,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,May 29,2003,73-347 Game Theory-Lecture 8,*,Static(or Simultaneous-Move)Games of Complete Information,Mixed Strategy Nash Equilibrium,May 29,2003,1,73-347 Game Theory-Lecture 8,Static(or Simultaneous-Move),Outline of Static Games of Complete Information,Introduction to games,Normal-form(or strategic-form)representation,Iterated elimination of strictly dominated strategies,Nash equilibrium,Review of concave functions,optimization,Applications of Nash equilibrium,Mixed strategy Nash equilibrium,May 29,2003,2,73-347 Game Theory-Lecture 8,Outline of Static Games of Com,Todays Agenda,Review of previous class,Mixed strategy Nash equilibrium in Battle of sexes,Use indifference to find mixed strategy Nash equilibria,May 29,2003,3,73-347 Game Theory-Lecture 8,Todays AgendaReview of previo,Mixed strategy equilibrium,Mixed Strategy:,A mixed strategy of a player is a probability distribution over the players strategies.,Mixed strategy equilibrium,A probability distribution for each player,The distributions are mutual best responses to one another in the sense of expected payoffs,May 29,2003,4,73-347 Game Theory-Lecture 8,Mixed strategy equilibriumMixe,Chris expected payoff of playing Opera:2,q,Chris expected payoff of playing Prize Fight:1-,q,Chris best response,B,1,(,q,):,Prize Fight(r=0),if,q1/3,Any mixed strategy(0,r1),if,q=1/3,Battle of sexes,Pat,Opera (,q,),Prize Fight (1-,q,),Chris,Opera (,r,),2,1,0,0,Prize Fight(1-,r,),0,0,1,2,May 29,2003,5,73-347 Game Theory-Lecture 8,Chris expected payoff of play,Pats expected payoff of playing Opera:,r,Pats expected payoff of playing Prize Fight:2(1-,r,),Pats best response,B,2,(,r,):,Prize Fight(q=0),if,r2/3,Any mixed strategy(0,q1),if,r=2/3,Battle of sexes,Pat,Opera (,q,),Prize Fight (1-,q,),Chris,Opera (,r,),2,1,0,0,Prize Fight(1-,r,),0,0,1,2,May 29,2003,6,73-347 Game Theory-Lecture 8,Pats expected payoff of playi,1,q,r,1,Chris best response,B,1,(,q,):,Prize Fight(r=0),if,q1/3,Any mixed strategy(0,r1),if,q=1/3,Pats best response,B,2,(,r,):,Prize Fight(q=0),if,r2/3,Any mixed strategy(0,q1),if,r=2/3,Battle of sexes,2/3,Three Nash equilibria:,(,(1,0),(1,0),),(,(0,1),(0,1),),(,(2/3,1/3),(1/3,2/3),),1/3,May 29,2003,7,73-347 Game Theory-Lecture 8,1qr1Chris best response B1(q),Expected payoffs:2 players each with two pure strategies,Player 1 plays a mixed strategy,(,r,1-,r,).,Player 2 plays a mixed strategy,(,q,1-,q,).,Player 1s expected payoff of playing,s,11,:,EU,1,(,s,11,(,q,1-,q,),)=,q,u,1,(,s,11,s,21,)+(,1-q,),u,1,(,s,11,s,22,),Player 1s expected payoff of playing,s,12,:,EU,1,(,s,12,(,q,1-,q,),)=,q,u,1,(,s,12,s,21,)+(,1-q,),u,1,(,s,12,s,22,),Player 1s expected payoff from her mixed strategy:,v,1,(,r,1-r,),(,q,1-,q,),)=,r,EU,1,(,s,11,(,q,1-,q,),)+(,1-r,),EU,1,(,s,12,(,q,1-,q,),),Player 2,s,21,(,q,),s,22,(1-,q,),Player 1,s,11,(,r,),u,1,(,s,11,s,21,),u,2,(,s,11,s,21,),u,1,(,s,11,s,22,),u,2,(,s,11,s,22,),s,12,(1-,r,),u,1,(,s,12,s,21,),u,2,(,s,12,s,21,),u,1,(,s,12,s,22,),u,2,(,s,12,s,22,),May 29,2003,8,73-347 Game Theory-Lecture 8,Expected payoffs:2 players ea,Expected payoffs:2 players each with two pure strategies,Player 1 plays a mixed strategy,(,r,1-,r,).,Player 2 plays a mixed strategy,(,q,1-,q,).,Player 2s expected payoff of playing,s,21,:,EU,2,(,s,21,(,r,1-r,),)=,r,u,2,(,s,11,s,21,)+(,1-r,),u,2,(,s,12,s,21,),Player 2s expected payoff of playing,s,22,:,EU,2,(,s,22,(,r,1-r,),)=,r,u,2,(,s,11,s,22,)+(,1-r,),u,2,(,s,12,s,22,),Player 2s expected payoff from her mixed strategy:,v,2,(,(,r,1,-r,),(,q,1-,q,)=,q,EU,2,(,s,21,(,r,1,-r,),)+(,1-q,),EU,2,(,s,22,(,r,1,-r,),),Player 2,s,21,(,q,),s,22,(1-,q,),Player 1,s,11,(,r,),u,1,(,s,11,s,21,),u,2,(,s,11,s,21,),u,1,(,s,11,s,22,),u,2,(,s,11,s,22,),s,12,(1-,r,),u,1,(,s,12,s,21,),u,2,(,s,12,s,21,),u,1,(,s,12,s,22,),u,2,(,s,12,s,22,),May 29,2003,9,73-347 Game Theory-Lecture 8,Expected payoffs:2 players ea,Mixed strategy equilibrium:2-player each with two pure strategies,Mixed strategy Nash equilibrium:,A pair of mixed strategies,(,(,r*,1-r*,),(,q*,1-q*,),),is a Nash equilibrium if,(,r*,1-r*,),is a best response to,(,q*,1-q,*),and,(,q*,1-q,*),is a best response to,(,r*,1-r*,),.That is,v,1,(,r*,1-,r*,),(,q*,1-,q*,),),v,1,(,r,1-,r,),(,q*,1-,q*,),),for all 0,r,1,v,2,(,(,r*,1-,r*,),(,q*,1-,q*,),v,2,(,(,r*,1-,r*,),(,q,1-,q,),for all 0,q,1,Player 2,s,21,(,q,),s,22,(1-,q,),Player 1,s,11,(,r,),u,1,(,s,11,s,21,),u,2,(,s,11,s,21,),u,1,(,s,11,s,22,),u,2,(,s,11,s,22,),s,12,(1-,r,),u,1,(,s,12,s,21,),u,2,(,s,12,s,21,),u,1,(,s,12,s,22,),u,2,(,s,12,s,22,),May 29,2003,10,73-347 Game Theory-Lecture 8,Mixed strategy equilibrium:2-,2-player each with two strategies,Theorem 1,(property of mixed Nash equilibrium),A pair of mixed strategies,(,(,r*,1-,r*,),(,q*,1-,q,*),),is a Nash equilibrium if and only if,v,1,(,r*,1-,r*,),(,q*,1-,q*,),),EU,1,(,s,11,(,q*,1-,q*,),),v,1,(,r*,1-,r*,),(,q*,1-,q*,),),EU,1,(,s,12,(,q*,1-,q*,),),v,2,(,(,r*,1-,r*,),(,q*,1-,q*,)