The Orthogonal Decomposition of Games and an Averaging Formula for the Shapley Value

Kleinberg, Norman L.; Weiss, Jeffrey H.

doi:10.1287/moor.11.1.117

Cited by 10 publications

(4 citation statements)

References 4 publications

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“…In the literature of game theory the concept of decomposition is used in different ways. For most of the cases the games with many payoff parameters are built from simpler games characterized by a significantly less number of parameters [see the works by Szép and Forgó [56], Kleinberg and Weiss [57], Sandholm [51], and Candogan et al [29]]. This gives us a deeper insight into the general properties of interactions described by the tools of game theory.…”

Section: Decomposition Of Two-player Gamesmentioning

confidence: 99%

Evolutionary potential games on lattices

2016

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Game theory provides a general mathematical background to study the effect of pair interactions and evolutionary rules on the macroscopic behavior of multi-player games where players with a finite number of strategies may represent a wide scale of biological objects, human individuals, or even their associations. In these systems the interactions are characterized by matrices that can be decomposed into elementary matrices (games) and classified into four types. The concept of decomposition helps the identification of potential games and also the evaluation of the potential that plays a crucial role in the determination of the preferred Nash equilibrium, and defines the Boltzmann distribution towards which these systems evolve for suitable types of dynamical rules. This survey draws parallel between the potential games and the kinetic Ising type models which are investigated for a wide scale of connectivity structures. We discuss briefly the applicability of the tools and concepts of statistical physics and thermodynamics. Additionally the general features of ordering phenomena, phase transitions and slow relaxations are outlined and applied to evolutionary games. The discussion extends to games with three or more strategies. Finally we discuss what happens when the system is weakly driven out of the "equilibrium state" by adding non-potential components representing games of cyclic dominance.

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Section: Decomposition Of Two-player Gamesmentioning

confidence: 99%

Evolutionary potential games on lattices

2016

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“…Observe that (27) holds if and only if u m , ν q −m (p) = 0 for all q −m ∈ E −m , where ν q −m is as defined in (26). Lemma 4.2 implies that {ν q −m } are basis vectors of ker D m .…”

Section: Decomposition Of Gamesmentioning

confidence: 99%

“…In this approach, the set of players is not made smaller or larger by the decomposition but the component games have simpler structure. Another method for decomposing the space of cooperative games appeared in [24,26,25]. In these papers, the algebraic properties of the space of games and the properties of the nullspace of the Shapley value operator (see [40]) and its orthogonal complement are exploited to decompose games.…”

Section: Introductionmentioning

confidence: 99%

Flows and Decompositions of Games: Harmonic and Potential Games

et al. 2011

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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.

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“…Alternative formulations of the Shapley value can be found, among others, in Kleinberg and Weiss[12], Rothblum[15] and Ruiz et al[16].…”

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confidence: 99%