Question

Suppose that two players are playing the following game. Player 1 can choose either Top or Bottom, and Player 2 can choose either Left or Right. The payoffs are given in the following table

		Player 2
Player 1		Left	Right
	Top	6 5	9 4
	Bottom	7 4	5 3

where the number on the left is the payoff to Player A, and the number on the right is the payoff to Player B.

A) (4 points) Does player 1 have a dominant strategy, and if so what is it?

B) (4 points) Does player 2 have a dominant strategy and if so what is it?

C) (4 points each) For each of the following strategy combinations, write TRUE if it is a Nash Equilibrium, and FALSE if it is not:

i) Top/Left

ii) Top/Right

iii) Bottom/Left

iv) Bottom Right

D) (4 points) What is Player 1’s maximin strategy?

E) (4 points) What is player 2’s maximin strategy?

F) (4 points) If the game were played with Player 1 moving first and player 2 moving second, using the backward induction method discussed in the class notes, what strategy will each player choose?

Answer 1

A) Player 1 does not have a dominant strategy because the choice of player 1 changes according to the choice of player 2. If player 2 plays left, then player 1 should play bottom but if player 2 plays right then player 1 should play top.

For a dominant strategy to exist player 1 would have to chose one strategy regardless of what player 2 does.

B) Player 2 has a dominant strategy of left because he always chooses left regardless of what player 1 is playing.

This strategy produces a better outcome for player 2 always.

C)

i) Top/Left is not a Nash Equilibrium because if player 2 plays left then player 1 would be better off playing bottom and it only takes one player wanting to deviate to ruin a potential Nash Equilibrium.

ii) Top/Right is not a Nash Equilibrium because given player 1's choice of top, player 2 would play left.

iii) Bottom/Left is a Nash Equilibrium because no player is better off deviating given the action of the other player.

Player 1 get max payoff from choosing bottom and player 2 has the dominant strategy to choose left. So bottom left is a Nash.

Given player 2's choice of left, player 1 would be worse off playing top (since 6 < 7) so player 1 also will not deviate.

iv) Bottom/Right is not a Nash Equilibrium because given player 1's choice of bottom, player 2 would be better of playing left (since 4 > 3).

D) Player 1's maximin strategy is to play top since this could lead to his maximum payoff of 9

E) Player 2's maximin strategy is to play left since this could lead to his maximum payoff of 5

F) Backward induction tells us that player 2 would play Left regardless of player 1's choice. This means player 1 will play bottom so that he gets a payoff of 7 instead of 6. Technically the strategy should be listed as (B,LL) meaning that player 1 chooses bottom and player 2 chooses left regardless of player 1's choice.

**Complete question is answered, please rate if you are satisfied**