Refinements of Bayesian Nash Equilibrium

Transcription

1 Refinements of Bayesian ash Equilibrium Joseph Tao-yi Wang /7/0 (Lecture, Micro Theory I)

2 Market Entry Game w/ Incomplete Information Example of many BE; some are less plausible than others: If Entrant s backing is Agent : Incumbent Agent : Entrant -, 40 3, 30 0, 60 0, 60

3 Market Entry Game w/ Incomplete Information If Entrant s backing is Agent : Incumbent Agent : Entrant -, 9 3, 30 0, 60 0, 60

4 Market Entry Game w/ Incomplete Information (0,60) (-,40) (3,30)

5 Market Entry Game w/ Incomplete Information Information Sets (0,60) (-,40) (3,30) (-,9) (3,30) (0,60)

6 BE if Player Chooses (0,60) (-,40) (3,30) (-,9) (3,30) (0,60)

7 BE if Player Chooses : Player s BR (0,60) (-,40) (3,30) (-,9) (3,30) (0,60)

8 BE if Player Chooses : Player s BR (0,60) (-,40) (3,30) (-,9) (3,30) (0,60)

9 BE if Player Chooses (0,60) (-,40) (3,30) (-,9) (3,30) (0,60)

10 BE if Player Chooses (0,60) (-,40) (3,30) (-,9) (3,30) (0,60)

11 BE if Player Chooses : Player s BR (0,60) (-,40) (3,30) (-,9) (3,30) (0,60)

12 BE if Player Chooses : Player s BR (0,60) (-,40) (3,30) (-,9) (3,30) (0,60)

13 BE if Player Chooses (0,60) (-,40) (3,30) (-,9) (3,30) (0,60)

14 Empty Threats Off the Equilibrium Path ot a Sensible Equilibrium If, Incumbent wouldn t want to ot SPE when p=0 Problem due to crazy beliefs that are: Off the Equilibrium Path: nodes that are not reached in equilibrium ot reached = Zero probability (discrete types only) On the Equilibrium Path: nodes that are reached in equilibrium

15 Perfect Bayesian Equilibrium A BE is a Perfect Bayesian Equilibrium (PBE) if at all nodes off the equilibrium path, there are strategies and beliefs consistent with Bayes Rule such that the strategies (both on and off the equilibrium path) are BR When p</, (, ) is not a PBE since when occurs (off-equilibrium path), is only a BR if p /.

16 Trembling-Hand Perfect Equilibrium To rule out crazy equilibrium, can perturb the BE by making them totally mixed: Consider a game with T stages Set of feasible actions at stage t is (finite) For the BE Consider a sequence of totally mixed strategies (sequence of trembles) All nodes are reached (and tested in the BE) o more crazy beliefs off the equilibrium path

17 Trembling-Hand Perfect Equilibrium A BE is Trembling-Hand Perfect (THP) if There exists some sequence of totally mixed strategy profiles (Converging to the equilibrium strategies) such that For all sufficiently large k, the equilibrium strategies are BR: ote: If a sequence of Logit-QRE converges to a BE, would the BE automatically be THP? QRE solves this by construct (already totally mixed )

18 BE if Player Chooses : ot THP (0,60) (-,40) (3,30) (-,9) (3,30) (0,60)

19 BE if Player Chooses : Indeed THP (0,60) (-,40) (3,30) (-,9) (3,30) (0,60)

20 Sequential Equilibrium A BE is a sequential equilibrium if Each strategy at each node is a BR When beliefs at each node are the limits of beliefs associated with trembles as the probability of trembles 0 ote: THP SE

21 Market Entry Game with Private Information (-,4) (3,3) (4,-) (3,3)

22 BE when p<5/6: (,, ) (-,4) (3,3) (4,-) (3,3)

23 BE when p<5/6: (,, ) (-,4) (3,3) (4,-) (3,3)

24 BE when p<5/6: (,, ) (-,4) (3,3) (4,-) (3,3) o Information Update!

25 BE when p<5/6: (,, ) (-,4) (3,3) (4,-) (3,3) But (,, ) is no longer an equilibrium if p>5/6!!

26 BE when p>5/6: ( ; Others Mix) (-,4) (3,3) (4,-) (3,3)

27 BE when p>5/6: ( ; Others Mix) (-,4) (3,3) (4,-) (3,3)

28 BE when p>5/6: ( ; Others Mix) (-,4) (3,3) (4,-) (3,3)

29 BE when p>5/6: ( ; Others Mix) (-,4) (3,3) (4,-) (3,3)

30 BE when p>5/6: ( ; Others Mix) (-,4) (3,3) (4,-) (3,3)

31 BE when p>5/6: ( ; Others Mix) (-,4) (3,3) (4,-) (3,3)

32 Modified Market Entry Game: ew Payoffs (-6,5) (-,4) (-,-6) (4,)

33 Separating Equilibrium: -, - (-6,5) (-,4) (-,-6) (4,)

34 Separating Equilibrium: -, - (-6,5) (-,4) (-,-6) (4,)

35 Separating Equilibrium: -, - (-6,5) (-,4) (-,-6) (4,)

36 Separating Equilibrium: -, - (-6,5) (-,4) (-,-6) (4,)

37 Separating Equilibrium: -, - (-6,5) (-,4) (-,-6) (4,)

38 Separating Equilibrium: -, - (-6,5) (-,4) (-,-6) (4,)

39 (-; -) is also Sequential! (-6,5) (-,4) (-,-6) (4,)

40 Pooling Equilibrium: (,, ) (-6,5) (-,4) (-,-6) (4,)

41 Pooling Equilibrium: (,, ) (-6,5) (-,4) (-,-6) (4,)

42 Pooling Equilibrium: (,, ) (-6,5) (-,4) (-,-6) (4,)

43 Pooling Equilibrium: (,, ) (-6,5) (-,4) (-,-6) (4,)

44 Pooling Equilibrium: (,, ) (-6,5) (-,4) (-,-6) (4,)

45 (,, ) is also a Sequential Equil.! Weak (-6,5) (-,4) Strong (-,-6) (4,)

46 (,, ) is also a Sequential Equil.! (,, ) is not ruled out by THP, and hence, is also a Sequential Equilibrium But why can t the type say, If I enter, I will be credibly signaling that I am, since if I were and chose to, my possible payoffs would be - or -6, smaller than 0 (equilibrium payoff if ). Seeing this, player s BR is Profitable for player to & signal

47 (Weak) Intuitive Criterion (Cho and Kreps) For first move player s action (not in PBE) Let be player i s payoff as type if he chooses but is believed to be type Let be this types PBE payoff The PBE fails the (Weak) Intuitive Criterion if, for some player of type, And, for all other types, (can t signal)

48 (Strong) Intuitive Criterion (for 3+ types) For first move player s action (not in PBE) Let be player i s payoff as type if he chooses but is believed to be type Let be this types PBE payoff The PBE fails the (Strong) Intuitive Criterion if, for some player of type, And, for all other types, (can t mimic)

49 Intuitive Criterion (Cho and Kreps) IC: I can credibly signal that I am high type Cause I gain (against PBE) if you believe me, and Weak IC: obody else can make similar claims ot only this claim, but any similar claim Stronger requirement of failure = er criterion Strong IC: obody else can make this claim Weaker requirement of failure = er criterion With only two types, and IC are the same

50 Intuitive Criterion (Cho and Kreps) In the previous Example, (,, ) fails the Intuitive Criterion If I enter, I will be credibly signaling that I am, since I gain if you believe me and if I were and chose to, my possible payoffs would be - or -6, smaller than 0 (PBE payoff if ). (,, ) meets Intuitive Criterion Such argument is not credible

51 Summary of 0. SPE under incomplete information: PBE Two special cases: SE and THP Different Types of PBE: Pooling Equilibrium Separating Equilibrium Semi-Pooling Equilibrium (MSE) Intuitive Criteria HW 0.: See handout