Bayesian Nash Equilibrium - UCLA Economics

1 Bayesian Nash EquilibriumIchiro ObaraUCLAF ebruary 1, 2012 Obara (UCLA) Bayesian Nash EquilibriumFebruary 1, 20121 / 28 Bayesian GameBayesian GameWe like to model situations where each party holds some privateinformation. For example,IA bidder does not know other bidders values in trader has some insider information about a recent technologicalinnovation by some (UCLA) Bayesian Nash EquilibriumFebruary 1, 20122 / 28 Bayesian GameBayesian GameBayesian GameABayesian Gameconsists ofplayer:a finite setNstate:a set action:a setAifor eachi Ntype:a setTifor eachi Nbelief:a functionpi:Ti ( j6=iTj)for eachi Npayoff:a functionui:A <for eachi (UCLA) Bayesian Nash EquilibriumFebruary 1, 20123 / 28 Bayesian GameBayesian GameInterpretation:I is a set of possiblestates of naturethat determine all physicalsetup of the game (payoffs).

2 ITiis the set ofi s privatetypesthat encode playeri sinformation/knowledge (ex. private signal).Ipiis playeri sinterim beliefabout the state and the other players are already familiar with the basic idea of BG: correlated Equilibrium andvariety of interpretations of mixed strategy (Harsanyi s purification argumentetc.) are some special cases. BG introduces incomplete information intogames in a very flexible (UCLA) Bayesian Nash EquilibriumFebruary 1, 20124 / 28 Bayesian GameBayesian GameBayesian games are often described more simply by eliminating belief onT i:pi(t i|ti) := pi( ,t i|ti)IPayoff onA T:ui(a,t) := ui(a, )pi( |t)wherepi( |t) =pi( ,t i|ti) pi( ,t i|ti).In this formulation, type encodes both payoff and (UCLA) Bayesian Nash EquilibriumFebruary 1, 20125 / 28 Bayesian GameExample: AuctionConsider the standard model of first price auction where each bidderonly knows her value and have a belief about the other bidders values.

3 This is a simple Bayesian game whereIthe set of players (bidders) isNIthe set of states isV1 .., VnIthe set of actions for bidderiisAi=<+Ithe set of types for bidderiisViIbidderi s interim belief ispi(v i|vi).Ibidderi s payoff isui(b,v) =1(bi maxj6=ibj)(vi bi).Obara (UCLA) Bayesian Nash EquilibriumFebruary 1, 20126 / 28 Bayesian GameCommon Prior Assumption (CPA)We almost always assume that all the interim beliefs are derived fromthe same prior. That is, we assumecommon Prior AssumptionA Bayesian Game (N, ,(Ai),(Ti),(pi),(ui)) satisfies thecommon priorassumptionif there existsp ( i NTi) such thatpi( ,t i|ti),ti Ti,i Nare all conditional distributions derived assumption is not entirely convincing, but it is a useful (UCLA) Bayesian Nash EquilibriumFebruary 1, 20127 / 28 Bayesian Nash EquilibriumBayesian Nash EquilibriumPlayeri s strategysiis a mapping fromTitoAi.

4 LetSibe the set ofplayeri s strategies. It is like a contingent plan of (s1,..,sn), playeri s interim expected payoff for typetiisE[ui((si(ti),s i( t i)), )|ti]:= t i T iui((si(ti),s i(t i)), )pi( ,t i|ti)A strategy profiles= (s1,..,sn) is aBayesian Nash Equilibriumiffor everyi N,siassigns an optimal action for eachtithatmaximizes playeri s interim expected (UCLA) Bayesian Nash EquilibriumFebruary 1, 20128 / 28 Bayesian Nash EquilibriumBayesian Nash EquilibriumHere is a formal Nash Equilibriums = (s 1,..,s n) Sis aBayesian Nash EquilibriumifE[ui((s i(ti),s i( t i)), )|ti] E[ui((ai,s i( t i)), )|ti]holds for everyai Aiandti Ti, for everyi (UCLA) Bayesian Nash EquilibriumFebruary 1, 20129 / 28 Bayesian Nash Bayesian Nash Equilibrium can be regarded as a Nash Equilibriumof some appropriately defined strategic interpretation is to regard each type as a distinct player and regardthe game as a strategic game among such i|Ti|players (cf.)

5 Definitionin O&R). Then a BNE can be regarded as a NE of this strategic that there exists common priorp. LetUi(s) =E[ui((si( ti),s i( t i)), )]be playeri s ex anteexpectedpayoff givens S. Then a BNEs is a NE of strategic game(N,(Si),(Ui)).Obara (UCLA) Bayesian Nash EquilibriumFebruary 1, 201210 / 28 Bayesian Nash that there are finite actions and finite types for each this case, the whole game can be regarded as a finite strategicgame (in either interpretation). In this setting, we can allow eachtype to randomize over actions as we did in mixed strategy NE. Thenwe can define a mixed strategy BNE and it follows immediately fromthe existence of MSNE that there exists a (UCLA) Bayesian Nash EquilibriumFebruary 1, 201211 / 28 ApplicationsExample 1: Second Price AuctionSuppose thatv= (v1,..,vn) is generated by common priorp ([0,1]n) in second price it is a dominant action to bid one s true value, bidders beliefsare not relevant for their decision.

6 Henceb = (b 1,..,b n), whereb i(vi) =vi, is a (UCLA) Bayesian Nash EquilibriumFebruary 1, 201212 / 28 ApplicationsExample 2: First Price AuctionAssume thatviis across bidders and follow the uniformdistribution on [0,1].We look for a symmetric BNE (b ,..,b ) in first price use guess and verify method : we assumeb(v) = vfor some ,then verify that this strategy is in fact optimal against itself for some .Obara (UCLA) Bayesian Nash EquilibriumFebruary 1, 201213 / 28 ApplicationsExample 2: First Price AuctionBidderi s expected payoff with valuevi [0,1] and bidbi [0,1] isPr(win|bi)(vi bi)=Pr(maxj6=ivj bi )(vi bi)=(bi )n 1(vi bi)The first order condition is:n 1 (bi )n 2(vi bi) (bi )n 1= 0n 1nvimaximizes the payoff givenvi(independent of ).Henceb (v) =n 1nvis the optimal bid for each bidder when all the otherbidders are using it.

7 That is, it is a symmetric BNE for every bidder tofollowb (v) =n (UCLA) Bayesian Nash EquilibriumFebruary 1, 201214 / 28 ApplicationsExample 2: First Price AuctionNext we consider general cumulative distribution functionF(still isassumed).Assume that the symmetric BNEb is strictly increasing and expected payoff for typevbidder isPr(win|b)(v b) =F(b 1(b))n 1(v b)The typevbidder s optimal bid b(v) is obtained from the following firstorder condition:(n 1)f(b 1( b(v)))F(b 1( b(v)))n 2b (b 1( b(v))(v b(v)) F(b 1( b(v)))n 1= 0 Obara (UCLA) Bayesian Nash EquilibriumFebruary 1, 201215 / 28 ApplicationsExample 2: First Price AuctionSince b(v) =b (v) in Equilibrium , this differential equation simplifies to(F(v)n 1) b (v)(v b(v)) F(v)n 1= this, we obtainIb (v) = v0(F(x)n 1) xdxF(v)n , by integration by parts,b (v) =v v0F(x)n 1dxF(v)n 1 IWe can verify that (1) this is in fact strictly increasing anddifferentiable and (2) second order condition is (UCLA) Bayesian Nash EquilibriumFebruary 1, 201216 / 28 ApplicationsExample 2.)

8 First Price AuctionBayesian Nash Equilibrium for the first price auctionIt is a Bayesian Nash Equilibrium for every bidder to follow the strategyb(v) =v v0F(x)n 1dxF(v)n 1for the first price auction with private (UCLA) Bayesian Nash EquilibriumFebruary 1, 201217 / 28 ApplicationsExample 3: Cournot Competition with Private CostConsider a Cournot model where each firm s cost is privateinformation and drawn from [0,1] according to the same CDFF independently. Letcbe the average that the inverse demand function isp(Q) = 3 s try to find a symmetric BNE (q ,..,q ).Obara (UCLA) Bayesian Nash EquilibriumFebruary 1, 201218 / 28 ApplicationsExample 3: Cournot Competition with Private CostFirmi s expected profit when its cost isciandqiis produced is i(qi,q ) =E[(3 qi (n 1)q ( c) ci)qi]From FOC, we obtainqi(ci) =3 (n 1)E[q ( c)] ci2 Taking the expectation and imposing symmetry, we haveE[q ( c)] =3 cn+ (q.)

9 ,q ), whereq (c) =3 cn+1 c c2, is the symmetric (UCLA) Bayesian Nash EquilibriumFebruary 1, 201219 / 28 ApplicationsExample 4: Double Auction (Chatterjee and Samuelson1983)One buyer and one s value and and seller s value is independently and uniformlydistributed on [0,1].Buyer s payoff isvb pwhen trading at pricep, 0 otherwise. Seller spayoff isp vswhen trading at pricep, 0 s strategy ispb: [0,1] [0,1] (price offer) and seller s strategyps: [0,1] [0,1] (asking price).Trade occurs at pricepb+ps2only ifpb ps. Otherwise no (UCLA) Bayesian Nash EquilibriumFebruary 1, 201220 / 28 ApplicationsExample 4: Double AuctionLook for a linear BNEp b(vb) =ab+cbvb,p s(vs) =as+ this linear strategy by the seller, the buyer s problem ismaxpb[vb 12{pb+as+pb2}]pb ascsFrom the first order condition, we obtainpb(vb) =23vb+13asObara (UCLA) Bayesian Nash EquilibriumFebruary 1, 201221 / 28 ApplicationsExample 4: Double AuctionSeller s problem ismaxps[12{ps+ps+ab+cb2} vs]ab+cb pscbFrom the first order condition, we obtainps(vs) =23vs+13(ab+cb)Obara (UCLA) Bayesian Nash EquilibriumFebruary 1, 201222 / 28 ApplicationsExample 4: Double AuctionMatching coefficients, we getab= 1/12,as= 1/4,cb=cs= 2 we find one BNE:p b(vb) =23vb+112p s(vs) =23vs+14 Obara (UCLA) Bayesian Nash EquilibriumFebruary 1, 201223 / 28 ApplicationsExample 4.

10 Double AuctionRemarkAn allocation is efficient if trade occurs whenevervb vs. So thisBNE does not generate an efficient this uniform distribution environment, however, this BNE is themost efficient one. Thus it is impossible to achieve the efficientallocation by any BNE in double inefficiency result is very general. The efficient allocation cannotbe achieved by ANY Bayesian Nash Equilibrium in ANY (UCLA) Bayesian Nash EquilibriumFebruary 1, 201224 / 28 ApplicationsExample 5: Global Game (Carlson and van Damme 1993)There are two investorsi= 1,2. Each investor chooses whether topurchase risky asset (RA) or safe asset (SA).Each investor s payoff depends on the state of economy <andthe other investor s Structure:I is uniformly distributed on<.IInvestoriobserves a private signalti= + i, where ifollowN(0, ).IGiventi, investoribelieves that iis distributed according toN(ti, )andt iis distributed according toN(ti, 2 ).