Dominance Method For Solving Mixed Strategy Game

The dominance method is a technique used in game theory to solve mixed strategy games. In a mixed strategy game, each player chooses a strategy randomly according to a probability distribution. The dominance method involves identifying and eliminating dominated strategies to determine the optimal mixed strategy for each player.

A dominant strategy is a strategy that is always worse than another strategy, regardless of the other player’s strategy. If a player has a dominant strategy, they will never choose that strategy in a rational game. Thus, dominated strategies can be eliminated from consideration, reducing the game to a smaller set of strategies.

To use the dominance method to solve a mixed strategy game, follow these steps:

Create a matrix representing the game and calculate the expected payoff for each player for each possible combination of strategies.

Identify any dominant strategies for each player. A dominated strategy can be eliminated from consideration, as the player will never choose that strategy in a rational game.

After eliminating dominated strategies, calculate the expected payoff for each player for each remaining strategy.

If there is a unique best response for each player, then the game has a unique mixed strategy Nash equilibrium.

If there is no unique best response for each player, repeat steps 2-4 until a Nash equilibrium is found.

For example, consider the following two-player game:

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| L | M | R |

—————-

U | 2 | 1 | 3 |

—————-

D | 1 | 3 | 2 |

—————-

To use the dominance method to solve this game, we first identify any dominant strategies. In this game, there are no dominant strategies for either player.

Next, we calculate the expected payoffs for each player for each strategy. Assuming each player chooses each strategy with equal probability, we get:

Player 1’s expected payoffs:

f Player 2 chooses L: (2+1)/2 = 1.5

If Player 2 chooses M: (1+3)/2 = 2

If Player 2 chooses R: (3+2)/2 = 2.5

Player 2’s expected payoffs:

If Player 1 chooses U: (2+1)/2 = 1.5

If Player 1 chooses D: (1+3)/2 = 2

If Player 1 chooses R: (3+2)/2 = 2.5

Since both players have a unique best response (Player 1 chooses R and Player 2 chooses M), the game has a unique mixed strategy Nash equilibrium.

In summary, the dominance method is a useful technique for solving mixed strategy games by eliminating dominated strategies and identifying best responses. It can help players determine the optimal strategy to use in a game and achieve a Nash equilibrium.