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 m Confess m 1 1 Prisoner 1 Confess
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 Prisoner 1 Confess 6,-10
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 contd 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 Mur 0 Prisoner 1 Confess 6,二6 Payoffs if prisoner 2 is Prisoner 2 altruistic Mum Confess Mum 1 4 Prisoner 1 Confess Q 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 contd ■So| ution 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 a (Confess if rational, Mum if altruistic)is prisoner 2's best response to prisoner 1,s Confess u 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 f Firm 1, Firm 2) Sets of strategies:S=[0,+∞),S2=[0,+∞) Payoff functions l1(qq2)=q1(-(q1+g2)c) (qpq2)=q2(-(q+q2)c) 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