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 crossderivatives (Monderer and Shapley in Games Econ Behav 14:124–143, 1996a). However, no analogous characterization is known for ordinal potential games. The present paper derives 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.