices gi en ade.
(iii) I τ( R
c1)< c2≤ˆp−1( ), he selle o e s ˆpR
c1(c2)<ˆp(c2) a e obse ing a ejec-
ion. As ≥ˆp(c2), no addi ional ade (compa ed o he benchma k) eme ges,
ne e heless he buye p o i s om he lowe p ices.
(i ) I ˆp−1( )< c2≤(ˆpR
c1)−1( ) he selle again o e s ˆpR
c1(c2)<ˆp(c2), bu now inducing
addi ional ade, as he benchma k o e would ha e been ejec ed while he ac ual
o e is no . This addi ionally gene a ed ade is he eason ha he in e im expec ed
p o i ∆ ˆ
UR( ;c1) a e obse ing a ejec ion, is inc easing in .
( ) I c2>(ˆpR
c1)−1( ), hen he second selle ’s o e will be ejec ed wi h and wi hou
in o ma ion disclosu e.
In P oposi ion 5.4 no assump ions on he second selle ’s dis ibu ion G2we e made. I
G2is s ic ly inc easing, i.e., i he suppo o g2is con ex, hen he condi ion in pa 2
simpli ies signi ican ly.
Co olla y 5.5. Le ( , g1, g2)be a sequen ial ading game, le ∈ V,c1∈ C1wi h
ˆp(c1)> , and le C2be con ex. Then ∆ˆ
UR( ;c1)>0i and only i c2> τ( )and
c2<ˆpR
c1−1( ).
We close his subsec ion wi h he ollowing example, which illus a es he indings in
P oposi ion 5.4.
Example 5.6. Conside he sequen ial ading game ( , g1
1, g2), whe e =g1
1=g2a e
he uni o m dis ibu ion on [0,1]. In his case we ha e ˆp(c1) = 1+c1
2and
∆ˆ
UA( ;c1) =
−c2
1
4i ≥1+c1
2
0 i < 1+c1
2,
∆ˆ
UR( ;c1) =
0 i ≥1+c1
2
(4 −c1−1)2
16 i 1+c1
4< < 1
2
(1−c1)(8 −c1−3)
16 ,i 1
2≤ < 1+c1
2.
17
-0.10 -0.05 0.00 0.05 0.10
∆ˆ
UA( ;c1)+∆ˆ
UR( ;c1)
0 0.25 0.5 0.75 1
c1= 0
c1=1
4
c1=1
2
Figu e 2: In e im e ec o he buye a e esul o i s ading game has been disclosed.
The o e all e ec o he in o ma ion disclosu e on he buye ’s wel a e is gi en by ∆ ˆ
UA+
∆ˆ
URand depic ed in Figu e 2 o se e al alues o c1. As shown in P oposi ion 5.4, he
e ec is posi i e o < ˆp(c1), whe e a discon inui y occu s. Fo ≥ˆp(c1), he e ec is
nega i e (s ic ly i c1>0) and independen o . Mo eo e , a buye wi h ixed alua ion
who is ha med by he in o ma ion exchange will be ha med mo e se e ely o la ge c1.
Simila ly, a buye (wi h ixed ) who bene i s om he in o ma ion exchange will p o i
less o la ge c1.
5.3 In e im Wel a e E ec s Be o e In o ma ion Disclosu e
We nex conside he expec ed wel a e e ec on he buye be o e he cos o he i s selle
has been e ealed, which is gi en by
∆ˆ
U( )≡EG1h∆ˆ
UA( ;·)+∆ˆ
UR( ;·)i=Z∞
−∞
g1(c1)∆ˆ
UA( ;c1)+∆ˆ
UR( ;c1)dc1.
In e ms o Figu e 2, his unc ion is a weigh ed a e age o he unc ions depic ed he e.
The ollowing example shows his unc ion o di e en dis ibu ions G1.
18
Example 5.7. Recall he sequen ial ading game ( , g1
1, g2) om Example 5.6, whe e all
dis ibu ions a e uni o m on [0,1]. Then,
∆ˆ
U1( )≡EG1
1h∆ˆ
UA( ;·)+∆ˆ
UR( ;·)i=
0 i < 1
4
(4 −1)3
48 i < 1
4≤ < 1
2
3
6− 2+ −1
4i ≥1
2
,
which is depic ed in Figu e 3b. Figu e 3a depic s he co esponding densi y g1
1(= g2= ).
Obse e ha ∆ ˆ
U1( )≡0 o su icien ly small . Mo e speci ically, i < ˆpR
c1c2=1
4,
he e is no ade a all and he in e im wel a e in he benchma k and he in o ma ion
exchange scena io a e bo h ze o. On he o he hand, ∆ ˆ
U1( )>0 o in e media y alues
o and ∆ ˆ
U1( )<0 o su icien ly high alua ions. This ollows as o all such
ha ˆpR
c1c2< < ˆpc1=1
2, he benchma k o e will be ejec ed wi h ce ain y and
he buye expec s a s ic ly posi i e p o i . Fo sligh ly abo e ˆpc1, he expec ed gain
ollowing a ejec ion s ill ou weighs he expec ed loss ollowing an accep ance, in pa icula
as he p obabili y o an accep ance is low. Howe e , e en hough, by P oposi ion 5.4, he
expec ed gain a e a ejec ion inc eases in , o la ge his e ec is ou weighed by he
inc easing p obabili y ha he i s o e is accep ed, which would esul in a loss.
Figu e 3c depic s he densi y unc ion g2
1wi h g2
1(c1)=4−8c1 o all c1∈0,1
2and
0 o he wise. The co esponding unc ion ∆ ˆ
U2is shown in Figu e 3d. Fo < ˆpc1=1
2,
he in e im expec ed gains a e o de ed: smalle c1, which a e mo e likely gi en g2
1 han
g1
1, imply lowe p ices by he second selle a e a ejec ion (Co olla y 4.7) and, hence,
highe gains (Figu e 2). Thus, ∆ ˆ
U2( ) inc eases as e han ∆ ˆ
U1( ) o < 1
2.
A hi d, maybe a bi pa hological, densi y is gi en by g3
1(c1) = 1 + sin 6πc1+π
12 o
all c1∈0,1
2and 0 o he wise, which is depic ed in Figu e 3e. The co esponding unc ion
∆ˆ
U3in Figu e 3 illus a es wo poin s: Fi s , gi en ∆ ˆ
U3<0, i is no mono onic. Tha
is, unlike Figu es 3b and 3d sugges , buye s wi h la ge do no au oma ically su e
mo e. Fu he mo e, ∆ ˆ
U3has se e al oo s. Hence, e en i some buye wi h alua ion
su e s a he in e im le el, he e migh be ano he buye wi h a la ge alua ion 0who
gains om he in o ma ion disclosu e.
All unc ions ∆ ˆ
Uj( ) o j∈ {1,2,3}a e no di e en iable o = ˆp(c1), indeed,
di e en iabili y a ha poin would equi e ha g1c1= 0. Bo h ∆ ˆ
U2and ∆ ˆ
U3can be
de i ed analy ically. The espec i e exp essions can be ound in Appendix B.
19
n
n
0 1
0 0.5 1
(a) PDF g1
1as in Example 5.7
n
n
0 0.25 0.5 0.75 1
-0.05 0 0.05
(b) ∆ˆ
U1( ) gi en PDF g1
1as in Example 5.7.
n
n
0 1
0 2 4
(c) PDF g2
1as in Example 5.7
n
n
0 0.25 0.5 0.75 1
-0.05 0 0.05
(d) ∆ˆ
U2( ) gi en PDF g2
1as in Example 5.7.
n
0 1
0 1 2
(e) PDF g3
1as in Example 5.7
∆U( )
0 0.25 0.5 0.75 1
-0.05 0 0.05
0
( ) ∆ˆ
U3( ) gi en PDF g3
1as in Example 5.7.
Figu e 3: In e im buye wel a e e ec s be o e in o ma ion disclosu e o di e en dis ibu ions.
20
As he p e ious example shows, he e is li le ha can be said abou ∆ ˆ
Uin e ms o
compa a i e s a ics. The e a e ce ain a eas hough, whe e c isp esul s can be ob ained.
P oposi ion 5.8. Le ( , g1
1, g2)and ( , g2
1, g2)be wo sequen ial ading games such ha
C1
1=C2
1and g1
1 i s -o de s ochas ically domina es g2
1.
1. Fo all < ˆp(c1
1)i holds ha 0≤∆ˆ
U1( )≤∆ˆ
U2( ).
2. Fo all > ˆp(c1
1)i holds ha ∆ˆ
U1( )≤∆ˆ
U2( )≤0.
5.4 Ex-An e Wel a e E ec s
F om an o e all wel a e pe spec i e, we ha e seen in Theo em 5.2 ha he disclosu e o
in o ma ion by he i s selle is bene icial, e en a he ex-pos le el. F om he buye ’s
pe spec i e his is no he case: ex pos , he e ec can be bo h posi i e o nega i e, and
he same is ue a he in e im s age. The buye ’s ex-an e e ec is gi en by
EFh∆ˆ
U(·)i=Z∞
−∞
( )∆ ˆ
U( )d ,
which again can be posi i e o nega i e as he ollowing example illus a es.
Example 5.9. Recall he sequen ial ading games ( , g1
1, g2) and ( , g2
1, g2) om Example
5.7. We ind EFh∆ˆ
U1(·)i=−1
768 <0 and EFh∆ˆ
U2(·)i=67
15360 >0
F om an ex-an e pe spec i e, he e is also no much ha can be said wi h espec o
changes in he dis ibu ions. Figu e 2 om Example 5.6 illus a es ha ∆ ˆ
UA+ ∆ ˆ
UR
only depends on he alues o c1bu no hei likelihood. I can also be seen ha i
g0
1 i s o de s ochas ically domina es g1 he e ec on ∆ ˆ
U( ) can be posi i e o nega i e
depending on . Hence, we can cons uc p obabili y dis ibu ions Fsuch ha a i s o de
s ochas ic shi om g1 o g0
1can ha e a posi i e o a nega i e e ec on EFh∆ˆ
U( )i.
5.5 When Cos s a e no Independen
Th oughou his pape , we ha e assumed ha he cos s o he wo selle s a e independen .
As hey p oduce a homogeneous good, his migh no be he case. Le ˜g(c1, c2) be a
join p obabili y densi y unc ion wi h ma ginal densi ies ˜g1(c1) and ˜g2(c2), and allow o
he possibili y ha ˜g(c1, c2)6= ˜g1(c1) ˜g2(c2). Conside now he sequen ial ading game
21
( , ˜g1,˜g2) as be o e. Clea ly, he equilib ium s a egies do no change compa ed o he
p e ious analysis, as hey do no depend on any dependency s uc u e be ween he wo
cos s. The ollowing co olla y ollows di ec ly om P oposi ion 5.1 and Theo em 5.2.
Co olla y 5.10. Le ( , ˜g1,˜g2)be a sequen ial ading game wi h ˜g(c1, c2)=0 o all
c2< c1. Then o all ∈ V,c1∈ C1, and c2∈ C2i holds ha
1. ˆu , c2; A
c1,˜g2= ˆu( , c2; , ˜g2)and ˆπ , c2; A
c1,˜g2= ˆπ( , c2; , ˜g2).
2. ˆu , c2; R
c1,˜g2≥ˆu( , c2; , ˜g2)wi h s ic inequali y i and only i bo h < ˆp(c1)
and τ( )< c2<ˆpR
c1−1( )hold.
Co olla y 5.10 shows ha i is no possible o he buye o ge ha med by he in o ma ion
exchange, i he cos o he second selle is (almos ) always highe han he cos o he
i s selle . Since, i is gua an eed ha he buye can no be ha med by he in o ma ion
exchange om his ex-pos pe spec i e, he e is no possibili y o damage o he buye
om all o he iming pe spec i es as well.
No e ha , analogous o Co olla y 5.3, i is s ill possible ha he second selle is ha med
by he in o ma ion exchange om an ex-pos pe spec i e.
6 Discussion and Conclusion
The objec i e o his pape was o analyze he wel a e e ec s o p i a e and unila e al
disclosu e o sensi i e in o ma ion in a sequen ial ba gaining con ex .
We de elop a sequen ial ading model in which one buye ba gains wi h wo di e en
selle s. Ou esul s show ha he buye p o i s om he p i a e and unila e al disclosu e
among he selle s i i e eals a ejec ion bu migh be ha med by a e ealed accep ance. I
is shown ha he buye canno be ha med bu migh p o i om he in o ma ion exchange,
i he cos o he selle ha ecei es he message is highe han he cos o he in o ma ion
p o iding selle . This esul holds om an ex-pos , in e im, and ex-an e pe spec i e. I
is also shown ha he exchange o in o ma ion imp o es he socie al wel a e by educing
ine iciencies and po en ially gene a ing addi ional ade.
Addi ionally, he buye can only be ha med by he in o ma ion exchange i he selle
in he second game holds he en i e ba gaining powe as assumed in he pape . To see
22
his, conside a e e sion o he ba gaining powe such ha he buye p oposes a ake-i -
o -lea e-i p ice in he second game. The second selle accep s e e y cos -co e ing o e
i espec i e o any in o ma ion exchange be ween he wo selle s. Thus, he unila e al
disclosu e o in o ma ion has no e ec on any playe s’ wel a e.
O e all, his pape s eng hens he heo e ical amewo k o assessing he wel a e
e ec s o in o ma ion exchanges by o e ing new insigh s and p o iding ools o assess
causali y o alleged damages. This does no only ques ion exis ing economic p inciples
as desc ibed abo e, bu also con ibu es o ques ions ha a e a he co e o ollow-on
damages claims, in which consume s mus demons a e a causal link be ween alleged
an icompe i i e p ac ices and hei claimed damages.
Re e ences
Ali, S.Nageeb, Kleine , And eas, and Zhang, Kun (2024): “F om Design o Disclosu e”. In:
a Xi 2411.03608.
Bagnoli, Ma k and Be gs om, Ted (2005): “Log-Conca e P obabili y and I s Applica ions”. In:
Economic Theo y 26(2), pp. 445–469.
Be gemann, Di k, B ooks, Benjamin, and Mo is, S ephen (2015): “The Limi s o P ice Dis-
c imina ion”. In: Ame ican Economic Re iew 105(3), pp. 921–957.
Be gemann, Di k and Mo is, S ephen (2019): “In o ma ion Design: A Uni ied Pe spec i e”. In:
Jou nal o Economic Li e a u e 57(1), pp. 44–95.
Blackwell, Da id (1953): “Equi alen compa isons o expe imen s”. In: The annals o ma he-
ma ical s a is ics 24(2), pp. 265–272.
de Oli ei a, Hen ique (2018): “Blackwell’s In o ma i eness Theo em using Diag ams”. In: Games
and Economic Beha io 109, pp. 126–131.
Eu opean Commission (2020): An i us : Commission ines e hylene pu chase s €260 million
in ca el se lemen .u l:h ps://ec.eu opa.eu/commission/p essco ne /de ail/
en/ip_20_1348 ( isi ed on 03/10/2025).
23
+Zˆp−1( )
τ( R
c1)
g2(c2)ˆp(c2)−ˆpR
c1(c2)dc2
+Z(ˆpR
c1)−1( )
ˆp−1( )
g2(c2) −ˆpR
c1(c2)dc2
≥0,
whe e he non-nega i i y o he h ee in eg ands ollows om ˆp(c2)≥ o all
c2≥τ( ), P oposi ion 4.6, and ˆpR
c1(c2)≤ o all c2≤ˆpR
c1−1( ), espec-
i ely. Since all in eg ands a e s ic ly posi i e in he in e io o he espec i e
in e als, ∆ ˆ
UR( ;c1)>0 i and only i G2ˆpR
c1−1( )> G2(τ( )). Finally,
he i s in eg al does no depend on , and he sum o he las wo in eg als
is weakly inc easing by he same easoning as be o e.
P oo o P oposi ion 5.8.1. Le < ˆp(c1
1) = ˆp(c2
1). Then ∆ ˆ
UA( ;c1) = 0 o all
c1∈ C1
1=C2
1. The e o e,
∆ˆ
Uj( ) = Z∞
−∞
gj
1(c1)∆ ˆ
UR( ;c1)dc1≥0 o j∈ {1,2}.
Conside now c1, c0
1∈ C1
1=C2
1wi h c1< c0
1and no e ha by P oposi ion 5.1
ˆu( , c2; R
c1, g2)≥ˆu( , c2; R
c0
1, g2) o all c2∈ C2and, hence, ∆ ˆ
UR( ;c1)≥∆ˆ
UR( ;c0
1).
Since g1
1 i s -o de s ochas ically domina es g2
1, we hus ha e
∆ˆ
U1( ) = Z∞
−∞
g1
1(c1)∆ ˆ
UR( ;c1)dc1≤Z∞
−∞
g2
1(c1)∆ ˆ
UR( ;c1)dc1= ∆ ˆ
U2( ).
2. Le > ˆp(c1
1). Then ∆ ˆ
UR( ;c1) = 0 o all c1∈ C1
1and, hus,
∆ˆ
Uj( ) = Z∞
−∞
gj
1(c1)∆ ˆ
UA( ;c1)dc1≤0 o j∈ {1,2}.
Again, P oposi ion 5.1 implies ha ∆ ˆ
UA( ;c1)≥∆ˆ
UA( ;c0
1) o all c1, c0
1∈ C1
1wi h
c1< c0
1and, hence, ∆ ˆ
U1( )≤∆ˆ
U2( ) by i s -o de s ochas ic dominance.
30
B Complemen a y Func ions Example 5.7
∆ˆ
U2( ; , g2
1) =
0,i ∈0,1
4
(3−4 )(4 −1)3
24 ,i ∈1
4,3
8
2−7
12 +11
128,i ∈3
8,1
2
−2
3 4+22
3 3−12 2+27
4 −463
384,i ∈1
2,3
4
−1
96,i ∈3
4,1
.
∆ˆ
U3( ; , g3
1)
=
0,i ∈0,1
4
cosπ(288 −71)
12 +6π(4 −1)(3π(4 −1)(2π(4 −1)+cos(π
12 ))+sin(π
12 ))−cos(π
12 )
1728π3,i ∈1
4,1
2
(72π2 2−5)cosπ(144 −71)
12 +288π3 3−1728π3 2−12π(3 −2) sinπ(144 −71)
12
1728π3
+12πsin(73π
12 )−24π(sin(73π
12 )−72π2) +cos(73π
12 )+4 cos(π
12 )−432π3
1728π3,i ∈1
2,1
31
