• 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

The almost perfect example of this type of games is chess. The game is of perfect information since both players can see all of the board and the moves that may be deducible by one are deducible by the other. This game can be describes in extensive form or normal form. For example, if one had to describe a game before it was played, it would require not one strategy, but a complete collection of strategies that allow for all contingencies caused by other opposing strategies.

We can assume are that the player prefers to win or at least bring the game to a draw, and that he can analyze the moves as far as possible (to the end of a game) and describe them. However, we cannot assume that there is any one good strategy from the beginning. We can describe the important points in this game as the following:

• There are two players
• They have interest in winning themselves
• The game is finite. This means that each move has a known number of countermoves (irrespective of their value to the strategy) until the very last.
• The game provides all the information to both players, and thus no surprises.

These factors define zero-sum games of perfect information. Analyzing these games allow us to generalize about certain properties irrespective of the game itself.

Now let us enumerate the strategies for black in the set B₁ to B_n and the strategies for white to be W₁ to W_n. We are lumping the individual moves into strategies, and thus the game is said to be in a normal form. This normal form game can be described in a matrix with W showing wins, D showing a draw, and L a loss, the beginnings of which are shown below.

However, this is simply a mathematical simplification. The actual game is played in extensive form, where the strategy develops like a tree. Above (at the beginning) is a decision tree with only two choices at each juncture expanded to the first 10 moves.

To answer the question in part about what strategy to chose and what the outcome will be, we can define three simple special cases for the normal form:

• A column is set W with all losses
• A row in set B with all wins
• At least one row has no losses and at least one column has no wins

Case 1 and 2 are easy: the players much simply chose the appropriate row/column to win. For case 3, both players can avoid losing by selecting the right rows and columns. Thus far the questions of what move and what outcome are easy to answer.

To contrast this with a game of imperfect information, let us compare to a game of coin toss. The matrix for the outcomes is shown below:

As we can see, the 3 cases are not in this matrix, and in fact, not in the matrix of any game lacking perfect information. As it turns out, Ernst Zermelo proved that for every zero-sum two person finite game of perfect information, the one of the three given cases must be true. Such games are called strictly determined, or that there must be a strategy that guarantees no losses.

1. Dauben, Joseph W "Game Theory." Microsoft Encarta. Microsoft Corp, 1999
2. Davis, Morton D. "Finite, Two-Person, Zero-Sum Games of Perfect Information" 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)