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Desirability and social rankings

Michele Aleandri, Felix Fritz, Stefano Moretti

TL;DR

This paper studies how to convert the partial desirability relation among players in multi-valued coalitional games into a total social ranking by axiomatizing several existing solutions. It provides unique characterizations for five social-ranking solutions—CP-majority, lex-cel, dual-lex, $L^{(1)}$, and $L^{(1)}_*$—via independent axiom sets that encode ceteris paribus comparisons, neutrality, anonymity, and coalition-size effects. The results reveal structural similarities and differences among the solutions, and a case study on a bicameral legislature demonstrates the practical implications of the axioms for ranking political parties. The analysis offers guidance on selecting a ranking method based on how strongly one wants to reward excellence in top coalitions versus considering the stability of CP-comparisons across coalition configurations.

Abstract

In coalitional games, a player $i$ is regarded as strictly more desirable than player $j$ if substituting $j$ with $i$ within any coalition leads to a strict augmentation in the value of certain coalitions, while preserving the value of the others. We adopt a property-driven approach to 'integrate' the notion of the desirability relation into a total relation by establishing sets of independent axioms leading to the characterization of solutionconcepts from the related literature. We focus on social ranking solutions consistent with the desirability relation and propose complementary sets of properties for the axiomatic characterization of five existing solutions: Ceteris Paribus (CP-)majority, lexicographic excellence (lex-cel), dual-lex, $L^{(1)}$ solution and its dual version $L^{(1)}_{*}$ . These characterizations reveal additional similarities among the five solutions and emphasize the essential characteristics that should be taken into account when selecting a social ranking. A practical scenario involving a bicameral legislature is studied.

Desirability and social rankings

TL;DR

This paper studies how to convert the partial desirability relation among players in multi-valued coalitional games into a total social ranking by axiomatizing several existing solutions. It provides unique characterizations for five social-ranking solutions—CP-majority, lex-cel, dual-lex, , and —via independent axiom sets that encode ceteris paribus comparisons, neutrality, anonymity, and coalition-size effects. The results reveal structural similarities and differences among the solutions, and a case study on a bicameral legislature demonstrates the practical implications of the axioms for ranking political parties. The analysis offers guidance on selecting a ranking method based on how strongly one wants to reward excellence in top coalitions versus considering the stability of CP-comparisons across coalition configurations.

Abstract

In coalitional games, a player is regarded as strictly more desirable than player if substituting with within any coalition leads to a strict augmentation in the value of certain coalitions, while preserving the value of the others. We adopt a property-driven approach to 'integrate' the notion of the desirability relation into a total relation by establishing sets of independent axioms leading to the characterization of solutionconcepts from the related literature. We focus on social ranking solutions consistent with the desirability relation and propose complementary sets of properties for the axiomatic characterization of five existing solutions: Ceteris Paribus (CP-)majority, lexicographic excellence (lex-cel), dual-lex, solution and its dual version . These characterizations reveal additional similarities among the five solutions and emphasize the essential characteristics that should be taken into account when selecting a social ranking. A practical scenario involving a bicameral legislature is studied.
Paper Structure (15 sections, 11 theorems, 106 equations, 2 tables)

This paper contains 15 sections, 11 theorems, 106 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Notations and preliminary notions
  3. Properties for social rankings
  4. Main results
  5. An axiomatic characterisation of the CP-majority
  6. An axiomatic characterization of the lex-cel
  7. An axiomatic characterization of the $L^{(1)}$ solution
  8. Comparison between the axioms
  9. Independence of the axioms for CP-majority
  10. Independence of the axioms for lex-cel
  11. Independence of the axioms for $L^{(1)}$
  12. Case study
  13. Final remarks
  14. Appendix
  15. Calculating the scores of the bicameral legislature in the Netherlands

Key Result

Lemma 4.1

Let $R$ be a social ranking satisfying Axioms N (Neu) and EC (EC) . Then for any $i,j \in N$ and $\succsim \in \hbox{\boldmath $\cal R$}({\cal F}(N))$ such that $|D_{ij}^{\succsim}| = |D_{ji}^{\succsim}|$ we have that $i \ I^\succsim \ j$.

Theorems & Definitions (42)

  • Example 2.1
  • Example 2.2
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.3: lex-cel bernardi:hal-02191137
  • Example 2.4
  • Definition 2.4: $L^{(1)}$ algaba:hal-03388789
  • Example 2.5
  • Example 3.1
  • ...and 32 more