Probability of Obtaining a Pure Strategy Equilibrium in Matrix Games with Random Payoffs

Abstract

If the payoffs in an mXn zero-sum matrix game are drawn randomly from a finite set of numbers, N, then the probability of obtaining a pure strategy equilibrium, p, will be a weighted sum of the probabilities of obtaining a pure strategy equilibrium, p_s, with s distinct payoffs, the weights, q_s being the probabilities of obtaining s distinct payoffs from N. However, as N → ∞ the probability q_mn → 1. In this limiting case p = P_mn Although p_mn has been derived by Goldman (1957) and Papavassilopoulos (1995), our method is more general. We show that p_mn= Σ_tp ^t_mn where p ^t_mn denotes the probability of obtaining a pure strategy equilibrium for the t^th (t = 1,...,s(= mn)) ordinal payoff, the ordinality being the rank when the payoffs are put in an ascending order.

Further, we introduce the notion of separation of arrays, S(r^k ,c^l), which is a necessary and sufficient condition for the equilibrium of an mXn zero-sum matrix game to be associated with a mixed strategy solution. This generalizes the notion of sepatation of diagonals for 2X2 zero-sum matrix games derived by Von Neumann and Morgenstern (1953).

It can be easily verified that as m or n increases p_mn decreases. Then given the importance of strong equilibrium, which is always a pure strategy equilibrium, a possible behaviourial interpretation is that players may prefer to play games with less number of strategies.