The Two-Person, Zero-Sum Game with Equilibrium Points

Zero-sum means the players have diametrically opposed interests.

Two strategies are said to be in equilibrium (they come in pairs, one for each player) if neither player gains by changing strategy unilaterally. The outcome corresponding to this pair of strategies is defined as the equilibrium point, which is considered stable because a player unilaterally picking a new strategy is hurt by the change.

For two-person, zero-sum games, there may be more than one equilibrium point, but it there is, they will all have the same payoff.

How to find the equilibrium point

Theoretic method

If equilibrium points exist, they are easy to find. For a given payoff matrix (A's choice is represented as rows, while B's as columns):

1. Since B would choose a strategy that yields the minimum value of any row A chooses, A should choose a strategy that yields the maximum of these minimum values; this value is called the maximin.
2. Since A would choose a strategy that yields the maximum value of any column B chooses, B should choose the column that minimizes these maximum values; this value is called the minimax.
3. If the minimax equals the maximin, the payoff is an equilibrium point and the corresponding strategies are an equilibrium strategy pair.

When an equilibrium point exists in a two-person, zero-sum game, it is called the solution. The reasons why equilibrium points are considered solutions are:

1. By playing his/her equilibrium strategy, a player will get at least the value of the game.
2. By playing his/her equilibrium strategy, an opponent can stop a player from getting any more than the value of the game.
3. Since the game is zero-sum, a player's opponent is motivated to minimize the player's payoff.

In games with equilibrium points, payoffs that are not associated with either equilibrium strategy have no bearing on the outcome. That is, the change to these values will not change player's strategies.

Simplification with domination

It is often possible to simply a game by eliminating dominated strategies.

Strategy A dominates strategy B if a player's payoff with A is

• always at lease as much as that of B (whatever other players do)
• at lease some of the time actually better than strategy B.

If all strategies but one are dominated for each player, the equilibrium point(s) can be calculated.

If there is no equilibrium point, a mixed strategy is required. Refer to the next chapter.