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A Coopetition Index for Coalitions in Simple Games

Michele Aleandri, Marco Dall'Aglio

TL;DR

This work defines coopetition indices for coalitions in monotone simple games by quantifying the coalition's attitude toward cooperation or competition toward the rest of the players. It introduces two index classes: a relative coopetition index $\mathcal{C}^v_{p,q}(S)$ and an absolute version $\widehat{\mathcal{C}}^v_{p,q}(S)$, built atop a probabilistic generalized value $\Phi^v_q(S)$ and distribution families $p$ and $q$, with sharp bounds $-\Phi^v_q(S) \le \mathcal{C}^v_{p,q}(S) \le \Phi^v_q(S)$. The paper derives concrete Banzhaf and Shapley-Owen variants, establishes key properties (e.g., null players neutralize coopetition, MWCs influence balance), and analyzes two canonical examples—the Apex game and symmetric majority games—showing frequent internal balance (zero coopetition). An electoral application using the 2015 Spanish election data demonstrates the indices’ diagnostic value for real-world coalition prospects, with PSOE-Podemos-UP exhibiting higher coopetition than PSOE-Ciudadanos in the analyzed setup. The authors close with open problems, including axiomatizations, TU extensions, and generalizing attitude notions beyond 2-partitions to richer coalition structures.

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.

A Coopetition Index for Coalitions in Simple Games

TL;DR

This work defines coopetition indices for coalitions in monotone simple games by quantifying the coalition's attitude toward cooperation or competition toward the rest of the players. It introduces two index classes: a relative coopetition index and an absolute version , built atop a probabilistic generalized value and distribution families and , with sharp bounds . The paper derives concrete Banzhaf and Shapley-Owen variants, establishes key properties (e.g., null players neutralize coopetition, MWCs influence balance), and analyzes two canonical examples—the Apex game and symmetric majority games—showing frequent internal balance (zero coopetition). An electoral application using the 2015 Spanish election data demonstrates the indices’ diagnostic value for real-world coalition prospects, with PSOE-Podemos-UP exhibiting higher coopetition than PSOE-Ciudadanos in the analyzed setup. The authors close with open problems, including axiomatizations, TU extensions, and generalizing attitude notions beyond 2-partitions to richer coalition structures.

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.
Paper Structure (16 sections, 13 theorems, 73 equations, 3 figures, 1 table)

This paper contains 16 sections, 13 theorems, 73 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation of the work
  3. Basic Notation
  4. Towards a New Index
  5. Balancing Cooperation and Competition in a Single Index
  6. Some important cases
  7. The Banzhaf Coopetition Indices
  8. The Shapley-Owen Coopetition Indices
  9. Some Common Properties
  10. Index properties
  11. Notable examples
  12. The Apex Game
  13. Symmetric Majority Games
  14. An Electoral Application
  15. Conclusion and Open Problems
  16. ...and 1 more sections

Key Result

Proposition 3.2

Fix $p$ and let $T\subset N$ be a losing coalition with $|T|\leq n-2$. For each coalition $S\subseteq N\setminus T$, $|S|\geq 2$, critical wrt $T$ and for all strictly positive probability distributions $p_S$ over the set $\Pi_2(S)$, i.e. $p_S(\pi)>0$ for all $\pi\in\Pi_2(S)$, then $i)$$\mathcal{A}^

Figures (3)

  • Figure 1: Graph of political affinities among the parties after the 2015 election.
  • Figure 2: Boxplots of the Shapley-Owen coopetition index arranged by coalition length (left) and connectedness (right).
  • Figure 3: Boxplots of the absolute Shapley-Owen coopetition index arranged by coalition length (left) and connectedness (right).

Theorems & Definitions (42)

  • Definition 2.1: Marichal et al. marichal2007axiomatic
  • Definition 2.2: Derks and Tijs derks2000merge
  • Example 2.3
  • Definition 2.4
  • Definition 2.5: Grabisch and Roubens grabisch1999axiomatic
  • Definition 2.6
  • Example 2.7: continues=ex:profitparadox
  • Example 2.8
  • Example 2.9: continues=ex:interparadox
  • Definition 3.1
  • ...and 32 more