Suppose that two players are playing the following game. Player 1 can choose either top or bottom, and Player 2 can choose either left of right. The payoffs are given in the following table
Player 2
Left Right
top 9,4 2,3
Player 1
Bottom 1,0 3,1
where the number on the left is the payoff to Player 1 and the number on the right is the payoff to player 2.
1) Determine the nash equilibrium of the game.
2) If each player plays their maximin strategy, what payoff will each of them receive?
Each player I think does not have a dominant strategy.
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To determine the Nash equilibrium of the game, we need to consider each player's best response to the other player's strategy.
1) To find Player 1's best response, we look at the payoffs for each choice by Player 2. If Player 2 chooses 'Left', Player 1's payoffs are 9 when they choose 'Top' and 1 when they choose 'Bottom'. If Player 2 chooses 'Right', Player 1's payoffs are 3 when they choose 'Top' and 0 when they choose 'Bottom'.
Since 9 is higher than 1, Player 1's best response to Player 2 choosing 'Left' is to choose 'Top'. Similarly, since 3 is higher than 0, Player 1's best response to Player 2 choosing 'Right' is to choose 'Top'.
Next, to find Player 2's best response, we look at the payoffs for each choice by Player 1. If Player 1 chooses 'Top', Player 2's payoffs are 9 when they choose 'Left' and 4 when they choose 'Right'. If Player 1 chooses 'Bottom', Player 2's payoffs are 1 when they choose 'Left' and 0 when they choose 'Right'.
Since 9 is higher than 4, Player 2's best response to Player 1 choosing 'Top' is to choose 'Left'. Similarly, since 1 is higher than 0, Player 2's best response to Player 1 choosing 'Bottom' is to choose 'Left'.
Therefore, the Nash equilibrium of the game is when Player 1 chooses 'Top' and Player 2 chooses 'Left' (9,4).
2) If each player plays their maximin strategy, they would choose the strategy that maximizes their minimum possible payoff.
For Player 1, their minimum possible payoff is 1 when they choose 'Bottom'. If Player 2 plays their maximin strategy, they would choose 'Left', resulting in a payoff of (1,1) for Player 1.
For Player 2, their minimum possible payoff is 4 when they choose 'Right'. If Player 1 plays their maximin strategy, they would choose 'Top', resulting in a payoff of (4,4) for Player 2.
So, if each player plays their maximin strategy, Player 1 would receive a payoff of 1, and Player 2 would receive a payoff of 4.