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

Koji Yokote () and Yukihiko Funaki ()
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Koji Yokote: Waseda University

Social Choice and Welfare, 2017, vol. 49, issue 1, No 8, 203 pages

Abstract: 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.

Date: 2017
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DOI: 10.1007/s00355-017-1056-6

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