Perfect Bayesian Equilibrium(cont) Requirement 3 Belief based on Bayes rule): At the information sets on the equilibrium path, beliefs are determined by Bayes rule and the players equilibrium strategies Requirement 4(Reasonable belief): At the information sets off the equilibrium path beliefs are determined by bayes rule and the players equilibrium strategies where possible Definition: A perfect Bayesian equilibrium consists of strategies and beliefs satisfying requirement 1 through 4
Perfect Bayesian Equilibrium (cont’) • Requirement 3 (Belief based on Bayes’ rule): At the information sets on the equilibrium path, beliefs are determined by Bayes’ rule and the players’ equilibrium strategies • Requirement 4 (Reasonable belief): At the information sets off the equilibrium path, beliefs are determined by Bayes’ rule and the players’ equilibrium strategies where possible • Definition: A perfect Bayesian equilibrium consists of strategies and beliefs satisfying requirement 1 through 4 6
PBE: EXample 1 1 R M 3 R”L R 2 02 If the play of the game reaches player 2's nonsigleton information Given player 2's belief the expected payoff from playing Ris UR=p 0+(1-p).1=1-p the expected payoff from playing L'is UL=p 1+(1-p). 2=2-p UR<UL for all p Subgame perfect Nash equilibrium: (L, L,)E p=1 Subgame perfect Nash equilibrium (R, R)are ruled out
PBE: Example 1 2 2 2 1 0 0 0 2 0 1 L M L’ R’ L’ R’ 1 1 3 R Subgame perfect Nash equilibrium : (L,L’) [p] [1-p] p=1 If the play of the game reaches player 2’s nonsigleton information Given player 2’s belief, the expected payoff from playing R’ is UR’ =p.0+(1-p).1=1-p the expected payoff from playing L’ is UL ’=p.1+(1-p).2=2-p UR’ < UL’ for all p Subgame perfect Nash equilibrium : (R,R’) are ruled out 7
PBE: EXample 2 A D 200 Information set is reached (D, L,R),p=1: Nash equilibrium .3.[1-p] (D,L, R), p=1: Subgame perfect Nash equilibrium R'L/R(D, L,R), p=1: Perfect Bayesian equilibrium Information set is not reached 30 0(A, L, L,),p=0: Nash equilibrium 2 32 Given player 3s belief p=0, player 3 play l, Given player 3 play L, player 2 play L Given player 2, 3 play (L, L), player 1 play A (A, L, L), p=0: NOT subgame Nash equilibrium Nash equilibrium of the only subgame is(L,R) (A, L, L), p=0: NoT Perfect Bayesian equilibrium Player3's belief p=0 conflicts with player 2 play L 8
PBE: Example 2 3 1 2 1 3 3 3 0 1 2 0 1 1 L R L’ R’ L’ R’ 2 D [p] [1-p] A 1 2 0 0 • Information set is reached (D, L, R’), p=1 : Nash equilibrium (D, L, R’), p=1 : Subgame perfect Nash equilibrium (D, L, R’), p=1 :Perfect Bayesian equilibrium • Information set is not reached (A, L, L’),p=0 : Nash equilibrium Given player 3’s belief p=0, player 3 play L; Given player 3 play L’, player 2 play L Given player 2, 3 play (L,L’), player 1 play A (A, L, L’),p=0 : NOT subgame Nash equilibrium Nash equilibrium of the only subgame is (L, R’) (A, L, L’), p=0 : NOT Perfect Bayesian equilibrium Player3’s belief p=0 conflicts with player 2 play L 8
PBE: EXample 3 1. If player 1's equilibrium strategy is A, requirement 4 may not determine 3's belief from player 2 A strategy( the player 3s information set is off equilibrium path 2. If player 2's strategy is A then requirement 4 puts 2 A no restrictions on 3 s belief (the player 3S R information set is off equilibrium path) 3 3. If 2s strategy is to play L with probability q1, R with probability g2, and A' with probability 1-q1-g2 RL R then requirement 4 dictates that 3s belief be p=g g1+g2) (the player 3's information set is on equilibrium path)
PBE: Example 3 3 L R L’ R’ L’ R’ 2 D [p] [1-p] 1 A A’ 1. If player 1’s equilibrium strategy is A, requirement 4 may not determine 3’s belief from player 2 strategy (the player 3 ‘s information set is off equilibrium path) 2. If player 2’s strategy is A’ then requirement 4 puts no restrictions on 3’s belief (the player 3 ‘s information set is off equilibrium path) 3. If 2’s strategy is to play L with probability q1 , R with probability q2 , and A’ with probability 1-q1 -q2 , then requirement 4 dictates that 3’s belief be p=q1 /(q1+q2 ) (the player 3 ‘s information set is on equilibrium path) 9
Signaling games A signaling game is a dynamic game of incomplete information involving two players: a Sender(S)and a Receiver(R) Timing of the game is as follows 1. Nature draws a type t, for the Sender from a set of feasible types T=t, .. t according to a probability distribution p(ti), where p(ti)>0 for every i and p(t,)+.+p(t)=1 2. The Sender observes t; and then chooses a message m; from a set of feasible M=m, 3. The Receiver observes m, (but not t]) and then chooses an action ak from a set of feasible actions A=(a,. axl 4. Payoffs are given by Us( m; ak) and UR( m; aR) 10
Signaling Games • A signaling game is a dynamic game of incomplete information involving two players: a Sender (S) and a Receiver (R) • Timing of the game is as follows: – 1. Nature draws a type t i for the Sender from a set of feasible types T={t1 ,…,t I } according to a probability distribution p(t i ), where p(t i )>0 for every i and p(t1 )+…+p(t I )=1 – 2. The Sender observes t i and then chooses a message mj from a set of feasible M={m1 ,…,. mJ } – 3. The Receiver observes mj (but not t i ) and then chooses an action ak from a set of feasible actions A={a1 ,…,aK} – 4.Payoffs are given by US (t i ,mj ,ak ) and UR(t i,mj ,ak ) 10