How to solve a two-person zero-sum game?

To solve a two-person zero-sum game, use the minimax theorem to find the optimal strategy.

A two-person zero-sum game is a game where one player's gain is the other player's loss. To solve such a game, we use the minimax theorem, which states that in a zero-sum game, the optimal strategy for one player is to minimize the maximum possible loss.

To apply the minimax theorem, we first create a payoff matrix that shows the possible outcomes for each player's strategies. We then find the maximum payoff for each row and the minimum payoff for each column. The optimal strategy for Player 1 is to choose the row with the highest minimum payoff, while the optimal strategy for Player 2 is to choose the column with the lowest maximum payoff.

For example, consider the following payoff matrix:

| | A | B | C |
|---|---|---|---|
| X | 2 | 1 | 3 |
| Y | 4 | 0 | 2 |
| Z | 1 | 3 | 2 |

To find the optimal strategy for Player 1, we first find the minimum payoff for each row:

- Row X: minimum payoff is 1
- Row Y: minimum payoff is 0
- Row Z: minimum payoff is 1

The highest minimum payoff is 1, which occurs in rows X and Z. Therefore, Player 1's optimal strategy is to choose either X or Z with equal probability.

To find the optimal strategy for Player 2, we find the maximum payoff for each column:

- Column A: maximum payoff is 4
- Column B: maximum payoff is 3
- Column C: maximum payoff is 3

The lowest maximum payoff is 3, which occurs in columns B and C. Therefore, Player 2's optimal strategy is to choose either B or C with equal probability.

By using the minimax theorem, we have found the optimal strategies for both players in this game.