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1、June 19, 2003,73-347 Game Theory-Lecture 22,1,Static (or Simultaneous-Move) Games of Incomplete Information,Introduction to Static Bayesian Games,June 19, 2003,73-347 Game Theory-Lecture 22,2,Outline of Static Games of Incomplete Information,Introduction to static games of incomplete information Nor
2、mal-form (or strategic-form) representation of static Bayesian games Bayesian Nash equilibrium Auction,June 19, 2003,73-347 Game Theory-Lecture 22,3,Todays Agenda,What is a static game of incomplete information? Prisoners dilemma of incomplete information Cournot duopoly model of incomplete informat
3、ion,June 19, 2003,73-347 Game Theory-Lecture 22,4,Static (or simultaneous-move) games of complete information,A set of players (at least two players) For each player, a set of strategies/actions Payoffs received by each player for the combinations of the strategies, or for each player, preferences o
4、ver the combinations of the strategies All these are common knowledge among all the players.,June 19, 2003,73-347 Game Theory-Lecture 22,5,Static (or simultaneous-move) games of INCOMPLETE information,Payoffs are no longer common knowledge Incomplete information means that At least one player is unc
5、ertain about some other players payoff function. Static games of incomplete information are also called static Bayesian games,June 19, 2003,73-347 Game Theory-Lecture 22,6,Prisoners dilemma of complete information,Two suspects held in separate cells are charged with a major crime. However, there is
6、not enough evidence. Both suspects are told the following policy: If neither confesses then both will be convicted of a minor offense and sentenced to one month in jail. If both confess then both will be sentenced to jail for six months. If one confesses but the other does not, then the confessor wi
7、ll be released but the other will be sentenced to jail for nine months.,June 19, 2003,73-347 Game Theory-Lecture 22,7,Prisoners dilemma of incomplete information,Prisoner 1 is always rational (selfish). Prisoner 2 can be rational (selfish) or altruistic, depending on whether he is happy or not. If h
8、e is altruistic then he prefers to mum and he thinks that “confess” is equivalent to additional “four months in jail”. Prisoner 1 can not know exactly whether prisoner 2 is rational or altruistic, but he believes that prisoner 2 is rational with probability 0.8, and altruistic with probability 0.2.,
9、June 19, 2003,73-347 Game Theory-Lecture 22,8,Prisoners dilemma of incomplete information contd,Given prisoner 1s belief on prisoner 2, what strategy should prison 1 choose? What strategy should prisoner 2 choose if he is rational or altruistic?,June 19, 2003,73-347 Game Theory-Lecture 22,9,Prisoner
10、s dilemma of incomplete information contd,Solution: Prisoner 1 chooses to confess, given his belief on prisoner 2 Prisoner 2 chooses to confess if he is rational, and mum if he is altruistic This can be written as (Confess, (Confess if rational, Mum if altruistic) Confess is prisoner 1s best respons
11、e to prisoner 2s choice (Confess if rational, Mum if altruistic). (Confess if rational, Mum if altruistic) is prisoner 2s best response to prisoner 1s Confess A Nash equilibrium called Bayesian Nash equilibrium,June 19, 2003,73-347 Game Theory-Lecture 22,10,Cournot duopoly model of complete informat
12、ion,The normal-form representation: Set of players: Firm 1, Firm 2 Sets of strategies: S1=0, +), S2=0, +) Payoff functions: u1(q1, q2)=q1(a-(q1+q2)-c), u2(q1, q2)=q2(a-(q1+q2)-c) All these information is common knowledge,June 19, 2003,73-347 Game Theory-Lecture 22,11,Cournot duopoly model of incompl
13、ete information,A homogeneous product is produced by only two firms: firm 1 and firm 2. The quantities are denoted by q1 and q2, respectively. They choose their quantities simultaneously. The market price: P(Q)=a-Q, where a is a constant number and Q=q1+q2. Firm 1s cost function: C1(q1)=cq1. All the
14、 above are common knowledge,June 19, 2003,73-347 Game Theory-Lecture 22,12,Cournot duopoly model of incomplete information contd,Firm 2s marginal cost depends on some factor (e.g. technology) that only firm 2 knows. Its marginal cost can be HIGH: cost function: C2(q2)=cHq2. LOW: cost function: C2(q2
15、)=cLq2. Before production, firm 2 can observe the factor and know exactly which level of marginal cost is in. However, firm 1 cannot know exactly firm 2s cost. Equivalently, it is uncertain about firm 2s payoff. Firm 1 believes that firm 2s cost function is C2(q2)=cHq2 with probability , and C2(q2)=
16、cLq2 with probability 1. All the above are common knowledge,June 19, 2003,73-347 Game Theory-Lecture 22,13,Cournot duopoly model of incomplete information contd,June 19, 2003,73-347 Game Theory-Lecture 22,14,Cournot duopoly model of incomplete information contd,June 19, 2003,73-347 Game Theory-Lecture 22,15,Cournot duopoly model of incomplete information contd,June 19, 2003,73-347 Game Theory-Lecture 22,16,Cournot duopoly model of incompl
