Supermodular Game

A strategic game $G=(I,(S_i),(g_i))$ is supermodular if each $S_i$ is a compact nonempty lattice in some Euclidean space, each $g_i$ is upper semicontinuous in $s_i$, and:

1. $g_i$ has increasing differences: for $s_i\ge s'_i$ and $s_{-i}\ge s'_{-i}$, $$ g_i(s_i,s_{-i})-g_i(s'_i,s_{-i}) \ge g_i(s_i,s'_{-i})-g_i(s'_i,s'_{-i}); $$
2. $g_i$ is supermodular in $s_i$: for every fixed $s_{-i}$, $$ g_i(s_i,s_{-i})+g_i(s'_i,s_{-i}) \le g_i(s_i\vee s'_i,s_{-i})+g_i(s_i\wedge s'_i,s_{-i}). $$

References

• [MFoGT, Section 4.12, Exercise 4(2)] Laraki, Renault, and Sorin, Mathematical Foundations of Game Theory. Topkis supermodular games.