Dissection of solutions in cooperative game theory using representation techniques

Hernández-Lamoneda, Luis; Juárez, Rubén; Sánchez-Sánchez, Francisco

doi:10.1007/s00182-006-0036-3

Cited by 21 publications

(27 citation statements)

References 5 publications

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“…(A somewhat different approach is taken in [10], but the results are essentially scaled versions of the maps above. )…”

Section: Game Theorymentioning

confidence: 99%

“…For example, using the insights above, we may easily verify that there must be precisely one linear symmetric solution concept that is also an efficient marginal value. It is the well-studied Shapley value [39], which is obtained by setting c 1 0 = · · · = c n−1 0 = 0, c n 0 = 1, and c 1 1 = · · · = c n−1 1 = 1 n−1 (see Section 3 of [22]; see also Section 4.3 of [10]).…”

Section: Game Theorymentioning

confidence: 99%

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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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“…(A somewhat different approach is taken in [10], but the results are essentially scaled versions of the maps above. )…”

Section: Game Theorymentioning

confidence: 99%

Section: Game Theorymentioning

confidence: 99%

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 is a subspace of games, whose values w(S; Q) depends only on the cardinality of S and on the structure of Q. 4 According to Theorem 1, U G is the largest subspace of G where S n acts trivially.…”

Section: Proposition 1 the Decomposition Of Rmentioning

confidence: 99%

A decomposition for the space of games with externalities

Sánchez-Pérez

2016

Int J Game Theory

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The main goal of this paper is to present a di¤erent perspective than the more 'traditional' approaches to study solutions for games with externalities. We provide a direct sum decomposition for the vector space of these games and use the basic representation theory of the symmetric group to study linear symmetric solutions. In our analysis we identify all irreducible subspaces that are relevant to the study of linear symmetric solutions and we then use such decomposition to derive some applications involving characterizations of classes of solutions.

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“…However, we can say more. For instance, it is entirely reasonable to ignore the piece of T s coming from S (3) , since it will only add the same amount to the score for each ranking r, so we will consider all such s to be equivalent 13 . What kind of s will give this?…”

Section: Representation Theory and Votingmentioning

confidence: 99%

“…Within the last five years, there have been several moves to go further and use the representation theory of the symmetric group to classify -usually with a view to first reframe previous results, then extend them in a more general framework. Orrison and his students ( [8]) have done so in voting theory (largely with a view to utilizing incomplete preferences), while recent work of Hernández-Lamoneda, Juárez, and Sánchez-Sánchez ( [13]) gives similar results in cooperative game theory. More recently, similar techniques have been used in joint work of Bargagliotti and Orrison in nonparametric statistics ( [4]).…”

Section: Introductionmentioning

confidence: 99%

The Borda Count, the Kemeny Rule, and the Permutahedron

Crisman¹

2014

The Mathematics of Decisions, Elections, and Games

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A strict ranking of n items may profitably be viewed as a permutation of the objects. In particular, social preference functions may be viewed as having both input and output be such rankings (or possibly ties among several such rankings). A natural combinatorial object for studying such functions is the permutahedron, because pairwise comparisons are viewed as particularly important.In this paper, we use the representation theory of the symmetry group of the permutahedron to analyze a large class of such functions. Our most important result characterizes the Borda Count and the Kemeny Rule as members of a highly symmetric one-parameter family of social preference functions.

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Dissection of solutions in cooperative game theory using representation techniques

Cited by 21 publications

References 5 publications

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

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

A decomposition for the space of games with externalities

The Borda Count, the Kemeny Rule, and the Permutahedron

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