Answer to Question #269216 in Microeconomics for lilu

Question

Find all Nash Equilibria using both pure and mixed strategies. (a)

(b)

H D

F B

HD (1,1) (0, 0) (0, 0) (1, 1)

FB (3,2) (1, 1) (0, 0) (2, 3)

Answer 1

Find the maximum element in each column of matrix A. These elements are underlined in matrix A. Their position corresponds to the acceptable situations of the 1st player when the second player has chosen strategy j, respectively.

Then, in each row of matrix B, let us select the largest element. These elements are underlined in matrix B. Their position will determine the 2nd player's acceptable situations when the first player has chosen strategy i, respectively.

Player A's payoff matrix:

1 0 0 1

1 0 0 1

Positions of maxima in columns of matrix A: (1,1), (2,1), (1,2), (2,2), (1,3), (2,3), (1,4), (2,4)

Payment matrix of player B:

3 1 0 2

2 1 0 3

Positions of maxima in the rows of matrix B: (1,1), (2,4)

Intersection of these two sets: (1;1), (2;4),

Thus, 2 Nash equilibrium situations were found (1;1), (2;4). These situations turned out to be Pareto optimal for both players.

In equilibrium situation (1;1) player 1 wins 1 unit, and player 2 wins 3 units.

In an equilibrium situation (2,4), Player 1 wins 1 unit and Player 2 wins 3 units.

The set of all realizable vectors of winnings for the game has the following form

As we see, there is only one Pareto-optimal situation: H1=, H2=.

Let's calculate the prices of the games for the matrix games: va=1, vb=1. Hence, the Nash utility function will take the form:

U=(H1-1)(H2-1)