A Theory of games with general complementarities

The current theory of games with strategic complementarities, also know as supermodular or quasisupermodular games, sets up a fundamental tool to investigate strategic interactions where diffuse Pareto-Edgeworth complementarities among the actions of the agents are present. It represents the basic analytical framework for modern industrial organization, and finds plenty of applications in strategic macroeconomics and in microeconomics at large.

However, the formal notion of complementarity that is employed in this theory is unsatisfactory because it bears too few connections with our intuitive idea of complementarity. This is the starting point of the present work.

We introduce a notion of complementarity that is more meaningful from an economic point of view, and investigate how far we can go from it. We define a new class of games, that we call games with general complementarities, and obtain for these games results in terms of existence of greatest and least Nash equilibria, comparative statics of these extremal equilibria, lattice structure of the Nash set, and rationalizability of Nash profiles.

Our games with general complementarities retain many of the basic properties of games with strategic complementarities. In the presence of this, our work represents a generalization of the latter class of games

Games with strategic complementarities are a mathematical construction that merges an order-based fixpoint theory, in the vein of Tarski, with an approach to comparative statics for payoffs defined on lattices, independently developed in operations research. To attain our results, we have generalized both the fixpoint part of the theory of games with strategic complementarities and its comparative statics part. These intermediary results have their own independent interest.

We present applications of our games with general complementarities to models of search and models of oligopoly, problems that are not tractable with standard tools.