Ordinal potentials in smooth games
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Ordinal potentials in smooth games Christian Ewerhart1 Received: 26 October 2018 / Accepted: 28 February 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 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. Keywords Ordinal potentials · Smooth games · Strategic complements and substitutes · Semipositive matrices JEL Classification C6 · C72 · D43 · D72
* Christian Ewerhart [email protected] 1
Department of Economics, University of Zurich, Schönberggasse 1, 8001 Zurich, Switzerland
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1 Introduction In a potential game (Rosenthal 1973; Monderer and Shapley 1996a), players’ preferences may be summarized in a single common objective function.1 Knowing whether a specific game admits a potential can be quite valuable. For example, the existence of a potential reduces the problem of finding a Nash equilibrium to a straightforward optimization problem. Therefore, it is of some interest to know the conditions under which a potential exists. Clearly, sufficient conditions are most desirable. However, necessary conditions are also important. After all, such conditions may help avoiding a futile search for a potential. Moreover, as will be shown, necessary conditions may be indicative of sufficient conditions as well.2 This paper considers smooth games, i.e., non-cooperative n-player games with the property that strategy spaces are non-degenerate compact intervals and payoff functions are twice continuously differentiable (Vives 1999). In the class of smooth games, we derive simple necessary conditions for the existence of a generalized ordinal potential.3 Certainly, a generalized ordinal potential cannot exist if it is possible to construct a strict improvement cycle (Voorneveld 1997), i.e., a finite circular sequence of strategy profiles with the property that moving to the next profile in the sequence amounts to one player strictly raising her payoff by a unilateral change in strategy. However, in general, identifying strict improvement cycles in smooth games is not straightforward. We therefore consider a specific path in the neighborhood of a fixed strategy profile xN∗ , and shrink it to virtually infinitesimal si
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