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Journal of Convex Analysis 30 (2023), No. 2, 481--498
Copyright Heldermann Verlag 2023

Existence and Statistical Estimation of Equilibria in Stochastic Electoral Competitions

Adib Bagh
Depts. of Mathematics and Economics, University of Kentucky, Lexington, U.S.A.

We establish the existence of a Nash equilibrium in a class of stochastic games representing electoral competitions between two candidates facing a continuum of voters. The underlying uncertainty in such games is due to the fact that candidates are uncertain about the fraction of the voters that will actually vote for them on election day. The payoff functions of the candidates are neither continuous nor concave. Therefore, we transform a game in this class into a new game with nicer geometric and continuity properties. We establish the existence of an equilibrium for the new game using a standard fixed point argument, and we finally show that an equilibrium of the new games is also an equilibrium of the original game. We further show the equilibria we found can be approximated statistically using non-parametric maximum likelihood estimators of the distribution of the random variable representing the uncertainty in the game.

Keywords: Stochastic voting models, Nash Equilibrium, log-concave random variables, non-parametric maximum likelihood estimators.

MSC: 91A80, 90C15, 62F12.

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