Ordinal Potentials in Smooth Games

While smooth exact potential games are easily characterized in terms of the crossderivatives of players’payo¤ functions, an analogous di¤erentiable characterization of ordinal or generalized ordinal potential games has been elusive for a long time. In this paper, it is shown that the existence of a generalized ordinal potential in a smooth game with multidimensional strategy spaces is crucially linked to the semipositivity (Fiedler and Pták, 1966) of a modi…ed Jacobian matrix on the set of interior strategy pro…les at which at least two …rst-order conditions hold. Our …ndings imply, in particular, that any generalized ordinal potential game must exhibit pairwise strategic complements or substitutes at any interior Cournot-Nash equilibrium. Moreover, provided that there are more than two players, the cross-derivatives at any interior equilibrium must satisfy a rather stringent equality constraint. The two conditions, which may be conveniently condensed into a local variant of the di¤erentiable condition for weighted potential games, are made explicit for sum-aggregative games, symmetric games, and two-person zero-sum games. For the purpose of illustration, the results are applied to classic games, including probabilistic all-pay contests with heterogeneous valuations, models of mixed oligopoly, and Cournot games with a dominant …rm.