复旦微观经济学Dynamic games of incomplete information

Junji Xiao (Lecture 5)
Micro
6 / 50
Perfect Bayesian equilibrium
Perfect Bayesian equilibrium (cont’ d)
Requirement 2 Given their beliefs, the players’strategies must be sequentially rational. That is, at each information set the action taken by the player with the move (and the player’ subsequent s strategy) must be optimal given the player’ belief at that information s set and the other players’subsequent strategies (where a "subsequent strategy" is a complete plan of action covering every contingency that might arise after the given information set has been reached).
Junji Xiao (Lecture 5)
Micro
10 / 50
Perfect Bayesian equilibrium
Perfect Bayesian equilibrium (cont’ d)
Requirement 3 At information set on the equilibrium path, beliefs are determined by Bayes’rule and the players’equilibrium strategies.
Junji Xiao (Lecture 5)
Micro
9 / 50
Perfect Bayesian equilibrium
Perfect Bayesian equilibrium (cont’ d)
De…nition For a given equilibrium in a given extensive-form game, an information set is on the equilibrium path if it will be reached with positive probability if the game is played according to the equilibrium strategies, and is o¤ the equilibrium path if it is certain not be reached if the game is played according to the equilibrium strategies (where "equilibrium" can mean Nash, subgame-perfect, Bayesian, or perfect Bayesian equilibrium).
Player 1
R0 0, 0 0, 1 1, 3
Nash equilibria: (L, L0 ) and (R, R 0 ). No subgame. SPNE: (L, L0 ) and (R, R 0 ).
Junji Xiao (Lecture 5)
Micro
5 / 50
Perfect Bayesian equilibrium
Junji Xiao (Lecture 5)
Micro
11 / 50
Perfect Bayesian equilibrium
Example (cont’ d)
Player 2’ belief must be p = 1: given player 1’ equilibrium strategy s s (namely L), player 2 knows which node in the information set has been reached. Suppose that there were a mixed strategy equilibrium in which player 1 plays L with probability q1 , M with probability q2 , and R with probability 1 q1 q2 . Then Requirement 3 would force player 2’ s belief to be p = q1 /(q1 + q2 ).
Player 2 L0 L 2, 1 M 0, 2 R 1, 3
Player 1
R0 0, 0 0, 1 1, 3
Junji Xiao (Lecture 5)
Micro
5 equilibrium
Example (cont’ d)
Player 2 L0 L 2, 1 M 0, 2 R 1, 3
Junji Xiao (Lecture 5)
Micro
5 / 50
Perfect Bayesian equilibrium
Perfect Bayesian equilibrium
Requirement 1 At each information set, the player with the move must have a belief about which node in the information set has been reached by the play of the game. For a nonsingleton information set, a belief is a probability distribution over the nodes in the information set; for a singleton information set, the player’ belief puts probability s one on the single decision node.
Advanced Microeconomics
Junji Xiao
Lecture 5
Junji Xiao (Lecture 5)
Micro
1 / 50
Road map
Road map
Dynamic games of incomplete information
Perfect Bayesian equilibrium Dynamic games of incomplete information Adverse Selection and Screening Signalling The Principal-Agent Problem
Junji Xiao (Lecture 5)
Micro
3 / 50
Perfect Bayesian equilibrium
Example (cont’ d)
Junji Xiao (Lecture 5)
Micro
4 / 50
Perfect Bayesian equilibrium
Example (cont’ d)
Junji Xiao (Lecture 5)
Micro
7 / 50
Perfect Bayesian equilibrium
Example (cont’ d)
Junji Xiao (Lecture 5)
Micro
8 / 50
Perfect Bayesian equilibrium
Example (cont’ d)
Given player 2’ belief, s
player 2’ expected payo¤ from playing R 0 : s p 0 + (1 p) 1 = 1 p.
player 2’ expected payo¤ from playing L0 : s p 1 + (1 p) 2 = 2 p.
Junji Xiao (Lecture 5)
Micro
12 / 50
Perfect Bayesian equilibrium
Perfect Bayesian equilibrium (cont’ d)
Requirement 4 At information set o¤ the equilibrium path, beliefs are determined by Bayes’rule and the players’equilibrium strategies where possible.
Junji Xiao (Lecture 5)
Micro
5 / 50
Perfect Bayesian equilibrium
Example (cont’ d)
Player 2 L0 L 2, 1 M 0, 2 R 1, 3
Player 1
R0 0, 0 0, 1 1, 3
Nash equilibria: (L, L0 ) and (R, R 0 ). No subgame. SPNE: (L, L0 ) and (R, R 0 ). However, (R, R 0 ) depends on a noncredible threat: if player 2 gets the move, then playing L0 dominates playing R 0 , so player 1 should not be induced to play R by 2’ threat to play R 0 if given the move. s To rule out (R, R 0 ), we impose the two requirements.