国外博弈论课件lecture8.ppt

May 29, 2003 73-347 Game Theory--Lecture 8 May 29, 2003 Lecture 8 Static (or Simultaneous-Move) Games of Complete Information Mixed Strategy Nash Equilibrium 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 Today’s Agenda Review of previous class Mixed strategy Nash equilibrium in Battle of sexes Use indifference to find mixed strategy Nash equilibria Mixed strategy equilibrium Mixed Strategy: A mixed strategy of a player is a probability distribution over the player’s 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 Battle of sexes Chris’ expected payoff of playing Opera: 2q Chris’ expected payoff of playing Prize Fight: 1-q Chris’ best response B1(q): Prize Fight (r=0) if q1/3 Opera (r=1) if q1/3 Any mixed strategy (0?r?1) if q=1/3 Battle of sexes Pat’s expected payoff of playing Opera: r Pat’s expected payoff of playing Prize Fight: 2(1-r) Pat’s best response B2(r): Prize Fight (q=0) if r2/3 Opera (q=1) if r2/3 Any mixed strategy (0?q?1) if r=2/3, Battle of sexes Chris’ best response B1(q): Prize Fight (r=0) if q1/3 Opera (r=1) if q1/3 Any mixed strategy (0?r?1) if q=1/3 Pat’s best response B2(r): Prize Fight (q=0) if r2/3 Opera (q=1) if r2/3 Any mixed strategy (0?q?1) if r=2/3 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 1’s expected payoff of playing s11: EU1(s11, (q, 1-q))=q×u1(s11, s21)+(1-q)×u1(s11, s22) Player 1’s expected payoff of playing s12: EU1(s12, (q, 1-q))= q×u1(s12, s21)+(1-q)×u1(s12, s22) Player 1’s expected payoff from her mixed strategy:v1((r, 1-r), (q, 1-q))=