There is a major result in Game Theory that provides for the existence of a equilibrium state or a solution. The minimax theorem establishes conditions on when the minimax and maximin of a function are equal. More precisely, the minimax theorem gives conditions on when
We have an optimal solution for the game with payout matrix
| 2 | 0 | 1 |
| 4 | -3 | 2 |
| 1 | -2 | 2 |
This represents a zero sum game, which means player 1 losing x implies player 2 gains x, and vice-versa. The rows are the payoff for player 1, a positive value means that player 2 loses the same amount.
To find an equilibrium we need to look at the rows for the minimum gain of player 1. This gives 0 in row 1, -3 in row 2, and -2 in row 3. Then we look at the columns for maximum loss of player 2. This gives -4 for column 1, 0 for column 2, and -2 for column 3. Notice 0 is the minimax and maximin. Hence this game has an equilibrium at 0. Player will pick strategy 1, and player 2 will pick strategy 2.