“Agreeing to Disagree” Type Results under Ambiguity

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“Agreeing to Disagree” Type Results under Ambiguity Adam Dominiak ? Virginia Polytechnic Institute Jean-Philippe Lefort † Universite Paris-Dauphine This Version: March 2, 2012 Abstract In this paper we characterize conditions under which it is impossible that non- Bayesian agents “agree to disagree” on their individual decisions. The agents are Choquet expected utility maximizers in the spirit of Schmeidler (1989, Econo- metrica 57, 571-587). Under the assumption of a common prior capacity distri- bution, it is shown that whenever each agent's information partition is made up of unambiguous events in the sense of Nehring (1999, Mat. Soc. Sci. 38, 197- 213), then it is impossible that the agents disagree on the common knowledge decisions, whether they are posterior capacities or posterior Choquet expecta- tions. Conversely, an agreement on posterior Choquet expectations - but not on posterior capacities - implies that each agent's private information consists of Nehring-unambiguous events. These results indicate that under ambiguity - contrary to the standard Bayesian framework - asymmetric information matters and can explain differences in common knowledge decisions due to the ambigu- ous nature of agents' private information. Keywords: Ambiguity, capacities, Choquet expected utility, unambiguous events, updating, asymmetric information, common knowledge, agreement theorem JEL-Codes: D70, D81, D82 ?Department of Economics, Virginia Polytechnic Institute and State University (Virginia Tech), Blacksburg, VA 24061-0316, USA, E-mail: dominiak@vt.

  • expected utility

  • agent

  • agents cannot

  • nehring-unambiguous

  • choquet expectations

  • bayesian framework

  • choquet expected

  • actions maximizing posterior

  • decisions

  • events


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“AgreeingtoDisagree”TypeResults
underAmbiguity
AdamDominiak

Jean-PhilippeLefort

VirginiaPolytechnicInstituteUniversite´Paris-Dauphine
ThisVersion:March2,2012

Abstract
Inthispaperwecharacterizeconditionsunderwhichitisimpossiblethatnon-
Bayesianagents“agreetodisagree”ontheirindividualdecisions.Theagentsare
ChoquetexpectedutilitymaximizersinthespiritofSchmeidler(1989,
Econo-
metrica
57
,571-587).Undertheassumptionofacommonpriorcapacitydistri-
bution,itisshownthatwhenevereachagent’sinformationpartitionismadeup
ofunambiguouseventsinthesenseofNehring(1999,
Mat.Soc.Sci.
38
,197-
213),thenitisimpossiblethattheagentsdisagreeonthecommonknowledge
decisions,whethertheyareposteriorcapacitiesorposteriorChoquetexpecta-
tions.Conversely,anagreementonposteriorChoquetexpectations-butnot
onposteriorcapacities-impliesthateachagent’sprivateinformationconsists
ofNehring-unambiguousevents.Theseresultsindicatethatunderambiguity-
contrarytothestandardBayesianframework-asymmetricinformationmatters
andcanexplaindifferencesincommonknowledgedecisionsduetotheambigu-
ousnatureofagents’privateinformation.

Keywords:
Ambiguity,capacities,Choquetexpectedutility,unambiguousevents,
updating,asymmetricinformation,commonknowledge,agreementtheorem

JEL-Codes:
D70,D81,D82


DepartmentofEconomics,VirginiaPolytechnicInstituteandStateUniversity(VirginiaTech),
Blacksburg,VA24061-0316,USA,E-mail:dominiak@vt.edu

Universite´Paris-Dauphine,LEDaPlaceduMare´chaldeLattredeTassigny,75775Pariscedex
16,France,E-mail:jplefort1@yahoo.fr

1Introduction

Inhiscelebratedarticle“AgreeingtoDisagree”,Aumann(1976)challengedtherole
thatasymmetricinformationplaysinthecontextofinterpersonaldecisionproblems
underuncertainty.PresupposingthatagentsareBayesianandshareanidenticalprior
probabilitydistribution,Aumannshowedthattheagentscannot“agreetodisagree”
ontheirposteriorbeliefs.Moreprecisely,wheneveragents’posteriorbeliefsforsome
fixedeventarecommonknowledge,thentheseposteriorsmustcoincide,despitethe
factthattheposteriorsmaybeconditionedondiverseinformation.Thisremarkable
resultimpliesthatwheneveragroupofagentscometocommonknowledgeofdecisions
thenthesedecisionsmustbemadeasiftherewherenoprivateinformationatall.In
thispaper,wescrutinizetheroleofasymmetricinformationamongnon-Bayesian
agents.Inessence,wedemonstratethatdifferencesincommonlyknowndecisionsare
possibleduetotheambiguouscharacterofagents’privateinformation.
WithintheBayesianframework,Aumann’simpossibilityresulthasbeenextended
tomoreabstractdecisionssuchasposteriorexpectations(GeanakoplosandSebenius
(1983))andactionsmaximizingposteriorexpectations(Milgrom(1981),Milgromand
Stokey(1982)andBacharach(1985)).These“agreeingtodisagree”typeresults,also
referredtoasprobabilisticagreementtheorems,areoftenviewedaspointingoutlimi-
tationsoftheexplanatorypowerofasymmetricinformation.Differencesinindividual
decisionscannotbeexplained
solely
bydifferencesinagents’privateinformation.Two
approacheshavebeenproposedinordertoovercometheselimitations.Inthefirstone,
Morris(1994,1995)advocatestodiscardthe“commonness”assumptionofpriorprob-
abilities.Thesecondapproach,suggestedbyMondererandSamet(1989),relieson
weakeningthenotionof“commonknowledge”.Although,bothoftheseapproaches
maintaintheBayesianparadigm.Inthispaper,wesuggestanalternativeapproach.
Wemaintaintheassumptionofcommonpriorbeliefsaswellasthenotionofcom-
monknowledge.Instead,weweakenthe“additivity”propertyofsubjectivebeliefsby
allowingtheagentstobenon-BayesianintheveinoftheChoquetexpectedutility
theoryofSchmeidler(1989).
InSchmeidler’stheorysubjectivebeliefsarerepresentedbyanormalizedandmono-

1

tone(but-non-necessarily-additive)setfunction,called
capacity
.Thenotionofcapac-
ityallowstoaccommodateambiguityandambiguityattitudesintothedecisionmaking
process.Ambiguityreferstosituationsinwhichprobabilitiesforsomeuncertainevents
areknown,whereasforothereventstheyareunknownduetomissingprobabilistic
information.Thelackofprobabilisticinformationandreactiontoit,asmanifestedfor
instanceinEllsberg’s(1961)typeexperiments,mayaffectagents’choicesintheway
thattheyareincompatiblewithsubjectiveexpectedutilitytheoryofSavage(1954).
Inthepresencenon-additivebeliefs,individualdecisionsaremadeonthebasisofmax-
imizingexpectedutilities,whicharecomputedbymeansofChoquet(1954)integrals.
Let’sconsiderafinitegroupofagentsfacingadynamicdecisionproblemunder
ambiguity.Theagentsshareacommonpriorcapacitydistributionoveranalgebra
ofeventsgeneratedbyafinitesetofstates.Moreover,eachagentisendowedwitha
partitionoverthesetofstateswhichrepresentshisprivateinformation.Therearetwo
stagesofplanning:anex-anteandaninterimstage.Attheex-antestageallagents
shareidenticalinformation.Attheex-poststage,eachagentreceiveshisprivateinfor-
mation.Conditionalontheirprivatesignals,theagentsrevisetheirpriorpreferences.
Posteriorpreferencesarederivedbyupdatingpriorcapacityandkeepingtheutility
functionunchanged.Therearemanyreasonableupdatingrulesfornon-additivebe-
liefs,withBayes’rulebeingonlyonealternative(seeGilboaandSchmeidler,1993).
However,ourresultsdonotdependuponwhichupdatingruleisused.Weonly
requirethatupdatingrulesrespectconsequentialism,apropertyintroducedbyHam-
mond(1988).Consequentialismrequiresthatposteriorpreferencesareonlyaffected
bytheconditioningevents,i.e.agents’privateinformationinoursetup.Counterfac-
tualevents,aswellasthepastdecisionhistory,areimmaterialforposteriordecisions
(seeHananyandKlibanoff(2007)).Onceposteriorpreferenceshavebeengenerated,
theagentsannouncetheirindividualdecisions.Anagreementondecisionsdesignates
situationsinwhichitisimpossiblethattheagentsdisagreeoncommonknowledge
posteriordecisions.Thedecisionsthatwefocusonare:posteriorcapacitiesforsome
fixedevent,posteriorChoquetexpectationsforagivenaction,andactionsmaximizing
posteriorChoquetexpectationsforagivensetoffeasibleactions.
Ourfirstobjectiveistocharacterizethepropertiesofeventsinagents’information

2

partitionswhichguaranteethatdisagreementsondecisionsareimpossible.Anatural
choiceforsucheventsareeventswhichareperceivedbytheagentsasbeingunam-
biguous.IntheBayesianframework,inwhichprobabilisticagreementtheoremsare
established,alluncertaineventsareunambiguous.Innon-Bayesiansetups,however,
someuncertaineventsmaybesubjectivelyseenasunambiguouswhileotherevents
areperceivedasambiguous.Recently,severalnotionsofrevealedunambiguousevents
havebeenproposed,e.g.,byNehring(1999)andbyZhang(2002).Westartour
analysisbyassumingthatonlytheeventswhichreflectagents’privateinformation
areunambiguous,whileothereventsmaybeambiguous.InturnsoutthatNehring’s
(1999)notionofunambiguouseventssufficestoruleoutpossibilitiesofdisagreements
oncommonknowledgedecisions.Moreprecisely,ifeachagent’sinformationpartition
ismadeupofNehring-unambiguouseventsthenitisimpossiblethatatsomestate
agents’decisionsarecommonknowledgeandtheyarenotthesame.Thesedecisions
canbeposteriorcapacities,posteriorChoquetexpectations,oractionsmaximizing
posteriorChoquetexpectations.However,disagreementsoncommonlyknownde-
cisionsmayoccurassoonasonedepartsfromthenotionofNehring-unambiguous
events.Especiallywhenadaptingaslightlyweakernotionofunambiguousevents,as
proposedbyZhang(2002),theagentsmay“agreetodisagree”ontheirindividualde-
cisions.Weexemplifyasituationinwhichadisagreementonposteriorbeliefsamong
twoagentswhoseinformationpartitionsaremadeupofZhang-unambiguousevents
occurs.Thatis,theagentscometoacommonknowledgeagreementoftheirposterior
capacitiesforsomefixedevent.Nevertheless,theseposteriorsdonotcoincide.
Nextwefocusonaconverseresult.Weconsidersituationsinwhichitisimpos-
siblethattheagents“agreetodisagree”ontheirdecisions.Animmediatequestion
thatarisesinthiscontextisifknowingthatdisagreementsareimpossiblecanone
infersomethingaboutthenatureoftheagents’privateinformation?Inprinciple,
theanswerisaffirmative.However,whatwemayinferaboutthenatureofagents’
privateinformationdependsonthetypeofdecisionsthattheagents“agreetoagree”
on.Assumingthatdisagreementsonposteriorcapacitiesareimpossible,wecanshow
thatnothingcanbeconcludedaboutthepropertiesoftheeventsinagents’informa-
tionpartitions.Thisisbecauseonecanalwaysfindacapacitydistributionandan

3

updatingruleforpriorbeliefssuchthatanagreementonposteriorbeliefsholdstrue.
Nevertheless,theeventsinagents’informationpartitionswillbeneitherNehring-
norZhang-unambiguousevents.However,whenanagreementisreachedonposterior
Choquetexpectations,andonactionsmaximizingposteriorChoquetexpectationsas
well,theneachagent’sprivateinformationmustbemadeupofNehring-unambiguous
events.
Thispaperisorganizedasfollows.Thenextsectionintroducesthecapacitymodel
ofSchmeidler(1989).First,theChoquetexpectedutilitypreferencesaredefined,and
then,thenotionofunambiguouseventsinthesenseofNehring(1999)andZhang
(2002)arepresented.InSection3,theChoquetexpectedutilitymodelisextendedto
dynamicchoicesituations.InSection4,weintroducethestandardepistemicframe-
workusedformodelinginterpersonaldecisionproblemswithdifferentialinformation.
InSection5,thesufficientconditionfortheimpossibilityof“agreeingtodisagree”on
individualdecisionsisestablished.Thissectionendswithanexampledemonstrat-
ingapossibilityofdisagreementaboutcommonknowledgeposteriorcapacities.In
Section6,thenecessaryconditionfortheimpossibilityofdisagreementonposteriors
Choquetexpectationsisestablishedandproven.Weclosethissectionwithabrief
discussiononthemeaningfulnessofconsequentialisminthecontextofinterpersonal
decisionproblemswithdifferentialinformation.InSection7,ano-speculativetrade
corollaryisestablished.Finally,weconcludeinSection8.

2StaticChoquetPreferences

Inthissectionwerecallthemaintenetsofthe
Choquetexpectedutility
theorypioneered
bySchmeidler(1989).WeconsiderafinitesetΩofstates.Anevent
E
isasubset
ofΩ.Let
A
=2
Ω
bethesetofallsubsetsofΩ.Forany
E

ΩwedenoteΩ
\
E
,
thecomplementof
E
,by
E
c
.Subjectivebeliefsoveruncertaineventsarerepresented
by
capacities
.Acapacity
ν
:
A→
R
isanormalizedandmonotonesetfunction,i.e.,
i
)
ν
(

)=0

(Ω)=1and
ii
)
ν
(
E
)

ν
(
F
)whenever
E

F

Ω.Capacitiesare
notrequiredtobeadditive,althoughtheymustsatisfythemonotonicityproperty.In
termsofqualitativebeliefs,monotonicityhasanaturalinterpretation;“larger”events,

4

withrespecttothesetinclusion,areregardedas“morelikely”.
Let
X
beasetofconsequences.Amapping
f


X
assigningconsequencesto
statesiscalledanaction.Let
F
beasetofallactions.Werefertoasubset
B⊂F
asasetoffeasibleactions.Forapairofactions
f,g
∈F
andanevent
E
∈A
,
denoteby
f
E
g
anactionthatassignstheconsequence
f
(
ω
)

X
toeachstate
ω
in
E
and
g
(
ω
)

X
toeachstate
ω

E
c
.Let
<
beapreferencerelationdefinedon
thesetofactions
F
.Apreferencerelation
<
issaidtoadmit
Choquetexpectedutility
representationifthereexistsavN-Mutilityfunction
u
:
X

R
andacapacity
ν
on
A
suchthatforany
f,g
∈F
:
ZZf
<
g

u

fdν

u

gdν.
(1)
ΩΩChoquetexpectedutilitypreferenceshavebeenjustifiedbehaviorallybySchmeidler
(1989),Gilboa(1987)andSarinandWakker(1992)foraninfinitestatespace.Im-
posingsomerichnessconditionsonthesetofoutcomesandallowingforafinitestate
space,Choquetexpectedutilitypreferenceshasbeenaxiomatizedby
?
,Nakamura
(1990)andChewandKarni(1994).
Inpresenceofnon-additivebeliefs,theexpectationsin(1)arecomputedbymeans
ofChoquetintegrals.Foragivenaction
f
,let
E
1
,...,E
n
denotethepartitionordered
fromthemosttotheleastfavorableevents,i.e.suchthat
uf
(
E
1
)
≥∙∙∙≥
uf
(
E
n
).
Therankingpositionofaneventexpressesitsfavorablenesswithrespecttoconse-
quencesassociatedwith
f
.The
Choquetintegral
of
f
withrespectto
ν
and
u
is
definedtobe:
Z
n
X

1
hi
u

fdν
=
uf
(
E
j
)

uf
(
E
j

n
)
νE
1
,...,E
j
+
uf
(
E
n
)(2)
Ω1=jForagivencapacity
ν
andanaction
f
onecandefinea
rank-dependentprobability
distribution
p

on
E
1
,...,E
n
,where


p

E
j
=
νE
1
,...,E
j

νE
1
,...,E
j

1
.
(3)

Theprobability
p

(
E
j
)of
E
j
canbeinterpretedasamarginalcapacitycontributionof
theevent
E
j
toevents
E
1
,...,E
j

1
yieldingbetterconsequences.Accordingly,(2)can
beequivalentlywrittenasanexpectedutilityof
f
withrespecttotherank-dependent

5

probabilitydistribution
p

and
u
:
Z
n
X

1
hi
u

fdp

=
uf
(
E
j
)
p

(
E
j
)
.
(4)
Ω1=jIngeneral,actionsgeneratingdistinctrankingpositionofstatesareevaluatedwithre-
specttodifferentrank-dependentprobabilitydistributions.Onlyactionswhichinduce
thesameorderingofevents
E
1
,...,E
n
,alsocalled
comonotonic
actions,arealways
evaluatedwithrespecttothesamerank-dependentprobabilitydistribution.
1
Inthefaceofambiguityitisimportanttolocalizeeventsthataresomehowunam-
biguous.Theintuitionbehindthenotionofunambiguouseventsis,thattheymust
supportsomekindofprobabilisticbeliefs.ForNehring(1999)ambiguityofanevent
iscloselyrelatedtoitsrankdependence.
2
Moreprecisely,Nehringcallsanevent
U
unambiguous,henceforthNehring-unambiguous,iftheprobability
p

(
U
)attachedto
theeventdoesnotdependontherankingpositionof
U
;orequivalently,itdoesnot
dependupontheact
f
beingevaluated.Accordingly,anevent
U
∈A
iscalledNehring-
unambiguousif
p

(
U
)=
p

(
U
)=
ν
(
U
)forall
f,g
∈F
.Let
A
NU
bethecollectionof
Nehring-unambiguousevents.
3
Anyeventin
A
NU
canbealsocharacterizedinterms
ofagivencapacity
ν
.Nehring(1999)showednamelythat
ν
isadditivelyseparable
acrossitsunambiguousevents.Thatis,
U
∈A
NU
ifandonlyifforall
E
∈A
:
ν
(
E
)=
ν
(
E

U
)+
ν
(
E

U
c
)
.
(5)

Frombehavioralpointofview,thenotionofNehring-unambiguouseventscanbe
alsocharacterizedbyapplyingSavage’s(1954)Sure-Thing-Principle(seeSarinand
Wakker(1992)andDominiakandLefort(2011)).Thatistosay,
U
∈A
NU
ifandonly
0ifforany
f,g,h,h
∈F
:

f
U
h
<
g
U
h

f
U
h
0
<
g
U
h
0
,
(6)
1
Formally,twoactions
f
and
g
arecalled
comonotonic
iftherearenowtwostates,
ω
and
ω
0
,such
that
f
(
ω
)
>g
(
ω
)and
f
(
ω
0
)
<g
(
ω
0
).
2
Recently,othernotionsofunambiguouseventshavebeensuggestedintheliterature,seefor
instanceEpsteinandZhang(2001),Zhang(2002)andGhirardato,Maccheroni,andMarinacci(2004).
3
Nehring(1999)provedthatforanycapacity
ν
theset
A
NU
isalwaysanalgebra.

6

and(6)isalsosatisfiedwhen
U
iseverywherereplacedby
U
c
.Otherwise,
U
iscalled
ambiguous.TheSure-Thing-Principleconstrainedtotheevents
U
and
U
c
guarantees
thattherankingacts
f
U
h
and
g
U
h
remainsunchangedwhateverarethecommon
consequencesassignedtostatesoutsideof
U
.
Zhang(2002)suggestedanalternativedefinitionofunambiguouseventsbyweak-
eningtheSure-Thing-Principle.Herefersanevent
U
tobeunambiguous,henceforth
Zhang-unambiguous,ifreplacingaconstantoutcome
x
outsideof
U
byanyothercon-
stantoutcome
x
0
doesnotchangetherankingofactsbeingcompared.Accordingly,
anevent
U
isZhang-unambiguousifandonlyifforanyaction
f,g
∈A
andforany
outcome
x,x
0

X
:
f
U
x
<
g
U
x

f
U
x
0
<
g
U
x
0
.
(7)
and(7)isalsotruewhen
U
iseverywherereplacedby
U
c
.Otherwise,
U
iscalled
ambiguous.Let
A
ZU
bethecollectionofall
Z
-unambiguousevents.Again,intermsof
capacitiesZhang(2002)showedthat
U
∈A
ZU
ifandonlyifforall
E
∈A
suchthat
c:U⊂Eν
(
E

U
)=
ν
(
E
)+
ν
(
U
)
.
(8)

Thus,theadditiveseparabilitypropertyof
ν
issatisfiedonlyonsubeventsoftheir
unambiguouscomplements.Itisworthtomentionthat
A
ZU
⊂A
NU
,since
A
ZU
doesnot
needtobeanalgebra.
4
Itisa
λ
-system,acollectionofeventsthatcontainsΩand
thatisclosedundercomplementsanddisjointunions,butnotunderintersections.

3DynamicChoquetPreferences

Inthesequel,weextendtheprevioussetuptodynamicchoicesituations.Indynamic
choiceproblemstherearetwostagesofplanning:exantestageandinterimstage.
Atinterimstage,agentsareinformedthatsomeeventhasoccurredandincorporate
thisinformationbyupdatingtheirpreferences.Whenbeliefsareprobabilisticand
preferencesareofexpectedutilitytype,thenpriorpreferencesareupdatedinthe
Bayesianway.Thatis,theconditionalpreferencesarederivedbyupdatingpriorbeliefs
4
Nehring(1999)provedthatforanycapacity
ν
theset
A
NU
isanalgebra.

7

withaccordancetoBayes’ruleandbyleavingtheutilityfunctionunchanged.However,
whenbeliefsarenon-additivetherearemanypossibleupdatingprocedures.Inthis
paper,weconstrainouranalysistoupdatingrulesgeneratingconditionalpreferences
whichrespect
consequentialism
andarerepresentablebyChoquetexpectedutilities
withrespecttoanupdatedcapacityandthesameunconditionalvN-Mutilityfunction.
Atex-antestage,whennoinformationisavailable,wedenoteby
<
theuncondi-
tionalChoquetexpectedutilitypreferences.Atinterimstage,anevent
E
hasbeen
observedandconditionalpreferencesaregenerated.Throughoutthepaper,weas-
sumedthatallconditioningeventsarenon-null,i.e.,
ν
(
E
)
>
0,anddenoteby
<
E
theconditionalpreferencesoverthesetofactions
F
.Suchaconditionalpreference
relationisviewedasgoverningdecisionsupontherealizationof
E
.AsMachina(1989)
observes,updatingofnon-expectedutilitypreferencesmayleadtoconditionalchoices
whichareaffectednotjustbytheconditioningevent
E
.Conditionalpreferencesmay
bealsoaffectedbystatesinthecounterfactualevent,
E
c
,aswellasbythethewhole
choicehistory,i.e.,bypriorchoicesandthefeasiblesetofactions.Suchaupdating
ruleisreferredtoasnon-consequantialist,apropertyintroducedbyHammond(1988).
Inthispaper,werequirethatupdatingrulesmaintain
consequentialism
.Thatis,
weconsiderupdatingrulesgeneratingconditionalChoquetpreferenceswhichdepend
only
ontheconditioningevents
E
byleavingtheforgoneuncertaintyaswellasthe
decisionhistoryimmaterialforfuturechoices.
5
Let
A
E
bethealgebraofeventsgener-
atedbyallsub-eventsofthetheconditionalevent
E
.Aconsequentialistupdatingrule
deliversconditionalpreferences
<
E
representablebyChoquetexpectedutilitieswith
respecttotheunconditionalvN-Mutilityfunction
u
andawell-definedconditional
capacity
ν
(
∙|
E
)on
A
E
,thatis,suchthatforall
f,g
∈F
:
ZZf
<
E
f

u

fdν
(
∙|
E
)

u

gdν
(
∙|
E
)
.
(9)
ΩΩdnaν
(
∙|
E
):
A
E

[0;1]
.
(10)
5
Frombehavioralpointofview,consequentialismstatesthatforanyaction
f,g
∈F
andevent
E
∈A
,whenever
f
(
ω
)=
g
(
ω
)forall
ω

E
then
f

E
g
.Themeaningfulnessofmaintaining
consequentialisminthecontextofinterpersonaldecisionproblemswillbefurtherdiscussedand
justifiedinSection6.

8

Therearemanyreasonablerevisionruleswhichguarantythatconditionalcapac-
itysatisfiestherequiredproperty(10).BesidetheBayesrule,proposedbyGilboa
andSchmeidler(1993),therearetwootherprominentrevisionrulesforcapacities;
theMaximum-Likelihoodupdatingrule,introducedbyDempster(1968)andShafer
(1976),theFull-BayesianupdatingrulesuggestedbyJaffray(1992)andWalley(1991),
andthe
h
-BayesianupdatingruleofGilboaandSchmeidler(1993).Itisworthto
mention,however,thattheresultsweobtainareindependentofwhichamongthe
consequentialsitupdatingruleisused.

4InterpersonalDecisionModel

Inthissectionwedescribeanepistemicframeworkinwhichagreementtheoremsare
established.Thereisafinitegroupofagents
I
indexedby
i
=1
,...,N
.Eachagent
i
is
endowedwithapartition
P
i
ofΩ.Thepartition
P
i
represents
i
’sprivateinformation.
Thatis,ifthetruestateis
ω
,then
i
isinformedoftheatom
P
i
(
ω
)of
P
i
towhich
ω
belongs.Intuitively,
P
i
(
ω
)isthesetofallstatesthatagent
i
considerspossibleat
ω
,
otherstatesaredeemedimpossibleat
ω
.Giventhisinformationstructureitissaid
thattheagent
i
knowsanevent
E
at
ω
if
P
i
(
ω
)

E
.Theeventthat
i
knows
E
,
denotedby
K
i
E
,isasetofallstatesinwhich
i
knows
E
,i.e.

K
i
E
=
{
ω

Ω:
P
i
(
ω
)

E
}
.
(11)

Anevent
E
iscommonknowledgeat
ω
ifeveryoneknows
E
at
ω
,everyoneknowsthat
everyoneknows
E
at
ω
,andsoon,adinfinitum.Theeventthateveryoneknowsan
event
E
iscapturedbyanoperator
K
1
:
A→A
definedas:
N\K
1
=
K
1
E
∩∙∙∙∩
K
n
E
=
K
i
E.
(12)
1=iAcommonknowledgeoperator
CK
:
A→A
isdefinedasaninfiniteapplicationof
theoperator
K
1
,i.e.
∞\CKE
=
K
1
E

K
1
K
1
E

K
1
K
1
K
1
K
1
E
∙∙∙
=
K
m
(
E
)
.
(13)
1=mAnevent
E
is
commonlyknown
at
ω
if
ω

CKE
.FollowingAumann(1976)and
Milgrom(1981),theconceptofcommonknowledgecanbeexpressedequivalentlyas

9