Linear-potential values and the 'Shapley family'

Reinoud A.M.G. Joosten, Hans Peters, Frank Thuijsman

Research output: Working paperProfessional

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Abstract

We generalize the potentials of Hart & Mas-Colell [1989] by intro-ducing a class of linear potentials for TU-games based on an idea of taxing and redistributing. To each potential an e¢ cient and additivevalue is associated, which attributes to each player his linearly modified contribution to the potential of the grand coalition of a so-calledtaxed game, plus an equal share in the tax revenues.The class of linear-potential values includes egalitarian and dis-counted Shapley values, but also weighted Shapley values. Using ourpotential we show that the class of consistent values in the sense of Hart& Mas-Colell [1989] can be extended. Furthermore, the Shapley fam-ily is enlarged by the classes of semi-egalitarian discounted weightedShapley values and equal-coalitional-improvement Shapley values.We investigate connections between restrictions on linear-potentialvalues and axioms, some of which lose independence, e.g., variants ofstandardness imply symmetry. We characterize several classes withinthe Shapley family by single axioms, such as symmetry and parameterdependent forms of egalitarianism, consistency and standardness, aswell as individual members by forms of consistency and standardness.
Original languageEnglish
Number of pages26
Publication statusPublished - 20 Dec 2013

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Shapley Value
Axioms
TU Game
Symmetry
Tax
Coalitions
Linearly
Attribute
Class
Family
Game
Restriction
Imply
Generalise
Form

Keywords

  • IR-88254

Cite this

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Linear-potential values and the 'Shapley family'. / Joosten, Reinoud A.M.G.; Peters, Hans; Thuijsman, Frank.

2013.

Research output: Working paperProfessional

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N2 - We generalize the potentials of Hart & Mas-Colell [1989] by intro-ducing a class of linear potentials for TU-games based on an idea of taxing and redistributing. To each potential an e¢ cient and additivevalue is associated, which attributes to each player his linearly modified contribution to the potential of the grand coalition of a so-calledtaxed game, plus an equal share in the tax revenues.The class of linear-potential values includes egalitarian and dis-counted Shapley values, but also weighted Shapley values. Using ourpotential we show that the class of consistent values in the sense of Hart& Mas-Colell [1989] can be extended. Furthermore, the Shapley fam-ily is enlarged by the classes of semi-egalitarian discounted weightedShapley values and equal-coalitional-improvement Shapley values.We investigate connections between restrictions on linear-potentialvalues and axioms, some of which lose independence, e.g., variants ofstandardness imply symmetry. We characterize several classes withinthe Shapley family by single axioms, such as symmetry and parameterdependent forms of egalitarianism, consistency and standardness, aswell as individual members by forms of consistency and standardness.

AB - We generalize the potentials of Hart & Mas-Colell [1989] by intro-ducing a class of linear potentials for TU-games based on an idea of taxing and redistributing. To each potential an e¢ cient and additivevalue is associated, which attributes to each player his linearly modified contribution to the potential of the grand coalition of a so-calledtaxed game, plus an equal share in the tax revenues.The class of linear-potential values includes egalitarian and dis-counted Shapley values, but also weighted Shapley values. Using ourpotential we show that the class of consistent values in the sense of Hart& Mas-Colell [1989] can be extended. Furthermore, the Shapley fam-ily is enlarged by the classes of semi-egalitarian discounted weightedShapley values and equal-coalitional-improvement Shapley values.We investigate connections between restrictions on linear-potentialvalues and axioms, some of which lose independence, e.g., variants ofstandardness imply symmetry. We characterize several classes withinthe Shapley family by single axioms, such as symmetry and parameterdependent forms of egalitarianism, consistency and standardness, aswell as individual members by forms of consistency and standardness.

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