# 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_{t}p _{mn} ^{t} where p _{mn} ^{t} 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.

## Keywords

Pure Strategy Matrix Game Strong Equilibrium Pure Strategy Equilibrium Pure Strategy Nash Equilibrium## Preview

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