Dynamic Games of Incomplete Information Chapter 4 Perfect

Dynamic Games of Incomplete Information -Chapter 4 Perfect Bayesian Equilibrium Game Theory--chapter 4 1

Outline of Dynamic Games of Incomplete Information n Introduction to Perfect Bayesian Equilibrium n Signaling games n Perfect Bayesian Equilibrium in Signaling games n Job-Market Signaling n Corporate investment and Capital Structure n Monetary Policy n Other Application of Perfect Bayesian Equilibrium n Refinements of Perfect Bayesian Equilibrium Game Theory--chapter 4 2

4. 1. Introduction to Perfect Bayesian Equilibrium. n Four equilibrium concepts: n Nash equilibrium (NE) difference relate n Subgame perfect Nash equilibrium (SPNE) difference relate n Bayesian Nash equilibrium (BNE) difference relate n Perfect Bayesian Equilibrium (PBE). Game Theory--chapter 4 3

Perfect Bayesian Equilibrium n Subgame concept SPNE The More general idea n n continuation game (后续博弈) Game Theory--chapter 4 PBE 4

Example 1 - SPNE n 我们的例子是完全信息的. 1 L R 2 2 2 A B 1 1 3 3 n 惟一的SPNE 是 (R, B). Game Theory--chapter 4 5

Example 2 -dynamic game of complete but imperfect information n 这里SPNE等同于 NE，因为这个博弈没有子博弈. 2 1 L R M (1, 3) 1 2 L’ R’ L’ (2, 1) (0, 0) (0, 2) L’ R’ R’ L 2, 1 0, 0 M 0, 2 0, 1 R 1, 3 (0, 1) n 注意：(L, L’) 和 (R, R’) 既是 SPNE, 又是 NE. 但 (R, R’) 是不可置信的威胁 Game Theory--chapter 4 6

Example 3 - SPNE n 剔除难以置信（implausible ）的均衡 (R, R’) 1 n 加入推断 L (2, 1) 1 -p R’ L’ (0, 0) (0, 2) Game Theory--chapter 4 (1, 3) M 2 p L’ R R’ (0, 1) 8

Bayes’ rule n 贝叶斯法则: Game Theory--chapter 4 12

To illustrate requirement 4 n 策略 (D, L, R’)和 player 3的推断 p=1是一个PBE和序贯均衡 n 策略 (A, L, L’)和 player 3的推断 p=0是PBE，不是序贯均衡 Game Theory--chapter 4 14

Perfect Bayesian equilibrium n 为了说明要求4的必要性, 我们把图 4. 1. 4 修改 为 图 4. 1. 5. Game Theory--chapter 4 15

Signaling game n 图 4. 2. 1 a 1 m 1 S m 2 t 1 [p] a 2 R N [1 -p] a 1 t 2 a 1 m 1 S Game Theory--chapter 4 m 2 a 2 19

Signaling game N n 扩展式 t 2 , 1 -p t 1 , p S S m 1 R a 1 a 2 m 2 a 1 m 2 R R a 2 a 1 Game Theory--chapter 4 R a 2 a 1 a 2 20

Signaling game n Sender和 Receiver 都有四个纯策略. n Sender的纯策略： n n (m 1, m 1), (m 1, m 2), (m 2, m 1), (m 2, m 2) (m 1, m 1), (m 2, m 2)：混同(pooling strategies)策略 (m 1, m 2), (m 2, m 1)：分离(separating strategies)策略 (m 1, with r plays m 1 1 -r play m 2 ): 杂合(hybrid strategies)策略 n receiver的纯策略： n (a 1, a 1), (a 1, a 2), (a 2, a 1), (a 2, a 2) n Sender的类型推断: p(t 1)=p, p(t 2)=1 -p Game Theory--chapter 4 21

Example 2 -啤酒或热狗（Beer&Quiche） 信号博弈 Game Theory--chapter 4 41

A Pooling equilibrium Game Theory--chapter 4 42

A Separating equilibrium Game Theory--chapter 4 43

A hybrid equilibrium Game Theory--chapter 4 44

Job Market Signaling Game Theory--chapter 4 45

Equilibrium (when mixed strategies are allowed) Game Theory--chapter 4 46

If t were common knowledge Game Theory--chapter 4 47

No need to imitate Game Theory--chapter 4 48

Want to imitate Game Theory--chapter 4 49

A pooling equilibrium (e*(L), e*(H) are the first best levels) Game Theory--chapter 4 50

A separating equilibrium Game Theory--chapter 4 51

An intuitive separating equilibrium (Cho-Kreps, 1987) Game Theory--chapter 4 52

Costless signaling: cheap talk n Heuristics: for cheap talk to be informative: Different sender-types have different preferences n Receiver’s preference depends on sender’s types n Receiver’s preferences are not completely opposed to sender’s n Used broadly in political economy and politics n Usually monetary transfer are not allowed n People (senders) are more heterogeneous n How to make yourself influential? n n Doing good rather than doing well Game Theory--chapter 4 53

Discussion n The refinement of PBEs n The importance of beliefs and correlating devices You are what you believe n What is dynamics, anyway? n Game Theory--chapter 4 54

Course review n Static Game of complete information n Dominant Strategy Equilibrium n Nash Equilibrium Pure strategy n Mixed strategy n n Dynamic game of complete information n Extensive representation n Game tree and subgame n Subgame perfection n Backward induction and limitations Game Theory--chapter 4 55

Course review n Static game of incomplete information n Incomplete information and imperfect information n Harsanyi transformation (nature and types) Bayesian Nash Equilibrium n Mechanism design (*) n n n Revelation principle screening Game Theory--chapter 4 56

Course review n Dynamic game of incomplete information n Perfect Bayesian Equilibrium n Requirements and definition § Continuation equilibrium § The importance of beliefs n n Pooling, separating, hybrid equilibrium Signaling game n n Costly signaling (job market signaling) Cheap talk (*) Game Theory--chapter 4 57

Course over Good Luck! Game Theory--chapter 4 58