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

Thomas Pitz, Vinicius Ferraz

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 Assemblée Nationale, where cordon sanitaire and double cordon scenarios produce sharp, interpretable power shifts.

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

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 Assemblée Nationale, where cordon sanitaire and double cordon scenarios produce sharp, interpretable power shifts.

Paper Structure

This paper contains 23 sections, 9 theorems, 59 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Games, Cohesion Structures and Probabilistic Values
  3. The Cohesion-Banzhaf Value
  4. Cohesion-weighted Banzhaf values
  5. Banzhaf-type cohesion axioms
  6. The Cohesion-Shapley--Shubik Value
  7. Shapley-type cohesion axioms
  8. Discussion and Outlook
  9. Scope and strength of the axioms
  10. Interpretation of the cohesion exponent
  11. Structural exclusion: pariah parties and economic equivalents.
  12. Application: The German Bundestag
  13. Historical illustration: the 1982 "Wende" and coalition instability
  14. Application: The French Assemblée Nationale
  15. Outlook and further applications.
  16. ...and 8 more sections

Key Result

Lemma 2.10

Results marked [L4] have been mechanically verified in the Lean 4 theorem prover; see Appendix app:formal-verification. Let $v$ be a simple game on $N$ (i.e. $v(S)\in\{0,1\}$ for all $S$, $v(\emptyset)=0$, $v(N)=1$) and let $i\in N$.

Figures (4)

  • Figure 1: Sensitivity of the Cohesion-Shapley index to the exponent $b$ in a three-player weighted majority game with players A (45%), B (35%), and C (20%), majority threshold 51%. Party A is ideologically isolated: $\kappa(\{A,B\})=0.20$ and $\kappa(\{A,C\})=0.05$. B acts as a bridge party cohesive with C: $\kappa(\{B,C\})=0.9$. At $b=0$ all three players receive equal Shapley shares of $1/3$. As $b$ increases, A's power collapses toward zero while B (bridge) and C (partner) each gain substantially, with B gaining faster due to its moderate cohesion even with the isolated player A.
  • Figure 2: Cohesion-Shapley power indices as a function of the exponent $b$, 21st German Bundestag (official result, 23 February 2025; CDU/CSU 208, AfD 152, SPD 120, Grüne 85, Linke 64 seats; threshold 316). At $b=0$ the index reduces to the classical Shapley--Shubik value. The dashed vertical line marks the canonical linear specification $b=1$. Left (Scenario A): pure ideology cohesion; SPD gains a slight structural advantage over AfD for all $b > 0$ due to its ideological proximity to CDU/CSU. Right (Scenario B): cordon sanitaire, AfD excluded from all coalitions ($\kappa = 0$, flat at zero); CDU/CSU and SPD dominate throughout.
  • Figure 3: Cohesion-Shapley power indices for the 17th French Assemblée Nationale (2024), bloc model (NFP 195, Ensemble 162, RN 139, LR 49, Others 32 seats; threshold 289). Left (Scenario A): pure ideology cohesion. Center (Scenario B): cordon sanitaire against RN. Right (Scenario C): double cordon against RN and NFP; no feasible majority exists and all indices equal zero.
  • Figure 4: Cohesion-Shapley power indices for the 17th French Assemblée Nationale (2024), party model (LFI 71, PS-Verts 124, Ensemble 162, LR 49, RN 139, Others 32 seats; threshold 289). Left (Scenario A): pure ideology cohesion. Center (Scenario B): cordon sanitaire against RN. Right (Scenario C): double cordon against RN and LFI; a viable PS-Verts--Ensemble corridor remains.

Theorems & Definitions (39)

  • Definition 2.1: Admissible Cohesion Structures
  • Remark 2.2
  • Remark 2.7
  • Remark 2.9
  • Lemma 2.10: Marginal contributions in monotone simple games [L4]
  • proof
  • Remark 2.11
  • Proposition 2.12: Marginal-contribution representation [L4]
  • proof
  • Remark 2.13
  • ...and 29 more