Chapter 5 Extensive Form Dynamic Games With Perfect

Chapter 5 Extensive Form (Dynamic) Games With Perfect Information (Theory)

Extensive Form Games with Perfect Information Entry Game: An incumbent faces the possibility of entry by a challenger. The challenger may enter or not. If it enters, the incumbent may either accommodate or fight. Payoff: - Challenger: u 1(Enter, Accommodate)=2, u 1(Out)=1, u 1(Enter, Fight)=0 - Incumbent: u 2(Out)=2, u 2(E, A)=1, u 2(E, F)=0

Extensive Form Games with Perfect Information Definition: an extensive form game consists of • the players in the game • when each player has to move • what each player can do at each of her opportunities to move • the payoff received by each player for each combination of moves that could be chosen by the players.

Extensive Form Games with Perfect Information Extensive Form Game Tree. Challenger Enter Stay Out Incumbent 1, 2 Fight Accommodate 0, 0 2, 1 - Nash Equilibria? - Solve via Backward Induction

Extensive Form Games with Perfect Information Normal Form (Simultaneous Move). Incumbent Accommodate Fight Enter 2, 1 0, 0 Stay Out 1, 2 Challenger

Extensive Form Games with Perfect Information Normal Form (Simultaneous Move). Incumbent Accommodate Fight Enter 2, 1 0, 0 Stay Out 1, 2 Challenger So we have two pure strategy NE, (enter, accommodate) and (stay out, fight). How come in the extensive form we only have one equilibrium by backward induction ?

Extensive Form Games with Perfect Information Definition: A strategy for a player is a complete plan of action for the player in every contingency in which the player might be called to act.

Extensive Form Games with Perfect Information Example (160. 1) Strategies of Player 2: E, F Strategies of Player 1: CG, CH, DG, DH 1 D C 2 E 2, 0 F 1 G 1, 2 3, 1 H 0, 0 “…a strategy of any player i specifies an action for EVERY history after which it is player i’s turn to move, even for histories that, if that strategy is followed, do not occur. ”

Extensive Form Games with Perfect Information Nash Equilibrium: each player must act optimally given the other players’ strategies, i. e. , play a best response to the others’ strategies. Problem: Optimality condition at the beginning of the game. Hence, some Nash equilibria of dynamic games involve non-credible threats.

Extensive Form Games with Perfect Information Consider a game G of perfect information consisting of a tree T linking the information sets i in I (each of which consists of a single node) and the payoffs at each terminal node of T. A sub-tree Ti is the tree beginning at information set i, and a subgame Gi is a sub-tree Ti and the payoffs at each terminal node Ti.

Extensive Form Games with Perfect Information A Nash equilibrium of G is subgame perfect if it specifies Nash equilibrium strategies in every subgame of G. In other words, the players act optimally at every point during the game. Ie, players play Nash Equilibrium strategies in EVERY subgame.

Extensive Form Games with Perfect Information Subgame Perfection - Every subgame perfect equilibrium (SPE) is a Nash equilibrium. - A subgame perfect equilibrium is a strategy profile that induces a Nash Equilibrium in every subgame.

Extensive Form Games with Perfect Information Example (172. 1) 1 C E D 2 2 2 F G H 3, 0 1, 1 J K I 2, 1 2, 2 1, 3

Extensive Form Games with Perfect Information Player 2 Optimal Strategies: (FHK), (FIK), (GHK), (GIK) Example (172. 1) 1 C E D 2 2 2 F G H 3, 0 1, 1 J K I 2, 1 2, 2 1, 3

Extensive Form Games with Perfect Information Player 2 Optimal Strategies: (FHK), (FIK), (GHK), (GIK) Example (172. 1) 1 Player 1 Best Responses: (C), (C or D, or E), (D) C E D 2 2 2 F G H 3, 0 1, 1 J K I 2, 1 2, 2 1, 3

Extensive Form Games with Perfect Information Player 2 Optimal Strategies: (FHK), (FIK), (GHK), (GIK) PE Strategy Pairs: (C, FHK), (C, FIK), (C, GHK), (D, GHK), (E, GHK), (D, GIK) 1 Player 1 Best Responses: (C), (C or D, or E), (D) C All S E D 2 2 2 F G H 3, 0 1, 1 J K I 2, 1 2, 2 1, 3

Extensive Form Games with Perfect Information Proposition (173. 1): Every finite extensive game with perfect information has a subgame perfect equilibrium.

Extensive Form Games with Perfect Information • Sequential Matching Pennies. • Preview of Repeated Games: Chain Store with N opponents.