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Extensive Form 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:

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Extensive form games with perfect information theory

Extensive Form GamesWith Perfect Information(Theory)


Extensive form games with perfect information
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


Extensive form games with perfect information1
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 information2
Extensive Form Games with Perfect Information

Extensive Form Game Tree.

Challenger

Stay Out

Enter

Incumbent

1,2

Accommodate

Fight

2,1

0,0

- Nash Equilibria?

- Solve via Backward Induction


Extensive form games with perfect information3
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


Extensive form games with perfect information4
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 ?


Extensive form games with perfect information5
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 information6
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

D

C

2

2,0

E

F

1

3,1

G

H

1,2

0,0


Extensive form games with perfect information7
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 information8
Extensive Form Games with Perfect Information

Consider a game G of perfect information consisting of a tree T linking the information sets i0 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 subgameGi is a sub-tree Ti and the payoffs at each terminal node Ti.


Extensive form games with perfect information9
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 information10
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 information11
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


Extensive form games with perfect information12
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


Extensive form games with perfect information13
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


Extensive form games with perfect information14
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


Extensive form games with perfect information15
Extensive Form Games with Perfect Information

Proposition: Every finite extensive game with perfect information has a subgame perfect equilibrium. (173.1 in Osborne)


Extensive form games with perfect information16
Extensive Form Games with Perfect Information

  • Sequential Matching Pennies.

  • Preview of Repeated Games: Chain Store with N opponents.


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