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.
Highlights
In this paper, we identify a class of games where every pure-strategy Nash equilibrium is essentially strict
We show that every Nash equilibrium is monotone
If an order structure is imposed on types, 212 our Theorem 1 can allow to tackle the issue. This follows a seminal idea in Riley (1979), where incomplete information and a form of the strict single crossing property are used to show existence of an evolutionarily stable strategy in the “war of attrition”. For this purpose, we restrict attention to a game Γ I that is symmetric, i.e., we focus on game Γ I S = I, ([0, 1]h, φ), A, u
Summary
Strict Nash equilibrium is a solution concept that possesses desirable features. In this paper, we identify a class of games where every pure-strategy Nash equilibrium is essentially strict. We consider games with an atomless space of players (or types, if the game if of incomplete information), and action sets that are second countable and satisfy a mild separation property.. To obtain existence of essen tially strict Nash equilibria, one can apply our result together with one of the many equilibrium existence theorems that the literature provides. We show the existence of an evolutionarily stable strategy in a general class of incomplete information games, and strict Nash equilibrium in a class of population games with heterogenous players. The “Appendix” collects one technical lemma (Lemma 1), its proof, and the proof of Proposition 2
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