Cohesion-Sensitive Power Indices: Representation Results for Banzhaf and Shapley Values

TL;DR

This paper introduces cohesion-sensitive power indices in cooperative game theory, accounting for coalition feasibility, and provides axiomatic characterizations with applications to legislative bodies.

Contribution

It develops a new class of power indices considering coalition feasibility, extending classical indices with cohesion structures and axiomatic foundations.

Findings

Cohesion-sensitive indices differ significantly from classical ones in legislative scenarios.

The framework is mechanically verified in Lean 4 with Mathlib.

Applied to German Bundestag and French Assemblée Nationale, revealing interpretable power shifts.

Abstract

In many applications of cooperative game theory -- from corporate governance and cartel formation to parliamentary voting -- not all winning coalitions are feasible. Ideological distances, institutional constraints, or pre-electoral agreements may render certain coalitions implausible. Classical power indices ignore this and weight all winning coalitions equally. We introduce cohesion structures to quantify coalition feasibility and axiomatically characterize two families of cohesion-sensitive power indices, represented as expected marginal contributions under Luce-type distributions. In the Banzhaf branch, coalition weights are a power transformation of cohesion; in the Shapley branch, additional axioms separate size from cohesion, recovering the classical size weights with cohesion acting within each size class. All results have been mechanically verified in Lean 4 with Mathlib. We illustrate the framework on the German Bundestag and the French Assembl\'ee Nationale, where cordon sanitaire and double cordon scenarios produce sharp, interpretable power shifts.