Fair‐ranking properties of a core selection and the Shapley value

Abstract

The fair‐ranking principle states that if all the underlying outside data of some coalition are the same in two games, then the relative positions of the players in the coalition should be the same in the two games. We investigate the implications of the principle in the context of transferable utility games. Our first axiom, coalitional fair‐ranking, requires that if the worth of a given coalition changes while the worths of other coalitions remain fixed, then the relative positions of all players in the given coalition should not be affected. We show that for games with at least three players, there is no core selection satisfying coalitional fair‐ranking. If the change is limited to the worth of the grand coalition, then there is a core selection satisfying the requirement. In addition, we present a new axiomatic characterization of the Shapley value on the basis of strong coalitional fair‐ranking: it requires that if a game is added to another game with two symmetric players, then the relative positions of the symmetric players should not be affected. The Shapley value is the only value satisfying strong coalitional fair‐ranking together with efficiency, the null player property and strategic equivalence.