On comparing equilibrium and optimum payoffs in a class of discrete bimatrix games

Abstract

In an m 1× m 2 bimatrix game, consider the case where payoffs to each player are randomly drawn without replacement, independently of payoffs to the other player, from the set of integers 1,2,…, m 1 m 2. Thus each player’s payoffs represent ordinal rankings without ties. In such ‘ordinal randomly selected’ games, assuming constraints on the relative sizes of m 1 and m 2 and ignoring any implications of mixed strategies, it is shown that payoffs to pure Nash equilibria (second-degree) stochastically dominate payoffs to pure Pareto optimal outcomes. Thus in such games where pure strategy sets do not differ much in size and payoffs conform with concave von Neumann-Morgenstern utility functions over ordinally ranked outcomes, players would prefer (ex ante) a ‘random pure strategy Nash equilibrium payoff’ to a ‘random pure Pareto optimal outcome payoff’.