## Mixed Strategies and Payoff Equilibrium

Let us consider two fencers. On an attack, each can either go left or right or move back. Dodging is effective in breaking an engagement and creating an opening. The decision is virtually instantaneous. The fencers are professionals and know each other's capabilities, which are reflected as chances of succeeding in the table below:

 Fencer B Left Right Avoid Fencer A Left 35% 10% 60% Right 45% 55% 65% Avoid 40% 10% 65%

Neither player has an clear superior strategy, since each one's decision hangs on the other's. However, we can analyze the underlying structure. It is clear that B should not avoid at any time, since he does better at all times going to the left. Likewise, A should always go to the right, which would confirm B's decision to go to the left.

This sort of a self-confirming decision pair is called an equilibrium strategy, the point from which changing strategies would cause no increase in payoff, negating any need to change strategies. The intersection is called the equilibrium point (45% in this case). There can be more than one such point, but their values are always equal. This is not true of non-zero-sum or n-person games.

Sometimes, however, there is a problem where there are no equilibrium points. In such cases, thinking about a move can cause it to be a consistent loss situation. In such a situation, a mixed strategy must be used to break even (reach zero-sum state). A mixed strategy is when a random method is used to select a pure strategy. It is independent of general knowledge of the strategy, since in non-equilibrium games it is only possible to reach a net null gain/loss using any strategy (against a good opponent).

1. Dauben, Joseph W "Game Theory." Microsoft Encarta. Microsoft Corp, 1999
2. Davis, Morton D. "The General, Finite, Two-Person, Zero-Sum Game" Game Theory, A Nontechnical Introduction. New York: Basic Books, Inc., 1930, pp 19-48
3. McCain, Roger A. "Game Theory: An Introductory Sketch." Hypertext. 1999 (August 1999)