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1. Introduction
This survey provides an overview of the theory of supermodular optimization and games, with a marked emphasis on accessibility, for as broad an audience as possible. Supermodular optimization is a new methodology for conducting comparative statics or sensitivity analysis, that is, it determines how changes in exogenous parameters affect endogenous variables in optimizing models.1 As such, the use of this methodology is pervasive in economics, and the conclusions thereby derived are often one of the main motivations behind the construction of a model. The main characteristic of this methodology is that it relies essentially on critical assumptions for the desired monotonicity conclusions and dispenses with superfluous assumptions that are often imposed only by the use of the classical method, which is based on the Implicit Function Theorem and includes smoothness, interiority, and concavity. The main insight is indeed quite simple. If, in a maximization problem, the objective reflects a complementarity between an endogenous variable and an exogenous parameter, in the sense that having more of one increases the marginal return to having more of the other, then the optimal value of the former will be increasing in the latter. In the case of multiple endogenous variables, then all of them must also be complements in order to guarantee that their increases are mutually reinforcing. This conclusion follows directly from the underlying complementarity relationship and is thus independent of the aforementioned superfluous assumptions. It thus holds even if there are multiple optimal values of the endogenous variable(s).
Is a new look at complementarity needed? Topkis (1998, p. 3) quotes Samuelson (1947) as asserting the following: "In my opinion the problem of complementarity has received more attention than is merited by its intrinsic importance," only for Samuelson to correct himself later in Samuelson (1974) by adding "The time is ripe for a fresh modern look at the concept of complementarity. The last word has not yet been said on this ancient preoccupation of literary and mathematical economists. The simplest things are often the most complicated to understand fully." It is hoped that this survey will convince the reader of the correctness of Samuelson's latter view.
Another major methodological breakthrough due to this framework of analysis is the theory of supermodular games, better known in...