耶鲁大学经济学系列讲义 Game Theory and Information Economics Dirk Bergemann （Department of Economics，Yale University） 博弈论与信息经济学 曹乾 整理 （东南大学 caoqianseu@） 东大青椒教育工作室制作
DirkBergemannDepartmentofEconomicsYaleUniversityGameTheoryandInformationEconomicsJanuary2006Springer-VerlagBerlinHeidelbergNewYorkLondonParisTokyoHongKongBarcelonaBudapest
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::-abriefhistory..........................................................................................:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::..................................................................................................................................................................................................................................:.............................:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::.....................................................................................................................:...........................................................................................................................................................................................................:MonopolisticCompetitionandtheDixit/Stiglitzmodel.....,.................................................................................................................................................................................................:::::::::::::::::::::::::::::::::::::::::::::::::::::::::(Tree)FormtoNormalForm.........................................................................................................:,;.............................................................:....................................................................................................................................:..............................................\NashProgram":AlternatingO ersandtheNashBargainingSolution........33
:::::::::::::::::::::::::::::::::::::::::::::::: nitelyRepeatedGames............................................................................................................................::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::.....................................................................'sLemonModel:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::............................................................................................................................................'sPriceSignal'sQuality..................................................................................................................................................................................................::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::.....................................................................................................................................................................................................................................................'sinformedprincipalproblem................................-MirrleesSingleCrossingCondition.........................................................................................................................................................................................................................................................................................................................................................::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::............................................................................................................................................................................................................................................................. niteoutcomesandactions...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................77
........................................................................:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::MechanismDesignwithOneAgent::::::::::::::::::::::::::::::..................................................................................................:Asymmetricinformation...................................................................................................................................................................................................................................................................................................................................................................................................................................................................:::::::::::::::::::::::::::::::::::::...............................................................................................................................................................................................................................................................:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ciency:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::............................................................................................................................................:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::......................................................................................................................109
,situationsinwhichanentiregroupofpeopleisa (auction,bargaining)involvegametheory,laborand -agentdecisionproblemswithinanorganization,manypersonmaycompeteforapromotion, sandtradepolicies,inmacroeconomics, , erenttheoriestodealwithnon-competitiveeconomicenvironments,itisusefultocomeupwithanencompassingtheoryofstrategicinteraction(gametheory)andthenseehowvariousnon-competitiveeconomicenvironments erenceifthegroupinteractsmorethanonce?
)(1913)mixedstrategies,:coreconceptualdevelopmenta)(1928)existenceinofzero-sumgamesb) rstphaseofapplicationsofgametheory(andinformationeconomics)inapplied :realintegrationofgametheoryinsightsinempiricalworkandinstitutionaldesign Formoreonthehistoryofgametheory,seeAumann'sentryon\GameTheory"(singleagent)(fewagents)(manyagents)
PartIStaticGamesofCompleteInformation
..,Oligopoly,PerfectCompetitionTodescribeastrategicsituationweneedtodescribetheplayers,therules,theoutcomes,andthepayo :(Duopoly).Two rms;constantmarginalcost:$1;no xedcost;totaldemandcurve:Q=13 PExample2:(Partnership).Twopartners;costofe ort:4;outputperpartnermakinge ort:=:(SealedBidSecondPriceAuction).Twobidders;i'.(Iftheybidthesame,eachhasa12chanceofgettingtheprizeandpayingthe(equal)bid).\players",thatis,eachplayer's\utility" rstwhatisagame?Itisasetofplayers:I=f1;2;:::;Ig;()asetofpossiblestrategiesforeachplayer8i;si2Si;() -o functionsforeachplayeri:ui:S1 S2 SI!R.()Apro leofpurestrategiesfortheplayersisgivenbyIs=(s1;:::;sI)2 Sii=1oralternativelybyseparatingthestrategyofplayerifromallotherplayers,denotedby i:
=(si;s i)2(Si;S i):Astrategypro lewecandeducethepayo (NormalFormRepresentation).Thenormalform Nrepresentagameas: N=fI;fSig;fuii( )gg:iExample1:(Duopoly).I=2;S1=S2=R+;u1(s)=u1(s1;s2)=s1(maxf0;13 s1 s2g 1)u2(s)=u2(s1;s2)=s2(maxf0;13 s1 s2g 1)Example2:(Partnership).Bimatrixrepresentation:E ortNoE ortE ort2,2-1,3NoE ort3,-10,0Thisisanexampleofstandardnormalformrepresentation:I=2;S1=S2=fE ort,NoE ortg;u1(E ort,E ort)=2;u2(NoE ort,E ort)= 1,etc...Example3:(SealedBidSecondPriceAuction).I=2;S1=S2=R+;8<v1 s2,ifs1>s2u11(s1;s2)=:2(v1 s2),ifs1=s20,ifs1<s28<v2 s1,ifs2>s1u12(s1;s2)=:2(v2 s1),ifs2=s10,ifs2< 'beliefabouteachother'sactionsarenotassumedtobecorrect,,sothateveryplayerisrational,everyplayerthinksthateveryplayerisrational,everyplayerthinksthateveryplayerthinksthateveryplayerisrational,possiblyadin ,theNashequilibriumconceptrequiresthateachplayer'schoicetobeoptimalgivenhisbeliefabouttheotherplayer'sbehavior,:Example2:(Partnership).
ortNoE ortE ort2,2-1,3NoE ort3,-10,0Whateveraction2chooses,1'sbestactionistochoosenoe \noe ort",eachplayerhasadominantstrategy(noe ort);butwhenplayerschoosetheirdominantstrategies,theoutcomesareine ,ifbothexerte ort,bothplayersarebettero .Thuseveninthesesimplesttypeofexamples,rationalityfailstoimplye nitionsofdomination. SomeNotation:Atypicalstrategypro leiss=(s1;:::;si 1;si;si+1;:::;sI)Writes i=(s1;:::;si 1;si+1;:::;sI) iforthesetofsuchpro les, i=S1 ::: Si 1 Si+1 ::: (s0i;s i)forastrategypro (si;s i)>ui(s0i;s i),foralls i2S iDe = (si;s i) ui(s0i;s i),foralls i2S i anduisi;s0 i>uis0i;s0 i,forsomes0 i2S iDe =-byde nition-ifsistrictlydominatess0i,,':(Partnership).\Noe ort"isstrictlydominant,so\Noe ort":(SealedBidSecondPriceAuction).=<v1 s2,ifs1>s2u11(s1;s2)=:2(v1 s2),ifs1=s20,ifs1<s2Ifs2 v1,u1(v1;s2)=0 u1(s1;s2)foralls12R+.Sov1givesatleastasmuchasanyotherstrategy,ifs2 <v1,u1(v1;s2)=v1 s2,so89<v1 s2,ifs1<s2=u11(v1;s2) u1(s1;s2)=:2(v1 s2),ifs1=s2; 0foralls12R+0,ifs1>s2Sincethisisnon-negativeforalls1,v1doesatleastaswellasanyotherstrategyifs2<=v1,thereexistss2suchthatu1(v1;s2)>u1(s1;s2).Consider rstthecasewheres1>v1;nowifv1<s2<s1,u1(s1;s2)=v1 s2<0=u1(v1;s2).Ontheotherhand,supposes1<v1;nowifs1<s2<v1,thenu1(s1;s2)=0<v1 s2=u1(v1;s2).,:ifs2 v1,anystrategys1withs1 s2givesthesameoptimalpayo of0.
[lecture2:]De ,:(Duopoly).Supposethats1+s2<(s1;s2)=s1(12 s1 s2)(Check!).Nowdu1ds=12 s12 ,wehave12 s2 2s1=0,.,s 1=6 ,ifs22[0;12),u16 12s2;s2>u1(s1;s2)foralls16=6 :foreveryactionofplayer2,player1hasadi :Example4:LeftMiddleRightUp1,01,20,1Down0,30,12,0\Middle"strictlydominates\Right".But\Middle"doesnotstrictlydominate\Left"and\Left"doesnotstrictlydominate\Middle",sothe\columnplayer"\rowplayer".Butsince\Right"isstrictlydominatedbysomeotherstrategy(\Middle"),:LeftMiddleUp1,01,2Down0,30,1But\Down"isstrictlydominatedinthisgame,so...LeftMiddleUp1,01,2\Left"isstrictlydominatedinthisgame,so...MiddleUp1,2Thisprocessisknownasiterateddeletionofstrictlydominatedstrategies.(Up,Middle)istheuniquestrategypro (IteratedStrictDominance).Theprocessofiterateddeletionofstrictlydominatedstrategiesproceedsasfollows:SetS0i= neSnirecursivelyby Sni=si2Sn 1 i@s0i2Sn 1i,(s0i;s i)>ui(si;s i),8s i2Sn 1 i;SetS1\1i=Sni:n=0ThesetS1iisthenthesetofpurestrategiesthatsurviveiterateddeletionofstrictlydominatedstrategies.
neaweakernotionofiteratedelimination,: 2 (3)(1)2
2 12;3 3;20;1() (2)10;01;34;2whichyieldsauniquepredictionofthegame,,howeverthisrunsintoadditionaltechnicalproblemsasorderandspeedintheremovalofstrategiesmattersfortheshapeoftheresidualgameasthefollowingexampleshows: 2 2 13;22;2() 11;10;0
10;01;1Ifweeliminate rst 1,then 2isweaklydominated,similarlyifweeliminate
1,then 2isweaklydominated,butifweeliminateboth,thenneither 2nor ;10;0M()1;12;1B0;02;1 (1984):\RationalizableStrategicBehaviorandtheProblemofPerfection,"Econometrica52,:(CoordinationFailure).Don'tInvestInvestInvest5,5-1,0Don't0,-10,:(a)Ifeachplayerhasastrictlydominantstrategy,therenoneedforanyfurtheranalysis:rationalplayersmustchoosethem.(Arguably)thesameifthereexistdominantstrategiesforeachplayer.(b)Ifauniquestrategypro lesurvivesiterateddeletionofstrictlydominatedstrategies,.(c)Thereforewewillneedtoassumemoreinordertomakeanypredictionatallin,say,:weareassumingmorethancommonknowledgeofrationalityintheanalysisthatfollows.
leofactionswereeachplayer':De nition:Strategypro les isaNashequilibriumif,foralli=1;:::;Iandallsi2Si, uis i;s i uisi;s iExample5:(CoordinationFailure).(Invest,Invest)and(Don'tInvest,Don'tInvest):Ifeachs iisadominantstrategy,thens :Ifs istheuniquestrategypro lesurvivingiterateddeletionofstrictlydominatedstrategies,thens , nition:Write i(s i)forplayeri'sbestresponse(s)tos : i(s i) argmaxui(si;s i)si2Si fsi2Si:ui(si;s i) ui(s0i;s i)foralls0i2SigLet (s) fs02S:s0i2 i(s i) isaNashequilibriumifandonlyifs 2 (s ).Example6:(Contributiontoapublicgood).Twoindividuals,individualihasincomewandchoosesgi2[0;w],'sprivateconsumptionw (g1;g2)= ln(w gi)+(1 )ln(g1+g2),where 2(0;1).Now(assuminginteriorsolution)wehaveatmaximum:du1= dg1w +1 =0g1g1+.(1 )w (1 )g1= g1+ g2so 1(g2)=(1 )w g2.(Technically, 1(g2)=f(1 )w g2g).Similarly, 2(g1)=(1 )w (see gure1),weseeg1=g2=g isuniqueNashequilibrium,whereg =(1 )w g , =(1 )w1+ .Whatisthee cientsymmetriclevelofcontribution?Choosegtomaximize(1 )ln(2g)+ ln(w g) tge w ge=0,(1 )(w ge)= =(1 )w>(1 )w1+ .
:Example3:(SecondPriceSealedBidAuction).8<fs1:s1>s2g,ifs2<v1 1(s2)=:R+,ifs2=v1fs1:s1<s2g,ifs2>v1(v1;v2)isone(outofmany)Nashequilibria(see gure2)[lecture3]:Example7:(MatchingPennies).HeadsTailsHeads1,-1-1,1Tails-1,11,-1NoNashequilibrium,aswede (deterministic). Expandthestrategyspace,andallowplayerstochoosep,;,playeri'smixedstrategysetisthesetofprobabilitydistributionsoverSi, i2 i,then i(si),wemustde neutilityovermixedstrategypro les.(Abusingnotationalittle),letX01ui(si; i)=ui(si;s i)@Y j(sj)As i2S ij6=iXX0Y1Iui( i; i)=ui(si; i) i(si)=ui(si;s i)@ Aj(sj)si2Sis2Sj=1Noticethat(withthisinterpretation),,payo nedexactlyasbefore,. isaNashequilibriumifandonlyifforalliand i2 i, ui i; i ui i; iIntherestofthiscourse,\strategy"meansmixedstrategyandNashequilibriummeans\mixedstrategyNashequilibrium".Ifthereisnorandomization,.
:19DominatedStrategiesRevisited:.Considerthefollowinggame:LeftRightUp4,0-2,0Middle0,00,0Down-2,04,0Purestrategy\Middle"\Middle":Theorem(Nash1950)Every ne : ! :sXk 0i 2ik=( 0i(si) i(si))si2SiLetvi( 0i; )=ui( 0i; 2 i) ck 0i ik(wherec>0) i( )=argmaxvi( 0i; ) 0i2 i ( )=If i( )gi=1Interpretation: isa\betterresponse". isnon-empty, 0i( rsttermislinear,andthusconcave,secondtermisnegativequadratic,andthusstrictlyconcave).Thus iisuniquelyde nedandcontinuousin .Thus isacontinuousfunctionon . = ( ),t hen isaNa i2 isuchthat =ui i; i ui i; i>:vi(" i+(1 ") i; ) vi( i; )=2" c"2k i ikwhichisstrictlypositivefor"su 'sFixedPointTheorem:SupposeXisanon-empty,compact,convexsubsetofRNandf:X! xedpoint,.,thereexistsx2Xwithx=f(x).NowNashequilibriumexistsbypoints1through3above. ThisprooffollowsGeanakoplos(1996):\NashandWalrasEquilibriumViaBrouwer,"CowlesFoundationDiscussionPaper#:Ifpayo sarecontinuous, s, sarisenaturallyineconomicsettings,-settinggames(tobediscussedfurthersinthenextsection).Wereturntothisissueattheendofthenextsection.
>0(withvaryingmarginalcost,entryas-sumptionsbecomecrucial;wewillreturntothislater).Demandx(p);assumedi erentiable,non-increasing,strictlydecreasingifx(p)>0,x(p)=0forallp (q)=minfp:q=x(p)[p c].Musthavepc=c,qcimplicitlyde nedbyp(qc)= ts:x(p)[p c]Chooseqtomaximizepro ts:q[p(q) c]FirstOrderCondition:MarginalRevenue=p(q)+qp0(q)=c=MarginalCostEquivalently,p c= qp0(q)= 1,pp where (qm)+qmp0(qm)=candpm=p(qm)Observethatpm>pcandqm< rmsQuantityCompetition(\CournotCompetition").Payo functions:2013I i(q1;:::::;qI)=q4ip@XqAj c5j=1Firstordercondition:0101Id i=p@XIqA0@Xj+qipqAj c=0dqij=1j=1Inasymmetricequilibrium,qi=qforalli(bycontinuity,asymmetricsolutiontothe rstorderconditionwillexist).SoifwewriteqI=Iqfortotalquantityproducedinoligopoly,wehaveqIsolving pqI+1qIp0qI==q1andqc=:howdoesqIvaryasIincreases....
(\BertrandCompetition").Payo Functions:( 1pp c,ifp pi=pi=xpgi(p1;:::::;pI)=#fi:0,ifpi>pwherep=minpiiApurestrategypro leisaNashequilibrium(foranyI 2)ifandonlyifp=candatleasttwo ,anysuchstrategypro leisanequilibrium(noincentivetodeviate).Let'sseewhatgoeswronginallotherpossiblecases.(i)p< rmsettingpriceequaltopismakinglossesandhasanincentivetoincreaseprice.(ii)p1=candpi>cforalli6=+"(forsome"su cientlysmall).(iii)p>candletnbethenumberof rmssettingpi==I,any rmnotsettingp i=phasanincentivetodeviatetopi=p "(increasingpro tsfrom0top " cxp ").Ifn=I,any rmhas anincentivetodeviatetopi=p "(increasingpro tsfrom1Ip cxptop " cxp ").Asanexercise, rms,butdi erentcosts:c1<<c2, rm2losesmoney;ifp>c1, rm1increasespro tsbychargingp ".Nowsupposethat rmschoosedi rmchoosingthelowerpricecanincreasepro tsbyraisingpricesby".Letpandpbethelowestpricechosenbythetwo =p=,ifp<c2, rm2losesmoney;ifp>c1, rm1increasespro tsbychargingp ".Nowsupposethatp6=,the rmwiththelowerlowestpricecanincreasepro tsbyraisingpricesby".12 SeeReny(1999):\OntheExistenceofPureandMixedStrategyNashEquilibriainDiscontinuousGames,"Econometrica67,1029-1056.[lecture4]:MonopolisticCompetitionandtheDixit/ :0u(m;x1;::::;xJ)=G@X1Jf(xj)A+mj=1whereG( )andf( ),weget rstorderconditions
@X1f(xj)Af0(xj)=pjj=1Thisgivesademandfunctionxj(p1;:::::;pJ).Each rmmaximizes(pj c)xj(p1;:::::;pJ)Each rmactslikeamonopolist,settingxxjj+dpj=dxjcdpjdpjpj c= xjpdxjjpjdpjLifebecomesevensimplerifthereareacontinuumofproducts: Z!u(m;x)=Gf(xj)dj+mj2[0;1]whereG( )andf( ),weget rstorderconditions Z!G0f(xj)djf0(xj)=pjj2[0;1]R Setx=j2[0;1]f(xj)(p0 1p jj;x) [f] rmmaximizes(pj c)xj(pj;x)Each rmactslikeamonopolist,settingxj+dxjpj=dxjcdpjdpjBysymmetry,wearelookingforalevelofoutputxandpricepsuchthatG0(x)f0(x)=pdxx+jp=dxjcdpjdpj[lecture9],-shapedaveragecostcurve: xedcostofentryK>0;variablecostfunctionc(q),withc(0)=0,c0 ( )>0andc00(q)> x(p)(where parameterizesthesizeofthemarket);inversedemandpq .Assumec( ),x( )andthusp( )aretwicecontinuouslydi =minK+c(q)qqwithqthecostminimizing = rstorderconditionoftheaveragecostminimizationis K+c(q)q2+c0(q)=
,(J ;p ;q )(i)SupplyEqualsDemand: x(p )=J q (ii)OptimalOutputChoice:q =argmaxp q c(q).,p =c0(q )(iii)ZeroPro t(FreeEntry)Condition:p q c(q ) K=0Note:thereisan\integerproblem."Butifthereisasolution,wemusthaveq =q,p =candJ = x(c) <c, >c, rmsmustmakestrictlypositivepro ts,violatingthezeropro tcondition. x(c)Ignoringtheintegerproblem,thereis\e ciententry,".,(J ;p;q )=q;c;qmaximizesZJq tpdt JK Jc(q) 0sincetotaloutputisproducedatminimumcost(c) :each rmsenters,:K>0. Stage2:Cournotcompetitionwithinversedemandfunctionpq .SupposethatthereisauniquesymmetricNashequilibriumoftheCournotgamewitheach rmpro-ducingoutputqJandeach rmearningoperatingpro ts J(.,ignoringsunkentrycost).Alsoassumethat (J ;p ;q )ischaracterizedbythreeproperties:(i)SupplyEqualsDemand: x(p )=J q (ii)NashOutputChoice: q =(J argm 1)q + qaxpq c(q) qJ q .,p+1J q p0q =c0(q ) (iii)EntryCondition: J K, J+1 KClearly,thereisine rmsenteringinequilibriumise cientnumberof (1)each rm'soutputisdecreasinginthenumberofentrants(qJisdecreasinginJ),\businessstealing";(2)totaloutputisincreasinginthenumberofentrants(JqJisincreasinginJ).ThenJ Jeff rmsenter.
:as !1,p !c.[lecture10](J ;p ;q ).Step1:let (Q)bea rm'sbestresponsewhentotaloutputofother rmsisQ;forsu cientlylarge , ( ) cientlyhighQ,,wehaveFOC: Q+ (Q )p+1Q+ p0 (Q ) (Q)=c0( (Q))Totallydi ,weobtain8> > 9>1 (Q )< p0Q+ 1+ 0 (Q)>> >=+1>> 2p00Q+ (Q) 1+ 0 (Q) (Q)>=c00( (Q)) 0 (Q)>: +1Q+ > (Q ) p0 0> (Q);Thus 0 (Q)= z(Q) Q+ (Q )z(Q)+1 p0 c00( (Q))where Q+ (Q )+1Q+ (Q)z(Q)=1p0 2p00 (Q)Wewillshowthatz(Q)<+ (Q) :Musthaveq J x(c) ,a ,:Nowobservethat x(c) qjp cj p x(c)p q =px(c) p(x(c)) !0,as !1
PartIIDynamicGamesofCompleteInformation
:Example1:(Chess).Example2:(Deterrence).An\entrant"isdecidingtoenteramarketcurrentlymonopolizedbyan\in-cumbent".First,theentrantdecideswhetherto\enter"or\stayout".Ifhestaysout,hispro tsare0,buttheincumbentgetspro ,theincumbentmustdecidewhetherto\ ght"or\accommodate".Iftheincumbent ghts,theentrantmakesnegativepro ts 1whiletheincumbentmakespro ,theentrantmakespro :(DividingaPrize).FirstPlayersplits$2intotwo(integer)piles(leftandright). :(Sequential,.\Stackelberg,"Duopoly).Asinourpreviousduopolyexample,butnowassumethat rm1choosesoutput rst,anditisobservedby (Tree)FormtoNormalFormNowastrategyforaplayerspeci ,:(Deterrence).\strategyset"equals\actionset".Thenormalform,then,isasfollows:FightAccommodateEnter-1,11,2StayOut0,30,3Example3:(DividingaPrize).Nowtheactionsforplayer1areA1=f(2;0);(1;1);(0;2)=fL;=:A1!,.,LLRforthestrategy\taketheleftpileif1chose(2;0),taketheleftpileif1chose(1;1),andtaketherightpileif2chose(0;2)".Nownormalformrepresentationis:LLLLLRLRLLRRRLLRLRRRLRRR(2;0)0;20;20;20;22;02;02;02;0(1;1)1;11;11;11;11;11;11;11;1(0;2)2;00;22;00;22;00;22;00;2Example4:(Sequential,.\Stackelberg,"Duopoly).Nowastrategyfor rm1isachoiceofoutputq12R+.Astrategyfor rm2isafunction,s2:R+!R+.Thuss2(q1)is rm2'schoiceofoutputif :u1(q1;s2)=(maxf0;13 q1 s2(q1)g 1)q1u2(q1;s2)=(maxf0;13 q1 s2(q1)g 1)s2(q1)
::(DividingaPrize).Player2'sBestResponses:1'sstrategy2'sbestresponse(2;0)fLLL;LLR;LRL;LRRg(1;1)fLLL;LLR;LRL;LRR;RLL;RLR;RRL;RRRg(2;0)fLLR;LRR;RLR;RRRgPlayer1'sBestResponses:2'sstrategy1'sbestresponseLLL(0;2)LLR(1;1)LRL(0;2)LRR(1;1)RLL(2;0),(0;2)RLR(2;0)RRL(2;0),(0;2)RRR(2;0)Nashequilibria:f(1;1);LLRgandf(1;1);:,;:(Deterrence).TwoNashequilibria:(Enter,Accommodate)and(StayOut,Fight). nition:Abackwardinductionequilibriumofaperfectinformationgameisastrategypro lewhereeachplayer'sstrategyisoptimalateverynodegiventhatheexpectsotherstofollowequilibriumstrategiesinthefuture.(Enter,Accommodate):(DividingaPrize).BackwardInductionequilibriaareNashequilibria,f(1;1);LLRgandf(1;1);:Inall nitehorizonperfectinformationgames:(1)thereexistsapurestrategybackwardsinductionequilibrium;(2)everybackwardinductionequilibriumisaNashequilibrium;(3)ifthegamehasnoties,thereisauniquebackwardinductionequilibrium;(4)ifthegameis\twoplayerconstantsum",allbackwardinductionequilibriagivethesamepayo :(Hex).See gure5.[lecture5]Example4:(SequentialDuopoly).Letus rst 'spro tsare: 2(q1;q2)=(maxf0;13 q1 q2)g 1)q2
:29Thisismaximize dbysettingq2=max 0;6 rm2'sstrategyiss2(q1)=m ax0;6 rm1'spro tsanticipatingthat 12, nn 1(q1)=max0;13 q1 max0;6 qoo 1 1q1 2=12 q1 6 q 1q1 2=6 q 1q12Ifq1>12, 1(q1)< q1=0,=, : Backwardinductionequilibriumstrategypro leis:q 1=6,s 2(q1)=max 0;6 q 12. Backwardinductionoutcomeis:q 1=6,q 2=::(FreeRiding).See :E ortNoE ortE ort2,2-1,3NoE ort3,-10,0Ingeneral,theextensiveformmustspecify: asetofplayers ateachpointinthegame:whichplayermoves,whatmoveshecanmake,whatheknowswhenhemoves payo ne\nodes"(whereactionsarechosen),\informationsets"(thesetofnodesthattheplayerchoosingcannotdistinguish),\histories"(completedescriptionofactionstodate)and\strategies"(functionsfromhistoriestoactions).:Example6:See gure7.
:ThisgamehasmanyNashequilibria,including(A;B;L)and(D;T;R).:LRT1,01,1B0,10,0ThishasuniqueNashequilibrium(T;R).AnticipatingNashplay, nition:A\subgame"isasegmentofagame(formally,asubsetofnodes) nition:StrategyPro (1and2): player1makesaproposalx2[0;1](.,heproposesthesplitxforhimself,1 xfortheplayer2). player2decideseithertoacceptorrejectthiso ,,,therolesarereversed: player2makesaproposalx2[0;1](.,heproposesthesplitxforhimself,1 xforplayer1). player1decideseithertoacceptorrejectthiso ,, 2(0;1)'sutilityfromanagreementunderwhichhegetsxinperiodt 1tis ers,(x1;:::;xtt)2[0;1] HtWriteH0=;forthe(empty)=[HtandHo=[,f1:He![0;1],andanacceptancerule,g1:Ho [0;1]!fA;,f2:Ho![0;1],andanacceptancerule,g2:He [0;1]!fA;,butbecauseitisofin nitelength,,theoddplayermightalwaysdemandeverythingandtheevenplayermightalwaysgiveit:f1(h)= 1,forallh2HeA,ig1(h;x)=fx=0Rallh2Hoandx2[0;1],ifx>,for0f2(h)=0,forallh2Hog2(h;x)=A,forallh2Heandx2[0;1]TheseconstituteaNashequilibrium(check!).Butsubgameperfectionhasalotofbiteinthisgame.
::ThefollowingstrategiesaretheuniquesubgameperfectNashequilibriumstrategies:f1(h)=1,forallh2He 1+ A,ifg1(h;x)=x 11+ Rorallh2Hoandx2[0;1],ifx>1,f1+ f2(h)=11+,forallh2Ho g2(h;x)=A,ifx 11+ Rh2Heandx2[0;1],ifx>1,forall1+ fromthatpointonis11+ .Alowerproposalwouldbeaccepted,givingalowerpayo .Ahigherproposalwouldberejected,givingpayo atmos 2t1+ (thediscountedvalueofwhattheproposerwouldbeo eredinthenextperiod). of 1+ (thediscountedvalueofbeingtheproposerinthenextperiod).Thereforeisrationaltoacceptexactlyo ersthatgivethisamount.[lecture6](formally,supremumandin mum)payo receivedbytheoddplayerinanySPNEofthein nitegame;letyandybethehighestandlowest(formally,supremumandin mum)payo ,xandxarealsothehighestandlowestpayo (discountedfromthatpointon)receivedbyeitherplayerafteranyhistoryinanySPNEofthecontinuation;similarly,.(i)y xand(ii)y (i),observethataplayerwouldalwaysacceptanyo erthatgavestrictlymorethan x(themosthecouldconceivablygetbyrejecting);sotheproposerwouldnevero erstrictlymorethat (ii),rejectionguaranteesaplayeratleast 1 ,part(i):aplayerwouldalwaysacceptanyo erthatgavestrictlymorethan x(themosthecouldconceivablygetbyrejecting);sotheproposerisguaranteed1 max(1 x; y) max 1 2 x; eitherbygettingcurrentproposalaccepted(givingatmost1 x)orbygettingitrejected(givingatmost y).Thesecondinequalityfollowsfromclaim1,part(i). 1 x 1 x,thenclaim3)x 2 x)x 2 x)x=0)(byclaim2)x=1, x<1 x, 1 x,byclaim4 1 (1 x),byclaim2 1 +2 xThusx 1 1 2=11+ .Byclaim2,x 1 x 1 1+ =11+ .Sox=x=11+ .Combinedwithclaim1,wehavey=y= 1+ .Itremainsonlytoverifythatthestrategiesdescribedabovearetheonlyonesgivingtheseuniqueequilibriumpayo s(check!).::F <2;assumeFisclosed, (x1;x2) (v1;v2).Utilityassignment
:(F;v)whereF\fx:x >:(DividingaDollarbetweentworiskneutralagents). F=(x1;x2)2<2+:x1+x2 1;v=(0;0):See :(DividingaDollarbetweenariskneutralandariskaverseagent).89<u1=x1,u2=px2,=F=:(u1;u2)2<2+:forsome(x1;x2)2<2+with;v=(0;0):x1+x2 ;1See selectsanoutcome, (F;v),foreverybargainingproblem(F;v).Bargainingproblem(F;v)issymmetricif(a)v1=v2and(b)F=f(x2;x1):(x1;x2):: (F;v) ciency:x (F;v)andx2F=)x= (F;v).:(F;v)symmetric) 1(F;v)= 2(F;v).:IfGisaclosed,convexsubsetofFand (F;v)2G,then (F;v)= (G;v).:Fixany(F;v)and 1; 22<++and
1;
22<;letw=( 1v1+
1; 2v2+
2)andG=f( 1x1+
1; 2x2+
2):(x1;x2) (G;v)=( 1 1(F;v)+
1; 2 2(F;v)+
2)Theorem(Nash):If satis esaxioms1through5,then (F;v)=argmax(x1 v1)(x2 v2)x2F;x v Uniquenessfollowsfromtransformingmaximandtoln(x1 v1)+ln(x2 v2).(F;v):x (x 1;x 2)=argmax(x1 v1)(x2 v2)x2F;x vBy(F;v)essential,x 1>v1andx 2> i=1x i vand
ii= vix i L(y)=( 1y2 v21y1+
1; 2y2+
2)=y1 v;x 1 v1x 2 v2andG=fL(y):(x )=(1;1)andL(v)=(0;0).Now89<z=L(y)an dgmax =arz1z2=y2argmaxx1 v1x2 v2z2G;z 0:z:x2Fx 1 v1x 2 v2;;x v=fz:z=L(x )g=(1;1)
: ThusG E=z2<2:z1+z2 ciencyandSymmetry: (E;(0;0))=(1;1).: (G;(0;0))= (E;(0;0))=(1;1).:L( (F;v ))= (G;(0;0))=(1;1);so (F;v)=x .Inexample1,theNashsolutionis112;,thee hciencyfriontieristhesetofpointswhereu1=1 (u22);thustheNashproduct(asafunctionofu2)isu21 (u22).Thisismaximizedwhereu2=p1andthusu31= lecorrespondstoplayer1receivingcashof13,\NashProgram":AlternatingO ersandtheNashBargainingSolution SeeMyerson(1991):GameTheory:AnalysisofCon
ict,chapter8, ersgamewheretheproposerpicksapointx=(x1;x2)2F, F=(x1;x2)2<2+:x1+x2 1;v=(0;0):wegetessentiallytheanalysiswehadbefore(inprinciple,playerscouldproposeanine cientpoint,buttheywouldnotdosoinequilibrium).Notethatas !1,theuniqueequilibriumgaveeachplayer12,.,,essentiallythesameargumentgoesthroughwhenthealternativeo ersgameisappliedtoanybargainingproblem;.,thereisauniquesubgameperfectequilibriumwhereeachplayer'sstrategyisstationary(.,isindependentofpasthistory).Onecanshowthatas !1,thisuniquesolutiongiveseachplayertheNashbargainingsolution.[lecture7]
:Example8:(TwiceRepeatedFreeRidingGame).See vethings:whattodoin rstperiodandwhattodoinsecondperiodaftereachofthe4histories(EE;EN;NE;NN).Thuswecouldwrite\EEENN"forplayer2'sstrategy\playEinthe rstperiod;playEinthesecondperiodifplayer1playsEinthe rstperiod;playNinthesecondperiodotherwise"ThisgamehasauniquesubgameperfectNashequilibrium:(NNNNN;NNNNN),.,eachplayermakesnoe :IfastagegamehasauniqueNashequilibrium,thentheuniquesubgameperfectNashequilibriumofthe ,:,15,01,0M0,54,40,0R0,10,03,:(L;L)and(R;R).Butconsiderthefollowingstrategyforeitherplayer: PlayMinthe (M;M)isplayedinthe rstperiod, rstperiodhistory,,theabovestrategyislet(M;f),wheref:fL;M;R2g!fL;M;Rgand f(h)=R,ifh=(M;M)L,ifh6=(M;M)Ifbothplayerschoosethisstrategy,, rstnotethatafterevery rstperiodhistory,players' rstperiod,hismaximumpayo is5+,hismaximumpayo is0++3.
: nitelyRepeatedGamesStagegamerepeatedin sdiscountedwithdiscountrate .FormalDescription:StageGame:players1;::;I;actionsetsA1;::;AI;stagegamepayo functionsg1;:::;gI;wheregi:A!R+andA=A1 :: :Ht=At =[(in nitelyrepeatedgame)t 1strategyforplayeriisafunctionsi:H!=S1 :: ritea(s)2A1forthein nitehistorygeneratedbystrategypro les,.,a(s)=a1(s);a2(s);::::wherea1I(s)=fsi(;)gi=1 Ia2(s)=sia1 (s)i=1 a3 I(s)=sia1(s);a2(s)i=1etc....Nowplayeri'spayo sfunctionisui:S!R,whereX1ut 1 i(s)=(1 ) giat(s)t=1Notethenorm1alization:ifplayeri'soneperiodpayo isxeveryperiod,hisin nitelyrepeatedgamepayo is(1 Pt 1 ) x==1Fact:(TheOneShotDeviationPrinciple)Inadiscountedin nitelyrepeated niteactiongame,strategypro -shotdeviationfromsiifs0i(ht)=s0i(ht)forallht6=:For niteactiongame,payo sarebounded,"-shotdeviation,s00iwhichgivesatleast" T cientlylarge,\T-shot"deviationgivesapositivegain,thentheremustexistalastdeviationwhichgivesastrictlypositivegain....:(In nitelyRepeatedPartnershipGame). ort(E)NoE ort(N)E ort(E)2,2-1,3NoE ort(N)3,-10,0Considerthefollowingtriggerstrategy: PlayEinthe rstperiod PlayEaftereveryhistoryinwhichEhasalwaysbeenplayedbybothplayersineveryperiodinthepast PlayNaftereveryhistoryinwhicheitherplayerhaseverplayedN
theequilibriumpath. Ontheequilibriumpath,followingthetriggerstrategy(.,playingE)givespayo stream(2;2;::::),(.,playingN)givespayo stream(3;0;0::::),andthusutility3(1 ).Thispaysifandonlyif <13. O theequilibriumpath,followingthetriggerstrategy(.,playingN)givespayo stream(0;0;::::),(.,playerE)givespayo stream( 1;0;0::::),andthusutility (1 ). 13.[lecture8]Fixstagegame(A;g). FeasiblePayo s: F=convx2RI:xi=gi(a)forsome gure11forthefeasiblepayo ( )foranin nitelyrepeatedgamewithdiscountrate ,andleta :Foranyv2Fwithv g(a ),thereexists <1;suchthatforall ,G( )hasasubgameperfectNashequilibriumwithpayo (a)=:8 <ai,ift=1stih=:ai,ifht=(a;::;a)a i,otherwiseEachtochecknoincentivetodeviateo (a) (1 )maxg(ai;a ai2Aii)+ gi(a ): [gi(a) gi(a )] (1 )maxg(ai;a i) gi(a)., maxgi(ai;a i) gi(a) =maxai2A i imaxg ga) gi(a )]Ai(ai;a i)i(a)+[gi(ai2iInth egeneralcas e(wherethereisnotapurestrategysupportingpayo v) ndasequenceofactionpro lesa1;a2;::::;aIsuchthat
:Nv=1XNg(an)n=1(canalwaysdothisifvisavectorofrationalnumbers;ifvincludesirrationalnumbers,wecanapproximateitarbitrarilyclosely).Nowconsiderthefollowingstrategy: +2n;::::andht=a1;a2;::;aN;a1;:::siht=ani,ift=n;N+n;Na i,otherwisePayo underthisstrategypro leis1 XNn 1 g N(an)1 n=1As !1,(butmessierargument)canestablishalowerboundof ,thistheoremessentiallycharacterizesthesetofallSPNE,sinceclearlynoplayercanbeforcedbelow0payo .Ingeneral, ,we' saregi(a1;a2)=ai(12 a1 a2).Question1:Forwhich doestriggerstrategyrevertingtoCournotoutputsupportcollusiveoutput?[Triggerstrategyis:(i)produce3inthe rstround;(ii)produce3if(3;3)wasalwaysplayedinthepast;(iii)produce4ifeitherplayereverproducedanythingotherthan3inthepast.]Triggerstrategiesisequilibriumif :\stick-and-carrot"strategies. Produce3inthe rstperiod. Produce3ifeitherboth rmsproduced3inthepreviousperiodorboth rmsproducedyinthepreviousperiod. Otherwise,=3andy6= =12< t=3(12 3 3)=18MaximumCollusiveDeviationPro t=814PunishmentPro t=y(12 2y)MaximumPunishmentDeviationPro t=maxq(12 y q)q =6 y 12 y 6 y 22 2=6 y2 Equilibr81iumpathgives:(18;18;:::).Bestdeviationgives4;y(12 2y);18;18;:::.Sonoincentivetodeviatefromequilibriumpathif:1(18 y(12 2y)) 81 .,36 24y+4y2 81 .,4y2 24y+27 .,(2y 3)(2y 9) .,y 3ory 922
\Acceptingpunishment"gives:(y(12 2y);18;18;:::).Bestdeviationgives:6 y 22;y(12 2y);18;18;:::.Sonoincentivetodeviatefrompunishmentif:1 (18 y(12 2y)) 6 y 2 y(12 2y)22 .,18 26 y y(12 2y).,18 72 12y+y2 12y+2y2 2 .,0 54 24y+.,5y2 48y+108 .,(5y 18)(y 6) ., y 65Combiningtheseconditions,werequire92 y sforthein nitelyrepeatedgameisinfactF=f(v1;v2):v1+v2 36g(see gure12).Onecanshowthatanypayo inthefeasiblesetwhereeachplayergetsstrictlymorethan0canbesupportedinsubgameperfectNashequilibriumThegeneralresultis: Minmaxpayo :vi=minmaxg;a i) i2 i i2 i(aii SIR=x2RI:xi>viforalliFolkTheorem:SupposeF\\SIR,th ereexists <1, suchthatif ,thein nitelyrepeatedgamehasasubgameperfectNashequilibriumwhereeachplayer'sequilibriumpayo isxi. Forsurveyofsuch\RepeatedGameswithPerfectMonitoring",seePearce(1992):\RepeatedGames:Co-operationandRationality,"inAdvancesinEconomicTheory:, :\RepeatedGameswithImperfectPrivateMonitoring,"whereplayersobserveimperfect(butpublic)signalsofothers'pastac-tions;somekeycontributionsareAbreu,PearceandStacchetti(1990).\TowardsaTheoryofDiscountedRepeatedGameswithImperfectMonitoring,"Econometrica58,1041-1063;andFudenberg,LevineandMaskin(1994).\TheFolkTheoremwithImperfectPublicInformation."Thereisaalsoabuddinglitera-tureon\RepeatedGameswithPrivateMonitoring",see
PartIIIStaticGamesofIncompleteInformation
43Harsanyi':Supposepayo sofatwoplayertwoactiongameareeither:HTH1,10,0T0,11,0orHTH1,00,1T0,01, sbutplayerIthinksthereisprobability thatpayo saregivenbythe rstmatrix,probability1 saregivenbythe rstmatrix,type2ifpayo :IIplaysHiftype1,Tiftype2;IplaysHif >12,Tif < uncertaintycanbecapturedbyasinglemoveof\nature" ,( nite)staticincompleteinformationgameconsistsof Players1;:::;I ActionssetsA1;:::;AI SetsoftypesT1;:::;TI Aprobabilitydistributionovertypesp2 (T),whereT=T1 ::: TI Payo functionsg1;:::;gI,eachgi:A T!RInterpretation:Naturechoosesapro leofplayerstypes,t (t1;:::;tI)2Taccordingtoprobabilitydistributionp(:). sgi(a;t).Astrategyisamappingsi:Ti!,andletS=S1 :: 'spaXyo functionoftheincompleteinformationgame,ui:S!R,isui(s)=p(t)gi(s(t);t)t2Twheres=(s1;:::;sI)ands(t)=(s1(t1);:::;sI(tI)).(Old)De nition:Strategypro les isapurestrategyNashequilibriumif uis i;s i uisi;s iforallsi2Siandi=1;::;IThiscanbere-writtenas:P p( t)gisi(ti);s i(t i);t Pp(t)gisi(ti);s i(t i);tt2Tt2Tforallsi2Siandi=1;::;IW ritingp(ti)=P p0ti;t iandp(t ijti) p(ti;t i)p,thiscanbere-writtenas:(ti)t0 i
44P p(t ijti)gis i(ti);s i(t i);t Pp(t ijti)giai;s i(t i);tt i2T it i2T iforallti2Ti,ai2Aiandi=1;:::;IExample:Duopoly 2(q1;q2;cL)=[(a q1 q2) cL]q2 2(q1;q2;cH)=[(a q1 q2) cH]q2 1(q1;q2)=[(a q1 q2) c]q1T1=fcg,T2=fcL;cHg,p(cH)= .Astrategyforplayer1isaquantityq 2:T2!R, 1;q 2(cH);q 2(cL).:q 2(cH)=argmax[(a q 1 q2) cH] 2(cH)=1(a q 1 cH)2Similarlyq 2(cL)=1(a q 21 cL)Wemustalsohave:q 1=argmax [(a q1 q 2(cH)) c]q1+(1 )[(a q1 q 2(cL)) c] 1=argmax[(a q1 q 2(cH) (1 )q 2(cL)) c] 1=1(a c q 2(cH) (1 )q 2(cL))2Solution: q 2(cH)=a 2cH+c+1(cH cL)36q 2(cL)=a 2cL+c (cH cL)36q 1=a 2c+ cH+(1 )cL3[lecture11]Example:Puri cationNSN1,-10,0S0,01,-1SupposeIhassomeprivatevaluetogoingN," U[0;x];IIhassomeprivatevaluetogoingN, U[0;x]
45Newgame:NSN1+",-1+ ",0S0, 1,-1BestResponses:IfIattachesprobabilityqtoIIgoingN,thenI'sbestresponseisNif"+q 1 ." 1 ,,thenII'sbestresponseisNif p (1 p). 2p "and ,respectively,,respectively: s1(")=N,if" (1 p)xS,if"<(1 p)x s2( )=N,if (1 q)xS,if <(1 q)xNows1isabestresponsetos2ifandonlyif(1 p)x=1 2q;s2isabestresponsetos1ifandonlyif(1 q)x=2p x2p =1 +xSo p= 1 x21 xq 2x1+x =1x 21 x x2 4 2 x1+x = 1x 21 xx2+4 2 x1+x !2+x+x2=4+x22 x+x24 +x2 1!21,asx!02Wesaythatthemixedstrategyequilibriumofthecompleteinformationgameis\puri ed" functions,g=(g1;:::;gI).Letplayerihaveatype i=( ai)a2A2R#A;letplayeri'stypebeindependentlydrawnaccordingtosmoothdensityfi( )withboundedsupp#Aort,say[ 1;1].Letplayeri'spayo inthe\"-perturbedgame"(thisisanincompleteinformationgame)begei(a; )=gi(a)+" aiTheorem(Harsanyi):"-perturbedwiththepropertythattheprobabilitydistributionoveractionsofeachplayerisconvergingtohisstrategyintheoriginalNashequilibrium.
PartIVDynamicGamesofIncompleteInformation
. Beliefsystem speci (zjh)fortheprobabilityofnodezgiveninformationseth. Strategypro le issequentiallyrationalwithrespecttobeliefsystem ifeachplayer'sstrategymaximizeshisexpectedutilityateveryinformationset,giventhestrategiesofotherplayersandthebeliefsovernodesatthatinformationset. Beliefsystem isconsistentwithstrategypro le ifbeliefsateachinformationsetreachedby aregeneratedby ,ifbeliefsystem isconsistentwith(In1;Accommodate)thenwemusthave (z1jh)=1and (z2jh)= (zjh)maytakeanyvaluein[0;1]andstillbeconsistentwith(Out;Fight).Observation:If isaNashequilibriumand isconsistentwith ,then . atthoseinformationsetsreachedby .[IdeaofProof],aninformationseth,andastrategy 0isuchthati'sexpectedutility(under )conditionalonreachinginfosethandfollowing 0iexceedsthatoffollowing 00iforplayerwhichequals 0iatinfosethandallinfosetsfollowingitandequals ( 00i; i)>ui( i; i),contradictingtheassumptionthat : nition:Astrategypro leandbeliefssystem,( ; ),isaweakperfectBayesianequilibriumif issequentiallyrationalwithrespectto and isconsistentwith .Strategypro le isaweakperfectBayesianequilibriumstrategypro leif( ; )isaweakperfectBayesianequilibriumforsomebeliefsystem .[lecture12]Terminology: ,,weakPBEre : theequilibriumo theequilibriumpath;(duetoKrepsandWilson1982) le 0has\fullsupport,"allinformationsetsarereachedandthusthereisauniquebelief
0suchthat 0isconsistentwith aresaidtobefullyconsistentwith ifthereexistsafullsupportsequence k,with k! ,suchthatuniquebeliefs kconsistentwith ksatisfy k! .De nition:Astrategypro leandbeliefssystem,( ; ),isasequentialequilibriumif . and isfullyconsistentgiven .Strategypro le issequentialequilibriumstrategypro leif( ; )isasequentialequilibriumforsomebeliefsystem .Itwouldbenicetotrytocapturedirectlysomeoftheintuitivepropertiesthatfullyconsistent mustsatisfy(withoutreferencetoperturbationsequence).Thisisapproachofintermediate\perfectBayesianequilibrium"[see,.,Gibbonsrequirement4]. If isasequentialequilibriumstrategypro le,then isasubgameperfectNashequilibrium. All niteperiod, niteactiongameshaveatleastonesequentialequilibrium(andthusatleastoneweakperfectBayesianequilibrium). SeeMyerson'sGameTheory:AnalysisofCon
ict,chapter4,
PartVInformationEconomics
,learning,Bayesianornot,becomesimportant, ,: smallnumberofparticipants institutionsmayberepresentedbyconstraints noncooperative(Bayesian)gametheory simpleassumptionsonbargaining:Principal-AgentparadigmWerefertothePrincipal-Agentparadigmasasettingwhereoneagent,calledthePrincipal,canmakeallthecontracto ers,andhencehas(almost)allbargainingpowerandasecondagent,calledtheAgent,canonlychoosewhethertoacceptorrejecttheo edtoaconstrainedoptimizationproblem,wherethePrincipalhasanobjectivefunction,andtheAgentsimplyrepresentsconstraintsonthePrincipal':(i)whethertheinformedoranuniformedagenthastheinitiative(makesthe rstmove,o erstheinitialcontract,arrangement;and(ii)whethertheuniformedagentisuncertainabouttheactionorthetype(information):mechanismdesign(socialchoice)\mechanismdesign"problemwhichencompassesauctions,bilateraltrade,publicgoodprovision,:privateinformationsocialchoice(type)ofagenti2I(allocation,outcome)f i2 igi2I !f: !X !x2X&si: i!Mig:M!X&%fmi2Migi2Imessages:fromagenttoprincipal
54Theupperpartofthediagramrepresentsthesocialchoiceproblem,,wheretheprincipalsattemptstoelicittheinformationbyannouncinganoutcomefunctionwhichmapsthemessage(information),thisthenrepresentsaBayesiangameinwhichtheycanin
,butclassicmodelsandresultstodemonstrate(i)howasymmetryofinformationchangesclassicale ciencyresultsofmarketsand(ii)howasymmetryofinformationchangesclassicalargumentsabouttheroleofpricesandtheequilibriumprocess.
' = svandub= bvwith b> s, sv pandthusbysellingtheobjecthesignalsthatv p() sThebuyerbuystheobjectif bE[v] p()andasheknowsthat()hastohold,hecanformaconditionalexpectation,that bE[vjp] p,p b p()2 sThusforthesaletooccur, b 2 s.()Thusunless,thetastesdi ersubstantially,themarketbreaksdowncompletely: marketmechanisminwhichalowerpricesincreasessalesfailstoworkasloweringthepricedecreasestheaveragequality,lowerpriceis\badnews". marketmaynotdisappearbutdisplaylowervolumeoftransactionthansociallyoptimal.
',,weventuredtheclaimthatastheamountofprivateinformationheldbythesellersdecreases,,namelythatintheexamplewestudied,foragivenconstellationofpreferencesbybuyerandseller,representedby band s, cdensityintheexample,: 1v~U 1";+"22 with"201;( )andhowita ectsthee xedpricepthatv p. sThebuyerbuystheobjectif bE[vjp] p,Theexpectedvalueisnowgivenby1E[vjp]=2 "+p s2andhenceforsaletooccur12 "+p sb p()2andinequalityprevailsif,providedthat"2[01;2)and2 s> b,p 11 2" b s22 s bConsidernextthee 1p= s+"()2inwhichcasetheexpectedconditionalvalueforthebuyeris 11 b s+"22orequivalently b s(1+2")()Thusastheamountofprivateinformation,measuredby",decreases,theine
'sPriceSignal'sQuality57evenifcondition()isnotmet,():12 "+p sb p()2asthelhsincreasesslowerinp,,whichwillnotinduceallsellerstoshowupatthemarket, howho wth iscana ( 1s2 "; 1s2+"].Ifwetheninsertthepriceasafunctionofx2( ";"],we nd1 s(12+x) 2 "+ sb 1 s+x22or 1 "+x b 1 s+x22Foragiven b; sand",wemaythensolvethe(in-)qualityforx,to ndthe(maximal)pricepwherethevolumeoftradeismaximized,or:x= b(1 ") s:2 s bThusxisincreasingin banddecreasingin";con rmingnowintermsofthevolumeoftradeourintuitionaboutprivateinformationande ,wecanverifythatforagiven"2[01;2),thevolumeoftradeispositive,butas"!12becomesarbitrarilysmallandconvergesasexpectedtozerofor b<2 'sPriceSignal'sQualityThissectionisbasedon?.= v pandthemonopolistcanprovidelowandhighquality:v=f0;1gatcost0<c0< >c1sothatitissociallye ,sayafraction .Observe toselltobothsegmentsofthemarket:p c1 (1 )(p c0)or p c1 (1 )c0.()Wecanthenmaketwoobservations: highqualityissuppliedonlyifpriceissu cientlyhigh,\highpricecansignalhighquality". ahigherfraction, ,ofinformedconsumersfavorse ciencyasitpreventsthemonopolistfromcuttingquality theinformationalexternalityfavorsgovernmentinterventionasindividualsonlytakeprivatebene tandcostintoaccount.
'\hiddeninformation"or\hiddenaction",whichwemaycall\searchgoods",wherewecanassertthequalitybyinspection,weconsidered\experiencegoods"(?,?),\credencegoods"(?).Inbothmodels,therewasroomforathirdparty,governmentorotherinstitution,,andimprovementinthesymmetryofinformationleadtoanimprovementinthee ,eitherthrough: costlysignalling optimalcontractingtoavoidmoralhazard,or optimalinformationextractionthroughamenuofcontract().
rstanalyzedby?.Naturechoosesaworkerstype(productivity)a2f1;(a).Fornotationalconvenience,wede nep,p(a=2):Theworkercanchooseaneducationallevele2R+.Thereareatwo rmswhocompeteinwagesw1; rmTheworkersutilityisu(w;e;a)=w rmisgivenbymin(a w2)p(aje)a2f1;2gWeobservethat@2u(w;e;a)=1@e@aa2>0andthustypeandstrategicvariablearecomplements(supermodular).:f1;2g!R+Apurestrategyforthe rmisafunctionw:R+!R+wherew(e)isthewageo ,dependingoneanddenotedbyp(aje).Theposteriorbeliefisamappingp^:R+![0;1]
,'(aje)isposteriorbelief,andhenceaprobabilityfunction,itisrequiredthat:X8e;9p(aje);(aje) 0;andp(aje)=1:() 2f1;2gMoreover,whenthe rmcanapplyBayeslaw,itdoesso, (a)=e;thenp(aje)=Pp(a):()fa0)je (a)=egp(a0Werefertoeducationalchoicewhichareselectedbysomeworker-typesinequilibrium, (a)=e,as\on-the-equilibriumpath".@awithe (a)=eas\o -the-equilibrium-path".Asbeforeitwillbesometimeseasiertorefertop(e),p(a=2je)andhencep(1je)=1 p(e):De (PBE).ApurestrategyPerfectBayesianEquilibriumisasetofstrategiesfe (a);w (e)gandposteriorbeliefsp(e)suchthat:;9p (aje); (aje) 0;and 2f1;2gp (aje)=1;;w i(e)=Pap (a je)a; ;e (a)2argmaxw (e) (a)=e;then:p(aje)=Pp(a):0fa0je (a)=egp(a)[ToBeAdded:NotionsofOnandOfftheEquilibriumPath]De (SeparatingPBE).ApurestrategyPBEisaseparatingequilibriumifa6=a0)e(a)6=e(a0).De (PoolingPBE).ApurestrategyPBEisapoolingequilibriumif8a;a0)e(a)=e(a0).[Lecture3]=e(1)=e(2)2[0;p]:(1)=0ande(2)2[1;2].
.(1)We ;1+p(e ) e 1+p(e) e()and8e;1+ep(e ) e 1+p(e) ()22Consider rstdownwarddeviations,<e ,then()requiresthatp(e ) p(e) e e:()Thenconsiderupwarddeviations,>e ,then()requiresthatp(e) p(e ) 1(e e ):()2Wecanthenaskforwhatlevelscanbothinequalitiesbesatis <e )p(e)=0e>e )p(e)=0;whichleavesuswithe<e )p e e()e>e ) p 1(e e ):2Wemaythenrewritetheinequalitiesin()ase<e )e p+e()e>e )e p+2eAstheinequalityhastoholdforalle,theassertingthat0 p e holds,followsimmediately.(2)-straints8e;1+p(e 1) e 1 1+p(e) eand8e;1+p(e 2) e 2 1+p(e) e:()22Asalongtheequilibriumpath,the rmsmustapplyBayeslaw,wecanrewritetheequationsas8e;1 e 1 1+p(e) eand8e;2 e 2 1+p(e) e:22Consider 1=(e) e:()
=e 2issupposedtobepartofaseparatingequilibrium,thenp(e 2)=()thate 2 1,forotherwisewewouldnotsatisfy1=p(e 2) e ,wewanttodetermineanupperboundfore ortotherwisehewouldmimicthelowerability,wecanrewrite()toobtain:1 p(e) 1(e 22 e);whichiseasiesttosatisfyifp(e)=0;andhence8e;2+e e 2whichimpliesthat:e 2 nestherangeofeducationalchoiceswhichcanbesupportedasanequi-libriumbutisnotacompleteequilibriumdescriptionaswehavenotspeci -bria,reducingthepredictiveabilityofthemodelandwemaylookatdi erentapproachestoreducedthemultiplicitly: re nedequilibriumnotion di erentmodel:: whendocostlysignalsmatterforParetoimprovements(orsimplyseparation):Spence-Mirrleessinglecrossingconditions whendocostlesssignalsmatter:,educationcouldalsobeinterpretedasanactofdisclosureofinformationthroughtheveri rmsforeducationalchoicelevelso , nedfollow?.De (EquilibriumDomination).GivenaPBE,themessageeisequilibrium-dominatedfortypea,ifforallpossibleassessmentsp(aje)(orsimplyp(e)):w (e (a)) e (a)>w(e) eaaorequivalently1+p (e (a)) e (a)>1+p(e) eaa
(IntuitiveCriterion).Iftheinformationsetfollowingeiso theequilibriumpathandeisequilibriumdominatedfortypea,thenp(aje)=0:()De (Uniqueness).TheuniquePerfectBayesianequilibriumoutcomewhichsatis estheintuitivecriterionisgivenbyfe 1=0;e 2=1;w (0)=1;w (1)=2g:()Thebeliefsarerequiredtosatisfy8<0;fore=0p (e)=:2[0;e];for0<e<11;fore -costseparatingequilibrium(or?equi-librium) rstshowthattherecan'tbeanypoolingequilibriasatisfyingtheintuitivecriterion, e (e +(1 p);e +2(1 p)).Anymessageeintheintervalisequilibriumdominatedforthelowproductivityworkeras1+p e >2 e;butanymessageintheintervalisnotequilibriumdominatedforthehighproductivityworker,as1+p e <2 e22andthusfore=e +(1 p),wehavep<1 (1 p),1p<1;(e;p(e))satis estheintuitivecriterion,wemusthavep(e)=1fore2(e +(1 p);e +2(1 p)),butthenpoolingisnotanequilibriumanymoreasthehighproductivityworkerhasapro tabledeviationwithanye2(e +(1 p);e +2(1 p)). 2>1,anye2(1;e 2)isequilibriumdominatedforthelowabilityworkeras1>2 e;butisnotequilibriumdominatedforthehighabilityworker,as2 e 2<2 (e)=1foralle2(1;e 2).Butthene 2>1,cannotbesupportedasanequilibriumasthehighabilityworkerhasapro tabledeviationbyloweringhiseducationalleveltosomee2(1;e 2), 2=!1:[Lecture4]
-Krepsintuitivecriterionmaybe,,supposethat,p,thepriorprobablitythataworkeroftypea=2ispresent,isarbitrarilylarge,(p !1).Inthatcase,itseemsaratherhighcosttopaytoincuraneducationcostofc(e=1)=12justtobeabletoraisethewagebyacommensuratelysmallamount w=2 (1+p) !0asp!,(suchasCho-Kreps'intuitivecriterion)'sinformedprincipalproblemInterestingly, ,nowtheworkersignsacontractwithhis/ (e)gdenotethecontingentwageschedulespeci erbywhichhecanseparatehimselffromalowabilityworkernmaxw(e) eofe;w(e)g2subjecttow(e1) e1 w(e2) e2(IC1)andw(e2) e2 w(e1) e1(IC2)22anda1 w(e1) 0(IR1)a2 w(e2) 0(IR2)Thustomakeincentivecompatibilityaseasyaspossiblehesuggestse1=0andw(e1=0)=(e2)=2,itfollowsthataftersettinge2=1,heindeedmaximizeshispayo .Supposeinsteadhewouldliketoo ee;w21+p w 0(IR1)whichwouldyieldw=1+ erentcasestoconsider:
+p 2 12:ahighproductivityworkerisbettero inthe\leastcost"separatingequilibriumthaninthee +p>2 12:ahighproductivityworkerisbettero inthee 12,thehighproductivityworkercannotdobetterthano eringtheseparatingcontract,,,hehaseverythingtolosebyo >12,theuniqueequilibriumcontractistheonewhere:w (e)=1+pforalle ,ifthe rmacceptsthiscontract,,onaveragethe rmbreaksevenbyacceptingthiscontract,providedthatitisas(ormore),ahighproductivityworkerstrictlypreferstoo ,alowproductivityworkerhaseverythingtolosefromo ,inthiscaseagainthisistheuniquecontracto :A R+andanarbitrarynumberofsignalsE R+withageneralquasilinearutilityfunctionu(t;e;a)=t+v(e;a)wherewerecallthattheutilityfunctionusedinSpencemodelwasgivenby:u(t;e;a)=t eaWenowwantaskwhenisitpossibleingeneraltosustainaseparatingequilibriumforallnagents,suchthata6=a0)e6=e0Supposewecansupportaseparatingequilibriumforalltypes,thenwemustbeabletosatisfyfora0>aandwithoutlossofgeneralitye0>e:t+v(e;a) 0t+v(e0;a),t 0t v(e0;a) v(e;a)()and
+v(e0;a0) t+v(e;a0),t 0t v(e0;a0) v(e;a0);()wheret,t(e)and0t,t(e0),andbycombiningthetwoinequalities,we ndv(e0;a) v(e;a) v(e0;a0) v(e;a0);()orsimilarly,againrecallthate<e0v(e;a0) v(e;a) v(e0;a0) v(e0;a):()Wethenwouldliketoknowwhataresu cientconditionsonv( ; )suchthatforeveryathereexistsesuchthat() :E A!Rhasincreasingdi erencesin(e;a)ifforanya0>a,v(e;a0) v(e;a) erencesin(e;a),ithasincreasingdi erencesin(a;e).Alternatively,wesaythefunctionvissupermodularin(e;a).Ifvissu cientlysmooth,thenvissupermodularin(e;a)ifandonlyif@2v=@e@a ;tcouldeitherbedeterminedexogeneously,asintheSpencesignallingmodelthroughmarketclearingconditions,orendogeneously,asinoptimallychosenbythemechanismdesigner,(a),,t(a),conditionalonthesortingallocationfa;e(a);t(a)gtobeincentivecompatible,iscontinuouslydi cientconditionforsorting(()and())tobeincentivecom-patibleforallt(a)(e;a)isstrictlysupermodular2.@v@e<( ; )istwicecontinouslydi (a^)+v(e(a^);a),whereaisthetruetypeofagentaande(a^)isthesignaltheinformedagentsendstomaketheuninformedagentbelieveheisoftypea^.(Su ciency)Fore(a)tobeincentivecompatible,a^mustlocallysolveforthe rstorderconditionsoftheagentata^=a,namelythetruetype:0t(a^)+@v(e(a^);a)de=0,ata^=a()@eda^ButFixthepaymentst(a)asafunctionofthetypeatobet(a)andsupposewithoutlossofgeneralitythatt(a)iscontinuouslydi ()de nesadi erentialequationfortheseparatinge ortlevelde0=t(a)()da@v(e(a);a)@e
,sayt(0)= rstorderconditionsisonlyanoptimalconditionforagentwithtypeaandnobodyelse,(a^)anddeda^areindependentofthetruetype,a,itfollowsthatif@2v(e;a)>0,@e@aforalleanda,then()canonlyidentifytruthtellingforagenta.(Necessity)Foranyparticulartransferpolicyt(a),wemaynotneedtoimposethesupermodularitycon-ditioneverywhere,anditmightoftenbysu cienttoonlyimposeitlocally,whereitishowevernecessarytoguaranteelocaltruthtelling,.@v(e(a^);a)>0@e@aate=e(a).However,aswearerequiredtoconsiderallpossibletransferst(a)witharbitrarypositiveslopes,wecanguaranteethatforeveryaandeveryethereissometransferproblemt(a)suchthate(a)=eby(),andhenceundertheglobalconditionont(a), (Spence-Mirrlees).ThefunctionUissaidtosatisfythe(strict)(increasing)nondecreasingint,andUy6=0,(x;y;t).De (Single-Crossing).ThefunctionUissaidtosatisfythesinglecrossingpropertyin(x;y;t)ifforall(x0;y0) (x;y)(x00;y0;t) U(x;y;t),thenU(x0;y0;t) U(x;y0;t)foral0lt>t;(x0;y0;t)>U(x;y;t),thenU(x0;y00;t)>U(x;y0;t)foral0lt>t;?-crossingorSpence-Mirrleescondition,whereMirrleesusedthedi erentialformforthe rsttimein?.Thenotionsofsupermodu-larityandsingle-crossingareexactlyformulatedin?andforsomecorrections?Someapplicationstosupermodulargamesareconsideredby?and?.Themathematicaltheoryis,interalia,dueto?and?.
,wehaveconsideredprivateinformationaboutsuchthingsasindividualpreferences,tastes,ideas,intentions,qualityofprojects,e ortcosts,etc,,suchasanindividual'shealth,theservicingandaccidenthistoryofacar,potentialandactualliabilitiesofa rm,earnedincome,etc,thatcanbecerti ,
ersthecontract,wedistinguishbetweenhiddeninformation-`adverseselection'andhiddenaction-`moralhazard'Themaintrade-o inadverseselctionisbetweene ortchoice,: \outcome"xandpossiblesome\signal"sbutnot\action"
uenceagentsactionbyo ect(nonobservable),Propertyinsurance( re,theft)FB:agentsactionisobservable(risksharing)SB:agentsactionisunobservable(risksharing-incentive);ahgatcostc2fcl;;xhgoccurrandomly,=Pr(xhjal)<ph=Pr(xhjah)Theprincipalcano erawage,contingentontheoutcomew2fwl;whgtotheagentandtheutilityoftheagentisu(wi) ciandoftheprincipalitisv(xi wi)whereweassumethatuandvarestrictlyincreasingandweaklyconcave.
,thentheprincipalcaninducetheagenttochoosethepreferredactiona byw(xi;a)= 1ifa6=a -aversepreference,;wiv(xh wh)+(1 pi)v(xl wl)ghgsubjecttopiu(wh)+(1 pi)u(wl) ci U;( ) new(xi;a ),wiThisisaconstrainedoptimizationproblem,andthe rstorderconditionsfromtheLagrangianL(wl;wh; )=piv(xh wh)+(1 pi)v(xl wl)+ (piu(wh)+(1 pi)u(wl) ci)aregivenbyV0(xi wi)= ;U0(wi)whichisBorch'(xi;a)=w(xi) ort,thentheincentiveconstraintis:phu(wh)+(1 ph)u(wl) ch plu(wh)+(1 pl)u(wl) clor(ph pl)(u(wh) u(wl)) ch cl()aspl!phwh wlmustincrease,(orindividualrationalityconstraint)phu(wh)+(1 ph)u(wl) ch U ()We rstshowthatbothconstraintswillbebindingiftheprincipalmaximizesmaxph(xh wh)+(1 ph)(xl wl)fwh;wlg
niteoutcomesandactions71subjectto()and().Fortheparticipationconstraint,principalcouldlowerbothpaymentsandbebettero .Fortheincentiveconstraint,subtract(1 ph)"u0(wh)fromwhandaddph"u0(wl)"su (1 ph)"fromu(wh)andaddph"tou(wl), "ph(1 1ph);u0(w 1h)u0(w>0l)sincewh>:u(wl)=U chpl phclph plandu(wh)=U chpl phclch clph +plph niteoutcomesandactionsSupposexi2fx1;:::;xIgandaj2fa1;:::aJgandtheprobabilitypij=Pr(xijaj)theutilityisu(w) aforagentandx wfortheprincipal.
(X)Imax(xi wi)pijfwIigi=1;ji=1giventhewagebilltheagentselectsajifandonlyifXIXIu(wi)pij aj u(wi)pik ak( k)i=1i=1andXIu(wi)pij aj U( )i=1Fixaj,thentheLagrangianis(X))IX(XIL(wij; ; )=(xi wi)pij+u(wi)(pij pik) (aj ak)i=(1ki=16=jX)I+ u(wi)pij aji=1Di erentiatingwithrespecttowijyields1X = + pik;8i()0k1u(wi)k6=pijjWitharisk-aversprincipalthecondition()wouldsimplybymodi edto:v0(xi wXi)= + u0k1 pik;8i()(wi)pkij6=jIntheabsenceofanincentiveproblem,or k=0,()statestheBorchruleofoptimalrisksharing:v0(xi wi);(w= i) ,`downward'bindingconstraints( k;k<j)andthe`upward'bindingconstraints( k;k>j).Suppose rstthenthattherewereonly`downward'bindingconstraints,=J,or k=0fork> cientformonotonicityiniwouldclearlybethatpikpijisdecreasinginiforallk<j.
,or8i<0i;8k<j:pik>pi0kpijpi0jorreversingtheratiopij<,forahigheroutcomexi0,itbecomesmorelikely,theactionpro lewashigh(j)ratherthanlow(k).Bymodifyingtheratioto:pi0k<pi0jpikpijwegetyetadi `parameter'afromtheobservationofthe`sample'xiasthefollowingtwostatementsareequivalent:ajisthemaximumlikelhoodestimatorofagivenxi,8kpik;p -Holmstrom[1986]:Theagentispunishedforoutcomesthatrevisebeliefaboutajdown,;conceptually,theprincipalisnotinferringanythingabouttheagent',theoptimalsharingrulere
ectspreciselythepricingofinference. 1 pik>0ifpik<pij pij ik1 p<0ifpik>,wenextgiveconditionswhichwillguaranteethat k=0forallk>:foraj<ak<aland 2[0;1]suchthatforak= aj+(1 )alwehave
Pij+(1 ) ,weknowthatforsomej<k, j>,thentheoptimalchoiceofakwouldbesameifweweretoconsiderAorfak;:::;;:::;,,considerfa1;:::;>(wi)pik ak<u(wi)pil al:i=1i=1andletj<kbeanactionwhere k>0andhenceXIXIu(wi)pik ak=u(wi)pij aj:i=1i=1Thenthereexists 2[0;1]suchthatak= aj+(1 )alandwecanapplyconvexity,by rstrewritingpikandusingthecumulativedistributionfunctionXIXIu(wi)pik ak=(u(wi) u(wi+1))Pik+u(wI) aki=1i=1Byrewritingtheexpectedvaluealsoforaj<ak<al,weobtainXI(u(wi) u(wi+1))Pik+u(wI) ak i=1 X!I (u(wi) u(wi+1))Pij+u(wI) aji=1 X!I+(1 )(u(wi) u(wi+1))Pil+u(wI) ali=1Thelaterisofcourseequivalentto X! IX!I u(wi)pij aj+(1 )u(wi)pil ali=1i=1Theinequalityfollowsfromtheconvexityandthefactthatwecouldassumewitobemonotone,sothatu(wi) u(wi+1) nalcommentonthemonotonicityofthetransferfunction:Perhapsagoodreasonwhythefunctionwimustbemonotonicisthattheagentmaybeabletoarti ,hewouldthenarti ciallyloweritfromxjtoxiwhenevertheoutcomeisxj.
ectthepayo estheprobabilitiespijsuchthatXIp^kj=qkipij;()i=1suchthatqki 0andXKqki=1k=(),itisgarbledbyastochasticmechanismthatisindependentoftheactionaj,^^kp^kj=xipij;k=1i=(inmatrixlanguage)byassumingthatx^=Q ,(p^;x^) ndanewwagescheduleforthe(p;x)problembasedonwbsuchthatajisalsoimplementablein(p;x)problem,(wi)=qkiu(wbk)()k=1WecanthenwriteXIX !IXKpiju(wi)=pijqkiu(wbk)i=1Xi=1k=1K=p^kju(wbk)k=1Buttheimplementationin(p;x)islesscostlyfortheprincipalthantheonein(p^;x^)() rstbestcontainsnoadditionalnoise.
,theprincipalcanobservesomeothersignal,sayy2Y=fy1;::::;yl;:::;yLg, edtobe1X ! = + wlk1 =jThusthecontractshouldintegratetheadditionalsignalyifthereexistssomexiandylsuchthatforajandak0plik6=plikplijpl0;ijbuttheinequalitysimplysaysthatxisnotasu cientstatisticfor(x;y)asitwouldbeifwecouldwritetheconditionalprobabilityasfollowsf(xi;yljaj)=h(xi;yl)g(xijaj):Thus,Hart-Holmstrom[1986]write:Theadditionalsignalswillnecessarilyenteranoptimalcontractifandonlyifita ectstheposteriorassessmentofwhattheagentdid;orperhapsmoreaccuratelyifandonlyifsin
\natural"=a+",where"isnormallydistributedwithzeromeanandvariance ,whiletheagenthasautilityfunction:U(w;a)= e r(w c(a))whereristhe(constant)degreeofabsoluteriskaversion(r= U00=U0),andc(a)=12ca2:Werestrictattentiontolinearcontracts:w= x+ :Aprincipaltryingtomaximizehisexpectedpayo willsolve:maxE(x w)a; ; subjectto:
( e r(w c(a))) U(w)anda2argmaxE( e r(w c(a)))awhereU(w)isthedefaultutilityleveloftheagent, nedasfollowsu(w)=E[u(x)]ThecertainyequivalentofanormallydistributedrandomvariablexunderCARApreferences,hencewwhichsolves e rw =E e rx hasaparticularlysimpleform,namelyw=E[x] 1r 2()2Thedi erencebetweenthemeanofrandomvariableanditscertainequivalentisreferredtoastheriskpremium:1r 2=E[x] (a)withrespecttoa,wherewb(a)isde nedby e rwb(a)=E( e r(w c(a)))Hence,theoptimizationproblemoftheagentisequivalentto:a2argmax fwb(a)g= 2argmax a+ 1ca2 r 2 222whichyieldsa = cInsertinga intotheparticipationconstraint 2 r + 1c 2 2=w c2c2
,2 =w +r 2 2 1 22cThisgivesustheagent'se ortforanyperformanceincentive .Theprincipalthensolves: 2 max (w +r 2 2+1 ) c22cThe rstorderconditionsare1 (r 2+ )=0;ccwhichyields: =11+rc 2E ortandthevariablecompensationsomponentthusgodownwhenc(costofe ort);r(degreeofriskaver-sion),and 2(randomnessofperformance)goup,;cor 2becomelarge,as !1 =w +1r 2 c22:(1+rc 2):?,?,?,?,?,?,?,?.[Lecture7]
PartVIMechanismDesign
,:(i)monopolisticpricediscrimination,(ii)optimaltaxation,(iii)thedesignofauctions,and(iv),truthtelling,\principal"(socialplanner,monopolist,etc.)whowouldliketoconditionheractiononsomeinformationthatisprivatelyknownbytheotherplayers,called\agents".Theprincipalcouldsimplyasktheagentsfortheirinformation,buttheywillnotreporttruthfullyunlesstheprincipalgivesthemanincentivetodoso,,theprincipalfacesatrade-o thatoftenresultsinane ',asopposedtousingaparticularmechanismforhistoricalorinstitutionalreasons.(Caveat:Sincetheobjectivefunctionoftheprincipalcouldjustbethesocialwelfareoftheagents,therangeofproblemwhichcanbestudiesisratherlarge.)(orequivalentwithacontinuumofin nitesimalsmallagents).Forexample,inaseconddegreepricediscriminationbyamonopolist,themonopolisthasincompleteinformationabouttheconsumer' ,,-stepgameofincompleteinformation,wheretheagents',theprincipaldesignsa\mechanism",\contract",or\incentivescheme".Amechanismisagamewheretheagentssendsomecostlessmessagetotheprincipal,, ed\reservationutility".Instep3,theagentswhoacceptedthemechanismplaythegamespeci rstclassiscalledthe\participation"or\individualrationality"constraint,,whatwewillcall\incentivecompatibility" cientoutcomestoarise:(i)whichallocationycanbeimplemented, cient,revenuemaximizing.
:MechanismDesignwithOneAgentOftenalsocalledself-selection,or\screening".Ininsuranceeconomics,ifainsurancecompanyo ersatari tailoredtotheaveragepopulation,thetari ,whocouldeitherhaveacoarseorasophisticatedtaste, erandatwhatprice?Themodelisgivenbytheutilityfunctionofthebuyer,whichisv( i;qi;ti)=u( i;qi) ti= iqi ti;i2fl;hg()where irepresentthemarginalwillingnesstopayforqualityqiandtiisthetransfer(price) isatis es0< l< h<1.()Thecostofproducingqualityq 0isgivenbyc(q) 0;c0(q)>0;c00(q)>0.()Theex-ante(prior)probabilitythatthebuyerhasahighwillingnesstopayisgivenbyp=Pr( i= h)Wealsoobservethatthedi erenceinutilityforthehighandlowvaluationbuyerforanygivenqualityqu( h;q) u( l;q)isincreasinginq.(ThisisknowastheSpence-Mirrleessortingcondition.).Ifthetasteparameter iwereacontinuousvariable,thesortingconditioncouldbewrittenintermsofthesecondcrossderivative:@2u( ;q)>0,@ @qwhichstatesthattaste tforthesellerfromabundle(q;t)isgivenby (t;q)=t c(q)
: erenttypeshavedi erentpreferences,theyshouldconsumedi iqi c(qi)gqiandthe rstorderconditionsyield:qi=q i()c0(q i)= i)q l<q cientsolutionistheequilibriumoutcomeifeitherthemonopolistcanperfectlydiscriminatebetweenthetypes( rstdegreepricediscrimination) ,themonopolistsetsti= iqi()andthensolvesforeachtypeseparately:max (ti;qi)()maxf iqi c(qi)g;fti;qigfti;qigusing().Likewisewithperfectcompetition,thesellerswillbreakeven,getzeropro tandsetpricesatti=c(q i): edimmediatelythatperfectdiscrim-inationisnowimpossibleas hq l tl=( h l)q l>0= hq h th()(1 )tl c(ql)+ (th (c(qh)))()ftl;ql;th;qhgsubjecttotheindividualrationalityconstraintforeverytype iqi ti=0(IRi)()andtheincentivecompatibilityconstraint iqi ti= iqj tj(ICi)(),,whichinturnallowsustosolveforqh;(i)IRlbinding,(ii)IChbinding,(iii)q^h q^l(iv)q^h=q h()
(i). hqh th= hql tl= lql tl()ICh h> lsupposethat lql tl>0,thenwecouldincreasetl;thbyaequalamount,satisfyalltheconstraintsandincreasethepro (ii)Supposenot,thenas hqh th> hql tl= lql tl=0() h> l(IRl)andthusthcouldbeincreased,againincreasingthepro toftheseller.(iii)Addinguptheincentiveconstraintsgivesus(ICl)+(ICl) h(qh ql)= l(qh ql)()andsince: h> l)q^h q^l=0:()NextweshowthatIClcanbeneglectedasth tl= h(qh ql)= l(qh ql):()Thisallowstosaythattheequilibriumtransfersaregoingtobetl= lql()andth tL= h(qh qL))th= h(qh qL)+ ,itisimmediatethatq^h=q handwecansolveforthelastremainingvariable,q^(1 p)( lql (c(ql))+p( h(q qh qL)+ lql c(q h)))glbutasq hisjustasconstant,theoptimalsolutionisindependentofconstanttermsandwecansimplifytheexpressionto:maxf(1 p)( lql c(ql)) p( h l)qlgqlDividingby(1 p)weget max lql c(ql) p( h l)qlql1 pforwhichthe rstorderconditionsare l c0(ql) p1 ( h l)ql=0pThisimmediatelyimpliesthatthesolutionq^l:
:MechanismDesignwithOneAgent()c0(q^l)< l()q^l<q landthequalitysupplytothelowvaluationbuyerisine cientlylow(withthepossibilityofcompleteexclusion).Considernexttheinformationrentforthehighvaluationbuyer,itisI(ql)=( h l) h)=( h l)qlanditalsorepresentsthedi erencebetweentheequilibriumutilityoflowandhightypeagent,orI( h)=U( h) U( l)Ifweextendthemodelandthinkofmorethantwotypes,thenwecanthinkoftheinformationrentasresultingfromtowadjacenttypes,say kand k ( k)=( k k 1)qk 1Astheinformationrentisalsothedi erencebetweentheagent'snetutility,wehaveI( k)=U( k) U( k 1);wellasweseenotquite,butmoreprecisely:I( k)=U( kj k) U( k 1j k);whereU( kj l)denotesingeneraltheutilityofanagentof(true)type l,whenheannouncesandpretendstobeoftype ,wecouldaskhowthenetutilityoftheagentoftype k 1! k:U( kj k) U( k 1j k)=( k k 1)qk 1 k k 1( k k 1)
:qk 1!qk;wegetU0( )=q( )whichisidenticalusingthespeci cpreferencesofourmodeltoU0( )=@u(q( ); ):@ Thuswehaveadescriptionoftheequilibriumutilityasafunctionofq( )aloneratherthan(q( );t( )).Thepursuitoftheadverseselectionproblemalongthislineisoftenreferredtoas\Mirrlees'trick".(x; ;t)=u(x; ) tandtheprincipalsV(x; ;t)=v(x; )+tandu; 2 R+.ThesocialsurplusisgivenbyS(x; )=u(x; )+v(x; ):Theuncertaintyaboutthetypeoftheagentisgivenby:f( );F( ).Weshallassumethe(strict)Spence-Mirrleesconditions@u>@2u0;>0:@ @ @ !y( )=(x( );t( )).De =(x;t)satis estruthtelling: u(x( ); ) t( ) uxb ; tb ;8 ;b 2 :De ( ),U( j )=u(x( ); ) t( );()andthenetutilityforagent ,misreprortingbytelling ^isdenotedby U ^j =uxb ; tb .
:MechanismDesignwithOneAgentWeareinterestedin(i)describingwhichcontractscanbeimplementedand(ii)( )=(x( );t( ))isincentivecompatibleifandonlyif::Z U( ) U(0)=u (x(s);s))ds;()(s)()canberestatedintermsof rstorderconditions:dU=@u@y+@u=0:()d @y@ @ ,or U( ) U ^j ;8 ;b ;whichis: U( )=Ub j =MUb +uxb ; uxb ;b ;andthus U( ) Ub uxb ; uxb ;b :()Asymmetricconditiongivesus U ^ U( ) ux( ); ^ u(x( ); ):()Combining()and(),weget: u(x( ); ) ux( ); ^ U( ) Ub uxb ; uxb ;b :Supposewithoutlossofgeneralitythat > ^,thenmonotonicityofx( )isimmediatefrom@2u.@ @xDividingby ^;andtakingthelimitas !b atallpointsofcontinuityofx( )yieldsdU( )=u (x( ); );d( )( )isnondecreasing,itcanonlyhaveacountablenumberofdiscontinuities,whichhaveLebesguemeasurezero,andhencetheintegralrepresentationisvalid,indepen-dentofcontinuitypropertiesofx( ),whichisincontrastwiththerepresentationofincentivecompatibilitybythe rstordercondition().
,thenthereexists andb . Ub j =U( ):()Supposewithoutlossofgeneralitythat ^.Theinequality()impliesthattheinequalityin()isreversed: uxb ; uxb ;b >U( ) Ub integratingandusing()wegetZ Z usxb ;sds>us(x(s);s)ds ^ ^andrearrangingZ h iubsx ;s us(x(s);s)ds>0() ^butsince@2u>0@ @xandthemonotonicityconditionimpliedthatthisisnotpossible,andhencewehavethedesiredcontradic-tion.[Lecture9] [v(x( ); )+t( )]y( )subjectto u(x( ); ) t( ) ux ^; t ^;8 ; ^andu(x( ); ) t( ) u ..
:,wecanomitthetransferpaymentsfromthecontrolproblemandconcentrateontheoptimalchoiceofxasfollows:maxE [S(x( ); ) U( )]()x( )subjecttodU( )=u (x( ); )()d andx( )nondecreasing()andU( )=u ;(),thatintegrationbyparts,usedinthefollowingformZZdU(1 F)=U(1 F)+UfandhenceZZUf= U(1 F)+dU(1 F)AsZ1Z1E1dU( )1 F( ) [U( )]=U( )f( )d = [U( )(1 F( ))]0+f( )d ()00d f( )wecanrewrite()andusing(),weget max1 F( )E S(x( ); ) u (x( ); ) U(0)()f( )subjectto()and().,weget (x; )=S(x; ) 1 F( )u (x; ):()f( ) (x; )isquasiconcaveandhasauniqueinteriormaximuminx2R+forall 2 :,thentherelaxedproblemissolvedaty=y( )if:
. x(x( ); )=0:( )satsify01Z t( )=u(x( ); ) @U(0)+u (x(s);s)ds)A0Theny=(x;t) @u=@ >0;theIRcontractcanberelatedasU(0)=u .@2 (x; )=@x@ -laxedprogramsatis erentiating x(x( ); )=0forall andhenceweobtain=)dx( )= x d xxandas xx 0hastobesatis edlocally,weknowthatx( )hastobeincreasing, rstorderconditionsandobtain rstresults:Sx(x( ); )=1 F( )uxf (x( ); )=0( )impliese cientprovisionfor =1andunderprovisionofthecontractedactivityforallbutthehighesttype =1:Toseethatx^( )<x ( ),,orx^( ) x ( ).Observe rstthatforallxand <1@ (x; )<@S(x; )()@x@xby:@2u(x; )>0.()@x@ Thusatx ( ),byde nition
:MechanismDesignwithOneAgent@S(x ( ); )=0@xandusing()@ (x ( ); )<@S(x ( ); )=0@x@xButas (x; )isquasiconcavethederivativewithrespecttoxsatis esthe(downwardsinglecrossingcondition),whichimpliesthatfor@ (x^( ); )=0@xtoholdx^( )<x ( ).The rstorderconditioncanbewrittenastorepresentthetrade-o :f( )Sx(x( ); )=[1 F( )]ux (x( ); ).increaseinjointsurplusincreaseinagentrentsConsiderthenthemonopolyinterpretation,where:max(1 F(p))(p c)pandtheassociated rstorderconditionisgivenby:f(p)(p c)=1 F(p)marginalinframarginalThematerialinthislecturesisbaseson?.[Lecture10]
: !cany( )beimplementedincentivecompatible? !whatistheoptimalchoiceamongincentive-compatiblemechanisms?Todayweshallintroduceseveralgeneralconcepts: revelationprinciple e ciency ,I=f1;:::;,mechanismdesigner,center, i2 i,histype,thatdetermineshispreferencesoverY,describedbyautilityfunctionui(y; )( ),butitcouldalsocontaininformationabouthisneighbor,competitors, :f: 1 :::: I!Y:Theproblemisthat =( 1;::; I),withMI= Mi=, :M1 ::: MI!Y:
-nismde nesagamewithincompleteinformationforwhichmustchooseanequilibriumconcept,: i! =(M1;::;MI;g( ))isacollectionofstrategysets(M1;:::;MI)andanoutcomefunctiong:M1 ::: MI!Y:Withaslightabuseofnotation,weusethesamenotation,Mi,,wehavethefollowingcommutingdiagram: 1 ::: f( )I!Y&%mg;c( )g( )M1 ::: MIwheremg;cisthemappingthatassociatestoeveryI-tupleoftruecharacteristics ,,orsocialchoicefunctionf( ) =(M1;::;MI;g( ))implementsthesocialchoicefunctionf( )ifthereisanequilibriumpro le(m 1( 1);:::;m I( I))ofthegameinducedby suchthatg(m 1( 1);:::;m I( I))=f( 1;:::; I).Theidenti cationofimplementablesocialchoicefunctionisat rstglanceacomplexproblembecausewehavetoconsiderallpossiblemechanismg( )(validforalloftheimplementationversionsabove),therevelationprinciple,simpli = iandg( )=f( ) ( i)= iforeach 2 .De ( )istruthfullyimplementable(orincentivecompatible)ifthedirectrevelationmechanism =( ;f( ))hasanequilibrium(m 1( 1);:::;m I( I))inwhichm i( i)= iforall i2 i,foralli.
:M!X TThebestresponsestrategyoftheagentisamappingm : !Msuchthatm ( )2argmaxfu(g(m); )gm2Mandheobtainstheallocationg(m ( ))()Thereforewesaythat =(g( );M)implementf,wherefsatis esg(m ( ))=f( )()forevery .Thusbasedontheindirectmechanism =(g( );M)wecande neadirectmechanismassuggestedby(): d=(f( ); ).(RevelationPrinciple).Ifthesocialchoicefunctionf( )canbeimplementedthroughsomemechanism,thenitcanbeimplementedthroughadirectrevelation(truthful)mechanism: d=(f( ); ).(g;M)beamechanismthatimplementsf( )throughm ( )betheequilibriummessage:f( )=g(m ( ))Considerthedirectmechanism d=(f( ); ).Ifitwerenottruthful,thenanagentwouldprefertoannounce 0ratherthan ,andhewouldget u(f( ); )<uf( 0; )butbyde nitionofimplementation,andmorepreciselyby(),thesewouldimplythat u(g(m ( )); )<ugm 0; whichisacontradictiontothehypothesisthatm ( )wouldbeanequilibriumofthegamegeneratedbythemechanism (g( );M):
cienttorestrictattentiontomechanismswhicho eramenuofcontracts:agentannounces ,andwillgety( )=(x( );t( ))()butmanymechanismsintherealworldareindirectinthesensethatgoodsareprovidedindi erentqualitiesand/ .How?O eringamenuofallocationsfx( )g 2 andlett(x),t(x( ))=t( )bethecorrespondingnonlineartari ,sothatthemenuisfx;t(x)gx2XTheproofthatthenonlineartari isimplem en tingthesame al ; 0suchthatx( )=x ( )6=t 0,thentheagentwouldhaveaninteresttomisrepresen th isstateoftheworld,andhencethedirectmechanismcouldn'( )=t ,thetransferfunctioncanbeuniquelyde nedasifx=x( ))t(x)=t( );whichleadstothenonlineartari .Thenotionofadirectmechanismmaythenrepresentanormativeapproach, i2[0;1].ThesetofallocationsisY=X =(xi;ti),wherexi2f0;1gdenotestheassignmentoftheobjecttobidderi:noif0,:ui(yi; i)= ixi tiThesecondpricesealedbidauctionimplementsthefollowingsocialchoicefunctionf( )=(x0( );x1( );x2( );t0( );t1( );t2( ))withx1( )=1;if 1 2;=0if 1< 2:x2( )=1;if 1< 2;=0if 1 2:x0( )=0;forall ;andt1( )= 2x1( )t2( )= 1x2( )t0( )=t1( )x1( )+t2( )x2( )Themessagemiisthebidfortheobject:
: i!Mi=R+Theoutcomefunctionisg:M!Ywith8>><x1=1;t1=m2xg=2=0;t2=;ifm01 m2>>:x1=0;t1=0fmx1<m22=1;t2=m;i1Avarietyofotherexamplescanbegiven:(Incometax).xistheagent'sincomeandtistheamountoftaxpaidbytheagent; istheagent'(PublicGood).xistheamountofpublicgoodsupplied,andtiistheconsumeri'smonetarycontributionto nanceit; ,.[Lecture12]
(VotingGame).-tionsonthecharacteristics,, lem =(m 1( 1);:::;m I( I))isadominantstrategyequilibriumofmechanism =(M;g( ))ifforalliandall i2 i:ui(g(m i( i);m i); i) ui(g(m0i;m i); i)forallm0i6=mi,8m i2M (RevelationPrinciple).Let =fM;g( )gbeamechanismthatimplementsthesocialchoicefunctionf( )forthedominatequilibriumconcept,thenthereexistsadirectmechanism 0=f ;f( )gthatimplementsf( ).(Dueto?.).Letm ( )=(:::;m i( i);:::)beanI tupleofdominantmessagesfor(M;g( )).De neg tobethecompositionofgandm .g ,g m org ( ),g(m ( )).()Byde nitiong ( )=f( ).Infact =( ;g ( ))( ),tha tisthe ^i; isuchthat u ig ^i; i; i>ui(g ( ); i):Butbythede nition()above,theinequalitycanberewrittenas uigm ^i; i; i>ui(g(m ( )); i):Butthiscontradictsourinitialhypothesisthat =(M;g( ))implementsf( )inadominantstrategyequilibrium,asm isobviouslynotanequilibriumstrategyforagenti,whichcompletestheproof.
nedinDe nition??-mentationinaveryrobustway,intermsofstrategiesandininformationalrequirementsasthedesignerdoesn'tneedtoknowP( ):De ( )isdictatorialifthereisanagentisuchthatforall 2 ,f( )2fx2X:ui(x; i) ui(y; i),wecanstatethecelebratedresultfrom?and?.(Gibbard-Satterthwaite).SupposethatXcontainsatleastthreeelements,andthatRi=Pforalli,andthatf( )=(x;t; i)=ui(x; i) cientallocationbyx ( ).ThenthegeneralizationoftheVickreyauctionsstates:?,?,?.(Vickrey-Clark-Groves).Thesocialchoicefunctionf( )=(x ;t1( );:::;tI( ))istruthfullyimplementableindominantstrategiesifforalli:2X32X3 ti( i)=4uj(x ( ); 4j)5 ujx i( i); 5j:()j6=ij6=,thenthereexist i;b i;and isuchthat uix b i; i; i+tbi i; i>vi(x ( i; i); i)+ti( i; i)() Substitutingfrom()fortbi i; iandti( i; i),thisimpliesthatXI XIujx b i; i; j>uj(x ( ); j);()j=1j=1whichcontradictsx ( )beinganoptimalpolicy:Thus,f( )-poste cientbutitmaynotsatisfybudgetbalanXce:ti( i)=.
,, les =(s 1(v1);:::;s I(vI))isaBayesianNashequilibriumofmech-anism =(S1;::;SI;g( ))ifforalliandallv2 : Ev iuigs i(vi);s i(v i);vijvi Ev iuigsi;s i(v i);vijviforallsi6=s i(v),8s i(v).=(v1;:::;vI)anddenotethedistributionovertypesbysymmetricf(vi);F(vi).Foragentitowinwithvaluev,heshouldhaveahighervaluethanalltheotheragents,whichhappenswithprobabilityGI 1(v),(F(v))andtheassociateddensityis2g(v)=I (I 1)f(v)(F(v)):Thisisoftenreferredtoasthe rstorderstatisticofthesampleofsizeI -Nashequilibriuminthe rstpriceauctionisgivenbyZv g(y)b (v)=ydy0G(v) 1agents,providedthatthehighestvaluationisisbelowv.(Thusthebidderactsifheweretomatchtheexpectedvalueofhiscompetitor.)
\inspiredguesses," ( )beingsought:(i)Alltypesofbiddersubmitbids:b(v)6=;forallv 0:(ii)Bidderswithgreatervaluesbidhigher:b(v)>b(z)forallv>z:(iii)Thefunctionb( )isdi ,manyfunctionsotherthanb ( )satisfyGuesses(i) (iii).Wenowshowthattheonlypossibleequilibriumsatisfying the m is b ( ).From(ii)and(iii),b( ),hasaninversefunction, w hichwedenoteas ( );thatsatis esb b^=b^forallnumbersb^intherangeofb( ):Valuev= b^isthevalueabiddermusthaveinordertobidb^whenusingstrategyb( ).Becauseb( )strictlyincreases,theprobabilitythatbidderiwinsifhebidsbis n 1Qb^=F b^,G( (b)):()Why?Well,observethatsinceb( )isstrictlyincreasingandtheotherbidders'valuesarecontinuo us lydistributed,wecanignoreties:anotherbidderjbidsthesameb^onlyifhisvalueispreciselyvj= b^;^isequaltotheprob-abilityoftheotherbiddersbiddingnomorethanb^ : Becauseb( )strictlyincreases,thisi sth e p robabilityn 1thateachotherbidder'svalueisnotmorethan b^:ThisprobabilityisG( (b))=F b^;theprobabilitythatthemaximumofthebidders'valueisnomorethan (b):Thus,theexpectedpro tofatypevbidderwhobidsbwhentheothersuseb( )is v;b^=v b^G b^:() Becauseof(iii),theinversefunction ( )isdi 0b^beitsderivativeatb^,thepartial derivativeof v;b^withrespecttob^is b(v;b)= G( (b))+(v b)g( (b)) 0(b):()Sinceb( )isanequilibrium,b(v)isanoptimalbidforatypevbidder,=b(v)maximizes (v;b):The rstordercondition b(v;b(v))=0holds: G( (b(v)))+(v b(v))g( (b(v)) 0(b(v))0:()Use (b(v))=vand 0(b(v))=1=b0(v)towritethisas G(v))+(v b(v))g(v)=0:()b0(v)Thisisadi erentialequationthatalmostcompletelycharacterizestheexactnatureofb( ):(v)b0(v)+g(v)=vg(v):()Sinceg(v)=G0(v),theleftsideisthederivativeofG(v)b(v):Sowecanintegratebothsidesfromanyv0toanyv(using\y"todenotethedummyintegrationvariable):ZvG(v)b(v) G(v0)b(v0)=yg(y)dy()0
>FromGuessA,alltypesbid,sowecantakev0!0:Weknowb(0) 0;asr=0:Hence,G(v0)b(v0)!0asv0!0:Takethislimitin(6:5)anddividebyG(v)toobtainZv g(y)b(v)=y0Gdy=b (v):()(v),wecanrewrite(??)usingintegrationbyparts:ZZu0v=uv uv0asZv I 1)b (v)=v F(ydy0F(v):(v)=(q(v);t(v))isincentivecompatibleifandonlyif::ZviUi(vi) Ui(0)=uvi(qi(si);si))dsi;()(si)(vi;qi)=viqiwhereqi=Pr(xi=1)(v)=(q(v);t(v))isincentivecompatibleifandonlyif::ZviUi(vi) Ui(0)=qi(si)dsi;()0
(si):(RevenueEquivalence).Givenanytwoauctionsmechanism(directornot)(v)(0),(andmoregeneral),itisenoughtofocusattentionondirectrevelationgameswhereplayerstruthfullyreporttheirtypesandanallocationischosen(perhapsrandomly) bi=qi(si): maxv)EvS(q(v);v) 1 F(uv(q(v);v) U(0)()f(v)andtheextensiontomanyagentsistrivial:"XI #max1 Fi(vEi)vS(q(v);v) uiv(qi(v);vi) Ui(0)()q(v)i=1fi(vi)andmakinguseoftheauctionenvironmentwegetZ1Z1 XI !"#vYImaxi)::qi(v)vi 1 Fi( fi(vi)dv1 dvIq(v)fi(vi)vi=11=0vI=0i=1whereq=(q1;:::;qI)()andXIqi 0;qi =1Buttheoptimizationproblemcanbesolvedpointwiseforeveryv=(v1;:::;vI):XI maxqi(v)vi 1 Fi(vi)qfi(vi)i=1
,followingMyerson,byv ithevirtualutilityofagentiv~i,vi 1 Fi(vi)fi(vi)Itisthenoptimaltosetthereservepriceforeveryagentsuchthatri=visatis es:vi 1 Fi(vi)=0fi(vi)
(OptimalAuction).Theoptimalpolicyq =(q 1;:::;q I)isthengivenby: i=0ifallv~i< ii=1if9v~i i>0)v~i=maxfv~1;:::;v~+Fi(ci)fi(ci)(withvaluationv1)andabuyer(withvaluationv2)whowishestoengageinatrade:virtualutilitynowbecomesidenticalto:ZZ v2 1 F2(v2) v1+F1(v1)f2(v2)f1(v1)pointwiseoptimizationoccurswhen v2 1 F2(v2) 1)v1+F1(v>0f2(v2)f1(v1)butforv2=v1thedi erenceofthevirtualutilitiesisnegative 2))v2 1 F2(v v1+F1(v1<0f2(v2)f1(v1)andthusingeneraline ciencyarediscussedin?.]
ciencyAgametheoristoramediatorwhoanalyzestheParetoe ciencyofbehaviorinagamewithincompleteinformationmustusetheperspectiveofanoutsider,sohecannotbasehisanalysisontheplayers'',,HolmstromandMyerson(1983)arguedthattheconceptofe ciencyforgameswithincompleteinformationshouldbeappliedtothemechanisms,ratherthantooutcomes,andthecriteriafordeterminingwhetheraparticularmechanism ise cientshoulddependonlyonthecommonlyknownstructureofthegame, nitionofParetoe ciencyinaBayesiancollective-choiceproblemis:Amechanismise cientifandonlyifnootherfeasiblemechanismcanbefoundthatmightmakesomeotherindividualsbettero andwouldcertainlynotmakeotherindividualsworseo .However,thisde ,wemustspecifywhatinformationistobeconsideredwhendeterminingwhetheranindividualis\bettero "or\worseo ."Amechanism
cannowbethoughtofascontigentallocationplan
: ! (Y)andconsequently
(yj )expressestheconditionalprobabilitythatyisrealizedgivenatypepro le .Onepossibilityistosaythatanindividualismadeworseo byachangethatdecreaseshisexpectedutilitypayo aswouldbecomputedbeforehisowntypeoranyotherindividuals'typearespeci ,wesaythatamechanism
isexanteParetosuperiortoanothermechanism ifandonlyifXXXXp( )
(yj )ui(y; ) p( ) (yj )ui(y; );8i2I;() 2 y2Y 2 neUi( j i)=pi( ij i) (yj )ui(y; ) i2 iy2Yandnoticethat
ciencyXp( ) (yj )ui(y; )=pi( i)Ui( j i):() 2 y2Y i2 iAnotherpossibilityistosaythatanindividualismadeworseo byachangethatdecreaseshisconditionallyexpectedutility,givenhisowntype(buttogiventhetypeofanyotherindividuals).Anoutsideobserver,whodoesnotknowanyindividual'stype,wouldthensaythataplayeri\wouldcertainlynotbemadeworseo (bysomechangeofmechanism)"inthissenseifthisconditionallyexpectedutilitywillnotbedecreased(bythechange)foranypossibletypeofplayeri:Thisstandardiscalledtheinterimwelfarecriterion,becauseitevaluateseachplayer'swelfareafterhelearnshisowntypebutbeforehelearnsanyotherplayer',wesaythatamechanism
isinterimParetosuperiortoanothermechanism ifandonlyifUi(
j i) Ui( j i);8i2I;8 i2 i;() byachangethatdecreaseshisconditionallyexpectedutility,\wouldcertainlynotbemadeworseo ",,wesaythatamechanism
isexpostparetosuperiortoanothermechanism ifandonlyifXX
(yj )ui(y; ) (yj )ui(y; );8i2I;8 2 ;()y2Yy2YandthisinequalityisstrictforatleastoneplayerinNandatleastonepossiblecombinationoftypes in suchthatp( )>0:Givenanyconceptoffeasibility,thethreewelfarecriteria(exante,interim,expost)giverisetothreedi erentconceptsofe ciencyForanysetofmechanisms (tobeinterpretedasthesetof\feasible"mechanismsinsomesense),wesaythatamechanism isexantee cientintheset ifandonlyif isin andthereexistsnoothermechanismvthatisin andisexanteParetosuperiorto :Similarly isinterime cientin ifandonlyif isin andthereexistsnoothermechanismvthatisin andisinterimParetosuperiorto ;and isexposte cientin ifandonlyif isin andthereexistsnoothermechanismvthatisin andisexpostParetosuperiorto . ciciencynotionscanbede ,,.,tothesetofincentivefeasiblemechanism,thatisallmechanismswhichsatisfytheincentivecompatibilitycondition.
, :RI!R: (x;y)2Xandforall I2A,8i;x iy)xFpy::x iy iz)ci(x)=1;ci(y)=2;ci(z)= esindependenceofirrelevantalternativesiffor Iand 0Iwiththepropertythat8i;8(x;y): ijfx;yg= 0ijfx;yg)Fjfx;yg=F0jfx;yg:'tsatsifyIIAforaswechangefromx iz iy;andy jx jz,tox iy iz;andy jz jx,:x 1y 1z;z 2x 2y;y 3z (x;y)2X;8 I2A,x iy)xFpy: 3andA= :A!Xassignsf( I)2Xforevery I2A.
