Null player neutrality in TU-games: Egalitarian and Shapley solutions

TL;DR

This paper introduces null player neutrality in TU-games, characterizes a family of solutions combining Shapley and equal division, and explores its implications for cooperative game solutions.

Contribution

It defines null player neutrality, characterizes solution families combining Shapley and equal division, and extends the class of egalitarian solutions in TU-games.

Findings

Family of solutions extends $eta$-egalitarian Shapley values to all real $eta$

Null player neutrality leads to a unique solution as equal division when replaced by its analogue

Characterization of solutions using efficiency, linearity, symmetry, and null player neutrality

Abstract

We introduce and study the axiom of null player neutrality in the context of cooperative games with transferable utility (TU-games). This axiom weakens the classical coalitional strategic equivalence: rather than requiring that augmenting a game by a null-player game leaves that player's payoff unchanged, it only requires that any change in payoff be independent of the specific augmenting game, provided both the null-player condition and the grand-coalition value are preserved. We show that efficiency, linearity, symmetry, and null player neutrality together characterize the family of all real linear combinations of the Shapley value and the equal division solution, a family that strictly extends the well-known class of $\alpha$-egalitarian Shapley values (convex combinations, $\alpha \in [0,1]$) to arbitrary $\alpha \in \mathbb{R}$. Replacing null player neutrality by its natural analogue for nullifying players uniquely pins down the equal division solution.