## The Value of a Random Game: The Advantage of Rationality

**William G. Faris****Robert S. Maier***Department of Mathematics, University of Arizona,**Tuscon, Arizona 85721, USA*

#### Abstract

Two players play against each other in a game with payoffs given by a random by matrix with mean zero. If one player adopts a uniform, purely random strategy, then his loss is limited by the law of averages to a quantity proportional to . On the other hand, if he plays an optimal strategy his losses will typically be considerably less. Numerical evidence is presented for the following conjecture: the standard deviation of the value of the game is asymptotically proportional to . This smaller loss exhibits the advantage of rationality over randomness. The rational player, moreover, tends as to employ a strategy vector that has half its components zero.