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 best-response dynamics has no cycles; and that all games with multiple equilibria have complementarities. As applications of my results, I show that: (1) generic 2 x 2 games either have no pure-strategy equilibria, or have complementarities; (2) generic two-player finite ordinal potential games have complementarities. (C) 2003 Elsevier Inc. All rights reserved.