Global Bifurcations in a Complementarity Game

Abstract

In this paper, we study the global dynamics of a complementarity game with effort cost externalities. Following Matsuyama (2002), we assume that identical players are simultanously engaged in two identical games, where the players’ efforts chosen in each of the games exhibit a strategic complementarity. Furthermore, there is a cost complementarity since marginal effort costs in each game depend on the same player’s effort level in the other game. Players are assumed to adapt their effort choices over time using a time-discrete gradient adjustment process. We demonstrate that multiple stable equilibria may occur and that asymmetric and symmetric equilibria may coexist. We characterize the shape and structure of the basins of attraction and describe the changes these basins undergo as the parameters which capture the complementarity and the externality vary. We find that in the model with nonnegative effort levels asymmetric equilibria are more likely. Consequently, heterogenous effort choices emerge endogenously despite the fact that players and games are symmetric.

Keywords: Complementarity games, dynamical systems, stability, Basins of attraction

JEL Classification: C62, C73