Table Of ContentStudies in Economic Theory
Editors
Charalambos D.Aliprantis
Purdue University
Department ofEconomics
West Lafayette,in47907-2076
USA
Nicholas C.Yannelis
University ofIllinois
Department ofEconomics
Champaign,il61820
USA
Titles in the Series
M.A.Khanand N.C.Yannelis(Eds.)
Equilibrium Theory
in Infinite Dimensional Spaces
C.D.Aliprantis,K.C.Border
and W.A.J.Luxemburg(Eds.)
Positive Operators,Riesz Spaces,and Economics
D.G.Saari
Geometry ofVoting
C.D.Aliprantisand K.C.Border
Infinite Dimensional Analysis
J.-P.Aubin
Dynamic Economic Theory
M.Kurz(Ed.)
Endogenous Economic Fluctuations
J.-F.Laslier
Tournament Solutions and Majority Voting
A.Alkan,C.D.Aliprantisand N.C.Yannelis
(Eds.)
Theory and Applications
J.C.Moore
Mathematical Methods for Economic Theory 1
J.C.Moore
Mathematical Methods for Economic Theory 2
M.Majumdar,T.Mitraand K.Nishimura
Optimization and Chaos
K.K.Sieberg
Criminal Dilemmas
M.Florenzanoand C.Le Van
Finite Dimensional Convexity and Optimization
K.Vind
Independence,Additivity,Uncertainty
T.Casonand C.Noussair(Eds.)
Advances in Experimental Markets
F.Aleskerovand B.Monjardet
Utility Maximization.Choice and Preference
N.Schofield
Mathematical Methods in Economics
and Social Choice
C.D.Aliprantis,K.J.Arrow,P.Hammond,
F.Kubler,H.-M.Wuand N.C.Yannelis(Eds.)
Assets,Beliefs,and Equilibria in Economic Dynamics
Dionysius Glycopantis
Nicholas C.Yannelis
Editors
Differential Information
Economies
123
Professor Dionysius Glycopantis
City University
Department ofEconomics
Northhampton Square
London EC1V OHB
Great Britain
Email:[email protected]
Professor Nicholas C.Yannelis
University ofIllinois
Department ofEconomics
330 Commerce West Building
Champaign,IL 61820,USA
Email:[email protected]
Parts ofthe papers ofthis volume have been published in the journal Economic Theory.
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Preface
It is not an exaggeration to say that all economic activity or all contracts among
individuals in a society are made under conditions of uncertainty or incomplete
information. Indeed, the need to introduce uncertainty in the classicalWalrasian
equilibriummodel,wasfeltbyArrowandDebreu(1954)anditisexplicitlymodeled
inChapter7oftheTheoryofValueofDebreu(1959).
InChapter7ofthewell-knowntreatiseofDebreu(1959),itissuggestedthat
once preferences and/or initial endowments are state dependent, i.e. they depend
onafinitenumberofstatesofnatureoftheworld,andagents,whoarecompletely
informed, maximize ex ante expected utility, then all the results on the existence
andoptimalityoftheWalrasianequilibriumcontinuetohold.
Radner (1968) went a step further, by introducing asymmetric or differential
informationintotheArrow–Debreumodel.Inparticular,heassignedtoeachagent,
in addition to his/her random initial endowment and random utility function, a
privateinformationset,whichisameasurablepartitionoftheexogenouslygiven
probabilitymeasurespace(whichdescribesthestatesofnatureoftheworld).
Radner (1968) noticed that if all net trades reflect the private information of
eachagent,(i.e.theyaremeasurablewithrespecttoσ-algebrathathis/herpartition
generates),then,againthestandardexistenceandoptimalityresultsoftheWalrasian
equilibriumconceptcontinuetohold,althoughsomepricesmightbenegative.This
Walrasian expectations equilibrium notion (or Radner equilibrium) is an ex ante
notion,andcapturestheideaofcontractsunderasymmetricinformation.
Arelatedconcept,called“rationalexpectationsequilibrium”(REE),wasalso
studied,e.g.,Kreps(1977),Radner(1979),Allen(1981),amongothers.TheREE
is an interim notion and in this set up agents maximize (interim) expected utility
functions,conditionedontheirownprivateinformationaswellastheinformation
thattheequilibriumpricesgenerate.
Given the fact that there is asymmetric information, a variety of equilibrium
concepts could be put forward.We find the condition of the measurability of an
allocationforREEasthebestwaytoproceed.Otherwisetheagentscalculatetheir
demandswithoutbeingcertainabouttheirinitialendowmentsrealization.
Unlike,theRadnerequilibrium,theREEneednotexistinwellbehavedeconomies,
asshownbyKreps(1977),andmaynotbeParetooptimal,unlessutilityfunctions
are state independent, and also it may not be Bayesian incentive compatible and
maynotbeimplementableasaperfectBayesianequilibrium1ofanextensiveform
game,(Glycopantis–Muir–Yannelis(2003)).
1Definitions of all concepts in this preface are given in the introductory chapter in this
volume.
VI Preface
Theabovetwonotions,i.e.,RadnerequilibriumandREEarenon-cooperative
solutionconcepts,i.e.,agentsmaximizeexanteorinterimutilityfunctionssubject
totheirownbudgetconstraintandtheirown,initialoreventual,privateinforma-
tionconstraint,independentlyofeachother,andwithoutsharingtheirownprivate
information.
In seminal papersWilson (1978) and Myerson (1984) introduced differential
informationinthecoreandtheShapleyvaluerespectively.Notice,thatoncecoop-
erationisallowed,thenabasicproblemwhicharisesishowtheprivateinformation
willbesharedamongtheagentsinacoalition.Forexample,poolingofinformation
maynotbethebestalternativeforanagentwhoiswellinformedandisexpected
tocooperatewithanon-wellinformedagent.Also,usingcommonknowledgein-
formationwithinacoalitionmaynotbesuchagreatideaforawellinformedagent
whocannottakeadvantageofhis/herfineprivateinformation.
To put it differently the issue of incentive compatibility of the information
asymmetriesbecomesarealproblemthatneedstobeaddressed.Afterall,agents
donotwanttobecheated,andatthesametimetheywouldliketowriteefficient
contracts. This of course poses the following question: Is it possible for agents
to write incentive compatible and Pareto optimal contracts? Let us answer this
questionbyconsideringasimpletwoagentsexample.
Example0.1 TherearetwoAgents,1and2,andthreeequallyprobablestatesof
naturedenotedbya,b,candonegoodperstatedenotedbyx.Theutilityfunctions,
initialendowmentsandprivateinformationsetsaregivenasfollows:
√
u1(w,x1)= x1, forw =a,b,c
√
u2(w,x2)= x2, forw =a,b,c
e1(a,b,c)=(10,10,0), F1 ={{a,b},{c}}
e2(a,b,c)=(10,0,10), F2 ={{a,c},{b}}.
Notice that a “fully”, pooled information, Pareto optimal, (i.e. a weak fine core
outcome)is
x1(a,b,c)=(10,5,5)
x2(a,b,c)=(10,5,5). (1)
However,thisoutcomeisnotincentivecompatiblebecauseiftherealizedstate
ofnatureisa,thenAgent1hasanincentivetoreportthatitisstatec,(noticethat
Agent2cannotdistinguishstateafromstatec)andbecomebetteroff.Inparticular,
Agent1willkeepherinitialendowmentintheevent{a,b}whichis10unitsand
receiveanother5unitsfromAgent2,instatec,(i.e.,u1(e1(a)+x1(c)−e1(c))=
u1(15)>u1(x(a))=10)andbecomesbetteroff.ObviouslyAgent2isworseoff.
Similarly,Agent2hasanincentivetoreportbwhenheobserves{a,c}
Thisexampledemonstratesthat“fullorexpostParetooptimality”isnotnec-
essarilycompatiblewithincentivecompatibility.
Most importantly, as it is known from Krasa–Yannelis (1994), the individual
measurabilityofallocationsintheonegoodcasecharacterizesincentivecompati-
bility.Thus,theonlycandidateintheaboveexampleforanincentivecompatible
Preface VII
allocation is the initial endowment which is dominated by the allocation in (1).
Consequently, full Pareto optimal and incentive compatible allocations need not
existastheaboveexampledemonstrates.2
Thus, if we were to produce positive existence results for cooperative solu-
tionconceptswhichguaranteeincentivecompatibilityweshouldnotinsistonfull
Paretooptimalitybutsome“constrainedinformational”Paretooptimality.Indeed,
bydefiningcoresandvaluesindifferentialinformationeconomiesimposingmea-
surabilityconstraints,oneisabletoproveexistenceandincentivecompatibilityof
cooperativesolutionconcepts,(e.g.Yannelis(1991)andKrasa–Yannelis(1994)).
Thefollowingexamplewillillustratetheroleoftheprivateinformationmea-
surabilityofanallocation.
Example0.2 TherearetwoAgents,1and2,twogoodsdenotedbyxandy and
twoequallyprobablestatesdenotedby{a,b}.Theagents’characteristicsare:
√
u1(w,x1,y1)= x1y1, forw =a,b
√
u2(w,x2,y2)= x2y2, forw =a,b
e1(a,b)=((10,0),(10,0)), F1 ={a,b}
e2(a,b)=((10,8),(0,10)), F2 ={{a},{b}}.
ThefeasibleallocationbelowisParetooptimal(interim,expostandexante).
((x1(a),y1(a)),(x1(b),y1(b)))=((5,4),(5,5))
((x2(a),y2(a)),(x2(b),y2(b)))=((15,4),(5,5)). (2)
However,theallocationin(2)aboveisnotincentivecompatiblebecauseifbisthe
realizedstateofnatureAgent2canreportstateaandbecomebetteroff,i.e.,
u2(e2(b)+(x2(a),y2(a))−e2(a))=u2((0,10)+(15,4)−(10,8))
=u2(5,6)>u2(x2(b),y2(b))=u2(5,5).
Noticethattheallocationin(2)isnotF1-measurable(i.e.,measurablewithrespect
to the private information ofAgent 1). Hence, an individually rational, efficient
(interim,exante,expost)withouttheF -measurability(i = 1,2)conditionneed
i
notbeincentivecompatible.
Observe that one can restore the incentive compatibility simply by making
theallocationin(2)aboveF -measurableforeachi,(i = 1,2).Inparticular,the
i
F -measurable allocation below is incentive compatible, and private information
i
(F -measurable)Paretooptimal.
i
(x1(a),y1(a)),(x1(b),y1(b))=((5,5),(5,5))
(x2(a),y2(a)),(x2(b),y2(b))=((15,3),(5,5)).
Theimportanceofthemeasurabilityconditioninrestoringincentivecompati-
bilityandofcourseguaranteeingtheexistenceofanoptimalcontractisobviousin
theaboveexampleandthisapproachwasintroducedbyYannelis(1991).
2This example is taken from Koutsougeras–Yannelis (1993). It should be noted that
Prescott-Townsend (1984) observed that the set of Pareto optimal and incentive com-
patibleallocationsmaynotbeconvexandthereforeneednotexist.SeealsoAllen(2003).
VIII Preface
It is worth pointing out that two important, new features that distinguish the
“partition approach” of modeling differential information (e.g., Radner (1968),
Wilson(1978),amongothers)andthemechanismdesignorHarsanyi-typemodeling
approach(e.g.,Myerson(1984),amongothers).
First, as the examples above indicated, initial endowments are random and
therefore the definition of incentive compatibility is different than the one found
inthemechanismdesignliteraturewhereinitialendowments,iftheyareexplicitly
stated,aretypicallyassumedtobeconstant.
Second,asthereaderwillobserve,inseveralofthepapersinthisvolumethe
incentivecompatibilityiscoalitionalratherthanindividual.Itisnotdifficulttosee
bymeansofexamplesthatcontractsthatareindividualincentivecompatiblemaynot
becoalitionalincentivecompatibleandthereforemaynotbeviable.Webelievethat
thecoalitionalincentivecompatibilityismoreappropriateformultilateralcontracts.
Thefollowingdemonstratesthis.
Example 0.3 Consider a three person differential information economy, with
Agents 1, 2, 3, two goods denoted by x,y, and the three equal probable states
aredenotedbya,b,c.Theagents’utilityfunctions,randominitialendowmentsand
privateinformationsetsareasfollows:
√
u (x ,y )= x y , i=1,2,3,
i i i i i
e1(a,b,c)=((20,0),(20,0),(20,0)), F1 ={a,b,c}
e2(a,b,c)=((0,10),(0,10),(0,5)), F2 ={{a,b},{c}}
e3(a,b,c)=((10,10),(10,10),(20,30)), F2 ={{a},{b},{c}}.
Theallocationbelowisindividualincentivecompatiblebutnotcoalitional.
((x1(a),y1(a)),(x1(b),y1(b)),(x1(c),y1(c)))=((10,5),(10,5),(12.5,7.5))
((x2(a),y2(a)),(x2(b),y2(b)),(x2(c),y2(c)))=((10,5),(10,5),(2.5,2.5))
((x3(a),y3(a)),(x3(b),y3(b)),(x3(c),y3(c)))=((10,10),(10,10),(25,25)).
(3)
NoticethatonlyAgent3cancheatAgents2and3instateaorb,byannouncingb
andarespectively,buthasnoincentivetodoso.Hence,allocation(3)isindividual
incentive compatible. However, Agents 2 and 3 can form a coalition and when
statecoccurstheyreporttoAgent1stateb.Thus,Agent1gets(10,5)insteadof
(12.5,7.5)andAgents2and3distributeamongthemselves2.5unitsofeachgood,
andclearlyarebetteroff.
Thereadermaywonderifthenewcooperativesolutionconceptsinadifferential
informationeconomyprovideanynewinsightsthatcannotbecapturedbytheREE
orWalrasianexpectationsequilibrium.Thefollowingexampledemonstratesthis.
Example 0.4 Consider a three person economy, withAgents 1, 2, 3, one good
denoted by x, and three equally probable states denoted by a,b,c. The agents’
utilityfunction,initialendowments,andprivateinformationsetsareasfollows:
√
u = x , i=1,2,3
i i
e1(a,b,c)=(5,5,0)), F1 ={{a,b},{c}}
Preface IX
e2(a,b,c)=(5,0,5), F2 ={{a,c},{b}}
e3(a,b,c)=(0,0,0), F2 ={{a},{b},{c}}.
TheallocationbelowisF -measurable(i=1,2,3)andcannotbeimprovedupon
i
byanyF -measurable,andfeasibleredistributionsoftheinitialendowmentsofany
i
coalition(thisistheprivatecore,Yannelis(1991)):
x1(a,b,c)=(4,4,1)
x2(a,b,c)=(4,1,4)
x3(a,b,c)=(2,0,0). (4)
Noticethattheallocationin(4)isincentivecompatibleinthesensethatAgent
3 is the only one who can cheatAgents 1 and 2 if the realized state of nature is
a. However,Agent 3 has no incentive to misreport state a since this is the only
state she gets positive consumption, and in any case one of Agents 1 or 2 will
be able to tell the lie. Neither is it possible, as it can be easily seen, to form a
coalition,profitabletobothmembers,andmisreportthestatetheyhaveobserved.
Finally,noticethatifAgent3had“bad”information,i.e.,F(cid:1) ={a,b,c},then,ina
3
privatecoreallocation,shegetszeroconsumptionineachstate.Thus,advantageous
informationistakenintoaccount.
Contrary to the private core allocation in (4) above, neither the REE nor the
Walrasianexpectationsequilibrium(orRadnerequilibrium)cancapturethisphe-
nomenon.BothconceptsignoreAgent3nomatterhowfineherprivateinformation
is, contrary to the private core which rewarded Agent 3 who used her superior
informationtomakeaParetoimprovementforthewholeeconomy.
TheaboveexamplesuggeststhattheREEmaynotbetheappropriateconcept
to capture contracts under asymmetric information and a new price expectations
equilibriummightbeneeded.Indeed,workinthisdirectionisrecentlybeingdone
byTourky–Yannelis(2003)whoareintroducingapersonalizedpriceexpectations
equilibriumnotion.ThisnotionexistsinsituationsthattheREEandRadnerequi-
libriumfailtoexistanditcancharacterizetheprivatecore.
Beforeweclose,wewouldliketoremarkthatdespitethefactthatthisbookhas
discussed some successful alternative equilibrium concepts, other than the REE
and Radner equilibrium, the issue of modeling continuum economies (perfectly
competitiveeconomiesinthesenseofAumann(1964))isstillopen.Theproblem
seemstocenterindefiningpreciselytheideaofeachagent’sprivateinformation
asbeingnegligible.
Acknowledgements.WearedeeplyindeptedtoourcoauthorAllanMuirforhisthougthful
commentsandsuggestions.WealsowishtothankRokoAliprantiswhoencouragedustoput
thisbookintheseriesStudiesinEconomicTheory.WernerMu¨llertheEditorofEconomics
inSpringer(nowboughtbyKluwer)andhisstaffwereextremelyhelpfulandpatientduring
thepreparationofthisvolume,andwesincerelyappreciatetheirkindassistance.Needlessto
saywithoutthecontributionofalltheauthorsthisvolumewouldneverhavebeenproduced.
X Preface
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