Dynamic Strategic Complements in Two Stage, 2x2 Games

Strategic complements are well understood for normal form games, but less so for extensive form games. There is some evidence that extensive form games with strategic complementarities are a very restrictive class of games (Echenique (2004)). We study necessary and sufficient conditions for strategic complements (defined as increasing best responses) in two stage, 2x2 games. We find that the restrictiveness imposed by quasisupermodularity and single crossing property is particularly severe, in the sense that the set of games in which payoffs satisfy these conditions has measure zero. Payoffs with these conditions require the player to be indifferent between their actions in two of the four subgames in stage two, eliminating any strategic role for their actions in these two subgames. In contrast, the set of games that exhibit strategic complements (increasing best responses) has infinite measure. This enlarges the scope of strategic complements in two stage, 2x2 games (and provides a basis for possibly greater scope in more general games). The set of subgame perfect Nash equilibria in the larger class of games continues to remain a nonempty, complete lattice. The results are easy to apply, and are robust to including dual payoff conditions and adding a third player. Examples with several motivations are included.