Example 2.2.1. A Simpler Payoff Matrix.
Consider the zero-sum game with payoff matrix in Table 2.2.2. For simplicity our payoff matrix contains only the payoffs and not the strategy names; but Player 1 still chooses a row and Player 2 still chooses a column.
Player 2 | ||
Player 1 | \((1, -1)\) | \((-0, 0)\) |
\((-1, 1)\) | \((-2, 2)\) |
If we know we are playing a zero-sum game, then the use of ordered pairs seems somewhat redundant: if Player 1 wins 1, then we know that Player 2 must lose 1 (win \(-1\)). Thus, if we KNOW we are playing a zero-sum game, we can simplify our notation by just using Player 1’s payoffs. The above matrix in Table 2.2.2 can be simplified as in Table 2.2.3.
Player 2 | ||
Player 1 | \(1\) | \(0\) |
\(-1\) | \(-2\) |