The single crossing conditions for incomplete preferences

We study the implications of the single crossing conditions for preferences described by binary relations. All restrictions imposed on the preferences are satisfied in the case of approximate optimization of a bounded-above utility function. In the context of the choice of a single agent, the transitivity of strict preferences ensures that the best response correspondence is increasing in the sense of a natural preorder; if the preferences are represented by an interval order, there is an increasing selection from the best response correspondence. In a strategic game, a Nash equilibrium exists and can be reached from any strategy profile after a finite number of best response improvements if all strategy sets are chains, the single crossing conditions hold w.r.t. pairs [one player’s strategy, a profile of other players’ strategies], and the strict preference relations are transitive. If, additionally, there are just two players, every best response improvement path reaches a Nash equilibrium after a finite number of steps. If each player is only affected by a linear combination of the strategies of others, the single crossing conditions hold w.r.t. pairs [one player’s strategy, an aggregate of the strategies of others], and the preference relations are interval orders, then a Nash equilibrium exists and can be reached from any strategy profile with a finite best response path.