Lecture 5 Dynamic Games of Incomplete Information Perfect Bayesian Equilibrium Signaling games Applications Job market signaling game Cheap talk game Investment financing Wage bargaining Reputation and cooperation
Lecture 5 Dynamic Games of Incomplete Information • Perfect Bayesian Equilibrium • Signaling Games • Applications – Job market signaling game – Cheap talk game – Investment financing – Wage bargaining – Reputation and cooperation 1
Dynamic games of Incomplete nformation: Example Player 2 R 2,10,0 R R Player1M0, 20, 1 2 R1,313 02 Nash equilibrium: (L, L), (R,R) Subgame perfect Nash equilibrium (L, L),(R,R)( no subgame) Perfect Bayesian equilibrium: L, L) and player 2's belief: player 1 play l probability =1(Whatever the belief, player 2 will play L')
Dynamic Games of Incomplete Information : Example 2 2 2 1 0 0 0 2 0 1 L M L’ R’ L’ R’ 1 1 3 R 2,1 0,0 0,2 0,1 1,3 1,3 L M R L’ R’ Player 1 Player 2 Nash equilibrium : (L,L’), (R,R’) Subgame perfect Nash equilibrium : (L,L’), (R,R’) (no subgame) Perfect Bayesian equilibrium: (L,L’) and player 2’s belief: player 1 play L probability =1 (Whatever the belief, player 2 will play L’ ) 2
Perfect Bayesian Equilibrium Requirement 1 Belief): At each information set, the player with the move must have a belief about which node in the information set has been researched 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's belief puts probability one on the single decision node
Perfect Bayesian Equilibrium • Requirement 1 (Belief) : At each information set, the player with the move must have a belief about which node in the information set has been researched 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’s belief puts probability one on the single decision node 3
Perfect Bayesian Equilibrium(cont) Requirement 2 (Play based on belief 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 players subsequent strategy) must be optimal given the player belief at that information set and the other player subsequent strategies
Perfect Bayesian Equilibrium (cont’) • Requirement 2 (Play based on belief) : 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’s subsequent strategy) must be optimal given the player belief at that information set and the other player’ subsequent strategies 4
Perfect Bayesian Equilibrium(cont) Definition For a given equilibrium in a given extensive-form game, an information set is on the equilibrium if it will be reached with positive probability if the game is played according to the equilibrium strategies. And is off the equilibrium if it is certain not to be reached if the game is played according to the equilibrium strategies
Perfect Bayesian Equilibrium (cont’) • Definition: For a given equilibrium in a given extensive-form game, an information set is on the equilibrium if it will be reached with positive probability if the game is played according to the equilibrium strategies. And is off the equilibrium if it is certain not to be reached if the game is played according to the equilibrium strategies 5
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
