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

  • 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)

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.

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Paper provided by Research Institute for Market Economy, Sogang University in its series Working Papers with number 1116.

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Length: 42 pages
Date of creation: 2011
Date of revision:
Handle: RePEc:sgo:wpaper:1116
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  1. Monderer, Dov & Shapley, Lloyd S., 1996. "Potential Games," Games and Economic Behavior, Elsevier, vol. 14(1), pages 124-143, May.
  2. Hofbauer, Josef & Sandholm, William H., 2009. "Stable games and their dynamics," Journal of Economic Theory, Elsevier, vol. 144(4), pages 1665-1693.e, July.
  3. Ui, Takashi, 2000. "A Shapley Value Representation of Potential Games," Games and Economic Behavior, Elsevier, vol. 31(1), pages 121-135, April.
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