Complementarity and supermodularity

I found a nice summary of key concepts in game theory, complementarity and supermodularity, which are especially important for auction and matching theory.


"Supermodularity and supermodular games" byXavier Vives
in the new palgrave dictionary of economics:


The below is quoted from Xavier's survey.

The basic idea of complementarity is that the marginal value of an action increases with the level of other actions available. The mathematical concept of supermodularity formalizes the idea of complementarity. The theory of monotone comparative statics and supermodular games provides the toolbox to deal with complementarities.

This theory, in contrast to classical convex analysis, is based on order and monotonicity properties on lattices. Monotone comparative statics analysis provides conditions under which optimal solutions to optimization problems change monotonically with a parameter.

The theory of supermodular games exploits order properties to ensure that the best response of a player to the actions of rivals increases with their level. The power of the approach is that it clarifies the drivers of comparative statics results and the need of regularity conditions; it allows very general strategy spaces, including indivisibilities and functional spaces such as those arising in dynamic or Bayesian games; it establishes the existence of equilibrium in pure strategies; it allows a global analysis of the equilibrium set when there are multiple equilibria, which has an order structure with largest and smallest elements; and finally, it finds that those extremal equilibria have strong stability properties and there is an algorithm to compute them.