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

Sung-Ha Hwang and Luc Rey-Bellet ()
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Luc Rey-Bellet: Department of Mathematics and Statistics, University of Massachusetts Amherst, Lederle Graduate Research Tower, MA 01003-9305, U.S.A

No 1116, Working Papers from Nam Duck-Woo Economic Research Institute, Sogang University (Former Research Institute for Market Economy)

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.

Keywords: normal form games; evolutionary games; potential games; zero-sum games; orthogonal decomposition; null stable games; stable games; replicator dynamics; Zeeman games; Hodge decomposition. (search for similar items in EconPapers)
JEL-codes: C72 C73 (search for similar items in EconPapers)
Pages: 42 pages
Date: 2011
New Economics Papers: this item is included in nep-gth
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