Consider the following games:
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 identify the pure strategy Nash equilibria, we need to find the combinations of strategies where no player has an incentive to deviate given the other players' choices.
Looking at the given games:
L C R
U 9.4 3.3 5.2
M 5.8 8.8 3.7
D 4.1 6.6 2.9
For player 1 (row player), if player 2 (column player) chooses C, player 1 prefers U (9.4 > 5.8 > 4.1). If player 2 chooses R instead, player 1 still prefers U (5.2 > 3.7 > 2.9). Therefore, U is a pure strategy Nash equilibrium for player 1.
For player 2, if player 1 chooses L, player 2 prefers M (8.8 > 6.6 > 3.3). If player 1 chooses D instead, player 2 prefers M (3.7 > 2.9 > 2.9). Therefore, M is a pure strategy Nash equilibrium for player 2.
There is no pure strategy Nash equilibrium for player 3 as there is no combination where player 3's payoffs are maximized.
Now, let's check if there is an equilibrium in dominant strategies. A dominant strategy is a strategy that yields the highest payoff regardless of the other players' choices.
For player 1, there is no dominant strategy as the optimal choice depends on the column player's choice.
For player 2, there is no dominant strategy as the optimal choice depends on the row player's choice.
For player 3, there is no dominant strategy as the optimal choice depends on both the row and column players' choices.
Next, let's see if there are strategies that survive iterated elimination of dominated strategies. A dominated strategy is a strategy that is always worse than another strategy regardless of the other players' choices.
In the given games, there are no dominated strategies for any player. Therefore, no strategies are eliminated through iterated elimination of dominated strategies.
Since there are no dominant strategies and no strategies that survive iterated elimination of dominated strategies, the choice of which Nash equilibrium to play depends on other factors such as preferences, risk aversion, or potential cooperation between players. Both U and M can be valid choices, depending on the specific circumstances.