Existence Of Mixed Strategy Equilibria Research Articles

Sufficient conditions for weak reciprocal upper semi-continuity in mixed extensions of games

We provide sufficient conditions for a game with discontinuous payoffs to be weakly reciprocally upper semi-continuous in mixed strategies. These conditions are imposed on the individual payoffs and not on their sum, and they can be readily verified in a large class of games even when the sum of payoffs in such games is not upper semi-continuous. We apply our result to establish the existence of mixed strategy equilibria in probabilistic voting competitions where candidates have very general utility functions as well as heterogeneous beliefs about the distribution of the voters.

On the Existence of Mixed Strategy Equilibria of Two Person Games with Incomplete Information

This paper is concerned with the existence of mixed strategy equilibria of games with incomplete information. Since the work (Milgrom and Weber 1985), it is common to assume that the regularity condition with respect to the prior that the prior is absolutely continuous with respect to the product of marginal probabilities. We offer an alternative set of assumptions without the regularity condition which ensure the existence of mixed strategy equilibria.

Optimal strategic reasoning with McNaughton functions

The aim of the paper is to explore strategic reasoning in strategic games of two players with an uncountably infinite space of strategies the payoff of which is given by McNaughton functions—functions on the unit interval which are piecewise linear with integer coefficients. McNaughton functions are of a special interest for approximate reasoning as they correspond to formulas of infinitely valued Lukasiewicz logic. The paper is focused on existence and structure of Nash equilibria and algorithms for their computation. Although the existence of mixed strategy equilibria follows from a general theorem (Glicksberg, 1952) [5], nothing is known about their structure neither the theorem provides any method for computing them. The central problem of the article is to characterize the class of strategic games with McNaughton payoffs which have a finitely supported Nash equilibrium. We give a sufficient condition for finite equilibria and we propose an algorithm for recovering the corresponding equilibrium strategies. Our result easily generalizes to n-player strategic games which don't need to be strictly competitive with a payoff functions represented by piecewise linear functions with real coefficients. Our conjecture is that every game with McNaughton payoff allows for finitely supported equilibrium strategies, however we leave proving/disproving of this conjecture for future investigations.

Equilibrium existence for zero-sum games and spatial models of elections

A theorem on existence of mixed strategy equilibria in discontinuous zero-sum games is proved and applied to three models of elections. First, the existence theorem yields a mixed strategy equilibrium in the multidimensional spatial model of elections with three voters. A nine-voter example shows that a key condition of the existence theorem is violated for general finite numbers of voters and illustrates an obstacle to a general result. Second, the theorem provides a simple and self-contained proof of Kramer's existence result for the multidimensional model with a continuum of voters [Kramer, G., 1978. Existence of electoral equilibrium. In: Ordeshook, P. (Ed.), Game Theory and Political Science. NYU Press, New York]. Third, existence follows for a class of multidimensional probabilistic voting models with discontinuous probability-of-winning functions.

Voting over income taxation

A major problem of the positive theory of income taxation is to explain why statutory income tax schedules in practice are marginal-rate progressive. While it is commonly believed that this is but a simple consequence of the fact that the number of relatively poor voters exceeds that of richer voters in general, putting this contention in a voting equilibrium context is not a trivial task. We do this here in the context of nonlinear taxation and attempt to provide a formal argument in support of this heuristic claim. We first establish the existence of mixed strategy equilibria and identify certain cases in which marginal-rate progressive taxes are chosen almost surely by the political parties. Unfortunately, we also find that if the tax policy space is not artificially constrained, the support of at least one equilibrium cannot be contained within the set of marginal-rate progressive taxes.

The existence of mixed-strategy equilibria in a price-setting oligopoly with convex costs

This paper shows that a mixed-strategy equilibrium will exist in an industry producing a homogeneous product with perfectly informed consumers if firms are identical, and have weakly-convex costs. This generalises an earlier result by Dasgupta and Maskin.