Question

8. Solving for dominant strategies and the Nash equilibrium Suppose Andrew and Beth are playing a game in which both must simultaneously choose the action Left or Right. The payoff matrix that follows shows the payoff each person will earn as a function of both of their choices. For example, the lower-right cell shows that if Andrew chooses Right and Beth chooses Right, Andrew will receive a payoff of 4 and Beth will receive a payoff of 6.

Beth

Left Right

Andrew Left 2,3 2,4

Right 3, 7   4, 6

The only dominant strategy in this game is for Andrew/Beth to choose left/right .

The outcome reflecting the unique Nash equilibrium in this game is as follows: Andrew chooses left/right and Beth chooses left/right.

Answer 1

The given payoff matrix is:

Player 1/2	Left	Right
Left	2,3	2,4
Right	3,7	4,6

Player 2 has no dominant strategy as his payoffs from playing left are not strictly greater than his payoffs from playing right. Player 1, however, has a dominant strategy of playing right as his payoffs from playing right are strictly greater than his payoffs from his payoffs from playing left. Therefore, player 1 will always choose right,

If player 1 chooses right, player 2's best response is to choose left to maximize his payoff. Similarly, if player 2 chooses left, player 1 chooses right to maximize his payoff. These best responses are in bold in the payoff matrix above.

Therefore, the Nash Equilibrium strategy of this game is player 1 chooses right and player 2 chooses left. The equilibrium outcome is 3 for player 1 and 7 for player 2. (3,7)