1、Static(or Simultaneous-Move)Games of Complete InformationMixed Strategy Nash EquilibriumMay 29,20031Outline of Static Games of Complete Information nIntroduction to gamesnNormal-form(or strategic-form)representation nIterated elimination of strictly dominated strategies nNash equilibriumnReview of c
2、oncave functions,optimizationnApplications of Nash equilibrium nMixed strategy Nash equilibrium May 29,20032Todays AgendanReview of previous classnMixed strategy Nash equilibrium in Battle of sexesnUse indifference to find mixed strategy Nash equilibriaMay 29,20033Mixed strategy equilibriumnMixed St
3、rategy:nA mixed strategy of a player is a probability distribution over the players strategies.nMixed strategy equilibriumA probability distribution for each playerThe distributions are mutual best responses to one another in the sense of expected payoffsMay 29,20034nChris expected payoff of playing
4、 Opera:2qnChris expected payoff of playing Prize Fight:1-qnChris best response B1(q):Prize Fight(r=0)if q1/3 Any mixed strategy(0r1)if q=1/3Battle of sexesPatOpera (q)Prize Fight (1-q)ChrisOpera (r)2,10,0Prize Fight(1-r)0 ,01,2May 29,20035nPats expected payoff of playing Opera:rnPats expected payoff
5、 of playing Prize Fight:2(1-r)nPats best response B2(r):Prize Fight(q=0)if r2/3Any mixed strategy(0q1)if r=2/3,Battle of sexesPatOpera (q)Prize Fight (1-q)ChrisOpera (r)2,10,0Prize Fight(1-r)0 ,01,2May 29,200361qr1nChris best response B1(q):nPrize Fight(r=0)if q1/3 nAny mixed strategy(0r1)if q=1/3nP
6、ats best response B2(r):Prize Fight(q=0)if r2/3 Any mixed strategy(0q1)if r=2/3Battle of sexes2/3Three Nash equilibria:(1,0),(1,0)(0,1),(0,1)(2/3,1/3),(1/3,2/3)1/3May 29,20037Expected payoffs:2 players each with two pure strategiesnPlayer 1 plays a mixed strategy(r,1-r).Player 2 plays a mixed strate
7、gy(q,1-q).nPlayer 1s expected payoff of playing s11:EU1(s11,(q,1-q)=qu1(s11,s21)+(1-q)u1(s11,s22)Player 1s expected payoff of playing s12:EU1(s12,(q,1-q)=qu1(s12,s21)+(1-q)u1(s12,s22)nPlayer 1s expected payoff from her mixed strategy:v1(r,1-r),(q,1-q)=r EU1(s11,(q,1-q)+(1-r)EU1(s12,(q,1-q)Player 2s2
8、1 (q)s22 (1-q)Player 1s11 (r)u1(s11,s21),u2(s11,s21)u1(s11,s22),u2(s11,s22)s12(1-r)u1(s12,s21),u2(s12,s21)u1(s12,s22),u2(s12,s22)May 29,20038Expected payoffs:2 players each with two pure strategiesnPlayer 1 plays a mixed strategy(r,1-r).Player 2 plays a mixed strategy(q,1-q).nPlayer 2s expected payo
9、ff of playing s21:EU2(s21,(r,1-r)=ru2(s11,s21)+(1-r)u2(s12,s21)Player 2s expected payoff of playing s22:EU2(s22,(r,1-r)=ru2(s11,s22)+(1-r)u2(s12,s22)nPlayer 2s expected payoff from her mixed strategy:v2(r,1-r),(q,1-q)=q EU2(s21,(r,1-r)+(1-q)EU2(s22,(r,1-r)Player 2s21 (q)s22 (1-q)Player 1s11 (r)u1(s1
10、1,s21),u2(s11,s21)u1(s11,s22),u2(s11,s22)s12(1-r)u1(s12,s21),u2(s12,s21)u1(s12,s22),u2(s12,s22)May 29,20039Mixed strategy equilibrium:2-player each with two pure strategiesnMixed strategy Nash equilibrium:nA pair of mixed strategies (r*,1-r*),(q*,1-q*)is a Nash equilibrium if(r*,1-r*)is a best respo
11、nse to(q*,1-q*),and(q*,1-q*)is a best response to(r*,1-r*).That is,v1(r*,1-r*),(q*,1-q*)v1(r,1-r),(q*,1-q*),for all 0 r 1v2(r*,1-r*),(q*,1-q*)v2(r*,1-r*),(q,1-q),for all 0 q 1Player 2s21 (q)s22 (1-q)Player 1s11 (r)u1(s11,s21),u2(s11,s21)u1(s11,s22),u2(s11,s22)s12(1-r)u1(s12,s21),u2(s12,s21)u1(s12,s2
12、2),u2(s12,s22)May 29,2003102-player each with two strategiesnTheorem 1(property of mixed Nash equilibrium)nA pair of mixed strategies(r*,1-r*),(q*,1-q*)is a Nash equilibrium if and only if v1(r*,1-r*),(q*,1-q*)EU1(s11,(q*,1-q*)v1(r*,1-r*),(q*,1-q*)EU1(s12,(q*,1-q*)v2(r*,1-r*),(q*,1-q*)EU2(s21,(r*,1-
13、r*)v2(r*,1-r*),(q*,1-q*)EU2(s22,(r*,1-r*)Player 2s21 (q)s22 (1-q)Player 1s11 (r)u1(s11,s21),u2(s11,s21)u1(s11,s22),u2(s11,s22)s12(1-r)u1(s12,s21),u2(s12,s21)u1(s12,s22),u2(s12,s22)May 29,200311Theorem 1:illustrationnPlayer 1:EU1(H,(0.5,0.5)=0.5(-1)+0.51=0EU1(T,(0.5,0.5)=0.51+0.5(-1)=0v1(0.5,0.5),(0.
14、5,0.5)=0.5 0+0.5 0=0nPlayer 2:EU2(H,(0.5,0.5)=0.51+0.5(-1)=0EU2(T,(0.5,0.5)=0.5(-1)+0.51=0v2(0.5,0.5),(0.5,0.5)=0.50+0.50=0Matching penniesPlayer 2H(0.5)T(0.5)Player 1H(0.5)-1,1 1,-1T(0.5)1,-1-1,1May 29,200312Theorem 1:illustrationnPlayer 1:v1(0.5,0.5),(0.5,0.5)EU1(H,(0.5,0.5)v1(0.5,0.5),(0.5,0.5)EU
15、1(T,(0.5,0.5)nPlayer 2:v2(0.5,0.5),(0.5,0.5)EU2(H,(0.5,0.5)v2(0.5,0.5),(0.5,0.5)EU2(T,(0.5,0.5)nHence,(0.5,0.5),(0.5,0.5)is a mixed strategy Nash equilibrium by Theorem 1.Matching penniesPlayer 2H(0.5)T(0.5)Player 1H(0.5)-1,1 1,-1T(0.5)1,-1-1,1May 29,200313nEmployees expected payoff of playing“work”
16、EU1(Work,(0.5,0.5)=0.550+0.550=50 nEmployees expected payoff of playing“shirk”nEU1(Shirk,(0.5,0.5)=0.50+0.5100=50nEmployees expected payoff of her mixed strategy v1(0.9,0.1),(0.5,0.5)=0.9 50+0.1 50=50Theorem 1:illustrationEmployee MonitoringManagerMonitor(0.5)Not Monitor(0.5)EmployeeWork(0.9)50,9050
17、,100Shirk(0.1)0 ,-10100,-100May 29,200314nManagers expected payoff of playing“Monitor”EU2(Monitor,(0.9,0.1)=0.990+0.1(-10)=80nManagers expected payoff of playing“Not”nEU2(Not,(0.9,0.1)=0.9100+0.1(-100)=80nManagers expected payoff of her mixed strategy v2(0.9,0.1),(0.5,0.5)=0.580+0.580=80Theorem 1:il
18、lustrationEmployee MonitoringManagerMonitor(0.5)Not Monitor(0.5)EmployeeWork(0.9)50,9050,100Shirk(0.1)0 ,-10100,-100May 29,200315nEmployeev1(0.9,0.1),(0.5,0.5)EU1(Work,(0.5,0.5)v1(0.9,0.1),(0.5,0.5)EU1(Shirk,(0.5,0.5)nManagerv2(0.9,0.1),(0.5,0.5)EU2(Monitor,(0.9,0.1)v2(0.9,0.1),(0.5,0.5)EU2(Not,(0.9
19、,0.1)n Hence,(0.9,0.1),(0.5,0.5)is a mixed strategy Nash equilibrium by Theorem 1.Theorem 1:illustrationEmployee MonitoringManagerMonitor(0.5)No Monitor(0.5)EmployeeWork(0.9)50,9050,100Shirk(0.1)0 ,-10100,-100May 29,200316nUse Theorem 1 to check whether(2/3,1/3),(1/3,2/3)is a mixed strategy Nash equ
20、ilibrium.Theorem 1:illustrationBattle of sexesPatOpera (1/3)Prize Fight (2/3)ChrisOpera (2/3)2,10,0Prize Fight(1/3)0 ,01,2May 29,200317Mixed strategy equilibrium:2-player each with two strategiesnTheorem 2 Let(r*,1-r*),(q*,1-q*)be a pair of mixed strategies,where 0 r*1,0q*1.Then(r*,1-r*),(q*,1-q*)is
21、 a mixed strategy Nash equilibrium if and only if EU1(s11,(q*,1-q*)=EU1(s12,(q*,1-q*)EU2(s21,(r*,1-r*)=EU2(s22,(r*,1-r*)nThat is,each player is indifferent between her two strategies.Player 2s21 (q)s22 (1-q)Player 1s11 (r)u1(s11,s21),u2(s11,s21)u1(s11,s22),u2(s11,s22)s12(1-r)u1(s12,s21),u2(s12,s21)u
22、1(s12,s22),u2(s12,s22)May 29,200318Use indifference to find mixed Nash equilibrium(2-player each with 2 strategies)nUse Theorem 2 to find mixed strategy Nash equilibrianSolve EU1(s11,(q*,1-q*)=EU1(s12,(q*,1-q*)nSolve EU2(s21,(r*,1-r*)=EU2(s22,(r*,1-r*)May 29,200319Use Theorem 2 to find mixed strateg
23、y Nash equilibrium:illustrationnPlayer 1 is indifferent between playing Head and Tail.EU1(H,(q,1q)=q(-1)+(1q)1=12qEU1(T,(q,1q)=q1+(1q)(-1)=2q1EU1(H,(q,1q)=EU1(T,(q,1q)12q=2q1 4q=2 This give us q=1/2Matching penniesPlayer 2H (q)T (1q)Player 1H (r)-1,1 1,-1T(1r)1,-1-1,1May 29,200320Use Theorem 2 to fi
24、nd mixed strategy Nash equilibrium:illustrationnPlayer 2 is indifferent between playing Head and Tail.EU2(H,(r,1r)=r 1+(1r)(-1)=2r 1EU2(T,(r,1r)=r(-1)+(1r)1=1 2rEU2(H,(r,1r)=EU2(T,(r,1r)2r 1=1 2r 4r=2 This give us r=1/2Hence,(0.5,0.5),(0.5,0.5)is a mixed strategy Nash equilibrium by Theorem 2.Matchi
25、ng penniesPlayer 2H (q)T (1q)Player 1H (r)-1,1 1,-1T(1r)1,-1-1,1May 29,200321nEmployees expected payoff of playing“work”EU1(Work,(q,1q)=q50+(1q)50=50 nEmployees expected payoff of playing“shirk”nEU1(Shirk,(q,1q)=q0+(1q)100=100(1q)nEmployee is indifferent between playing Work and Shirk.n50=100(1q)nq=
26、1/2Use Theorem 2 to find mixed strategy Nash equilibrium:illustrationEmployee MonitoringManagerMonitor(q)Not Monitor(1q)EmployeeWork (r)50,9050,100Shirk(1r)0 ,-10100,-100May 29,200322nManagers expected payoff of playing“Monitor”EU2(Monitor,(r,1r)=r90+(1r)(-10)=100r10nManagers expected payoff of play
27、ing“Not”EU2(Not,(r,1r)=r100+(1r)(-100)=200r100nManager is indifferent between playing Monitor and Not 100r10=200r100 implies that r=0.9.nHence,(0.9,0.1),(0.5,0.5)is a mixed strategy Nash equilibrium by Theorem 2.Use Theorem 2 to find mixed strategy Nash equilibrium:illustrationEmployee MonitoringMan
28、agerMonitor(q)Not Monitor(1q)EmployeeWork (r)50,9050,100Shirk(1r)0 ,-10100,-100May 29,200323nUse Theorem 2 to find a mixed Nash equilibriumUse Theorem 2 to find mixed strategy Nash equilibrium:illustrationBattle of sexesPatOpera (q)Prize Fight (1-q)ChrisOpera (r)2,10,0Prize Fight(1-r)0 ,01,2May 29,2
29、00324nUse Theorem 2 to find a mixed Nash equilibriumUse Theorem 2 to find mixed strategy Nash equilibrium:illustrationExamplePlayer 2L (q)R (1-q)Player 1T (r)6,42,6B(1-r)3 ,36,1May 29,200325SummarynMixed strategiesnMixed Nash equilibriumnFind mixed Nash equilibriumnNext timen2-player game each with a finite number of strategiesnReading listsnChapter 1.3 of Gibbons and Cha 4.3 of OsborneMay 29,200326
