Question

In: Economics

8. Solving for dominant strategies and the Nash equilibrium Suppose Andrew and Beth are playing a...

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.

Solutions

Expert Solution

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)


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