Coopetitive Index: a measure of cooperation and competition in coalition formation
Michele Aleandri, Marco Dall'Aglio
TL;DR
The paper addresses how to quantify the internal cooperation versus competition within coalitions in monotone transferable utility games by extending the coopetition index to all non-empty coalitions, including singletons. It introduces internal and external probability distributions, defines the attitude and coopetition index, and derives an absolute, scale-free version that lies in $[-1,1]$, connecting these measures to generalized semivalues like the Shapley value. It provides explicit formulas for notable indices (Banzhaf, Uniform Shapley, Shapley–Owen) and develops axiomatic characterizations (Linearity, Symmetry over pure bargaining games, External Null Player Neutrality, and null-player contraction variants) that uniquely determine the Uniform Shapley and Shapley–Owen coopetition indices. The work thus offers a versatile, interpretable toolkit for assessing coalition cohesion and strategic interactions in TU-games with potential applications in economics and political science.
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.