Algebraic structure of games

Kleinberg, Norman L.; Weiss, Jeffrey H.

doi:10.1016/0165-4896(85)90005-8

Cited by 4 publications

(3 citation statements)

References 1 publication

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“…Thus, P corresponds to the set of normalized games, which have globally consistent pairwise comparisons. Due to (25), the pairwise comparisons of games do not have locally inconsistent components, thus Theorem 3.1 implies that H corresponds to the set of normalized games, which have globally inconsistent but locally consistent pairwise comparisons. Hence, from the perspective of the Helmholtz decomposition, the flows generated by games in P and H are gradient and harmonic flows respectively.…”

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%

“…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. This approach does not necessarily simplify the analysis of games but it leads to an alternative expression for the Shapley value [25]. Our work is on decomposition of noncooperative games, and different from the above references since we explicitly exploit the properties of noncooperative games in our framework.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Decomposition Of Gamesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Flows and Decompositions of Games: Harmonic and Potential Games

et al. 2011

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Representation theory of the symmetric group in voting theory and game theory

Crisman¹,

Orrison²

2017

Algebraic and Geometric Methods in Discrete Mathematics

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This paper is a survey of some of the ways in which the representation theory of the symmetric group has been used in voting theory and game theory. In particular, we use permutation representations that arise from the action of the symmetric group on tabloids to describe, for example, a surprising relationship between the Borda count and Kemeny rule in voting. We also explain a powerful representation-theoretic approach to working with linear symmetric solution concepts in cooperative game theory. Along the way, we discuss new research questions that arise within and because of the representation-theoretic framework we are using.

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Voting on cyclic orders, group theory, and ballots

Crisman,

Holleran,

Martin

et al. 2024

Mathematical Analyses of Decisions, Voting and Games

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A cyclic order may be thought of informally as a way to seat people around a table, perhaps for a game of chance or for dinner. Given a set of agents such as { A , B , C } \{A,B,C\} , we can formalize this by defining a cyclic order as a permutation or linear order on this finite set, under the equivalence relation where A ≻ B ≻ C A\succ B\succ C is identified with both B ≻ C ≻ A B\succ C\succ A and C ≻ A ≻ B C\succ A\succ B . As with other collections of sets with some structure, we might want to aggregate preferences of a (possibly different) set of voters on the set of possible ways to choose a cyclic order. However, given the combinatorial explosion of the number of full rankings of cyclic orders, one may not wish to use the usual voting machinery. This raises the question of what sort of ballots may be appropriate; a single cyclic order, a set of them, or some other ballot type? Further, there is a natural action of the group of permutations on the set of agents. A reasonable requirement for a choice procedure would be to respect this symmetry (the equivalent of neutrality in normal voting theory). In this paper we will exploit the representation theory of the symmetric group to analyze several natural types of ballots for voting on cyclic orders, and points-based procedures using such ballots. We provide a full characterization of such procedures for two quite different ballot types for n = 4 n=4 , along with the most important observations for n = 5 n=5 .

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Algebraic structure of games

Cited by 4 publications

References 1 publication

Flows and Decompositions of Games: Harmonic and Potential Games

Flows and Decompositions of Games: Harmonic and Potential Games

Representation theory of the symmetric group in voting theory and game theory

Voting on cyclic orders, group theory, and ballots

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