A new basis and the Shapley value

Yokote, Koji; Funaki, Yukihiko; Kamijo, Yoshio

doi:10.1016/j.mathsocsci.2015.12.003

Cited by 12 publications

(11 citation statements)

References 12 publications

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“…In this case, {ū T } T ⊆N :|T |=1 , {e T } T ⊆N :|T |≥2 is a basis of this space. Another interesting basis can also be found in Yokote et al[15].…”

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confidence: 67%

A new axiomatization of a class of equal surplus division values for TU games

2018

RAIRO-Oper. Res.

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In this paper, we propose a variation of weak covariance named as non-singleton covariance, requiring that changing the worth of a non-singleton coalition in a TU game affects the payoffs of all players equally. We establish that this covariance is characteristic for the convex combinations of the equal division value and the equal surplus division value, together with efficiency and a one-parameterized axiom treating a particular kind of players specially. As special cases, parallel axiomatizations of the two values are also provided.

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“…In this case, {ū T } T ⊆N :|T |=1 , {e T } T ⊆N :|T |≥2 is a basis of this space. Another interesting basis can also be found in Yokote et al[15].…”

mentioning

confidence: 67%

A new axiomatization of a class of equal surplus division values for TU games

2018

RAIRO-Oper. Res.

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“…(vii) The Yokote basis: (see Yokote et al (2013); Yokote (2015)) it is a basis of the set of games, which is defined by…”

Section: Bases and Linear Transforms On Set Functionsmentioning

confidence: 99%

“…Hence, the interaction and Banzhaf interaction transforms can be seen as extensions of these values. The duality between transforms and bases permits to easily solve the so-called "inverse problem", well-known in game theory (see, e.g., Kleinberg and Weiss (1985), Dragan (1991Dragan ( , 2006 and Yokote et al (2013)): given a game v on X, find all games v ′ having the same Shapley value (or any other linear value: Banzhaf, egalitarian, etc. ), i.e., Φ…”

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confidence: 99%

Bases and Transforms of Set Functions

Grabisch

2016

Studies in Fuzziness and Soft Computing

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International audienceThe paper studies the vector space of set functions on a finite set X, which can be alternatively seen as pseudo-Boolean functions, and including as a special cases games. We present several bases (unanimity games, Walsh and parity functions) and make an emphasis on the Fourier transform. Then we establish the basic dual-ity between bases and invertible linear transform (e.g., the Möbius transform, the Fourier transform and interaction transforms). We apply it to solve the well-known inverse problem in cooperative game theory (find all games with same Shapley value), and to find various equivalent expressions of the Choquet integral

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“…This paper is a complement study to Yokote et al (2013). Yokote et al (2013) introduced a new basis that has two desirable properties.…”

Section: Introductionmentioning

confidence: 99%

“…This paper is a complement study to Yokote et al (2013). Yokote et al (2013) introduced a new basis that has two desirable properties. First, when we express a game by a linear combination of the basis, the coefficients related to singletons coincide with the Shapley value.…”

Section: Introductionmentioning

confidence: 99%

Several Bases of a Game Space and an Application to the Shapley Value

Funaki¹,

Yokote²

2019

Handbook of the Shapley Value

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We introduce several bases of the set of TU games. Given a coalition T , Yokote et al. (2013) introduced the commander game in which a coalition including 1 player in T obtains payoff. On the other hand, Shapley (1953) introduced the unanimity game in which a coalition including all players in T obtains payoff. We consider the intermediate between the two games. We introduce a game in which a coalition including k players in T obtains payoff, where 1 ≤ k ≤ |T |. We show that, if there is a specific relationship between the size of coalition T and k, we can construct a new basis. By using the new basis, we give sufficient conditions under which the Shapley value coincides with the prenucleolus.

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A new basis and the Shapley value

Cited by 12 publications

References 12 publications

A new axiomatization of a class of equal surplus division values for TU games

A new axiomatization of a class of equal surplus division values for TU games

Bases and Transforms of Set Functions

Several Bases of a Game Space and an Application to the Shapley Value

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