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.
