Static (or Simultaneous-Move) Games of Incomplete Information
Introduction to Static Bayesian Games
73-347 Game Theory--Lecture 22
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
73-347 Game Theory--Lecture 22
Today’s Agenda
What is a static game of incomplete information?
Prisoners’ dilemma of incomplete information
Cournot duopoly model of incomplete information
73-347 Game Theory--Lecture 22
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.
73-347 Game Theory--Lecture 22
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.
Static games of incomplete information are also called static Bayesian games
73-347 Game Theory--Lecture 22
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
Confess
-6 , -6
0 , -9
Confess
Prisoner 1
-9 , 0
-1 , -1
Mum
Mum
73-347 Game Theory--Lecture 22
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 , and altruistic with probability .
Prisoner 2
Payoffs if prisoner 2 is altruistic
Confess
-6 , -10
0 , -9
Confess
Prisoner 1
-9 , -4
-1 , -1
Mum
Mum
73-347 Game Theory--Lecture 22
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?
Prisoner 2
Payoffs if prisoner 2 is rational
Confess
-6 , -6
0 , -9
Confess
Prisoner 1
-9 , 0
-1 , -1
Mum
Mum
Prisoner 2
Payoffs if prisoner 2 is altruistic
Confess
-6 , -10
0 , -9
Confess
Prisoner 1
-9 , -4
-1 , -1
Mum
Mum
73-347 Game Theory--Lecture 22
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
73-347 Game Theory--Lecture 22
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
73-347 Game Theory--Lecture 22
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
73-347 Game Theory--Lecture 22
Cournot duopoly model of
incomplete information cont’d
Firm 2’s marginal cost depends on some factor (. technology) that only firm 2 knows. Its marginal cost can be
HIGH: cost function: C2(q2)=cHq2.
LOW: cost function: C2(q2)=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 2’s cost. Equivalently, it is uncertain about firm 2’s payoff.
Firm 1 believes that firm 2’s cost function is
C2(q2)=cHq2 with probability , and
C2(q2)=cLq2 with probability 1–.
All the above are common knowledge
73-347 Game Theory--Lecture 22
Cournot duopoly model of
incomplete information cont’d
73-347 Game Theory--Lecture 22
Cournot duopoly model of
incomplete information cont’d
73-347 Game Theory--Lecture 22
Cournot duopoly model of
incomplete information cont’d
73-347 Game Theory--Lecture 22
Cournot duopoly model of
incomplete information cont’d
73-347 Game Theory--Lecture 22
Cournot duopoly model of
incomplete information cont’d
73-347 Game Theory--Lecture 22
Cournot duopoly model of
incomplete information cont’d
73-347 Game Theory--Lecture 22
Summary
What is a static game of incomplete information?
Prisoners’ dilemma of incomplete information
Cournot duopoly model of incomplete information
Next time
More examples
Bayesian Nash equilibrium
Reading lists
Chapter of Gibbons
73-347 Game Theory--Lecture 22
Lecture 22
