## The Utility Theory

The whole existence of the minimax theorem is based on the assumption that opponents will play to maximize winnings at all times. It is a simple fact of life that players will not always act by the best strategy, and has little to do with the content of the theory. It is a sort of aversion to risk determined by the personality of the player or by characteristics of a group the player belongs to. It is not a lack of insight into the situation, and to explain this we need a mechanism that relates the goals of the player, whether real or illusory, to the strategy used to gain this objective. Simply knowing the structure of the game is not enough.

Enter the theory of utility, a vehicle for mathematically conveying the "wants" of the player. A utility function describes the preferences of a person. For example, is a person would chose to get three objects A, B, and C in that order of preference, the utility function may assign them utility values ("utiles" according to Davis, pp. 54) of 6, 8, and 12. Now if a contest with A, B and C as prizes has a value of 8 utiles, then the person is indifferent to a choice between the contest and B. Furthermore, the function can be used to evaluate all contests that eventually lead to A, B, and C.

When von Neumann and Morgenstern designed their utility definition, they also asserted that any contest giving those prizes away must have an utility equal to the weighted average of the utilities of the prizes. For example, if there is a 30% chance of winning A and B and 40% chance of winning C, then the contest's utility is 9.

To make an utility function, it is necessary to ask many simple questions, establish a baseline or common denominator, and calculate relate utilities from there. However, there are cases where preferences are intransitive: the values from the simple questions put together cause a contradiction. For example, is the persons preferences are A < B, B < C, but C > A, this contradicts the logical interpretation that A < C. In such cases, an utility cannot be set. Needless to say, where wants are involved, context and experience matter, but in many cases a generalized baseline can be calculated for a large group, which allows us to use minimax on solid footing on the utility values, which produce zero sums in games involving "wants" regardless of the players final goals.

1. Dauben, Joseph W "Game Theory." Microsoft Encarta. Microsoft Corp, 1999
2. Davis, Morton D. "Utility Theory" Game Theory, A Nontechnical Introduction. New York: Basic Books, Inc., 1930, pp 49-64
3. McCain, Roger A. "Game Theory: An Introductory Sketch." Hypertext. 1999 (August 1999)