Extensive Form Games

With Perfect Information(Extensions)

Allowing for Simultaneous Moves

•New type of game where players make simultaneous decisions in a sequential environment.

•For example, player I moves first choosing C or D. Then players II and III move simultaneously, each choosing E or F.

•8 Histories of the game: (C, (E,E)), (C, (E,F)), (C, (F,E)), (C,(F,F)), (D, (E,E)), (D, (E,F)), (D, (F,E)), (D,(F,F)).

Allowing for Simultaneous Moves

•Definition: An extensive game with perfect information and simultaneous moves consists of: 1) a set of players, 2) a set of sequences (terminal histories), 3) actions available to each player at each possible move in the game, and 4) preferences for each player over each set of terminal histories (ie, payoffs)

•Definition: A strategy of player i in an extensive game with perfect information and simultaneous moves is a function that assigns to each history, h, after which i is one of the players whose turn it is to move, an action in Ai(h). Ie, the set of actions available to player i after history h.

Allowing for Simultaneous Moves

• Example 1: Variant of Battle of the Sexes.

U2

U2

CP

CP

3,1

0,0

0,0

1,3

2,2

I

Book Concert

I

II

Allowing for Simultaneous Moves

•Nash Equilibrium: Same definition as before.

•Subgame Perfect Nash Equilibrium: Same definition as before.

•What are the NE of the BoS game?

•What are the SPNE of the BoS game?

Allowing for Simultaneous Moves

Player 2

U2 CP

Player 1

Book, U2 2,2 2,2

Book, CP 2,2 2,2

Concert, U2 3,1 0,0

Concert, CP 0,0 1,3

Allowing for Simultaneous Moves

Player 2

U2 CP

Player 1

Book, U2 2,2 2,2

Book, CP 2,2 2,2

Concert, U2 3,1 0,0

Concert, CP 0,0 1,3

Allowing for Simultaneous Moves

•Nash Equilibia:

•( (Concert,U2), U2) ( (player 1 top node, simul. game), player 2 simul. game)

•( (Book,U2), CP)

•( (Book,CP), CP)

•SPNE

•( (Concert,U2), U2)

•( (Book,CP), CP)

In the subgame following player 1 choosing to go to the concert, players must play Nash Equilibrium strategies. So they have to go to the same concert.

Illustration of Firm Entry

•Consider an industry that is currently monopolized by a single firm (the incumbent). A second firm, (the challenger) considers entry.

•Demand P(Q) = - Q.

•Costs C1(q1) = cq1 for the incumbent, C2(q2) = c q2 - f for the challenger.

•If the challenger stays out, the incumbent gets the market to himself and obtains the monopoly level of profits.

•If the challenger enters, the firms play the simultaneous Cournot game.

•What is the SPNE of this game?

Illustration of Firm Entry

•Incumbent monopoly:

* qm = (1/2)( - c)

m = (1/4)( - c)2

•Cournot equilibrium profits: = (1/9)( - c)2

= (1/9)( - c)2 - f

•So now draw the game in extensive form with simultaneous moves.

Illustration of Firm Entry

Cournot Game

(0,m)

Challenger

Stay Out Enter

( (1/9)(-c)2-f, (1/9)(-c)2 )

In Cournot, each firm chooses some positive quantity so we can’t draw the game matrix.

Cournot Payoffs

Illustration of Firm Entry•If fixed costs are small, there is a unique SPNE in which the challenger enters and firms set their Cournot quantities.

•If fixed costs are large, there is a unique SPNE in which the challenger stays out the the incumbent captures the whole market.

•If (1/9)( - c)2 = f, the game has two SPNE (both of the two above).

•What about a threat by the incumbent to “flood the market” upon entry? Is this credible?

•What about Bertrand Competition upon entry? Homework 4!