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