• Zero-sum two person games of perfect information
• Why these games are strictly determined
• Generalized forms of zero-sum games
• Mixed strategy and payoff equilibrium
• Von Neumann's minimax theorem
• The utility theorem
• Non-zero-sum games

We can show indirectly that one player must have a winning strategy once he reaches the winning position defined by the fact that at that position he is executing a winning strategy. Without going into the mathematics, we can prove the existence of this winning strategy. Let us call out players A and B, and begin with the assumption that the game is not strictly determined but finite like a coin toss. This means that like the coin toss, there is one strategy where A can win, and the converse. From this, we can say that the initial move cannot be part of A's winning strategy, since this guarantees a victory for player A. We can also say that the initial move cannot be one that guarantees A's defeat, since that would guarantee a win for B. therefore the winning position is not the first one.

Since the first move was non-deterministic, we can say the same about the next move, and all the ones after that. Thus the game can never end, since by definition in a non-strictly determined game neither player is guaranteed a win at a position if a previous position were not already determining to the game. Thus the game cannot be finite, which contradicts the assumption that the game is finite. Thus the game must be strictly determined, or there must be from the beginning a determined winning strategy for one player or another.

Let us not confuse this with the fact that a player may never actually come upon the winning strategy. For example, in chess it is hard for players to actually compute the extensive form of the game, especially given time constraints, but also due the sheer computing power necessary to compute the geometrically increasing strategies. Thus, having given no consideration to the fact that there are such things as prefferences, non-logical strategies, common sense, attacks, and commitments, a game theorist would make a poor player indeed. Nonetheless, these facts become useful when we apply game theory to other situations, since is such cases, it is better to have a poor player than none at all.

1. Dauben, Joseph W "Game Theory." Microsoft Encarta. Microsoft Corp, 1999
2. Davis, Morton D. "The One Person Game" Game Theory, A Nontechnical Introduction. New York: Basic Books, Inc., 1930, pp 8-18
3. McCain, Roger A. "Game Theory: An Introductory Sketch." Hypertext. 1999 (August 1999)