A characterization of strategic complementarities

I characterize games for which there is an order on strategies such that the game has strategic complementarities. I prove that, with some qualifications, games with a unique equilibrium have complementarities if and only if Cournot bestresponse dynamics has no cycles; and that all games with multiple equilibria have complementarities. As applications of my results, I show: 1. That generic 2X2 games either have no pure-strategy equilibria, or have complementarities. 2. That generic two-player finite ordinal potential games have complementarities. JEL Classification: C62, C71.