A Coopetition Index for Coalitions in Simple Games

Abstract: In simple games, larger coalitions typically wield more power, but do all players align their efforts effectively? Consider a voting scenario where a coalition forms, but needs more voters to pass a bill. The cohesion of the new group of voters hinges on whether all the new members can proficiently collaborate with the existing players to ensure the bill's passage or if subgroups form that pursue an independent alternative, thus generating antagonism among the new voters. This research introduces two classes of coopetition indices -- one relative and one absolute, the latter ranging from -1 to 1, to measure agents' preferences for cooperation (when positive) or competition (when negative) with the remaining players. These indices, together with a generalized group value, provide a comprehensive picture of the relevance and the cohesion of groups. We discuss the relationship with similar group indices and provide proper coopetition Banzhaf and Shapley-Owen types of indices. By applying our indices to the apex game and symmetric majority games, we observe that cooperation and competition frequently balance each other out, leading to null values for the Shapley-Owen and Banzhaf coopetition indices. An electoral application with real world data is also considered.