“Agreeing to Disagree” Type Results under Ambiguity

Adam Dominiak

Description

Niveau: Supérieur, Doctorat, Bac+8
“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

Sujets

• Lefort
• State university system
• Expected utility hypothesis
• Decision

Informations

• Publié par: mijec
• Nombre de lectures: 11
• Langue: English

Extrait

“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:[e-mail]
†
Universite´Paris-Dauphine,LEDaPlaceduMare´chaldeLattredeTassigny,75775Pariscedex
16,France,E-mail:[e-mail]

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-