Examples and exercises on Nash equilibrium in games in which each player has finitely many actions

Procedure

Check each action pair to see if it has the property that each player's action maximizes her payoff given the other players' actions.

Example: coordination between players with different preferences

Two firms are merging into two divisions of a large firm, and have to choose the computer system to use. In the past the firms have used different systems, I and A; each prefers the system it has used in the past. They will both be better off if they use the same system then if they continue to use different systems.

We can model this situation by the following two-player strategic game.

Player 2

I A

Player 1 I

2,1 0,0

0,0 1,2

A

To find the Nash equilibria, we examine each action profile in turn.

(I,I): Neither player can increase her payoff by choosing an action different from her current one. Thus this action profile is a Nash equilibrium.
(I,A): By choosing A rather than I, player 1 obtains a payoff of 1 rather than 0, given player 2's action. Thus this action profile is not a Nash equilibrium. [Also, player 2 can increase her payoff by choosing I rather than A.]
(A,I): By choosing I rather than A, player 1 obtains a payoff of 2 rather than 0, given player 2's action. Thus this action profile is not a Nash equilibrium. [Also, player 2 can increase her payoff by choosing A rather than I.]
(A,A): Neither player can increase her payoff by choosing an action different from her current one. Thus this action profile is a Nash equilibrium.

We conclude that the game has two Nash equilibria, (I,I) and (A,A).

Example: players with opposing preferences

An established firm and a newcomer to the market of fixed size have to choose the appearance for a product. Each firm can choose between two different appearances for the product; call them X and Y. The established producer prefers the newcomer's product to look different from its own (so that its customers will not be tempted to buy the newcomer's product) while the newcomer prefers that the products look alike.

We can model this situation by the following two-player strategic game.

Player 2

X Y

Player 1 X

2,1 1,2

1,2 2,1

Y

To find the Nash equilibria, we examine each action profile in turn.

(X,X): Firm 2 can increase its payoff from 1 to 2 by choosing the action Y rather than the action X. Thus this action profile is not a Nash equilibrium.
(X,Y): Firm 1 can increase its payoff from 1 to 2 by choosing the action Y rather than the action X. Thus this action profile is not a Nash equilibrium.
(Y,X): Firm 1 can increase its payoff from 1 to 2 by choosing the action X rather than the action Y. Thus this action profile is not a Nash equilibrium.
(Y,Y): Firm 2 can increase its payoff from 1 to 2 by choosing the action X rather than the action Y. Thus this action profile is not a Nash equilibrium.

We conclude that the game has no Nash equilibrium!

Exercise

Find the Nash equilibria of the following strategic game.

Player 2

L R

Player 1 T

2,2 0,0

0,0 1,1

B

[Solution]