On comparing equilibrium and optimum payoffs in a class of discrete bimatrix games
Each \(m_1\times m_2\) payoff matrix in the bimatrix games studied here is formed by a random permutation of the integers from 1 to \(M=m_1m_2\). The author is interested in comparing a ``random pure strategy Nash equilibrium payoff'' with a ``random pure Pareto optimal outcome payoff'', in a certain sense of those terms, ignoring the possibility of mixed strategies. The payoff numbers are regarded as simply giving ordinal rankings of the outcomes, and except in the \(2\times 2\) case another restriction is made in order to get a dominance relation between the two above-mentioned types of random payoff: that the players have concave von Neumann-Morgenstern utility functions over ordinally ranked pure outcomes. The main theorem states that in this case, if \(m_1>2\sum_{n=1}^{M}{1\over n}\) then expected payoff for a random (in the sense considered) pure strategy Nash equilibrium dominates that for a random pure Pareto optimal outcome. In particular the conclusion is valid when \(m_1=m_2\) or \(m_1\geq m_2\geq 9\). (When \(m_1=m_2 \geq 9\) the inequality of the main theorem holds; if \(m_1=m_2<9\), an alternate more complicated sufficient condition is satisfied).
scientific article; zbMATH DE number 1406426