A Non-Robustness in the Order Structure of the Equilibrium Set in Lattice Games

The order and lattice structure of the equilibrium set in games with strategic complements do not survive a minimal introduction of strategic substitutes: in a lattice game in which all-but-one players exhibit strategic complements (with one player exhibiting strict strategic complements), and the remaining player exhibits strict strategic substitutes, no two equilibria are comparable. More generally, in a lattice game, if either (1) just one player has strict strategic complements and another player has strict strategic substitutes, or (2) just one player has strict strategic substitutes and has singleton-valued best-responses, then without any restrictions on the strategic interaction among the other players, no two equilibria are comparable. In such cases, the equilibrium set is a non-empty, complete lattice, if, and only if, there is a unique equilibrium. Moreover, in such cases, with linearly ordered strategy spaces, the game has at most one symmetric equilibrium. Several examples are presented.