. 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 9 4 2 5
Bottom 1 0 3 1
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) Does player 1 have a dominant strategy, and if so what is it?
B) Does player 2 have a dominant strategy and if so what is it?
C) Determine the Nash equilibrium of this game. If there are no equilibrium in pure strategies, then say so. Otherwise, indicate what the equilibrium are by the strategies that would be chosen (i.e. Top/Left or Bottom/Right).
D) If each player plays their maximin strategy, what payoff will each of them receive?
To analyze this game, we will consider dominant strategies, Nash equilibrium, and maximin strategy.
A) To determine if Player 1 has a dominant strategy, we compare the payoffs for each choice. Looking at the first column (Left), Player 1 gets a higher payoff of 9 when choosing Top compared to 1 when choosing Bottom. Similarly, in the second column (Right), Player 1 gets a higher payoff of 5 when choosing Top compared to 0 when choosing Bottom. Therefore, Player 1 does have a dominant strategy, which is to choose Top.
B) To determine if Player 2 has a dominant strategy, we compare the payoffs for each choice. Looking at the first row (Top), Player 2 gets a higher payoff of 9 when choosing Left compared to 2 when choosing Right. Similarly, in the second row (Bottom), Player 2 gets a higher payoff of 4 when choosing Left compared to 5 when choosing Right. Therefore, Player 2 does not have a dominant strategy.
C) To find the Nash equilibrium, we need to identify the combination of strategies where neither player can unilaterally improve their payoff. In this game, there is no Nash equilibrium in pure strategies because there is always a better response for one player given the other player's strategy. For example, if Player 2 chooses Left, Player 1 is better off choosing Top. If Player 2 chooses Right, Player 1 is better off choosing Top as well. Similarly, if Player 1 chooses Top, Player 2 is better off choosing Left, and if Player 1 chooses Bottom, Player 2 is better off choosing Left as well.
D) The maximin strategy for each player is to choose the strategy that ensures the highest possible payoff under the assumption that the opponent will play their strategy to maximize their own payoff. For Player 1, the maximum payoffs considering the worst-case scenario (maximin) are 9 and 4 when choosing Top. For Player 2, the maximum payoffs considering the worst-case scenario (maximin) are 4 and 5 when choosing Left. Therefore, if each player plays their maximin strategy, Player 1 will receive a payoff of 4, and Player 2 will receive a payoff of 4 as well.