L
C
R
U
9.4
3.3
5.2
M
5.8
8.8
3.7
D
4.1
6.6
2.9
Determine the pure strategy Nash equilibria
Is there an equilibrium in dominant strategies?
Are there strategies that survive iterated elimination of dominated strategies?
Which of the Nash equilibria should be chosen
To determine the pure strategy Nash equilibria, we need to identify any combination of strategies where no player can unilaterally deviate to improve their outcome.
Looking at the matrix, we can see that (M, U) is a pure strategy Nash equilibrium. If player L deviates to strategy C, they would decrease their payoff from 9.4 to 3.3. If player R deviates to strategy C, they would decrease their payoff from 5.2 to 3.7. Therefore, no player has an incentive to deviate from (M, U) and it is a pure strategy Nash equilibrium.
In terms of dominant strategies, a strategy is dominant if it always provides a higher payoff regardless of the other player's strategy. In this case, none of the strategies for any player is dominant. Therefore, there is no equilibrium in dominant strategies.
To check if there are strategies that survive iterated elimination of dominated strategies, we need to eliminate any dominated strategies. Looking at the matrix, we can see that strategy C for player L is dominated by strategy M. In any scenario where player R chooses strategy C, player L would always get a higher payoff by choosing strategy M. Therefore, strategy C for player L can be eliminated. Similarly, strategy C for player R is dominated by strategy U. In any scenario where player L chooses strategy C, player R would always get a higher payoff by choosing strategy U. Therefore, strategy C for player R can be eliminated. After eliminating these dominated strategies, we are left with the following reduced matrix:
L
M
R
U
9.4
3.3
5.2
M
5.8
8.8
3.7
D
4.1
6.6
2.9
There are no other dominated strategies remaining, so no further elimination is necessary. Therefore, the strategies that survive iterated elimination of dominated strategies are (U, M) and (M, U).
Finally, we need to determine which of the Nash equilibria should be chosen. Both (U, M) and (M, U) are pure strategy Nash equilibria, meaning that they are stable outcomes. The choice between these equilibria would depend on other factors such as preferences, risk tolerance, and potential payoff differences.