# Weighted potential and consistency: a unified approach to values for TU-games (Revised version)

Theo Driessen, T. Radzik, R.G. Wanink

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## Abstract

A value on the set {\bf G} of all transferable utility games is said to have a weighted potential representation if there exists a potential function $P: {\bf G} \to {\mathbb{R}}$, associated with a collection of weights, such that, for every game and every player, an appropriately chosen weighted extension of the player's marginal contribution (with respect to the potential) agrees with the player's value in the game. The main theorem states that a value has a weighted potential representation if and only if the value satisfies the well-known efficiency, linearity and symmetry axioms, together with two additional conditions involving a unique collection of constants that describes such a value. Particularly, this unified approach provides various weighted potential representations for the Shapley value, the solidarity value as well and several egalitarian values based on some kind of equity principle. For the class of values that have a weighted potential representation, it is further stated that such a value is consistent (or possesses the reduced game property) with respect to an appropriately chosen type of reduced game (that can be regarded as a generalized version of the known reduced game \`a la Hart and Mas-Colell). Finally, an axiomatic characterization of every value with a weighted potential representation is given in terms of two axioms, namely the corresponding consistency (reduced game) axiom together with a minor axiom referring to some kind of standardness for two-person games.
Original language Undefined Enschede University of Twente, Department of Applied Mathematics Published - 1998

• MSC-90D12
• EWI-3265
• MSC-90D40