1Lectures 12-13Incomplete InformationStatic Case14.12 Game Theory Muhamet YildizRoad Map1. Examples2. Bayes’ rule3. Definitions1. Bayesian Game2. Bayesian Nash Equilibrium4. Mixed strategies, revisited5. Economic Applications1. Cournot Duopoly2. Auctions3. Double Auction2Incomplete informationWe have incomplete (or asymmetric) information if one player knows something (relevant) that some other player does not know.An ExampleNatureHigh pLow 1-pFirmHireWWorkShirkDo not hire(1, 2)(0, 1)(0, 0)Do nothireHireWWorkShirk(1, 1)(-1, 2)(0, 0)3The same exampleNatureHigh pLow1-pFirmhireWWorkShirk(1, 2)(0, 1)(0, 0)Do notHireWWorkShirk(1, 1)(-1, 2)Another ExampleNatureHigh 0.5Low .5SellerpBBuyBuyDon’t(p, 2-p)(0, 0)(0, 0)p’pBBuyDon’t(p, 1-p)(p’, 1-p’)(0, 0)p’Don’tDon’tBuyBB(p’,2-p’)(0,0)What would you askif you were to choosep from [0,4]?4Same “Another Example”NatureHigh 0.5Low .5SellerpBBuyBuyDon’t(p, 2-p)(0, 0)(0, 0)p’BBuyDon’t(p, 1-p)(p’, 1-p’)(0, 0)Don’tDon’tBuyBB(p’,2-p’)(0,0)What would you askif you were to choosep from [0,4]?NatureHigh 0.5Low .5Bayes’ RuleProb(A and B)•Prob(A|B) = Prob(B)• Prob(A and B) = Prob(A|B)Prob(B) = Prob(B|A)Prob(A)Prob(B|A)Prob(A)•Prob(A|B) = Prob(B)5Example• Prob(Work|Success) = µp/[µp + (1−µ)(1-p)]• Prob(Work|Failure) = (1-µ)p/[µ(1−p) + (1−µ)p]SuccessFailureWorkµShirk1−µp1-p1-pp0 0.2 0.4 0.6 0.8 100.10.20.30.40.50.60.70.80.91µP(w|S),P(W|F)P(W|S) P(W|F)6Bayesian Game (Normal Form)A Bayesian game is a list G = {A1,…,An;T1,…,Tn;p1,…,pn;u1,…,un}where •Aiis the action space of i (aiin Ai)•Tiis the type space of i (ti)•pi(t-i|ti) is i’s belief about the other players•ui(a1,…,an;t1,…,tn) is i’s payoff.An ExampleNatureHigh pLow 1-pFirmHireWWorkShirkDo not hire(1, 2)(0, 1)(0, 0)Do nothireHireWWorkShirk(1, 1)(-1, 2)(0, 0)TFirm={tf};TW= {High,Low}AFirm= {Hire, Don’t}AW= {Work,Shirk}pF(High) = ppF(Low) = 1-p7Bayesian Nash equilibriumA Bayesian Nash equilibrium is a Nash equilibrium of a Bayesian game. Given any Bayesian game G = {A1,…,An;T1,…,Tn;p1,…,pn;u1,…,un} a strategy of a player i in a is any function si:Ti → Ai;A strategy profile s* = (s1*,…, s1*) is a Bayesian Nash equilibrium iff si*(ti) solvesi.e., si*is a best response to s-i*.() ( ) ( ) ()()()iiiTtnniiiiiiAattpttstsatstsuiiii|;,...,,,,...,*1*11*11*1max−∈++−−∈∑−−An ExampleNatureHigh pLow 1-pFirmHireWWorkShirkDo not hire(1, 2)(0, 1)(0, 0)Do nothireHireWWorkShirk(1, 1)(-1, 2)(0, 0)TFirm={tf};TW= {High,Low}AFirm= {Hire, Don’t}AW= {Work,Shirk}pF(High) = p >1/2pF(Low) = 1-psF* = Hire, sF* (High) = WorksF* (Low) = ShirkAnother equilibrium?8Stag Hunt, Mixed Strategy(6,6)(0,4)(4,0)(2,2)Mixed Strategies• t and v are iid with uniform distribution on [−ε,ε].• t and v are privately known by 1 and 2, respectively, i.e., are types of 1 and 2, respectively.• Pure strategy: –s1(t) = Rabbit iff t > 0;–s2(v) = Rabbit iff t > 0.• p = Prob(s1(t)=Rabbit|v) = Prob(t > 0) = 1/2.• q = Prob(s2(v)=Rabbit|t) = 1/2.6,60,4+v4+t,02+t,2+vU1(R|t) = t +2q+4(1-q) = t + 4 – 2qU1(S|t) = 6(1-q);U1(R|t) > U1(S|t) Ù t+4–2q > 6(1-q)Ù t > 6-6q+2q-4 = 2 – 4q =