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Decompositions of two player games: potential, zero-sum, and stable games

Author

Listed:
  • Sung-Ha Hwang

    () (Department of Economics, Sogang University, Seoul)

  • Luc Rey-Bellet

    () (Department of Mathematics and Statistics, University of Massachusetts Amherst, Lederle Graduate Research Tower, MA 01003-9305, U.S.A)

Abstract

We introduce several methods of decomposition for two player normal form games. Viewing the set of all games as a vector space, we exhibit explicit orthonormal bases for the subspaces of potential games, zero-sum games, and their orthogonal com- plements which we call anti-potential games and anti-zero-sum games, respectively. Perhaps surprisingly, every anti-potential game comes either from the Rock-Paper- Scissors type games (in the case of symmetric games) or from the Matching Pennies type games (in the case of asymmetric games). Using these decompositions, we prove old (and some new) cycle criteria for potential and zero-sum games (as orthogonality relations between subspaces). We illustrate the usefulness of our decomposition by (a) analyzing the generalized Rock-Paper-Scissors game, (b) completely characteriz- ing the set of all null-stable games, (c) providing a large class of strict stable games, (d) relating the game decomposition to the decomposition of vector elds for the replicator equations, (e) constructing Lyapunov functions for some replicator dynam- ics, and (f) constructing Zeeman games -games with an interior asymptotically stable Nash equilibrium and a pure strategy ESS.

Suggested Citation

  • Sung-Ha Hwang & Luc Rey-Bellet, 2011. "Decompositions of two player games: potential, zero-sum, and stable games," Working Papers 1116, Research Institute for Market Economy, Sogang University.
  • Handle: RePEc:sgo:wpaper:1116
    as

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    File URL: ftp://163.239.156.99/wpaper/HSH_RIME_2011-16.pdf
    File Function: First version, 2011
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    References listed on IDEAS

    as
    1. Ui, Takashi, 2000. "A Shapley Value Representation of Potential Games," Games and Economic Behavior, Elsevier, vol. 31(1), pages 121-135, April.
    2. Hofbauer, Josef & Sandholm, William H., 2009. "Stable games and their dynamics," Journal of Economic Theory, Elsevier, vol. 144(4), pages 1665-1693.4, July.
    3. Monderer, Dov & Shapley, Lloyd S., 1996. "Potential Games," Games and Economic Behavior, Elsevier, vol. 14(1), pages 124-143, May.
    Full references (including those not matched with items on IDEAS)

    More about this item

    Keywords

    normal form games; evolutionary games; potential games; zero-sum games; orthogonal decomposition; null stable games; stable games; replicator dynamics; Zeeman games; Hodge decomposition.;

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games

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