Outline of Static Games of Incomplete Information Introduction to static games of incomplete information Normal-form (or strategic-form) representation of static Bayesian games Bayesian Nash equilibrium ■Auction
Outline of Static Games of Incomplete Information ◼ Introduction to static games of incomplete information ◼ Normal-form (or strategic-form) representation of static Bayesian games ◼ Bayesian Nash equilibrium ◼ Auction 2
Today's Agenda What is a static game of incomplete information? Prisoners'dilemma of incomplete information Cournot duopoly model of incomplete information Battle of sexes of incomplete information First-price auction 3
Today’s Agenda ◼ What is a static game of incomplete information? ◼ Prisoners’ dilemma of incomplete information ◼ Cournot duopoly model of incomplete information ◼ Battle of sexes of incomplete information ◼ First-price auction 3
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 over the combinations of the strategies All these are common knowledge among all the players
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 over the combinations of the strategies ◼ All these are common knowledge among all the players. 4
Static (or simultaneous-move)games of INCOMPLETE information Payoffs are no longer common knowledge Incomplete information means that >At least one player is uncertain about some other player's payoff function (type) Static games of incomplete information are also called static Bayesian games
Static (or simultaneous-move) games of INCOMPLETE information ◼ Payoffs are no longer common knowledge ◼ Incomplete information means that ➢ At least one player is uncertain about some other player’s payoff function (type) ◼ Static games of incomplete information are also called static Bayesian games 5
Prisoners'dilemma of complete information Two suspects held in separate cells are charged with a major crime. However,there is 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 will be released but the other will be sentenced to jail for nine months. Prisoner 2 Mum Confess Mum -1 -1 -9 Q Prisoner 1 Confess 0 -9 -6,-6 6
Prisoners’ dilemma of complete information Prisoner 2 Mum Confess Prisoner 1 Mum -1 , -1 -9 , 0 Confess 0 , -9 -6 , -6 6 ◼ Two suspects held in separate cells are charged with a major crime. However, there is 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 will be released but the other will be sentenced to jail for nine months
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 he 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. Payoffs if prisoner 2 is Prisoner 2 altruistic Mum Confess Mum -1， -1 -91 -4 Prisoner 1 Confess -9 -6,-10 7
Prisoners’ dilemma of incomplete information Payoffs if prisoner 2 is altruistic Prisoner 2 Mum Confess Prisoner 1 Mum -1 , -1 -9 , -4 Confess 0 , -9 -6 , -10 7 ◼ Prisoner 1 is always rational (selfish). ◼ Prisoner 2 can be rational (selfish) or altruistic, depending on whether he is happy or not. ◼ If he 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
Prisoners'dilemma of incomplete information cont'd ■ Given prisoner 1's belief on prisoner 2,what strategy should prison 1 choose? ■ What strategy should prisoner 2 choose if he is rational or altruistic? Payoffs if prisoner 2 is Prisoner 2 rational Mum Confess Mum -1 -1 -9, 0 Prisoner 1 Confess 0 -9 -6，-6 Payoffs if prisoner 2 is Prisoner 2 altruistic Mum Confess Mum -1, -1 -9 -4 Prisoner 1 Confess 0 -9 =6,-10
Prisoners’ dilemma of incomplete information cont’d Payoffs if prisoner 2 is rational Prisoner 2 Mum Confess Prisoner 1 Mum -1 , -1 -9 , 0 Confess 0 , -9 -6 , -6 8 ◼ Given prisoner 1’s belief on prisoner 2, what strategy should prison 1 choose? ◼ What strategy should prisoner 2 choose if he is rational or altruistic? Payoffs if prisoner 2 is altruistic Prisoner 2 Mum Confess Prisoner 1 Mum -1 , -1 -9 , -4 Confess 0 , -9 -6 , -10
Prisoners'dilemma of incomplete information cont'd ■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 1's best response to prisoner 2's choice(Confess if rational,Mum if altruistic). (Confess if rational,Mum if altruistic)is prisoner 2's best response to prisoner 1's Confess A Nash equilibrium called Bayesian Nash equilibrium
Prisoners’ dilemma of incomplete information cont’d ◼ 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 1’s best response to prisoner 2’s choice (Confess if rational, Mum if altruistic). ◼ (Confess if rational, Mum if altruistic) is prisoner 2’s best response to prisoner 1’s Confess ◼ A Nash equilibrium called Bayesian Nash equilibrium 9
Cournot duopoly model of complete information The normal-form representation: -Set of players: Firm 1,Firm 2) Sets of strategies:S=[0,+oo),S2=10,+o0) >Payoff functions: u(qw92)=q(a-(91+92-C), u2(9wq2)=q2(-(q1+42c) All these information is common knowledge 10
Cournot duopoly model of complete information ◼ 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 10
Cournot duopoly model of incomplete information A homogeneous product is produced by only two firms:firm 1 and firm 2.The quantities are denoted by q,and 42,respectively. They choose their quantities simultaneously. The market price:P(O)=a-2,where a is a constant number and Q=q1+q2. Firm 1's cost function:Ci(q1)=cq1. All the above are common knowledge 11
Cournot duopoly model of incomplete 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 1’s cost function: C1 (q1 )=cq1 . ◼ All the above are common knowledge 11