Characterizing robust solutions in monotone games

In game theory, p-dominance and its set-valued generalizations serve as important robust solution concepts. We show that in monotone games, (which include the broad classes of supermodular games, submodular games, and their combinations) these concepts can be characterized in terms of pure strategy Nash equilibria in an auxiliary game of complete information. The auxiliary game is constructed in a transparent manner that is easy to follow and retains a natural connection to the original game. Our results show explicitly how to map these concepts to corresponding Nash equilibria thereby identifying a new bijection between robust solutions in the original game and equilibrium notions in the auxiliary game. Moreover, our characterizations lead to new results about the structure of entire classes of such solution concepts. In games with strategic complements, these classes are complete lattices. More generally, they are totally unordered. We provide several examples to highlight these results.