1005.2405v2 - Flows and Decompositions of Games: Harmonic...

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Unformatted text preview: Flows and Decompositions of Games: Harmonic and Potential Games Ozan Candogan, Ishai Menache, Asuman Ozdaglar and Pablo A. Parrilo * Abstract In this paper we introduce a novel flow representation for finite games in strategic form. This representation allows us to develop a canonical direct sum decomposition of an arbitrary game into three components, which we refer to as the potential , harmonic and nonstrategic components. We analyze natural classes of games that are induced by this decomposition, and in particular, focus on games with no harmonic component and games with no potential component. We show that the first class corresponds to the well-known potential games . We refer to the second class of games as harmonic games , and study the structural and equilibrium properties of this new class of games. Intuitively, the potential component of a game captures interactions that can equivalently be represented as a common interest game, while the harmonic part represents the conflicts between the interests of the players. We make this intuition precise, by studying the properties of these two classes, and show that indeed they have quite distinct and remarkable characteristics. For instance, while finite potential games always have pure Nash equilibria, harmonic games generically never do. Moreover, we show that the nonstrategic component does not affect the equilibria of a game, but plays a fundamental role in their efficiency properties, thus decoupling the location of equilibria and their payoff-related properties. Exploiting the properties of the decomposition framework, we obtain explicit expressions for the projections of games onto the subspaces of potential and harmonic games. This enables an extension of the properties of potential and harmonic games to nearby games. We exemplify this point by showing that the set of approximate equilibria of an arbitrary game can be characterized through the equilibria of its projection onto the set of potential games. Keywords: decomposition of games, potential games, harmonic games, strategic equivalence. * All authors are with the Laboratory for Information and Decision Systems (LIDS), Massachusetts Institute of Technology. E-mails: { candogan, ishai, asuman, parrilo } @mit.edu . This research is supported in part by the National Science Foundation grants DMI-0545910 and ECCS-0621922, MURI AFOSR grant FA9550-06-1-0303, NSF FRG 0757207, by the DARPA ITMANET program, and by a Marie Curie International Fellowship within the 7th European Community Framework Programme. arXiv:1005.2405v2 [cs.GT] 25 Jun 2010 1 Introduction Potential games play an important role in game-theoretic analysis due to their desirable static prop- erties (e.g., existence of a pure strategy Nash equilibrium) and tractable dynamics (e.g., convergence of simple user dynamics to a Nash equilibrium); see [32, 31, 35]. However, many multi-agent strate- gic interactions in economics and engineering cannot be modeled as a potential game.gic interactions in economics and engineering cannot be modeled as a potential game....
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.

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1005.2405v2 - Flows and Decompositions of Games: Harmonic...

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