国外博弈论课件lecture(13).ppt

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1、May 20, 2003,73-347 Game Theory-Lecture 2,1,Static (or Simultaneous-Move) Games of Complete Information,Dominated Strategies Nash Equilibrium,May 20, 2003,73-347 Game Theory-Lecture 2,2,Outline of Static Games of Complete Information,Introduction to games Normal-form (or strategic-form) representati

2、on Iterated elimination of strictly dominated strategies Nash equilibrium Review of concave functions, optimization Applications of Nash equilibrium Mixed strategy equilibrium,May 20, 2003,73-347 Game Theory-Lecture 2,3,Todays Agenda,Review of previous class Dominated strategies Iterated elimination

3、 of strictly dominated strategies Nash equilibrium,May 20, 2003,73-347 Game Theory-Lecture 2,4,Review,The normal-form (or strategic-form) representation of a game G specifies: A finite set of players 1, 2, ., n, players strategy spaces S1 S2 . Sn and their payoff functions u1 u2 . un where ui : S1 S

4、2 . SnR.,All combinations of the strategies. A combination of the strategies is a set of strategies, one for each player,May 20, 2003,73-347 Game Theory-Lecture 2,5,Review,Static (or simultaneous-move) game of complete information Each players strategies and payoff function are common knowledge amon

5、g all the players. Each player i chooses his/her strategy si without knowledge of others choices. Then each player i receives his/her payoff ui(s1, s2, ., sn). The game ends.,May 20, 2003,73-347 Game Theory-Lecture 2,6,Solving Prisoners Dilemma,Confess always does better whatever the other player ch

6、ooses Dominated strategy There exists another strategy which always does better regardless of other players choices,May 20, 2003,73-347 Game Theory-Lecture 2,7,Definition: strictly dominated strategy,May 20, 2003,73-347 Game Theory-Lecture 2,8,Example,Two firms, Reynolds and Philip, share some marke

7、t Each firm earns $60 million from its customers if neither do advertising Advertising costs a firm $20 million Advertising captures $30 million from competitor,May 20, 2003,73-347 Game Theory-Lecture 2,9,2-player game with finite strategies,S1=s11, s12, s13 S2=s21, s22 s11 is strictly dominated by

8、s12 if u1(s11,s21)u1(s12,s21) and u1(s11,s22)u1(s12,s22). s21 is strictly dominated by s22 if u2(s1i,s21) u2(s1i,s22), for i = 1, 2, 3,May 20, 2003,73-347 Game Theory-Lecture 2,10,Definition: weakly dominated strategy,May 20, 2003,73-347 Game Theory-Lecture 2,11,Strictly and weakly dominated strateg

9、y,A rational player never chooses a strictly dominated strategy. Hence, any strictly dominated strategy can be eliminated. A rational player may choose a weakly dominated strategy.,May 20, 2003,73-347 Game Theory-Lecture 2,12,Iterated elimination of strictly dominated strategies,If a strategy is str

10、ictly dominated, eliminate it The size and complexity of the game is reduced Eliminate any strictly dominated strategies from the reduced game Continue doing so successively,May 20, 2003,73-347 Game Theory-Lecture 2,13,Iterated elimination of strictly dominated strategies: an example,Player 1,Player

11、 2,Middle,Up,Down,Left,Right,May 20, 2003,73-347 Game Theory-Lecture 2,14,Example: Tourists & Natives,Only two bars (bar 1, bar 2) in a city Can charge price of $2, $4, or $5 6000 tourists pick a bar randomly 4000 natives select the lowest price bar Example 1:Both charge $2 each gets 5,000 customers

12、 and $10,000 Example 2: Bar 1 charges $4, Bar 2 charges $5 Bar 1 gets 3000+4000=7,000 customers and $28,000 Bar 2 gets 3000 customers and $15,000,May 20, 2003,73-347 Game Theory-Lecture 2,15,Example: Tourists & Natives,Payoffs are in thousands of dollars,May 20, 2003,73-347 Game Theory-Lecture 2,16,

13、One More Example,Each of n players selects a number between 0 and 100 simultaneously. Let xi denote the number selected by player i. Let y denote the average of these numbers Player is payoff = xi 3y/5,May 20, 2003,73-347 Game Theory-Lecture 2,17,One More Example,The normal-form representation: Play

14、ers: player 1, player 2, ., player n Strategies: Si =0, 100, for i = 1, 2, ., n. Payoff functions: ui(x1, x2, ., xn) = xi 3y/5 Is there any dominated strategy? What numbers should be selected?,May 20, 2003,73-347 Game Theory-Lecture 2,18,New solution concept: Nash equilibrium,The combination of stra

15、tegies (B, R) has the following property: Player 1 CANNOT do better by choosing a strategy different from B, given that player 2 chooses R. Player 2 CANNOT do better by choosing a strategy different from R, given that player 1 chooses B.,May 20, 2003,73-347 Game Theory-Lecture 2,19,New solution conc

16、ept: Nash equilibrium,The combination of strategies (B, R) has the following property: Player 1 CANNOT do better by choosing a strategy different from B, given that player 2 chooses R. Player 2 CANNOT do better by choosing a strategy different from R, given that player 1 chooses B.,May 20, 2003,73-3

17、47 Game Theory-Lecture 2,20,Nash Equilibrium: idea,Nash equilibrium A set of strategies, one for each player, such that each players strategy is best for her, given that all other players are playing their equilibrium strategies,May 20, 2003,73-347 Game Theory-Lecture 2,21,Definition: Nash Equilibri

18、um,May 20, 2003,73-347 Game Theory-Lecture 2,22,2-player game with finite strategies,S1=s11, s12, s13 S2=s21, s22 (s11, s21)is a Nash equilibrium if u1(s11,s21) u1(s12,s21), u1(s11,s21) u1(s13,s21) andu2(s11,s21) u2(s11,s22).,May 20, 2003,73-347 Game Theory-Lecture 2,23,Finding a Nash equilibrium: c

19、ell-by-cell inspection,Player 1,Player 2,Middle,Up,Down,Left,Right,May 20, 2003,73-347 Game Theory-Lecture 2,24,Summary,Dominated strategies Iterated elimination Nash equilibrium Next time Nash equilibrium Best response function Reading lists Sec 1.1.C and 1.2.A of Gibbons and Sec 2.6-2.8 of Osborne,