Coopetitive Index: a measure of cooperation and competition in coalition formation

TL;DR

This paper generalizes the coopetition index to a broader class of TU-games, providing a universal measure for cooperation and competition in coalitions, with explicit formulas, axiomatic characterizations, and meaningful comparisons.

Contribution

It introduces an absolute coopetition index for all non-empty coalitions in TU-games, connecting it to classical semivalues and providing axiomatic characterizations.

Findings

Derived explicit formulas linking coopetition to semivalues.

Established axiomatic characterizations for index variants.

Facilitated meaningful comparison of coalition tendencies.

Abstract

We extend the coopetition index introduced by Aleandri and Dall'Aglio (2025) for simple games to the broader class of monotone transferable utility (TU) games and to all non-empty coalitions, including singletons. The new formulation allows us to define an absolute coopetition index with a universal range in [-1,1], facilitating meaningful comparisons across coalitions. We study several notable instances of the index, including the Banzhaf, Uniform Shapley, and Shapley-Owen coopetition indices, and we derive explicit formulas that connect coopetition to classical semivalues. Finally, we provide axiomatic characterizations of the Uniform Shapley and Shaple--Owen versions, showing that each is uniquely determined by linearity, symmetry over pure bargaining games, external null player neutrality, and a contraction axiom reflecting its internal distribution. These results position the coopetition index as a versatile tool for quantifying the cooperative and competitive tendencies of coalitions in TU-games.