Question

3. Consider the following game in normal form. Player 1 is the "row" player with strate- gies a, b, c, d and Player 2 is the "column" player with strategies w, x, y, z. The game is presented in the following matrix: W Z X y a 3,3 2,1 0,2 2,1 b 1,1 1,2 1,0 1,4 0,0 1,0 3,2 1,1 d 0,0 0,5 0,2 3,1 с Find all the Nash equilibria in the game in pure strategies.

Answer 1

Answer :

Assuming each player (1 and 2) is trying to maximize their corresponding payoff shown in the matrix.

1: When Player 1 plays a, Player 2 will play w (since w gives a payoff of 3 which is the highest in the row of strategy a)

Similarly when Player 2 plays w, Player 1 will play a (since a gives a payoff of 3 which is the highest in the column of strategy w)

Thus (a, w) is a nash equilibrium

2 : When Player 1 plays b, Player 2 will play z (since the payoff of 4 is the highest in the row of strategy b)

Now when Player 2 will play z, Player 1 will play d (since the payoff of 3 is the highest in the column of z)

Thus not a nash.

3 : When Player 1 plays c, Player 2 will play y (since payoff of 2 is highest in the row of c)

Similarly when Player 2 plays y, Player 1 will play c (since payoff of 3 is the highest in the column of y)

Thus (c, y) is another nash equilibrium

4 : When Player 1 plays d, Player 2 will play x (since 5 is the highest payoff in the row of d)

When Player 2 plays x, Player 1 will play a (since 2 is the highest payoff in the column of x)

Thus not a nash.

Therefore there are two pure strategies nash equilibria in this game = (a, w) and (c, y)