国外博弈论课件lecture10.ppt

June 2, 2003 73-347 Game Theory--Lecture 10 June 2, 2003 Lecture 10 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 Example: Rock, paper and scissors (Exercise 139.1 of Osborne) How to find mixed strategy Nash equilibria in a 2-player game each with a finite number of strategies Example: Rock, paper and scissors Each of two players simultaneously announces either Rock, or Paper, or Scissors. Paper beats (wraps) rock Rock beats (blunts) scissors Scissors beats (cuts) paper The player who names the winning object receives $1 from her opponent If both players name the same choice then no payment is made Example: Rock, paper and scissors Can you guess a mixed strategy Nash equilibrium? Mixed strategy Nash 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,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 ?1 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 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-r*))v2((r*, 1-r*), (q*, 1-q*)) ? EU2(s22, (r*, 1-r*)) Mixed strategy equilibrium: 2-player each with two strategies Theorem 2 Let ((r*, 1-r*), (q*