Dynamic games of incomplete informationTwo-period reputation gameSlide 3Slide 4Spence’s education gameSlide 6Slide 7Basic signaling gameIdea behind Perfect Bayesian equilibriumPerfect Bayesian EquilibriumThe repeated public good gameSlide 12Analysis of second periodSlide 14Equilibrium of the gameSequential equilibrium: PreliminariesSequential equilibriumSome properties of sequential equilSequential equilibrium vs PBECournot competition: incomplete infoSlide 21Slide 22Complete info benchmarkRevealing costs to a rivalSlide 25Slide 26Example: Signaling willingness to paySlide 28Lemons: Problem of quality uncertaintySlide 30Signaling quality through warrantiesSlide 32Dynamic games of incomplete information.Two-period reputation game•Two firms, i =1,2, with firm 1 as ‘incumbent’ and firm 2 as ‘entrant’•In period 1, firm 1 decides a1={prey, accommodate}•In period 2, firm 2 decides a2={stay, exit}•Firm 1 has two types: sane (wp p) or crazy (wp 1-p) •Sane firm has D1/ P1 if it accommodates/preys, D1> P1 •However, being monopoly is best, M1> D1 •Firm 2 gets D2/ P2 if firm 1 accommodates/preys, with D2>0> P2 •How should this game be played?Two-period reputation game•Key idea: Unless it is crazy, firm 1 will not prey in second period. Why?•Of course, crazy type always preys. What will sane type do?•Two kinds of equilibria:1. Separating equilibrium- different types of firm 1 choose different actions2. Pooling equilibrium- different types of firm 1 choose the same action•In a separating equilibrium, firm 2 has complete info in second period: μ(θ=sane| a1= accommodate)=1, and μ(θ=crazy| a1= prey)=1•In a pooling equilibrium, firm 2 can’t update priors in the second period: μ(θ=sane| a1= prey)=pTwo-period reputation game•Separating equil: -Sane firm1 accommodates, 2 infers that firm 1 is sane and stays in. -Crazy firm1 preys, 2 infers that firm 1 is crazy and exits.-Above equil is supported if: δ(M1- D1)≤ D1- P1 •Pooling equil: -Both types of firm 1 prey, firm 2 has posterior beliefs μ(θ=sane| a1= prey)=p & μ(θ=sane| a1= accommodate)=1, and stays in iff accommodation is observed -Pooling equil holds if: δ(M1- D1)> D1- P1 -Also pooling equil requires: pD2+(1-p)P2≤0Spence’s education game•Player 1 (worker) chooses education level a1≥0•Private cost of education a1 is a1/θ, θ is ability•Worker’s productivity in a firm is θ•Player 2 (firm) minimizes the difference of wage (a2) paid to player 1 and 1’s productivity θ•In equilibrium, wage offered, a2(a1)=E(θ|a1) •Let player 1 have two types, θ/ & θ//, wp p/ & p//•Let σ/ & σ// be equilibrium strategies, with: a1/є support(σ/) and a1//є support(σ//) •In equilibrium, a2(a/1)-a/1/ θ/ ≥ a2(a//1)-a//1/ θ/ and a2(a//1)-a//1/ θ// ≥ a2(a/1)-a/1/ θ//, implying, a//1≥a/1Spence’s education game•Separating equilibrium: -Low-productivity worker reveals his type and gets wage θ/. He will choose a/1=0-Type θ// cannot play mixed-strategy-a2(a/1)-a/1/ θ/ ≥ a2(a//1)-a//1/ θ/ gives, a//1≥ θ/(θ//-θ/)-a2(a//1)-a//1/ θ// ≥ a2(a/1)-a/1/ θ//, gives a//1≤ θ//(θ//-θ/)-Thus, θ/(θ//-θ/) ≤ a//1≤ θ//(θ//-θ/)-Consider beliefs: {μ(θ/|a1)=1 if a1 ≠ a//1, μ(θ/|a//1)=0}-With these beliefs, (a/1=0, a//1) with θ/(θ//-θ/) ≤ a//1≤ θ//(θ//-θ/), is a separating equilibrium- In fact, there are a continuum of such equilibria!Spence’s education game•Pooling equilibrium:-Both types choose same action,-The wage is then-Consider beliefs,- With these beliefs, is the pooling equil education level iff for each θ, θ/≤aaa~//1/1//////12)~(ppaa 1~a/~1//////app }~ if ,1)({111/aaa Basic signaling game•Player 1 is sender and player 2 is receiver•Player 1’s type is θєΘ, 2’s type is common knowledge•I plays action a1є A1, 2 observes a1and plays a2є A2.•Spaces of mixed actions are A1 and A2 •2 has prior beliefs, p, about 1’s types•Strategy for 1 is a distribution σ1(.|θ) over a1 for type θ•2’s strategy is distribution σ2(.|a1) over a2 for each a1•Type θ’s payoff to σ1(.|θ) when 2 plays σ2(.|a1) is:•Player 2’s ex-ante payoff to σ2(.|a1) when 1 plays σ1(.|θ) is:),,()()(),,(211122112111 2aauaaaua a),,()()()(),,(212122112121 2aauaaapua aIdea behind Perfect Bayesian equilibrium•Since 2 observes 1’s action before moving, he should use this fact before he moves•Thus 2 should update priors about 1’s type p to form posterior distribution μ(θ|a1) over Θ•This is done by using Baye’s rule•Extending idea of subgame perfection to Bayesian equil requires 2 to maximize payoff conditional on a1.•Conditional payoff to σ2(.|a1) is 2),,()()(),)(.,()(212122112121aaauaaaaauaPerfect Bayesian Equilibrium•A PBE of a signaling game is a strategy profile σ* and posterior beliefs μ(|a1) such that:1.2.3. and μ(|a1) is any probability distribution on Θ if ),,,(maxarg)(.,*211*11u),,,()(maxarg)(.,21211*212auaaa0)()( if ,)()()()()(///1*1//1*1/1*11apapapa0)()( //1*1/apThe repeated public good game•Two players i=1,2 decide whether to contribute in periods t=1,2•The stage game is•Each player’s cost ci is private knowledge•It is common knowledge that ci is distributed on [ , ] with distribution P(.). Also, <1< •The discount factor is δ1 \ 2Contribute Not contributeContribute 1-c1, 1-c21-c1, 1Not contribute1, 1-c20, 0ccccThe repeated public good game•One shot game:-The unique Bayesian equilibrium is the unique solution to c*=1-P(c*)-The cost of contributing equals probability that opponent won’t contribute-Types ci ≤ c* contribute, others don’t•In repeated version, with action space {0, 1}, a strategy for player i is a pair (σ0i(1| ci), σ1i(1| h1, ci)) corresp to 1st/ 2nd period prob of contributing where history is h1 є {00, 01, 10, 11}•In period 1, i contributes iff ci ≤ c^. In a symmetric PBE 1ˆ0 ,ˆˆˆ21 ccccAnalysis of second period•Neither player contributed:-Both players learn that rival’s cost exceeds -Posterior beliefs are and P(ci |00)=0 if ci ≤ c^ .-In a (symm) 2nd period equil each player contributes iff -In period 2, type