# Strict Nash equilibria in non-atomic games with strict single crossing in players (or types) and actions

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## Abstract

In this paper, we study games where the space of players (or types, if the game is one of incomplete information) is atomless and payoff functions satisfy the property of strict single crossing in players (types) and actions. Under an additional assumption of quasisupermodularity in actions of payoff functions and mild assumptions on the player (type) space—partially ordered and with sets of uncomparable players (types) having negligible size—and on the action space—lattice, second countable and satisfying a separation property with respect to the ordering of actions—we prove that every Nash equilibrium is essentially strict. Furthermore, we show how our result can be applied to incomplete information games, obtaining the existence of an evolutionary stable strategy, and to population games with heterogeneous players.

## Keywords

Single crossing Strict Nash Pure Nash Monotone Nash Incomplete information ESS## JEL Classification

C72## Notes

### Acknowledgments

We are particularly indebted to an associate editor and four anonymous referees for their useful suggestions that helped us to improve the paper. All mistakes remain ours.

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