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

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.