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

2. 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: u1(Enter, Accommodate)=2, u1(Out)=1, u1(Enter, Fight)=0
- Incumbent: u2(Out)=2, u2(E,A)=1, u2(E,F)=0

3. 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.

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

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

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

7. 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.

8. Extensive Form Games with Perfect Information
Example (160.1)
Strategies of Player 2: E, F
Strategies of Player 1: CG,CH,DG,DH
“…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.” 1 C D 2
2,0 E F 1
3,1 G H
1,2
0,0

9. 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.

10. 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.

11. 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.

12. 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.

13. Extensive Form Games with Perfect Information
Example (172.1) 1 C E D 2 2 2 F G J K H I
3,0
1,0
1,3
2,2
2,1
1,1

14. Extensive Form Games with Perfect Information
Example (172.1)
Player 2 Optimal Strategies:
(FHK),(FIK),(GHK),(GIK) 1 C E D 2 2 2 F G J K H I
3,0
1,0
1,3
2,2
2,1
1,1

15. Extensive Form Games with Perfect Information
Player 2 Optimal Strategies:
(FHK),(FIK),(GHK),(GIK)
Example (172.1) 1
Player 1 Best Responses:
(C),(C),(C or D, or E),(D) C E D 2 2 2 F G J K H I
3,0
1,0
1,3
2,2
2,1
1,1

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

17. Extensive Form Games with Perfect Information
Proposition (173.1): Every finite extensive game with perfect information has a subgame perfect equilibrium.

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