4. Beliefs at all info sets off the equilibrium path are determined by Bayes' Rule & the players' equilibrium strategies where possible.

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1 A. Perfect Bayesian Equilibrium B. PBE Examples C. Signaling Examples Context: A. PBE for dynamic games of incomplete information (refines BE & SPE) *PBE requires strategies to be BE for the entire game & for every continuation game + forces us to consider strategy profiles & beliefs which are reasonable on the equilibrium path (or off). ef: A PBE consists of a strategy profile (σ) and system of beliefs (μ): [σ, μ]. To be PBE, a candidate strategy profile must satisfy the following: 1. At each info set, the player who moves must assign a belief (i.e., probability μ k ) to each node in the info set. a. Singleton info set player has belief Pr=1 on that 1 decision node 2. Sequentially rationality: σ has players best- responding at each info set, given the beliefs & other players' subsequent strategies. 3. Beliefs at all info sets on the equilibrium path are determined by Bayes' Rule & the players' equilibrium strategies. Pr(player is at decision node x given a certain history of play h & strategy profile σ) is: 4. Beliefs at all info sets off the equilibrium path are determined by Bayes' Rule & the players' equilibrium strategies where possible. Bullets 1-3 weak PBE refines BE Bullets 1-4 PBE refines BE & SPE (ote: There usually won t be a diff b/w weak & not) Method for Solving PBE: Guess & Check 1. Propose a candidate strategy profile σ and/or the behavioral strategy 1 for 1 player. 2. Calculate other players' beliefs μ based on the profile above. 3. Calculate the BR of other players conditional on these calculated beliefs. 4. Check if any of the players have an incentive to deviate (Because you calculated the BR of the other players, you may only need to check the incentives of the 1 st player). 5. If possible, calculate beliefs consistent w/behavioral strategy profile off the equilib path. 1 Behavioral strategy: a strategy which assigns to each info set a probability distribution over the choices of that set. 1

2 B. PBE Examples Ex B.1 (2, 0, 0) R (0,1,1) A R 1- p L (0,1,2) L p R (3,3,3) L 2 1) # Proper Subgames? 1 2) ormal Form for P2- P3 game? L R L 2, 1 3,3 R 1,2 1,1 (1,2,1) 3) SPE for entire game? [,L,R ] 4) nique? Yes. L Ss R for P2 & P3 has strict preference only 1 E for P2- P3 a. P- 1 has strict pref. over this E (for ) and payoffs for A 5) What beliefs about P3 support this? p=1 PBE [,L,R, p=1] 6) PBE or Weak PBE? PBE 7) Consider behavioral strategy profile [A,L,L, p=0] a. Payoffs? (2,0,0) b. Is this an E? i. Yes. o player has an incentive to deviate. 1. Why only need to check P1 to see this? P2 was a strictly dominant strategy, & P3 follows his belief about p=0. c. Is it weakly PBE? i. Consider non- singleton info sets first: 1. Given beliefs (p=0), is P3 best responding by playing L? Yes. ii. Are P1 & P2 best responding? 1. P1 deviation to? payoff 1<2 2. P2 deviation to R?- - > payoff 0=0 iii. Weakly PBE! (i.e., bullets 1-3 ) iv. SPE? o. What s going on here? 1. Weak PBE doesn t constrain off- equ- path beliefs (bullet 4). Beliefs in [A,L,L, p=0] are inconsistent with P2 playing dominant strategy L. 2. Add bullet 4 P3 s belief must be p=1. L doesn t satisfy sequential rationality. ot PBE Ex B.2: Focusing on beliefs: we add strategy A for P2 w/a ending the game. Consider 2 strats: 1. [A,A,L, p=0] a. P3 info set still off- eq- path, but P3 s beliefs are now unconstrained by P2s strategy. P3 can have any beliefs he wants. 2. [A, μ 1 L+ (1- μ 1 )R + 0A, L, p=0] a. P3 info set still off- eq- path, but P2 s mixed strategy constrains P3 s beliefs. To satisfy bullet 4, need p 0 & p=μ 1.

3 C. Signalling Examples Ex C.1. An abstract version of the signaling model. 2 types of Senders: t {t 1, t 2 }, & the ex- ante probability distribution is Pr(t= t 1 ) = ½. 2 signals that any type of Sender can send: m {L, R}, Receiver s beliefs are denoted [p; 1 - p] & [q; 1 - q], respectively for R's 2 info sets Receiver has 2 actions at each info set: a {u, d} Payoffs: 1,3 2,1 4,0 p L S t1 3 R q ½ 0,0 R.1 R.2 2,4 ½ 1,0 0,1 1-p 1-q L R 1. What are the strategy spaces of each player? S Sender = {LL; LR; RL; RR} S Receiver = {uu; ud; du; dd} 2. Find all pure strategy PBE. Consider all Sender pooling strategies, then all Sender separating strategies. S t2 Pooling on L (i.e., LL) 1. Receiver's beliefs for the info set on the equilibrium path: p=½=1- p 2. Sequential rationality Receiver best responds to LL by playing u (strict domnt) since: 1,2 R (u p=½, LL)= ½ (3+4)=7/2 > ½ = ½(0+1) = R (d p=½, LL) 3. Sender receives either 1 or 2 (depending on type). oes either type have an incentive to deviate? Clearly, type t 2 does not want to deviate & send signal R, since she'll get a lower payoff. But type t 1 can get a higher payoff of 2 if she sends the signal R and the Receiver plays u. Therefore, off the equilibrium path, we need to have the Receiver playing d, which is consistent with all beliefs where q < 2/3: preferred to as long as: R (d q) > R (u q R (d q) = 0q + (1- q)2 = 2-2q > q = 1q + (1- q)0 = R (u q) [LL, ud, p = ½, q < 2/3] is a PBE.

4 Pooling on R (or RR) 1. Receiver's beliefs for this info set on the equilibrium path are q= ½= 1 q. 2. Sequential rationality Receiver best responds to RR by playing d since: 3. We could go through the same steps as above, or note that Sender type t 1 receives 0 with the behavioral strategy profile, and is guaranteed at least a payoff of 1 if she deviates. We know from above that Receiver's best response to the signal L is to play u Sender type t 1 will get 1. no PBE (or equilibrium) when the Sender pools on R. Separating both of receiver s info sets are on equilib path use Bayes rule to figure out beliefs for each + players strategies Separating LR *note: LR refers to strategy of (t1,t2) 1) Beliefs: LR p=1, q=0 a. Receiver s best response: ud (R.1, R.2) sender payoff=1 for both S t1 & S t2 i. Sender Incentive to eviate? Right now, gets 1. S t1 : switch to R 0<1 S t1 won t deviate 2. S t2 : switch to L 2>1 S t2 deviates ot a PBE Separating RL 1) Beliefs: RL p=0, q=1 a. Receiver s best response: uu S t1 payoff=2, S t2 payoff=2 i. Sender Incentive to eviate? Right now, gets 1. S t1 : switch to L 1<2 S t1 won t deviate 2. S t2 : switch to R 1=1 S t2 won t deviate b. either sender type deviates [RL, uu, p=0, q=1] is a PBE Ex C.2 Mark Buehrle's throwing arm is either in good or bad shape, with Pr(t = Good) = ½. Value to the ChiSox is $20 million if Buehrle's arm is in good shape and worth $0 if it's in bad shape. Buehrle knows what his type is, but the ChiSox do not. The ChiSox can offer Buehrle an $8 million dollar contract or offer him nothing. Buehrle can send a signal to the ChiSox organization by choosing to take a physical which would reveal the state of his arm for $1 million, or send a signal by doing nothing at all. ChiSox's beliefs as to Buehrle's arm is denoted by p. Assume that Buehrle's utility is monotonically increasing in the offer he receives from the ChiSox. ote: if testing was not possible, the ChiSox would offer him an $8 million dollar contract. 4

5 Payoffs: 8,12 7,12 MB good o Test Test Sox.2 p 0,0 ½ -1,0 Sox.1 8, -8 ½ 7,-8 1-p 5 Sox.3 o Test Test 0,0 MB bad What are the strategy spaces of each player? S Buerhle = {TT; T; T; } -1,0 S ChiSox = {;o; o; o; ;ooo} Find all pure strategy PBE. Pooling on no test 1) ChiSox Beliefs: p=½ a. ChiSox best response given these beliefs & pooling on no test: i. Sox.1: expected value of offer is 2, vs not offer at 0 offer ii. Sox.2 ffer iii. Sox.3 ot ffer b. Buehrle Incentive to eviate? o. Test is strictly dominated for both types. c. either sender type deviates [o Test, o Test; o; p=½] is a PBE Pooling on test : Could go through steps above, but note: Buehrle's bad type will get a payoff of - $1 mill (since the ChiSox know exactly where they are on the game tree). If the bad type deviates & plays no test, he's assured of at least 0 regardless of ChiSox beliefs. Pooling on test can t be an equilibrium. Separating with MB bad playing no test and MB good playing test : 1) ChiSox's beliefs are such that p = 0. a. ChiSox BR: offer if signal is test, no offer if the signal is no test. b. Buehrle Incentive to eviate? o i. MB good : play no test no offer at payoff 0<7 ii. MB bad : play test no offer at payoff - 1<0 c. [(test, no test); (no offer, offer, no offer); p = 0] is a PBE. f the 2 PBEs, 1 seems odd. Which? (separating) We ll talk about an equilib refinement that would rule this one out when we get to the Spence education model (Cho- Kreps Intuitive Criterion & ominance Refinements)