In the class of smooth non-cooperative games, exact potential games and weighted potential games are known to admit a convenient characterization in terms of cross-derivatives (Monderer and Shapley, 1996a). However, no analogous characterization is known for ordinal potential games. The present paper derives simple necessary conditions for a smooth game to admit an ordinal potential. First, any ordinal potential game must exhibit pairwise strategic complements or substitutes at any interior equilibrium. Second, in games with more than two players, a condition is obtained on the (modified) Jacobian at any interior equilibrium. Taken together, these conditions are shown to correspond to a local analogue of the Monderer-Shapley condition for weighted potential games. We identify two classes of economic games for which our necessary conditions are also sufficient.
ordinal potentials smooth games strategic complements and substitutes semipositive matrices