Although the minimax theorem is an important result, it does not directly tell us how to find Nash equilibria, as it is not obvious how to determine all of the max-min or min-max strategies.

We begin by analysing two simple examples of zero sum games. Our main tool will be the Equality of Payoffs Theorem.

Example 3.8

Consider the game of Matching Pennies which we originally considered in Example 2.11.

Strategy 1 Strategy 2
Strategy 1 \(-1\) \(1\)
Strategy 2 \(1\) \(-1\)

Example completed by hand in the lecture

The next example (a slight variation) shows that the theory allows us to determine optimal strategies that are not obvious in advance.

Example 3.9

Consider the game

Strategy 1 Strategy 2
Strategy 1 \(3\) \(-1\)
Strategy 2 \(-1\) \(9\)

Example completed by hand in the lecture

Example 3.10

Consider the game

Strategy 1 Strategy 2 Strategy 3
Strategy 1 \(1\) \(2\) \(3\)
Strategy 2 \(3\) \(1\) \(2\)
Strategy 3 \(2\) \(3\) \(1\)

Example completed by hand in the lecture

Example 3.11

Consider the game

Strategy 1 Strategy 2 Strategy 3
Strategy 1 \(-2\) \(2\) \(-1\)
Strategy 2 \(1\) \(1\) \(1\)
Strategy 3 \(3\) \(0\) \(1\)

Example completed by hand in the lecture