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Ordinal potentials in smooth games

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  • Christian Ewerhart

Abstract

A non-cooperative n-player game is called smooth if strategy spaces are non-degenerate compact intervals and payoff functions are twice continuously differentiable. In the class of smooth games, exact potential games are known to admit a convenient characterization in terms of cross-derivatives (Monderer and Shapley, 1996a). This characterization extends easily to weighted potential games. 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.

Suggested Citation

  • Christian Ewerhart, 2017. "Ordinal potentials in smooth games," ECON - Working Papers 265, Department of Economics - University of Zurich, revised Jul 2018.
  • Handle: RePEc:zur:econwp:265
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    File URL: http://www.econ.uzh.ch/static/wp/econwp265.pdf
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    References listed on IDEAS

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    More about this item

    Keywords

    Ordinal potentials; smooth games; strategic complements and substitutes; semipositive matrices;

    JEL classification:

    • C6 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D43 - Microeconomics - - Market Structure, Pricing, and Design - - - Oligopoly and Other Forms of Market Imperfection
    • D72 - Microeconomics - - Analysis of Collective Decision-Making - - - Political Processes: Rent-seeking, Lobbying, Elections, Legislatures, and Voting Behavior

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