Characterizing Monotone Games

Solution concepts in games of strategic heterogeneity (GSH), which include games of strategic complements (GSC) as a special case, have been shown to possess very useful properties, such as the existence of highest and lowest serially undominated strategies, and the equivalence of the stability of equilibria and dominance solvability. The main result of this paper gives necessary and sufficient conditions for when a very general class of games, referred to as games of mixed heterogeneity (GMH), can be transformed into GSH in such a way so that these properties are preserved, allowing us to draw the same strong conclusions about solution sets in games that are not originally GSH. This is achieved by reversing the orders on the actions spaces of a given subset of players. Our second main result shows, rather surprisingly, that under mild conditions on the underlying ordering of action spaces, the reversal of orders is the only way in which such a transformation can be achieved. Applications to aggregate games, market games, and crime networks are given.