Monotonicity implies linearity: characterizations of convex combinations of solutions to cooperative games

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    3 Citations (Scopus)

    Abstract

    The purpose of this study is to provide a comprehensive characterization of linear solutions to cooperative games by using monotonicity. A monotonicity axiom states an increase in certain parameters of a game as a hypothesis and states an increase in a player’s payoff as a conclusion. We focus on various parameters of a game and introduce new axioms. Combined with previous results, we prove that efficiency, symmetry and a monotonicity axiom characterize (i) four linear solutions in the literature, namely, the Shapley value, the equal division value, the CIS value and the ENSC value, and (ii) a class of solutions obtained by taking a convex combination of the above solutions. Our methodological contribution is to provide a new linear algebraic approach for characterizing solutions by monotonicity. Using a new basis of the linear space of TU games, we identify a class of games in which a solution that satisfies monotonicity is linear. Our approach provides some intuition for why monotonicity implies linearity.

    Original languageEnglish
    Pages (from-to)171-203
    Number of pages33
    JournalSocial Choice and Welfare
    Volume49
    Issue number1
    DOIs
    Publication statusPublished - 2017 Jun 1

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    Values
    intuition
    efficiency
    Monotonicity
    Cooperative game
    Linearity
    literature
    Axiom
    TU game
    Symmetry
    Axioms
    Shapley value
    Intuition

    ASJC Scopus subject areas

    • Social Sciences (miscellaneous)
    • Economics and Econometrics

    Cite this

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    abstract = "The purpose of this study is to provide a comprehensive characterization of linear solutions to cooperative games by using monotonicity. A monotonicity axiom states an increase in certain parameters of a game as a hypothesis and states an increase in a player’s payoff as a conclusion. We focus on various parameters of a game and introduce new axioms. Combined with previous results, we prove that efficiency, symmetry and a monotonicity axiom characterize (i) four linear solutions in the literature, namely, the Shapley value, the equal division value, the CIS value and the ENSC value, and (ii) a class of solutions obtained by taking a convex combination of the above solutions. Our methodological contribution is to provide a new linear algebraic approach for characterizing solutions by monotonicity. Using a new basis of the linear space of TU games, we identify a class of games in which a solution that satisfies monotonicity is linear. Our approach provides some intuition for why monotonicity implies linearity.",
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