Chiba, Sao i
A icle
In o ma ion ansmission and coun e ailing biases in
o ganiza ions
Games
P o ided in Coope a ion wi h:
MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: Chiba, Sao i (2024) : In o ma ion ansmission and coun e ailing biases in
o ganiza ions, Games, ISSN 2073-4336, MDPI, Basel, Vol. 15, Iss. 3, pp. 1-25,
h ps://doi.o g/10.3390/g15030018
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Ci a ion: Chiba, S. In o ma ion
T ansmission and Coun e ailing
Biases in O ganiza ions. Games 2024,
15, 18. h ps://doi.o g/10.3390/
g15030018
Academic Edi o s: Ul ich Be ge and
Ma co A. Ma ini
Recei ed: 11 Ma ch 2024
Re ised: 2 May 2024
Accep ed: 16 May 2024
Published: 22 May 2024
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games
A icle
In o ma ion T ansmission and Coun e ailing Biases
in O ganiza ions
Sao i Chiba
Facul y o Economics, Kyo o Sangyo Uni e si y, Mo oyama, Kamigamo, Ki a-ku, Kyo o 603-8555, Japan;
[email p o ec ed]
Abs ac : A decision make (DM) mus choose be ween wo p ojec s o decide on no p ojec . The
expec ed bene i s o hese p ojec s a e co ela ed. The DM seeks ad ice om an agen wi h p i a e
in o ma ion abou he p ojec s’ bene i s. Howe e , he agen ’s di e gen p e e ences o p ojec s
and lack o conside a ion o he DM’s implemen a ion cos s may in oduce wo ypes o biases:
p ojec bias, a o ing he agen ’s p ojec , o pande ing bias, a o ing he p ojec p e e ed by he
DM. Ou indings e eal ha p ojec co ela ion leads o hese biases coun e ailing each o he ,
acili a ing he ansmission o in o ma ion. The agen ypically ecommends a p ojec based on
p i a e in o ma ion o dissuade he DM om choosing no p ojec , as his would be de imen al o he
agen . Addi ionally, we explo e op imal delega ion wi hin o ganiza ions. In con as o he p e ailing
li e a u e ad oca ing o delega ion o biased agen s o enhanced in o ma ion elici a ion, ou s udy
sugges s limi ed bene i s in he con ex o p ojec co ela ion.
Keywo ds: bias; cheap alk; co ela ion; ou side op ions
JEL Classi ica ion: C72; D23; D83
1. In oduc ion
A signi ican challenge in o ganiza ions and ma ke s a ises when he decision make
(DM) elies on ad ice om a mo e in o med agen . An ex ensi e li e a u e, da ing back
o C aw o d and Sobel (1982) [
1
], in es iga ed he c edibili y o “cheap alk” in si ua ions
wi h con lic s o in e es be ween he DM and he agen . Che, Dessein, and
Ka ik (2013) [2]
con ibu ed o unde s anding s a egic communica ion by inco po a ing an ou side op ion
in o a cheap alk model, examining he economics o pande ing. This pape in oduces
a no el conside a ion: in s a egic communica ion h ough cheap alk, how does he
co ela ion be ween p ojec bene i s in luence in o ma ion ansmission amids mul iple
dimensions o con lic s o in e es be ween playe s? Ou key insigh is ha p ojec
co ela ion leads o p ojec bias ( a o ing he agen ’s p ojec ) and pande ing bias ( a o ing
he DM’s p e e ed p ojec ), o se ing each o he and acili a ing in o ma ion ansmission.
In o ganiza ional decision making, a c ucial conce n is elici ing in o ma ion when
in e es s among s akeholde s, such as he i m’s CEO and local manage s, di e ge. We
explo e whe he con lic s o in e es consis en ly hinde in o ma ion ansmission and iden i y
scena ios whe e misalignmen may be bene icial. Ou model simpli ies o ganiza ional
dynamics in o a p incipal–agen ela ionship, whe e a DM seeks ad ice om an agen
who p i a ely knows he bene i s o wo p ojec s o he op ion o no p ojec . Con lic ing
p e e ences and biases, like p ojec bias o pande ing bias, independen ly impede in o ma ion
ansmission. Howe e , he nega i e co ela ion o bene i s be ween he wo p ojec s o se s
hese biases, collec i ely acili a ing in o ma ion ansmission.
Real-wo ld examples demons a e he ele ance o ou s udy. Conside a conglome a e
ha decides o in es in one o wo p ojec s o o no in es in any p ojec a all ( he ou side
op ion). I he a ailable op ion is be ween wo p ojec s ela ed o he ai line and oil
Games 2024,15, 18. h ps://doi.o g/10.3390/g15030018 h ps://www.mdpi.com/jou nal/games
Games 2024,15, 18 2 o 25
indus ies, espec i ely, we can expec nega i e co ela ion in payo s o he wo p ojec s
1
.
Bu , be ween wo p ojec s ela ed o he ai line and ho el indus ies, espec i ely, a posi i e
co ela ion can be assumed because bo h indus ies complemen each o he . I he a ailable
op ion is be ween a wind powe p ojec and a sola powe p ojec , he e is a posi i e
co ela ion because go e nmen egula ions a e o boos /supp ess he ma ke as a whole
2
.
I i is be ween an oil ene gy p ojec and a enewable ene gy p ojec , we may expec a
nega i e co ela ion. I we compa e p ojec s ela ed o Mac OS X and Macbook, espec i ely,
he e is a posi i e co ela ion be ween he payo s om he wo p ojec s
3
, while be ween
p ojec s ela ed o Mac OS X and Windows ope a ion sys em, espec i ely, he e may be
a nega i e co ela ion. The assump ion o he co ela ion in each example is easonable
based on inhe en p ojec ela ionships, con ibu ing o he p ac ical ele ance o ou s udy.
To cap u e hese eal-wo ld se ings, we allow he DM o choose be ween p ojec s 1
and 2 o op o no p ojec , incu ing he ull implemen a ion cos . The agen possesses
pe ec and p i a e in o ma ion on he s a e, de e mining he bene i s (excluding cos s)
o each p ojec . The e a e ou s a es: s a e 1, whe e bo h p ojec s yield la ge bene i s,
s a e 2, whe e p ojec 1 yields la ge bene i s and p ojec 2 yields small bene i s, s a e 3,
whe e p ojec 2 yields la ge bene i s and p ojec 1 yields small bene i s, and s a e 4, whe e
bo h p ojec s yield small bene i s. A e obse ing he s a e, he agen sends a cheap alk
message o he DM, who hen makes a choice. Ex an e, playe s may ha e di e en p ojec
p e e ences (p ojec bias). The DM’s bias is owa d p ojec 1, expec ing la ge bene i s based
on he common p io . The agen ’s bias may align o di e om he DM. Due o he DM’s
bias, he ou side op ion is less likely when p ojec 1 is ecommended. To a oid he ou side
op ion, he agen migh ecommend p ojec 1, e en i p ojec 2 yields mo e bene i s. This
scena io is a o m o pande ing bias iden i ied by CDK [2].
The main esul om ou analysis is ha he co ela ion be ween he bene i s o he
wo p ojec s coun e ac s pande ing bias, impac ing in o ma ion ansmission and wel a e.
This coun e ac ing e ec is s onge wi h a mo e highly nega i e co ela ion be ween he
p ojec s’ bene i s. Le us explain he in ui ion behind his esul . Conside a scena io whe e
he s a e is almos ce ainly ei he s a e 2 o 3. I he DM and agen biases di e , he agen
has a small incen i e o hide in o ma ion. Fo ins ance, in s a e 2, he agen may wan
p ojec 2, bu hides his o educe he p obabili y o he DM selec ing he ou side op ion. In
s a e 3, e ealing his in o ma ion inc eases he ou side op ion p obabili y, bu he agen
p e e s p ojec 2. Hence, he agen is willing o e eal in o ma ion in ei he s a e. Ye , i
bo h playe s a e biased owa d p ojec 1, bo h biases cause he agen o hide s a e 3 and
always epo s a e 2 o he DM. On he con a y, wi h a highly posi i e co ela ion, he DM
chooses be ween p ojec 1 and he ou side op ion in ei he s a e. Consequen ly, he agen
has no incen i e o e eal s a e 4, leading he DM o choose he ou side op ion, ega dless
o he p ojec bias. The posi i e co ela ion weakens he p ojec bias’s abili y o coun e ac
he pande ing bias.
Nex , we explo e an impo an ex ension o eal-wo ld applica ions be o e concluding
in Sec ion 5. One signi ican ex ension in ol es de e mining he op imal o ganiza ional
s uc u e in he esou ce alloca ion p oblem aced by a DM esponsible o unding
p ojec s. As he p incipal, he DM con ols he o ganiza ion’s esou ces and makes he
inal in es men decisions on p ojec s, aiming o e ec i ely elici he agen ’s in o ma ion.
We assess wo mechanisms o op imizing o ganiza ional s uc u e: non-delega ion and
e o-based delega ion. While delega ion implies a loss o con ol, an unin o med p incipal
may lead o in o ma ion loss. Ou indings sugges ha , in scena ios wi h a subs an ial
p ojec bias, e o-based delega ion enhances in o ma ion ansmission in decision making,
bene i ing bo h pa ies. Ou baseline model seamlessly accommoda es hese ex ensions,
and ou key insigh s on nega i e co ela ion o se ing p ojec and pande ing biases emain
obus , ensu ing e ec i e in o ma ion ansmission.
The subsequen sec ions o he a icle a e o ganized as ollows: Sec ion 2 e iews he
ela ed li e a u e, Sec ion 3in oduces a model wi h disc e e p ojec s and an ou side op ion,
emphasizing he impac o co ela ion in p ojec payo s on communica ions. Sec ion 4
Games 2024,15, 18 3 o 25
explo es op imal delega ion, and Sec ion 5p o ides concluding ema ks. P oo s a e
a ailable in Appendix A.
2. Rela ed Li e a u e
Ou wo k is ela ed o cheap alk models in CDK [
2
] and CS [
1
]. CDK ocuses on he
economics o pande ing, whe e he agen may bias his ecommenda ions owa ds he DM’s
condi ionally be e looking p ojec . The agen has an incen i e o dis o in o ma ion o
dissuade he DM om choosing he ou side op ion. The mo e aluable he op ion, he
s onge his incen i e becomes. In con as o CDK, ou model allows o co ela ion among
p ojec s, esul ing in non-mono onic in o ma ion ansmission in p ojec bias, while keeping
pande ing bias cons an . CS only examined p ojec bias, whe eas ou s udy del es in o he
in e ac ions be ween pande ing and p ojec biases. CS e ealed a mono onic ela ionship
be ween p ojec bias and in o ma ion ansmission, while ou indings showcase a non-
mono onic ela ionship.
Chiba and Leong (2013 [
3
] and 2015 [
4
]) s udied cheap alk models wi h coun e ailing
biases. In hei 2013 s udy, an ou side op ion was in oduced o a uni o m quad a ic case in
CS. Thei 2015 s udy ea u ed a model wi h wo p ojec s and an ou side op ion, whe e he
payo s o he wo p ojec s we e pe ec ly nega i ely co ela ed. Bo h s udies un eiled a
non-mono onic ela ionship be ween in o ma ion ansmission and p ojec bias, bu did
no allow o an examina ion o he impac o co ela ion on in o ma ion ansmission.
Mo eo e , hese wo ks do no p o ide an explana ion o why p ojec bias coun e ac s
pande ing bias. In ou model, which in ol es disc e e p ojec s, we conside he ull ange
o possible p ojec payo co ela ions. This enables us o discuss he e ec o p ojec
co ela ion on he non-mono onic ela ionship be ween in o ma ion ansmission and
p ojec bias.
In con as o ou s udy, which explo es he in e ac ion be ween p ojec and pande ing
biases, Chak abo y and Ha baugh (2007 [
5
] and 2010 [
6
]) examined o he ypes o coun e ailing
biases in cheap alk models. In hei models, he decision make (DM) de e mines he
in es men alloca ion o mul iple p ojec s, and he mul i-dimensional s a e de e mines he
op imal in es men amoun o each p ojec . Howe e , hey did no s udy he impac o
co ela ion among p ojec s’ bene i s o he p esence o an ou side op ion. Thei ocus was
on iden i ying condi ions ha mi iga e he nega i e impac o p ojec bias on in o ma ion
ansmission. Chak abo y and Ha baugh (2007) showed ha , i bo h playe s’ payo
unc ions a e supe modula , hey can each an ag eemen on he compa a i e anking
o p ojec s, i espec i e o he le el o p ojec bias. This leads o an equilib ium, whe e
he agen con eys he compa a i e anking o p ojec s o he DM. In Chak abo y and
Ha baugh (2010), he agen has s a e-independen linea p e e ences, seeking o induce he
la ges possible p ojec in each dimension. They ound ha , i he DM’s payo unc ion is
quasi-con ex, an in o ma i e equilib ium can be es ablished. In his equilib ium, he agen
pa i ions he mul i-dimensional s a e space in o mul iple egions, causing he agen o be
indi e en among he DM’s esponses in all egions o he pa i ion.
B andenbu ge and Polak (1996) [
7
], in addi ion o CDK, conside ed he logic o pande ing,
whe e an in es men manage o a i m has incen i es o skew in es men decisions owa ds
a p ojec ha he ma ke belie es is likely o succeed. Howe e , hey did no inco po a e
s a egic communica ions in o hei model. Heidhues and Lage lo (2003) [
8
] in es iga ed
simila pande ing issues in elec o al compe i ion wi h mul i-agen models.
Ou wo k is connec ed o he li e a u e explo ing he impac o p ojec bias on
in o ma ion ansmission, as examined by Landie , S ae , and Thesma (2009) [
9
]. They
in es iga ed a cos ly signaling model ea u ing an in o med manage and an unin o med
wo ke . The manage delibe a ely chose he subop imal p ojec o he agen o encou age
op imal e o om he agen du ing p ojec implemen a ion. Ou model is simila o hei s
in he con ex o a e o-based delega ion model, especially when he co ela ion be ween
p ojec s is nega i e.
Games 2024,15, 18 4 o 25
Ou model is ela ed o communica ion games, pa icula ly hose in ol ing e o s ages,
whe e an ou side op ion ep esen ing a e o o a p oposal can be chosen. In Ma hews
(1989) [
10
] and Shimizu (2013 [
11
], 2017 [
12
]), he DM selec s a p oposal based on he agen ’s
message, and he agen can ei he accep o e o i . Unlike ou pape , hey pe mi ed he
agen , a he han he DM, o choose an ou side op ion. In legisla i e p ocedu e models
wi h a closed ule by Gilligan and K ehbiel (1987) [
13
] and K ishna and Mo gan
(2001) [14]
,
he agen ecommends a p ojec , and he DM decides whe he o au ho ize o e o i . Unlike
ou e o-based delega ion game, hei models do no conside coun e ailing biases o
co ela ion be ween p ojec payo s. Chiba and Leong (2023) [
15
] s udied he ela ionship
be ween he bene i s o delega ion and he le el o p ojec bias, wi hou conside ing how he
in e ac ion o p ojec and pande ing biases leads o his esul . Blume and Boa d (
2013) [16]
also s udied a cheap alk model wi h an ou side op ion, bu hei main ocus di e s om
ou s. In hei model, playe s a e language-cons ained a di e en le els, esul ing in
di e en language usage.
3. Model
Ou model is closely ela ed o cheap models wi h an ou side op ion in CDK [
2
], as
well as Chiba and Leong (2015) [
4
], using he same e minology om he wo pape s, unless
o he wise speci ied.
3.1. Se up
An o ganiza ion iden i ies wo po en ial p ojec s—p ojec s 1 and 2. The o ganiza ion
can ei he p oceed wi h a p ojec ,
P∈ {
1, 2
}
, o choose no p ojec ( he ou side op ion),
P=∅.
The o ganiza ion has wo playe s: an unin o med decision make (DM) and an
in o med agen (A). The agen possesses in o ma ion: he/she obse es a wo-dimensional
s a e o he wo ld θ= (θ1,θ2)de ined as ollows:
θ=
(1, 1)w.p. 1+ρ
4
(1, l)w.p. 1−ρ
4
(l, 1)w.p. 1−ρ
4
(l,l)w.p. 1+ρ
4
(1)
whe e l∈(0, 1)and ρ∈(−1, 1).
Bo h playe s a e isk-neu al and aim o maximize payo s. I he DM selec s a p ojec ,
bo h playe s’ payo s depend on he s a e, he p ojec , and he cos o implemen a ion. On
he con a y, i he DM chooses he ou side op ion, each playe ’s payo (o expec ed payo )
is ze o. The payo o playe j∈{DM,A}is deno ed as Uja,θ,cj:
Uj(P,θ,cj) =
θ1−cji P=1
bjθ2−cji P=2
0 i P=∅
(2)
whe e
bDM ∈(l
, 1
)
,
bA∈(l
, 1
/l)
, and
cA=
0. The pa ame e
cDM
is d awn om a uni o m
dis ibu ion wi h suppo [0, 1].
The p ojec se , he dis ibu ions o
θ
and
cDM
, and he pa ame e s
(ρ,cA,bDM,bA)
a e common knowledge. The pa ame e
cDM
is publicly obse ed jus be o e he DM’s
decision making. The s a e
θ
is only obse able o he agen . Be o e he DM chooses
P
,
he agen sends a cheap alk message
m∈M
o he DM, whe e
M
is any la ge space (e.g.,
M=R2
++).
Timeline:
S ep 1.
Na u e selec s he s a e θ. This is p i a ely and pe ec ly obse ed by he agen .
S ep 2.
The agen sends a cheap alk message
m
. The DM obse es his message wi hou
noise.
Games 2024,15, 18 5 o 25
S ep 3.
The DM’s p ojec cos cDM is de e mined and publicly obse ed.
S ep 4.
The DM decides whe he o implemen a p ojec
P∈1, 2
o he ou side op ion o
no p ojec P=∅.
S ep 5.
Bo h playe s’ payo s a e ealized, and he game concludes.
This model inco po a es wo dimensions o biases—p ojec bias and pande ing bias—and
aco ela ion be ween he bene i s o he wo p ojec s.
The p ojec bias pe ains o he di e ence in p e e ences o e p ojec s, akin o he
“p e e ence simila i y pa ame e ” (
b
) in CS [
11
]. We in e p e
|bDM −bS|
as he le el o
p ojec bias, assuming bDM ∈(l, 1)and bA∈(l, 1/l)as men ioned abo e.
Assump ion
bDM <
1 implies ha he DM is ex an e biased owa d p ojec 1, while
l<
bDM
implies ha he DM does no always s ic ly p e e p ojec 1 o e p ojec 2 unde pe ec
in o ma ion. The DM s ic ly p e e s p ojec 1 o p ojec 2 gi en
θ∈{(1, 1),(1, l),(l,l)}
,
while he DM s ic ly p e e s p ojec 2 o e p ojec 1 gi en θ= (l, 1).
On he o he he hand, he agen can be ex an e biased owa d ei he p ojec : he agen
is ex an e biased owa d p ojec 1 and p ojec 2 i
bA<
1 and
bA>
1, espec i ely. The
agen is ex an e indi e en be ween he wo p ojec s i
bA=
1. Assump ion
bA∈(l
, 1
/l)
implies ha he agen does no always s ic ly p e e one p ojec o e he o he unde
pe ec in o ma ion
4
. The p e e ence anking o he agen unde pe ec in o ma ion is
dependen on he pa ame e . I
bA∈(l
, 1
)
, he agen s ic ly p e e s p ojec 1 o e p ojec 2
gi en
θ∈{(1, 1),(1, l),(l,l)}
, while he agen s ic ly p e e s p ojec 2 o e p ojec 1 gi en
θ= (l
, 1
)
. I
bA∈(
1, 1
/l)
, he agen s ic ly p e e s p ojec 1 o e p ojec 2 gi en
θ= (
1,
l)
,
while he agen s ic ly p e e s p ojec 2 o e p ojec 1 gi en θ∈{(1, 1),(l, 1),(l,l)}.
The pande ing bias is ela ed o he p e e ence o he ou side op ion o no p ojec ,
ep esen ed by
cj
, he cos incu ed by playe
j
when a p ojec is implemen ed. Fo
simplici y,
cA=
0, and
cDM
ollows a uni o m dis ibu ion wi h suppo
[
0, 1
]
. The agen
ne e inds he ou side op ion op imal, while he DM may p e e he ou side op ion in
ce ain s a es.
The pa ame e
ρ
is he co ela ion coe icien be ween
θ1
and
θ2
. Addi ionally, o bo h
playe s, ρis he co ela ion coe icien be ween he bene i s o he wo p ojec s5.
While ou se up is di e en in some aspec s, i is essen ial o no e ha
ρ=
0 simila o
CDK’s model wi h mul iple p ojec s and con inuous s a es, and
ρ=−
1 is simila o Chiba
and Leong’s (2015) [
4
] model wi h wo s a es and wo p ojec s (see Figu e 1). The alue o
ρ
depends on he pai o p ojec s being compa ed in a uni o m quad a ic example o CS,
whe e he e a e con inuous p ojec s and con inuous s a es.
𝜌
Chiba & Leong
(2015)
CDK
(2013)
-1 0 1
This pape
Figu e 1. Co ela ions in models
We choose no o adop a se up di ec ly compa able o exis ing li e a u e like CDK [
2
]
and CS [1]. He e a e he easons o ou depa u e:
Fi s , we conside ou s a es o wo p ojec s, de ia ing om CDK, CS, and Chiba
and Leong (2015) [
4
], who assumed con inuous s a es and ei he disc e e o con inuous
p ojec s. This choice helps explain he impac o he co ela ion be ween he bene i s o he
Games 2024,15, 18 6 o 25
wo p ojec s on in o ma ion ansmission. Addi ionally, ou se up isola es he pa ame e
ρ
,
in luencing he co ela ion o he wo p ojec s’ bene i s o bo h playe s wi hou a ec ing
he mean o a iance o a p ojec o any playe .
Nex , in CDK’s model o pande ing, p ojec cos s a e p ede e mined and publicly known
6
.
Howe e , we in oduce an analysis o he in e ac ion be ween pande ing bias and
p ojec bias. The assump ion o con inuous p ojec cos s is mo e gene al om a heo e ical
s andpoin , al hough ou main esul emains unchanged e en wi h he ixed cos assump ion.
3.2. Resul s
The solu ion concep employed is Pe ec Bayesian Equilib ium (PBE). The agen ’s
s a egy is ep esen ed by a unc ion
q(m|θ)
, linking each s a e
θ
wi h a message dis ibu ion
used by he agen in ha s a e. The DM’s s a egy is deno ed as
P(m
,
cDM)
, associa ing he
agen ’s message
m
and he DM’s cos
cDM
wi h he DM’s decision
P
. The DM’s belie is
cap u ed by a unc ion µ(θ|m), whe e µ(θ|m)≥0 and
Z
θ∈{1,l}2
µ(θ|m) = 1,
e lec ing he DM’s pos e io as a unc ion o mby Bayes’ ule.
The in e ac ion be ween p ojec and pande ing biases can ei he ein o ce o coun e ac
each o he , impac ing in o ma ion ansmission and wel a e. C ucially, we demons a e
ha his coun e ac ing e ec hinges on he co ela ion in p ojec bene i s. As he co ela ion
coe icien
ρ
app oaches
−
1, he p ojec bias exe s a s onge in luence, opposing he
di ec ion o he pande ing bias. Consequen ly, a la ge p ojec bias can enhance in o ma ion
ansmission and imp o e wel a e.
Ou ini ial lemma, akin o CDK’s Lemma 1 (CDK [
2
], p. 57), s a es ha he agen
consis en ly a o s a p ojec o e he ou side op ion. The agen ac ically selec s messages
o maximize he p obabili y o his/he p e e ed p ojec being chosen, ei he by e ealing
his/he p e e ence o wi hholding in o ma ion en i ely. Consequen ly, he agen needs a
mos wo messages.
Lemma 1. E e y PBE is equi alen o one whe e he agen ’s s a egy in ol es a mos wo messages.
To s eamline ou analysis, we adop a bina y message se
M={1, 2}
. Wi hou loss o
gene ali y, we ocus on equilib ia whe e he agen ’s s a egy complies wi h:
q(1|θ=(1, l)) ≥q(1|θ=(l, 1)) (3)
This implies ha he agen mo e equen ly ecommends p ojec 1 in scena ios whe e i is
supe io o bo h playe s han in he e e se si ua ion.
The subsequen lemma adap s CDK’s Lemma 2 om hei disc e e p ojec and con inuous
s a e model (CDK [2], pp. 57–58) o ou disc e e s a e and disc e e p ojec amewo k.
Lemma 2. Fo any PBE, he ollowing s a emen s hold:
(1)
I q(1|θ=(l, 1)) >0holds, hen,
q(1|θ=(1, 1)) =q(1|θ=(1, l)) =q(1|θ=(l,l)) =1 (4)
holds.
(2)
I q(1|θ= (1, 1)) ∈(0, 1)o q(1|θ= (l,l)) ∈(0, 1)holds, hen
q(1|θ= (1, l)) = 1and q(1|θ= (l, 1)) = 0(5)
hold.
Games 2024,15, 18 7 o 25
(3)
I q(1|θ=(1, l)) <1holds, hen
q(1|θ=(1, 1)) =q(1|θ=(l, 1)) =q(1|θ=(l,l)) =0 (6)
holds.
Lemma 2 es ablishes ha , i he agen sends
m=
1 wi h a posi i e p obabili y gi en
θ= (l
, 1
)
, hen he agen sends
m=
1 o su e gi en any
θ∈{(1, 1),(1, l),(l,l)}
. Simila ly,
i he agen sends
m=
2 wi h a posi i e p obabili y gi en
θ= (
1,
l)
, hen he agen sends
m=
2 o su e gi en any
θ∈{(1, 1),(l, 1),(l,l)}
. I he agen mixes be ween wo messages
gi en a leas ei he o
θ= (
1, 1
)
and
θ= (l
,
l)
, hen, he/she sends
m=
1 o su e gi en
θ= (1, 1)and m=2 o su e gi en θ= (l,l), espec i ely.
We will de ine he ypes o equilib ia based on CDK’s e minology:
De ini ion 1.
(1) In a u h ul equilib ium (T),
q(
1
|θ=θ′) =
1 o any
θ′∈{(1, l),(1, 1),(l,l)}
and q(1|θ= (l, 1)) = 0and P(m,cDM)∈{m,∅} o any m.
(2) In a pande ing- owa d-1 equilib ium (P1),
q(
1
|θ=θ′) =
1 o any
θ′∈{(1, l),(1, 1),(l,l)}
and q(1|θ= (l, 1)) ∈(0, 1)and P(m,cDM)∈{m,∅} o any m.
(3)
In a pande ing- owa d-2 equilib ium (P2),
q(
1
|θ= (
1,
l)) =
1,
q(
1
|θ=θ′)∈(
0, 1
)
o
some θ′∈{(1, 1),(l,l)}and q(1|θ= (l, 1)) = 0and P(m,cDM)∈{m,∅} o any m.
(4)
In a pande ing- owa d-2 equilib ium’ (P2’),
q(
1
|θ= (
1,
l)) =
1and
q(
1
|θ=θ′) =
0 o
any θ′∈{(1, 1),(l,l),(l, 1)}and P(m,cDM)∈{m,∅} o any m.
(5)
In a ze o equilib ium (Z), q(1|θ=θ′) = 1 o any θ′and P(m,cDM)∈{1, ∅} o any m.
In a ze o equilib ium (Z), he agen does no e eal any in o ma ion. The emaining
equilib ia a e pa i ion equilib ia, whe e he agen pa i ions he s a e space in o wo pa s
and discloses which pa i ion he s a e θbelongs o (see Figu e 2).
T u h ul- elling
equilib ium (T)
(1, 1)
(1, l)
(l, 1)
(l, l)
𝜃!
𝜃"
0Pande ing- owa d-1
equilib ium (P1)
(1, 1)
(1, l)
(l, 1)
(l, l)
𝜃!
𝜃""
0
Pande ing- owa d-2
equilib ium (P2)
(1, 1)
(1, l)
(l,1)
(l, l)
𝜃!
𝜃"
0Pande ing- owa d-2
equilib ium’ (P2’)
(1, 1)
(1, l)
(l, 1)
(l, l)
𝜃!
𝜃"
0
Figu e 2. In o ma ion pa i ions in di e en ypes o equilib ia
The i s ou equilib ia, namely T, P1, P2, and P2’, a e e e ed o as in o ma i e
equilib ia. Equilib ium messages a e in e p e ed in wo ways: indica ing a anking be ween
he wo p ojec s o ecommending a speci ic p ojec . In an in o ma i e equilib ium, he
agen can induce ei he p ojec by sending
m=
1 (
m=
2) wi h a posi i e p obabili y. In
a ze o equilib ium, he e is no in o ma ion ansmission, and he agen can only induce
p ojec 1 o he ou side op ion, ega dless o he message sen .
Games 2024,15, 18 8 o 25
Acco ding o he i s in e p e a ion, in T, he agen consis en ly discloses he ue
anking o he decision make (DM). As explained in he p e ious sec ion, unde pe ec
in o ma ion, he DM s ic ly p e e s p ojec 1 o e p ojec 2 gi en
θ∈{(1, 1),(1, l),(l,l)}
,
while he DM s ic ly p e e s p ojec 2 o e p ojec 1 gi en
θ= (l
, 1
)
. Hence, in T, gi en
any s a e, he agen u h ully ecommends a p ojec ha he DM should p e e . The e o e,
we call his a u h ul equilib ium. This equilib ium is no a ull e ela ion equilib ium. As
we will show, a ull e ela ion equilib ium does no exis in his model.
In P1, he agen ecommends p ojec 1, which is ex an e p e e ed by he DM, mo e
o en han in T. Hence, we call his a pande ing- owa d-1 equilib ium. In P2 and P2’, he
agen ecommends p ojec 2 mo e o en han in T. Hence, we call he wo equilib ia a
pande ing- owa d-2 equilib ium and a pande ing- owa d-2 equilib ium’, espec i ely.
In he second in e p e a ion, he agen uses he /his in o ma ion on he s a e and
ecommends ei he p ojec in an in o ma i e equilib ium, whe eas he agen always
ecommends p ojec 1 ega dless o he /his in o ma ion in a ze o equilib ium.
Based on Blackwell’s in o ma i eness, he agen ansmi s mo e in o ma ion o he DM
in any o T, P1, P2, and P2’ han in Z because he in o ma ion ansmi ed in Z is cons uc ed
by ga bling o he in o ma ion ansmi ed in any o T, P1, P2, and P2’. Howe e , we canno
compa e he in o ma i eness among T, P1, P2, and P2’ in Blackwell’s sense.
The subsequen lemma p esen s a cha ac e iza ion o all equilib ia o ixed pa ame e s
and compa es hei wel a e.
Lemma 3. Fo any ixed pa ame e s, he ollowing s a emen s hold:
(1)
The e exis a mos wo ypes o equilib ia, an in o ma i e equilib ium (T, P1, P2, o P2’) and
a ze o equilib ium (Z). Mo eo e , Z always exis s.
(2)
The e is a unique in o ma i e equilib ium i T, P1, o P2’ exis s.
(3)
I an in o ma i e equilib ium exis s, he in o ma i e equilib ium makes bo h playe s be e o
han Z.
Lemma 3 shows ha mul iple ypes o in o ma i e equilib ia do no exis oge he .
Only possible mul iplici y is one ype o in o ma i e equilib ium (T, P1, P2, o P2’) and a
ze o equilib ium (Z). When T, P1, o P2’ exis s, his is he unique in o ma i e equilib ium.
Howe e , when P2 exis s, he e can be mul iple P2s. Mo eo e , he in o ma i e equilib ium
is be e han a ze o equilib ium o bo h playe s.
Because o Lemma 3, i he e is T, P1, o P2’, we ocus on he unique in o ma i e
equilib ium. I he e is P2, meaning ha mul iple P2s can exis , we will ocus on he one
ha maximizes he DM’s ex an e expec ed payo . O he wise, we conside Z. Acco dingly,
he ollowing esul p o ides compa a i e s a ics ac oss pa ame e s7.
P oposi ion 1. Le
B(ρ,l):=2+ (1+ρ)l
3+ρ,
D(bDM,ρ,l):=1+2(l+1)(1−bDM)2
ρbDM(bDM +1)(1−l) + (3+5l−(5+3l)bDM)bDM
,
d(ρ,l):=1−(1−ρ)(1−l)
l+lρ+2.
Then, o any ixed bDM,ρ, and l:
(1)
A u h ul equilib ium (T) exis s o bA∈hB(ρ,l)·l
bDM ,B(ρ,l)
bDM i;
(2)
A pande ing- owa d-1 equilib ium (P1) exis s o bA∈l+l2
2bDM ,B(ρ,l)·l
bDM ;
Games 2024,15, 18 15 o 25
DM’s ex-an e
expec ed payo
Fo 𝝆 = 𝟎. 𝟖
𝑏%& 𝐷 𝑏%&, 𝜌, 𝑙 𝑏$
2.0
0.30
0.28
DM’s ex-an e
expec ed payo
Fo 𝝆 = 𝟎. 𝟒"
𝑏$
𝑏%& 𝐷 𝑏%&, 𝜌, 𝑙
0.30
0.28
2.0
Fo 𝝆 = −𝟎. 𝟒
DM’s ex-an e
expec ed payo
𝑏$
𝑏%& 𝐷 𝑏%&, 𝜌, 𝑙
0.32
0.30
0.28
2.0
Fo 𝝆 = −𝟎. 𝟖
DM’s ex-an e
expec ed payo
𝑏$
𝑏%& 1.0' 2.0
0.32
0.30
0.28
Figu e 7. DM’s ex-an e expec ed payo in he mos in o ma i e equilib ium (
bDM =
0.6,
l=
0.5, and
a ious ρ).
Agen ’s ex-an e
expec ed payo
Fo 𝝆 = 𝟎. 𝟖
𝑏%&'𝐷 𝑏%&, 𝜌, 𝑙 𝑏$
2.0
0.8
0.6
Agen ’s ex-an e
expec ed payo
Fo 𝝆 = 𝟎. 𝟒
𝑏%&'𝐷 𝑏%&, 𝜌, 𝑙 𝑏$
2.0
0.8
0.6
Fo 𝝆 = −𝟎. 𝟒
Agen ’s ex-an e
expec ed payo
𝑏%&'𝐷 𝑏%&, 𝜌, 𝑙 𝑏$
2.0
0.8
0.6
Fo 𝝆 = −𝟎. 𝟖
Agen ’s ex-an e
expec ed payo
𝑏%&'
𝑏$
1.0' 2.0
1.0
0.8
0.6
Figu e 8. Agen ’s ex-an e expec ed payo in he mos in o ma i e equilib ium (
bDM =
0.6,
l=
0.5,
and a ious ρ).
4. The E ec s o Co ela ions on Ve o-Based Delega ion
This sec ion explo es he possibili y o decision make s (DM) o delega e choices o
agen s, aiming o alle ia e he nega i e impac o wo-dimensional bias. Ou delega ion
model, akin o exis ing li e a u e on delega ion, can be linked o he amewo k o incomple e
con ac s (G ossman and Ha , 1996 [
17
]; Ha and Moo e, 1990 [
18
]). Ou model del es in o
op imal delega ion in he con ex o wo-dimensional bias, explaining how he in e ac ion
o he p ojec and pande ing biases in luences he bene i s o delega ion o he DM.
Games 2024,15, 18 16 o 25
The DM, ac ing as he p incipal, con ols he o ganiza ion’s esou ces and makes
decisions. The p incipal’s in e es lies in elici ing he agen ’s in o ma ion o decision
making. To explo e he op imal o ganiza ional s uc u e, we conside wo mechanisms:
non-delega ion (communica ion) (Nd) and e o-based delega ion (Vd). Full delega ion is
excluded om ou discussion because, in ou model, se ing he DM bea s he en i e p ojec
implemen a ion cos
10
. Like CDK [
2
], his pape ocuses on he si ua ions unde which
pande ing incen i es a e s ong (because only he DM incu s he p ojec implemen a ion
cos s) and examines how he le el o co ela ion a ec s he DM’s bene i o delega ing
he selec ion o he p ojec in he p esence o wo-dimensional biases ( he pande ing and
p ojec biases).
Delega ing decision-making implies a loss o con ol, bu an unin o med p incipal
(DM) holding decision-making au ho i y ine i ably esul s in a loss o in o ma ion. Milg om
and Robe s’s “delega ion p inciple” sugges s ha decision-making powe should be wi h
an in o med agen , like di isional manage s in ou case s udy (Milg om and Robe s,
1992 [19]). The ques ion is, unde wha ci cums ances does delega ion bene i he DM?
In he case o non-delega ion (Nd), playe s engage in he game desc ibed in Sec ion 3.
The imeline is as ollows:
S ep 1.
Na u e selec s he s a e θ, p i a ely and pe ec ly obse ed by he agen .
S ep 2. The agen sends a cheap alk message,
m
, which he DM obse es wi hou any noise.
S ep 3.
The DM’s p ojec cos , cDM, is de e mined and publicly obse ed.
S ep 4.
The DM decides whe he o implemen p ojec
P
om he se
{1, 2}
o choose he
ou side op ion o no p ojec (P=∅).
S ep 5.
Bo h playe s’ payo s a e ealized.
In Vd, he agen is au ho ized o choose be ween p ojec s, bu he p incipal can e o
he agen ’s choice in a o o he ou side op ion. This adjus men changes S ep 4 unde
non-delega ion (Nd) in o S ep 4’:
S ep 4’.
The p incipal chooses P om he se {m,∅}.
Simila o Sec ion 3, we conside a bina y message se ,
M={1, 2}
. The agen ’s
message is deno ed as
md
. The exp ession
qd(θ)
ep esen s he p obabili y ha he agen
sends
md∈{1, 2}
o each s a e
θ
. The exp ession
Pdc,md
is he p incipal’s s a egy,
and
µd(md)
is he p incipal’s pos e io belie . Ou ocus is on equilib ia whe e he agen ’s
s a egy sa is ies condi ion (3) in Sec ion 3.
P e iously, we de ined a ious ypes o equilib ia, and hey emain applicable o
bo h delega ion mechanisms. These include a u h ul equilib ium (T), a pande ing-
owa d-1 equilib ium (P1), a pande ing- owa d-2 equilib ium (P2), a pande ing- owa d-2
equilib ium’ (P2’), and a ze o equilib ium (Z). Sec ion 3discusses he esul s unde Nd.
Unde Vd, condi ions (8) and (9) a e no longe necessa y. In Vd, he agen is no longe
equi ed o pe suade he p incipal (DM) o ag ee o he agen ’s p ojec anking.
Wi h ixed alues o
ρ
,
bDM
, and
l
, e o-based delega ion (Vd) enhances in o ma ion
ansmission compa ed o non-delega ion (Nd) when he e is a signi ican p ojec bias,
deno ed as |bA−bDM|.
Figu e 9illus a es he exis ence o each equilib ium ype o
l=
0.5 and
ρ=−
0.4
unde Vd. Assuming
bDM
is small enough ha
bDM <d(ρ
,
l)
, o
bA>D(bDM,ρ,l)
,
in o ma ion agg ega ion occu s unde Vd, whe eas no in o ma i e equilib ium is p esen
unde Nd. The imp o emen is depic ed by (Z→)P2 and (Z→)P2′in Figu e 9.
Games 2024,15, 18 17 o 25
𝑏-
! ",$ $
%!"
Z
T
𝑏&'
! ",$
%!"
!
𝑑 𝜌, 𝑙
P2
P1
.
%!"! ",$
(Z®)𝐏𝟐 P2’
(Z®)𝐏𝟐’
𝟎. 𝟔 𝟎. 𝟖% 𝟎. 𝟗% 𝟏. 𝟎
𝟐. 𝟎
𝟏. 𝟓
𝟏. 𝟎
(0.5, 0.5)
Figu e 9. The mos in o ma i e equilib ium unde e o-based delega ion (Vd) (
l=
0.5 and
ρ=−
0.4)
(The imp o emen on he mos in o ma i e equilib ium unde no-delega ion (Nd) is desc ibed by
(Z→)P2 and (Z→)P2′).
Nex , se
l
o a ixed alue. Then,
d(ρ
,
l)
s ic ly inc eases wi h
ρ
. Consequen ly,
e o-based delega ion (Vd) enhances he expec ed payo s o bo h playe s compa ed o
non-delega ion (Nd) ac oss a b oade ange o bDM as ρinc eases.
The i s pa o his obse a ion is e iden om he unc ional o m o
d(ρ
,
l)
. The e o e,
e o-based delega ion (Vd) enhances in o ma ion ansmission o e non-delega ion (Nd)
o a mo e ex ensi e se o pa ame e s. Figu e 10 illus a es equilib ia ypes unde e o-
based delega ion (Vd) and demons a es imp o ed in o ma ion ansmission ac oss a ious
ρ alues, wi h l ixed a 0.5.
Fo 𝝆 = 𝟎. 𝟖
𝑏#$
𝑏*
T
Z
P2
P2’
(Z®)𝐏𝟐
(Z®)𝐏𝟐’
P1
𝑑 𝜌, 𝑙
(0.5, 0.5)
1.0
1.5
1.0
Fo 𝝆 = −𝟎. 𝟖
𝑏#$
𝑏*
Z
T
P2
P2’
(Z®)𝐏𝟐,𝐏𝟐’
P1
𝑑 𝜌, 𝑙
(0.5, 0.5)
1.0
1.5
1.0
Fo 𝝆 = 𝟎. 𝟒
𝑏#$
𝑏*
(Z®)𝐏𝟐
Z
TP2
P2’
(Z®)𝐏𝟐’
P1
(0.5, 0.5) 1.0
𝑑 𝜌, 𝑙
1.5
1.0
Fo 𝝆 = −𝟎. 𝟒
𝑏#$
𝑏*
(Z®)𝐏𝟐,𝐏𝟐’
Z
T
P2
P2’
P1
𝑑 𝜌, 𝑙
(0.5, 0.5)
1.0
1.5
1.0
Figu e 10. The mos in o ma i e equilib ium unde Vd (l=0.5 and a ious ρ).
Games 2024,15, 18 18 o 25
Skipping he de ails due o hei ob ious na u e, inc eased in o ma ion ansmission
esul ing om delega ion consis en ly bene i s he agen . Howe e , his is no always he
case o he DM. Ye , we obse e ha , pa icula ly o
bA
nea and abo e
D(bDM
,
ρ
,
l)
,
enhanced in o ma ion ansmission due o delega ion also bene i s he DM.
Recall ha , unde Nd, o any ixed
bDM <d(ρ
,
l)
, he e is discon inui y a
bA=
D(bDM
,
ρ
,
l)
in he le el o in o ma ion ansmission. This discon inui y leads o a sudden
d op in he DM’s ex an e expec ed payo s.
Fi s , we es ablish:
VDM,P2(ρ,bA,bDM,l)−VDM,Z(ρ,bA,bDM,l)>0
a bA=D(bDM,ρ,l).
Fu he mo e, conce ning bA,VDM,Z(ρ,bA,bDM,l) emains cons an , whe eas
VDM,P2(ρ,bA,bDM,l)is con inuous. Hence, he e exis s ϵ>0 such ha :
VDM,P2(ρ,bA,bDM,l)−VDM,Z(ρ,bA,bDM,l)>0
o bA∈(D(bDM,ρ,l),D(bDM,ρ,l) + ϵ).
Ve o-based delega ion (Vd) enhances he DM’s ex an e expec ed payo compa ed o
no-delega ion (Nd), pa icula ly o
bA
wi hin he ange
(D(bDM
,
ρ
,
l)
,
D(bDM
,
ρ
,
l) + ϵ)
.
This con i ms he second pa o ou obse a ion.
G ea e po en ial o imp o emen exis s when he co ela ion is close o 1. In simple
e ms, non-delega ion can be e ec i e when p ojec payo s exhibi nega i e co ela ion.
This aligns wi h ou obse a ion in Sec ion 3 ha nega i e co ela ion o se s he p ojec
and pande ing biases.
This inding de ia es om exis ing li e a u e. Dessein (2002) [
20
] examined a model
wi h con inuous p ojec s and s a es, demons a ing ha e o-based delega ion s ic ly
domina es non-delega ion i and only i he p ojec bias is small. Mylo ano (2008) [
21
]
sugges ed ha , wi h he op imal choice o he de aul p ojec , Vd can eplica e any op imal
ou come unde Nd o la ge bias. Con a y o CDK [
2
], which compa ed delega ion egimes
based on compa a i e s a ics wi h espec o he p incipal’s payo o he ou side op ion,
ou model sugges s ha , e en when communica ion is no in luen ial, delega ion can be
s ic ly p e e ed o e communica ion by he DM.
5. Conclusions
The inclusion o he ou side op ion in oduces wo di e en dimensions o biases
be ween he playe s, in ol ing p ojec and pande ing biases. A s ong nega i e co ela ion
be ween p ojec payo s ampli ies he coun e ailing impac o he p ojec bias on pande ing
bias. Consequen ly, he p ojec bias exhibi s a non-mono onic ela ionship wi h in o ma ion
ansmission and he decision make ’s (DM) ex an e expec ed payo s.
When looking a delega ion in he p esence o hese biases, ou indings di e om
wha is in he exis ing li e a u e. Ve o-based delega ion imp o es in o ma ion ga he ing
in decision making, especially when he p ojec bias is signi ican . Also, an inc ease in
co ela ion be ween p ojec payo s expands he ange whe e his imp o emen is seen.
Fu u e esea ch should explo e he bes delega ion me hod when coun e ailing biases
a e p esen . We need o check i he conclusions made by Gilligan and K ehbiel (1987) [
13
],
K ishna and Mo gan (2001) [
14
], and Ma in (1997) [
22
] abou closed- ule dominance o e
open- ule s ill apply when he e a e coun e ailing biases among mul iple agen s and he
p incipal. Fu u e s udies should also y o endogenize he pande ing bias (c . Ran aka i
(2012) [23]).
Las ly, we a e in e es ed in adding an agen o he cu en model. Ba aglini (2002) [
24
],
Le y and Razin (2007) [
25
], and Amb us and Takahashi (2008) [
26
] in es iga ed cheap alk
models wi h wo agen s and a wo-dimensional s a e space. In hei models, he s a e, he
DM’s choice, he agen s’ op imal choices, and he biases o he agen s a e all de ined on
wo-dimensional euclidean space, and each playe is be e o as he DM’s choice is close
Games 2024,15, 18 19 o 25
o he /his op imal choice. They showed ha ull e ela ion is possible in a la ge s a e space
and when biases a e no la gely di e en gi en small s a e spaces. On he con a y, ou
choice se and ou wo-dimensional biases a e no de ined in hei ways. I is no i ial
ha hei conclusions hold in ou model.
Funding: This esea ch was unded by Japan Socie y o he P omo ion Science (no. 16K03549,
20K01544, 24K04799), Kyo o Sangyo Uni e si y Publica ion G an s, he Join Resea ch P og am o
KIER (Kyo o Uni e si y), and he Kyo o Uni e si y Founda ion.
Da a A ailabili y S a emen : Da a a e a ailable upon he eques o he co esponding au ho .
Acknowledgmen s: We hank Ba Lipman, Takashi Shimizu, and Min-Hung Tsay as well as semina
pa icipan s a 10 h Japan-Taiwan-Hong-Kong Con ac Theo y Con e ence, 2016 DC con e ence
in Japan, 2016 Lisbon Mee ings in Game Theo y and Applica ions, 2016 Mee ings o he Japanese
Economic Associa ion, 2018 Eu opean Mee ings o Econome ic Socie y, and 2018 EARIE o help ul
commen s. Resea ch assis ance by Dengwei Qi is app ecia ed. This pape was o iginally i led
“In o ma ion Agg ega ion and Coun e ailing Biases in O ganiza ions.” All emaining e o s a e
ou own.
Con lic s o In e es : The au ho decla es no con lic s o in e es .
Appendix A
Appendix A.1. P oo o Lemma 1
Gi en θ=(θ1,θ2), he agen ’s expec ed payo by sending a bi a y message m′is:
θ1·P (DM implemen s p ojec 1 gi en m=m′)
+bA·θ2·P (DM implemen s p ojec 2 gi en m=m′).
I he agen wan s o induce p ojec 1 gi en some
θ
, he agen p e e s a message ha
induces he DM o selec p ojec 1 wi h he highes p obabili y. Mul iple messages ha
induce p ojec 1 su i e only i hese messages induce p ojec 1 wi h he same p obabili y.
The a gumen is simila o he case when he agen wan s o induce p ojec 2. Hence, e e y
equilib ium ou come is eplica ed by an equilib ium in which he agen uses he same
numbe o messages as he numbe o p ojec s implemen ed on he equilib ium pa h. Tha
is, we need o use a mos wo messages o cons uc an equilib ium in his model.
Appendix A.2. P oo o Lemma 2
Suppose ha he DM implemen s he p ojec ecommended by he agen . Then, i he
agen sends m=1, he p obabili y o DM implemen ing p ojec 1 is gi en by:
E[θ1|m=1].
I he agen sends
m=
2, he p obabili y o he DM implemen ing p ojec 2 is gi en by:
bDM ·E[θ2|m=2].
In he ollowing p oo s, we use hese esul s:
(1)
Suppose ha q(1|θ=(l, 1)) >0 holds. Tha is, gi en θ=(l, 1):
θ1·E[θ1|m=1]
| {z }
The agen ’s expec ed payo i he/she sends m=1
≥(bA·θ2)·(bDM ·E[θ2|m=2])
| {z }
The agen ’s expec ed payo i he/she sends m=2
.
Then, s ic inequali y should hold gi en any θ∈{(1, 1),(1, l),(l,l)}. Tha means:
q(1|θ=(1, 1)) =q(1|θ=(1, l)) =q(1|θ=(l,l)) =1
Games 2024,15, 18 20 o 25
(2)
Suppose ha
q(
1
|θ= (
1, 1
)) ∈(
0, 1
)
o
q(
1
|θ= (l
,
l)) ∈(
0, 1
)
holds. Tha is, gi en
any θ∈{(1, 1),(l,l)}:
θ1·E[θ1|m=1] = bA·θ2·bDM ·E[θ2|m=2].
Then, he s ic inequali y:
θ1·E[θ1|m=1]>bA·θ2·bDM ·E[θ2|m=2].
should hold gi en
θ= (
1,
l)
. Tha means
q(1|θ=(1, l)) =
1. In addi ion, he
s ic inequali y:
θ1·E[θ1|m=1]<bA·θ2·bDM ·E[θ2|m=2].
should also hold gi en θ= (l, 1). Tha means q(1|θ=(l, 1)) =0.
(3)
Suppose ha q(1|θ=(1, l)) <1 holds. Tha is, gi en θ=(1, l):
θ1·E[θ1|m=1]≤bA·θ2·bDM ·E[θ2|m=2].
Then, s ic inequali y should hold gi en any θ∈{(1, 1),(l, 1),(l,l)}. Tha means:
q(1|θ=(1, 1)) =q(1|θ=(l, 1)) =q(1|θ=(l,l)) =0
Appendix A.3. P oo o Lemma 3
(1)
F om Lemma 2, T, P1, P2, and P2’ a e he only ypes o equilib ia ha can exis . We
can also show ha he DM’s belie
E[θ1|m=
1
]
is la ge in P2’ han in P2, in P2 han
in T, and in T han in P1, espec i ely. Co espondingly, he DM’s belie
E[θ2|m=
2
]
is la ge in P1 han in T, in T han in P2, and in P2 han in P2’, espec i ely.
Conside he se o pa ame e s
bDM
and
bA
gi en which T exis s. In T, he agen
s ic ly p e e s sending m=1 o m=2 gi en θ= (1, l), ha is:
1·E[θ1|m=1]>l·bA·bDM ·E[θ2|m=2].
We canno cons uc P2 o P2’, which equi es
1·E[θ1|m=1]≤l·bA·bDM ·E[θ2|m=2].
wi h la ge
E[θ1|m=
1
]
and smalle
E[θ2|m=
2
]
. Simila ly, in T, he agen p e e s
sending m=2 o m=1 gi en θ= (l, 1), ha is:
l·E[θ1|m=1]≤1·bA·bDM ·E[θ2|m=2].
We canno cons uc P1, which equi es
1·E[θ1|m=1] = l·bA·bDM ·E[θ2|m=2].
wi h smalle E[θ1|m=1]and la ge E[θ2|m=2].
Simila ly, we can show ha P1, P2, and P2’ do no exis oge he .
The second sen ence ollows om he p ope y o cheap alk models: he e always
exis s an equilib ium whe e no in o ma ion is e ealed (a babbling equilib ium). In
his model, wi hou in o ma ion, he DM ne e implemen s p ojec 2. Thus, a ze o
equilib ium always exis s.
(2) F om he de ini ion, i T, P1, o P2’ exis s, he agen does no mix be ween he messages
gi en any s a e, and hence, his is he only in o ma i e equilib ium. On he o he
hand, i P2 exis s, he e can be mul iple P2s. P2 equi es
E[θ1|m=1] = bA·bDM ·E[θ2|m=2]
Games 2024,15, 18 21 o 25
wi h la ge
E[θ1|m=
1
]
and smalle
E[θ2|m=
2
]
han T. Bu , P2 allows he agen
o mix be ween he wo messages gi en
θ∈{(1, 1),(l,l)}
, and he agen can mix
di e en ly gi en he wo s a es.
(3)
The wel a e compa ison is i ial. I he e is an in o ma i e equilib ium, bo h playe s
p e e p ojec 2 o e p ojec 1 gi en a leas one s a e. Howe e , in a ze o equilib ium,
only p ojec 1 is implemen ed. Bo h playe s should be be e o in an in o ma i e
equilib ium han in a ze o equilib ium.
Appendix A.4. P oo o P oposi ion 1
To ind he exis ence condi ions o each in o ma i e equilib ium, we compu e he
DM’s belie s consis en wi h he agen ’s message s a egies. Then, we ind pa ame ic
assump ions sa is ying condi ions (8) and (9) ( he DM obeys he agen ’s ecommenda ion)
and condi ions (10) and (11) ( he message s a egy is incen i e-compa ible o he agen ) in
Sec ion 3.
In a u h ul equilib ium (T), he agen ’s message s a egy is:
m=1 w.p. 1 i θ∈{(1, l),(1, 1),(l,l)}
m=2 w.p. 1 i θ= (l, 1)
The DM’s consis en belie s a e:
E[θ1|m=1] = 2+ (1+ρ)l
3+ρ,
E[θ2|m=1] = 2l+ (1+ρ)
3+ρ,
E[θ1|m=2] = l,
E[θ2|m=2] = 1.
These belie s sa is y condi ions (8) and (9).
Nex , condi ion (10) binds gi en θ= (1, 1):
1·2+ (1+ρ)l
3+ρ≥(1·bA)·(1·bDM)
⇔bAbDM ≤2+ (1+ρ)l
3+ρ=B(ρ,l).
Condi ion (11) binds gi en θ= (l, 1):
(1·bA)·(1·bDM)≥l·2+ (1+ρ)l
3+ρ
⇔bAbDM ≥l·2+ (1+ρ)l
3+ρ=l·B(ρ,l).
The e o e, P oposi ion 1 (1) holds.
In P1, he agen ’s message s a egy is:
m=1 w.p. 1 i θ∈{(1, l),(1, 1),(l,l)},
m=1 w.p. qand m=2 w.p. 1 −qi θ= (l, 1)
o some q∈(0, 1). The DM’s consis en belie s a e:
E[θ1|m=1] = 2+ (1+ρ+q(1−ρ))l
3+ρ+q(1−ρ),
E[θ2|m=1] = 2l+ (1+ρ+q(1−ρ))
3+ρ+q(1−ρ),
E[θ1|m=2] = l,
E[θ2|m=2] = 1.
Games 2024,15, 18 22 o 25
Condi ion (11) should hold wi h equali y gi en θ= (l, 1):
(1·bA)·(1·bDM) = l·2+ (1+ρ+q(1−ρ))l
3+ρ+q(1−ρ)
⇔qP1:=(2+ (1+ρ)l)l−bDMbA(3+ρ)
(1−ρ)(bDMbA−l2)
We can show qP1∈(0, 1) o
l+l2
2bDM
<bA<B(ρ,l)l
bDM
.
Condi ions (8)–(10) do no bind. P oposi ion 1(2) holds.
F om Lemma 3, P2 equi es
E[θ1|m=1] = bA·bDM ·E[θ2|m=2].
As exp ession (12) shows, he DM’s belie s de e mine he DM’s ex an e expec ed payo ,
independen o how he agen mixes he wo messages gi en each o
θ∈{(1, 1),(l,l)}
.
Hence, we compu e he simple P2, whe e he agen ’s message s a egy is
m=1 w.p. 1 i θ= (1, l),
m=1 w.p. qand m=2 w.p. 1 −qi θ∈{(1, 1),(l,l)},
m=2 w.p. 1 i θ= (l, 1)
o some q∈(0, 1). The DM’s consis en belie s a e:
E[θ1|m=1] = 1−ρ+q(1+ρ)(1+l)
1−ρ+2q(1+ρ),
E[θ2|m=1] = (1−ρ)l+q(1+ρ)(1+l)
1−ρ+2q(1+ρ),
E[θ1|m=2] = (1−ρ)l+ (1−q)(1+ρ)(1+l)
1−ρ+2(1−q)(1+ρ),
E[θ2|m=2] = 1−ρ+ (1−q)(1+ρ)(1+l)
1−ρ+2(1−q)(1+ρ).
Condi ion (10) should hold wi h equali y gi en θ= (1, 1):
E[θ1|m=1] = bAbDME[θ2|m=2]
⇔1−ρ+q(1+ρ)(1+l)
1−ρ+2q(1+ρ)=bAbDM
1−ρ+ (1−q)(1+ρ)(1+l)
1−ρ+2(1−q)(1+ρ).
The pa ame e
q
sa is ying he las equali y is uniquely de e mined. Le
qP2
deno e such a
unique q. We can show ha
qP2∈(0, 1) o bA∈B(ρ,l)
bDM
,1
bDMB(ρ,l).
Bu , condi ion (9) addi ionally equi es:
bDM ·E[θ2|m=2]≥E[θ1|m=2]
⇔q≥1−(bDM −l)·(1−ρ)
(1−bDM)(1+ρ)(1+l)=:q.
Hence,
qP2≥q
is he necessa y condi ion o he exis ence o P2, which holds i and only i
bA≤D(bDM,ρ,l),
Games 2024,15, 18 23 o 25
whe e we ind D(ρ,l,bDM)by plugging q=q(bDM,l,ρ)in o
1−ρ+q(1+ρ)(1+l)
1−ρ+2q(1+ρ)=bDMbA
1−ρ+ (1−q)(1+ρ)(1+l)
1−ρ+2(1−q)(1+ρ).
In summa y, P2 exis s o
qP2∈(0, 1) o bA∈B(ρ,l)
bDM
,min1
bDMB(ρ,l),D(bDM,ρ,l).
We can also show ha (9) binds and
D(bDM,ρ,l)<1
bDMB(ρ,l)
i bDM <d(ρ,l). P oposi ion 1(3) holds.
In P2’, he agen ’s message s a egy is
m=1 w.p. 1 i θ= (1, l)
m=2 w.p. 1 i θ∈{(1, 1),(l,l),(l, 1)}.
The DM’s consis en belie s a e:
E[θ1|m=1] = 1,
E[θ2|m=1] = l,
E[θ1|m=2] = 1+ρ+2l
3+ρ,
E[θ2|m=2] = 2+ (1+ρ)l
3+ρ=B(ρ,l).
These belie s sa is y condi ions (8) and (9) only i
bDM ≥d(ρ,l)
. Condi ion (11) binds gi en
θ= (1, 1):
bAbDMB(ρ,l)≥1⇔bA≥1
bDMB(ρ,l).
Fo bA<1/l, (10) holds gi en θ= (1, l). P oposi ion 1(4) holds.
Fo
bDM <d(ρ
,
l)
, (9) is iola ed e en i we can cons uc an incen i e-compa ible
message s a egy o he agen . The e is discon inui y in he sense ha , as we inc ease
bDM
,
he in o ma i e equilib ium shi s om P2 o Z, skipping P2’ because he DM’s bias is oo
s ong o obey.
The non-mono onici y o he DM’s ex an e expec ed payo di ec ly ollows om
claims (1)–(5) in P oposi ion 1 and exp ession (12) in Sec ion 3. We can compu e he ex an e
expec ed payo o he DM using he abo e-men ioned esul s.
Fo example,
VDM,T(ρ,bA,bDM,l) = (2+ (1+ρ)l)2
8(3+ρ)+(1−ρ)b2
DM
8,
VDM,P2′(ρ,bA,bDM,l) = 1−ρ
8+(2+(1+ρ)l)2b2
DM
8(3+ρ)
,
VDM,Z(ρ,bA,bDM,l) = (1+l)2
8.
Appendix A.5. P oo o Co olla y 1
This esul ollows om P oposi ion 1 and exp ession (13) in Sec ion 3. We can compu e
he ex an e expec ed payo o he DM using he abo e-men ioned esul s.
Games 2024,15, 18 24 o 25
Fo example,
VA,T(ρ,bA,bDM,l) = (2+ (1+ρ)l)2
4(3+ρ)+(1−ρ)bDMbA
4,
VA,P2′(ρ,bA,bDM,l) = 1−ρ
4+(2+ (1+ρ)l)2bDMbA
4(3+ρ),
VA,Z(ρ,bA,bDM,l) = (1+l)2
4.
Appendix A.6. P oo o Sec ion 4
Unde Vd, condi ions (8) and (9) in Sec ion 3do no need o hold any mo e. Hence,
om P oposi ion 1, he equilib ium ou come is di e en only o
bDM <d(ρ,l)
and
bA≥D(bDM,ρ,l).
Fo bDM <d(ρ,l), he e is no discon inui y in he sense ha P2 exis s o
bA∈B(ρ,l)
bDM
,1
bDMB(ρ,l)
and P2’ exis s o
bA∈1
bDMB(ρ,l),1
l.
No es
1
“O e he pas 50 ading sessions, he e has been a s ong in e se co ela ion be ween he ai line index and oil p ices.”
See h ps://www. eu e s.com/a icle/us-usa-ai lines-s ocks/ ising-oilp ices-help-g ound-u-s-ai line-s ocks-could-make- hem-
cheap-idUSKCN1IM24C (accessed on 30 No embe 2023).
2
The e has been a posi i e co ela ion be ween wind and sola powe since 2009. See h ps://ieeexplo e.ieee.o g/s amp/s amp.
jsp?a numbe =7110436 (accessed on 30 No embe 2023). F om 2002 o 2012, he a e age annual g ow h a es o wind and sola
powe we e 26.1% and 50.1%, espec i ely. See h p://www.ene gies- enou elables.o g/obse e /h ml/in en ai e/pd /15e-
in en ai e-Chap01-Eng.pd (accessed on 30 No embe 2023).
3
Apple ecei es a 30% commission on so wa e bough h ough he Mac App S o e. See h ps://discussions.apple.com/ h ead/70
32123 (accessed on 30 No embe 2023).
4
Fo example, i
bA≤l
, he agen p e e s p ojec 1 gi en any s a e. I
bA≥
1
/l
, he agen p e e s p ojec 2 gi en any s a e. We
elimina e hese si ua ions om ou analysis.
5
Fo each playe
j∈{DM,A}
, a co ela ion coe icien o he wo p ojec s’ bene i s
co (θ1
,
bjθ2)
is
ρ
. Bu , ega dless o
ρ
, he
uncondi ional means o p ojec 1 and p ojec 2 a e
E[θ1] = 1+l
2
and
E[bjθ2] = bj(1+l)
2
, and hei uncondi ional a iances a e
Va [θ1] = (1−l)2
2and Va [bjθ2] = b2
j(1−l)2
2.
6
CDK [
2
] did no conside he p ojec cos ; ins ead, hey examined di e en payo s esul ing om selec ing he ou side op ion o
di e en playe s: a posi i e payo o he p incipal and a ze o payo o he agen .
7Lemma 3 and P oposi ion 1 a e closely ela ed o Chiba and Leong (2015) [4].
8
Fo e y la ge
bDM
, he e is no a ea o Z below he a ea P1 since we assume
bA>l
. I we elax his assump ion, he a ea Z
appea s below he a ea P1 o any le el o bDM <1.
9
Since we assume
bA<
1
/l
, he e is no a ea o Z abo e P2’ o
bDM >d(ρ
,
l)
. I we elax ou assump ion and conside
bA≥
1
/l
,
hen he in o ma i e equilib ium exis ing e en ually changes om P2’ o none (i.e., only Z exis s).
10 We can also show ha ull delega ion always makes he DM wo se o han e o-based delega ion gi en ou model se ing.
Re e ences
1. C aw o d, V.; Sobel, J. S a egic In o ma ion T ansmission. Econome ica 1982,50, 1431–1451. [C ossRe ]
2. Che, Y.-K.; Dessein, W.; Ka ik, N. Pande ing o Pe suade. Am. Econ. Re . 2013,103, 47–79. [C ossRe ]
3.
Chiba, S.; Leong, K. Cheap Talk wi h Ou side Op ions. Depa men o Managemen , Uni e si à Ca’ Fosca i Venezia Wo king
Pape No. 16/2013. A ailable online: h p://ss n.com/abs ac =2332689 (accessed on 10 Ma ch 2024).
4. Chiba, S.; Leong, K. An Example o Con lic s o In e es as Pande ing Disincen i es. Econ. Le . 2015,131, 20–23. [C ossRe ]
5. Chak abo y, A.; Ha baugh, R. Compa a i e Cheap Talk. J. Econ. Theo y 2007,132, 70–94. [C ossRe ]
6. Chak abo y, A.; Ha baugh, R. Pe suasion by Cheap Talk. Am. Econ. Re . 2010,100, 2361–2382. [C ossRe ]
