On ordered equilibria in games with increasing best responses

This paper investigates conditions under which games have totally ordered equilibria, which has implications for the stability as well as the Pareto-ranking of equilibria. We first show that when best responses are strongly increasing, all games with up to three players as well as games with symmetric best responses have totally ordered equilibria. Furthermore, the same results hold when best responses are non-decreasing and equilibria are strict. Non-decreasingness and strong increasingness of best responses are implied by payoffs satisfying the single-crossing and strict single-crossing properties, and hence the former assumptions are weaker than what is assumed in games of strategic complements. Nevertheless, we show that even when the stronger assumption of increasing differences is satisfied, non-symmetric games with more than three players generally do not have totally ordered equilibria.