Question

Find all of the pure and mixed strategy Nash equilibrium of the following game:

Answer 1

Answer:

1. Pure strategy Nash equilibrium:

In this game, the pure strategy Nash equilibria are:
(a) (Top, Left) with payoff (5 10), and
(b) (Middle, Center) with payoff (8, 8).
  

reason: These are the two cells in the maatrix that coincide the two players' best responses to each other's moves. This is illustrated in the following matrix where the best response of each player is underlined. The cells where both payoffs are underlined represent the Nash equilibrium. Player 1's best responses are underlined in red, and Player 2's in blue.

Center Player 2 Left 5, 10 3,7 2,5 Top Middle Bottom Player 1 4,4 8,8 Right 2,1 6,5 2,2 2,7

2. Mixed strategy Nash equilibrium:

We can reduce this 3X3 matrix to 2X2 matrix by eliminating strictly dominated strategies of the players. Player 1 will never play Bottom. So we can eliminate this row. Similarly, Player 2 will never play Right. this column can be eliminated too. So we are left with the following 2X2 matrix:

  

Player 1 q 1-9 Player 2 Left Center 5, 10 4,4 3,7 8,8 p Top Middle 1-p

Let the probability of Player 1 moving Top be p, and Middle (1-p). Then his probability of choosing when he is indifferent between moving Top and Middle will be equal:

5q + 4(1-q) = 3q + 8(1-q); so, q = 2/3; (1-q) = 1/3

Similarly for Player 2: let the probability of Player 2 moving Left be p and Center (1-p). To make him indifferent at moving between Left and Center:

10p + 7(1-p) = 4p + 8(1-p); so, p = 4/7; (1-p) = 3/7

Thus, Player 1 will move Top with probability 4/7, and Middle with probability 3/7. Player 2 will move Left with probability 2/3 and Center with probability 1/3.

(Solution: Equilibrium solution will be (Top, Left) with probability 8/21 (4/7 * 2/3 = 8/21))