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arxiv: 2606.31955 · v1 · pith:R3T7TWJKnew · submitted 2026-06-30 · 💰 econ.TH

Comparison games and ranking of players

Daniela Bubboloni , Stefano Moretti This is my paper

Pith reviewed 2026-07-01 01:55 UTC · model grok-4.3

classification 💰 econ.TH
keywords coalitional gamessemivaluesBanzhaf valueShapley valueplayer rankingcomparison gamescoalitional networkspseudo-games
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The pith

A framework using coalitional networks computes semivalues and player rankings from pairwise coalition comparisons alone.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method for assessing player importance when only relative strengths between pairs of coalitions are available instead of absolute values. It establishes that coalitional networks can represent every coalitional pseudo-game and handle cases with incomplete or inconsistent comparison data. Within this structure, semivalues including the Banzhaf and Shapley values remain expressible as weighted sums of differences drawn from the given comparisons. A derived player's score then reduces the task of ranking players to comparisons of average coalitional worth rather than marginal contributions. This shift yields new interpretations that are especially direct for the Banzhaf value.

Core claim

Comparison games are modeled through coalitional networks that encode relative pairwise coalition comparisons; these networks represent all coalitional pseudo-games and allow semivalues to be recovered exactly as weighted sums of differences in specific comparisons, while a player's score, defined via average coalitional worth, directly determines the induced ranking of players.

What carries the argument

Coalitional networks, which organize limited or heterogeneous pairwise coalition comparisons so that semivalues remain recoverable as weighted sums of differences.

If this is right

  • Semivalues can be calculated without ever knowing absolute coalition worths, using only the supplied pairwise comparisons.
  • Player rankings follow immediately from the player's score, which averages coalitional worth instead of marginal contributions.
  • The same numerical values and rankings are obtained for Banzhaf and Shapley even when many comparisons are absent or heterogeneous.
  • Interpretations of semivalues shift from marginal increments to direct comparisons of coalition strengths.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may extend to other solution concepts if they also admit weighted-sum expressions over coalition comparisons.
  • Domains that collect pairwise strength judgments, such as voting systems or team performance data, could apply the networks directly.
  • Empirical checks on small pseudo-games with deliberately removed comparisons would test how often the recovered rankings remain stable.

Load-bearing premise

Any set of relative pairwise coalition comparisons can be arranged into a coalitional network that still lets semivalues be written exactly as weighted sums of those differences, for every pseudo-game and every pattern of missing or varying data.

What would settle it

A concrete pseudo-game together with a coalitional network on its coalitions such that the Banzhaf value (or Shapley value) cannot be recovered as the prescribed weighted sum of the network's comparison differences.

Figures

Figures reproduced from arXiv: 2606.31955 by Daniela Bubboloni, Stefano Moretti.

Figure 1
Figure 1. Figure 1: the coalitional network N R of Example 1. Consider now the Net-Outdegree pseudo-game (X, CN ) induced by the capacity c of the network N R. We have that the Net-Outdegree pseudo-game (X, CN ) is as follows (sets’ brackets are omitted): C N (∅) = C N (1) = 0, CN (2) = 1, CN (3) = −1, C N (1, 2) = −1, CN (1, 3) = 1, CN (2, 3) = 0, CN (1, 2, 3) = 0. Example 2. Suppose that k ≥ 2 teachers teach for a class X o… view at source ↗
read the original abstract

This work addresses the problem of assessing player importance in coalitional settings where the available information concerns the relative strength between pairs of coalitions, rather than the absolute worth of each coalition. We introduce a novel framework that is flexible enough to represent all coalitional pseudo-games and, through the use of coalitional networks, naturally accommodates scenarios with limited or heterogeneous coalition comparisons. Importantly, this framework still enables the computation of semivalues of pseudo-games, such as the Banzhaf and Shapley values, that can be expressed as weighted sums of differences in specific coalition comparisons, thus offering interpretations beyond traditional approaches. Furthermore, for ranking players rather than computing exact numerical attributions, we introduce the concept of a player's score, which simplifies the process of determining rankings based on semivalues, and shifts the perspective from average marginal contribution to average coalitional worth. This turns out to be particularly enlightening for the Banzhaf value.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper introduces a framework for assessing player importance in coalitional pseudo-games when only relative pairwise comparisons between coalitions are available, rather than absolute coalition worths. It claims that coalitional networks can represent all such pseudo-games, accommodate limited or heterogeneous comparison data, and still permit exact computation of semivalues (Banzhaf, Shapley) as weighted sums of differences from specific comparisons. For ranking purposes, it defines a 'player's score' based on average coalitional worth, which simplifies semivalue-based rankings and offers a new perspective particularly for the Banzhaf value.

Significance. If the central representation and exact weighted-sum expressions hold, the contribution lies in extending semivalue theory to comparison-based data while preserving computational tractability. The player's score concept provides a direct ranking tool without needing full numerical attributions. This could matter in applied settings (e.g., tournament or network data) where absolute values are unavailable. The manuscript does not appear to supply machine-checked proofs or reproducible code, but the framework is presented as newly constructed from comparison data without evident circularity.

minor comments (2)
  1. The abstract states that the framework 'still enables the computation of semivalues... as weighted sums of differences,' but without the full derivations or examples in the provided text it is not possible to verify the exact weighting or the handling of arbitrary missing comparisons.
  2. Notation for coalitional networks and the player's score should be introduced with explicit definitions and small illustrative examples early in the manuscript to aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for acknowledging the potential contribution of the coalitional-network framework to semivalue computation under comparison data. The recommendation is listed as uncertain, yet the report contains no enumerated major comments. Accordingly, we have no specific points to rebut or revise at this stage and remain available for any further questions the editor or referee may wish to raise.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper introduces a novel framework for coalitional pseudo-games using coalitional networks derived directly from pairwise comparison data. Semivalues are expressed as weighted sums of differences in coalition comparisons as a stated consequence of the network structure, without any reduction to fitted parameters, self-definitions, or load-bearing self-citations. The player's score is presented as a simplification for ranking, not a tautological renaming. No load-bearing step reduces by construction to its inputs; the framework is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; all details on representation of pseudo-games and network structure are absent.

pith-pipeline@v0.9.1-grok · 5674 in / 1071 out tokens · 33075 ms · 2026-07-01T01:55:40.907821+00:00 · methodology

discussion (0)

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