Ordinal Potentials in Smooth Games

Christian Ewerhart

doi:10.2139/ssrn.3054604

Abstract

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.

Highlights

In a potential game (Rosenthal, 1973; Monderer and Shapley, 1996a), players’ preferences may be summarized in a single common objective function.1 Knowing if a speci...c game admits a potential can be quite valuable
No analogous characterization is known for ordinal potential games
As will be shown, necessary conditions may be indicative of su¢cient conditions as well

Summary

Introduction

In a potential game (Rosenthal, 1973; Monderer and Shapley, 1996a), players’ preferences may be summarized in a single common objective function. Knowing if a speci...c game admits a potential can be quite valuable. The analysis starts by considering strict improvement cycles that involve two players only In this case, the existence of a generalized ordinal potential is shown to imply that the product of the slopes of any two players’ mutual local best-response functions (or, more generally, the product of the corresponding cross-derivatives) must be nonnegative at any strategy pro...le at which at least two ...rst-order conditions hold in the interior. It turns out that the existence of a particular strict improvement cycle corresponds precisely to the property that the modi...ed Jacobian is semipositive (Fiedler and Pták, 1966).6 We transform these conditions in several steps, using a powerful recursive characterization of semipositivity due to Johnson et al (1994), as well as the technique of ‡ipping around individual strategy spaces (Vives, 1990; Amir, 1996).

Preliminaries

Strict improvement cycles involving two players

Boundary equilibria and related issues19

Contests

Concluding remarks