Supermodularity

Table 4.1. Diameter of the supermodular aggregates at the top surface of the PPO-CS2 membrane...

If quasi-concavity of the players payoffs cannot be verified, there is an alternative existence proof that relies on Tarski s (1955) fixed point theorem and involves the notion of supermodular games. The theory of supermodular games is a relatively recent development introduced and advanced by Topkis (1998). 

Supermodularity essentially means complementarity between any two strategies and is not linked directly to either convexity, concavity or even continuity. (This is a significant advantage when forced to work with discrete strategies, e.g., Cachon 2001.) However, similar to concavity/convexity, supermodular-... 

Theorem 3. In a supermodular game there exists at least one NE. 

Supermodular games approach. In some situations, supermodular games provide a more convenient tool for comparative statics. 

Theorem 11. Consider a collection of supermodular games on RP parameterized by a parameter a. Further, suppose d Kijdxida > 0 for all i. Then the largest and the smallest equilibria are increasing in a. 

Roughly speaking, a sufficient condition for monotone comparative statics is supermodularity of players payoffs in strategies and a parameter. Note that, if there are multiple equilibria, we cannot claim that every equilibrium is monotone in a rather, a set of all equilibria is monotone in the sense of Theorem 10. A convenient way to think about the last Theorem is through the... 

Topkis, D. M. 1998. Supermodularity and complementarity. Princeton University Press. 

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