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arxiv: 2605.18228 · v1 · pith:2H4OUNNInew · submitted 2026-05-18 · 💰 econ.TH

Consistency, unanimity, and the Borda rule in social ranking

Takahiro Suzuki , Stefano Moretti , Rachel Ruell\'e This is my paper

Pith reviewed 2026-05-19 23:56 UTC · model grok-4.3

classification 💰 econ.TH
keywords Borda rulesocial rankingconsistencyunanimityneutralitycoalitional contextssocial choice
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The pith

A specific Borda-type social ranking solution is characterized by weak consistency, closeness to unanimity on linear symmetric domains, neutrality, and independence of perfunctory participation.

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

The paper adapts the Borda rule from voting into social ranking, a framework that judges individuals by the performance rankings of the coalitions they join. It examines solutions that meet consistency and closeness to unanimity, then isolates one particular Borda version through the addition of neutrality and independence from adding irrelevant coalitions. A reader would care because the result supplies an axiomatically grounded ordinal method for scoring personal competence inside group settings. If the characterization holds, it supplies a concrete procedure for ranking people according to coalition outcomes without needing numerical utilities.

Core claim

Among several possible definitions of the Borda rule as a social ranking solution, one is characterized by a weak version of consistency, closeness to unanimity under the linear and symmetric domain, neutrality, and independence of perfunctory participation. This yields a new Borda-type SRS for evaluating the competence of individuals in coalitional contexts.

What carries the argument

The Borda rule adapted as a social ranking solution, which scores individuals from the performance rankings of their coalitions and is isolated by the listed axioms.

If this is right

  • The identified Borda-type SRS becomes the unique solution satisfying the four properties together.
  • Neutrality guarantees that only coalition performance data, not individual labels, determine the final ranking.
  • Independence of perfunctory participation ensures the ranking stays unchanged when an irrelevant worst-class coalition is added.
  • Closeness to unanimity on linear domains forces the solution to respect clear preference orderings within coalitions.

Where Pith is reading between the lines

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

  • The same axiomatic approach could be tested on team-performance data to rank employees or athletes by group outcomes.
  • Relaxing the linear-domain restriction might produce alternative rules or require new axioms to restore uniqueness.
  • Similar consistency and unanimity properties could isolate other classical voting rules when transplanted into social ranking.
  • The method offers a template for ordinal ranking tasks in multi-agent systems where agents are evaluated via coalition results.

Load-bearing premise

The characterization of the specific Borda solution requires restricting closeness to unanimity to the linear and symmetric domain.

What would settle it

A concrete counter-example outside the linear symmetric domain in which another social ranking solution satisfies weak consistency, neutrality, and independence of perfunctory participation yet differs from the identified Borda rule would refute the uniqueness claim.

read the original abstract

The social ranking is a recently proposed framework for evaluating the power of individuals according to the performance ranking of their coalitions. Although its origin can be traced to the classical power indices in simple games, social ranking approaches carry out this evaluation within the ordinal framework of social choice theory. This article introduces the Borda rule into social ranking. Specifically, we focus on two essential properties of the Borda rule--consistency and closeness to unanimity--and investigate the social ranking solutions (SRSs) satisfying these properties. Among several possible definitions of the Borda rule as an SRS, we characterize one of such solutions by (a weak version of) consistency, closeness to unanimity (under the linear and symmetric domain), neutrality (i.e., names of the individuals do not matter), and independence of perfunctory participation (i.e., adding a perfunctory coalition into the worst class of the coalitional ranking does not affect the social ranking). We therefore propose a new Borda-type SRS for evaluating the competence of individuals in coalitional contexts.

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

1 major / 2 minor

Summary. The manuscript introduces the Borda rule into the social ranking framework, which evaluates individual power from ordinal coalition performance rankings. It focuses on consistency and closeness to unanimity, and characterizes one specific Borda-type social ranking solution (SRS) as the unique rule satisfying a weak consistency axiom, closeness to unanimity restricted to the linear and symmetric domain, neutrality, and independence of perfunctory participation.

Significance. If the characterization holds under the stated axioms and domain, the result supplies a normative justification for selecting this particular Borda extension among possible variants, adapting classical social-choice properties to the coalitional ordinal setting. This contributes to the literature on social rankings by providing an axiomatic foundation that could guide applications in competence or power evaluation.

major comments (1)
  1. [Main characterization result (§4)] The central characterization (main result, likely Theorem in §4) establishes uniqueness only after restricting closeness to unanimity to the linear and symmetric subdomain. This restriction appears load-bearing: the proof strategy uses it to exclude other candidate Borda-type SRSs, yet the manuscript does not supply either a counterexample showing that uniqueness fails on the full domain or a substantive argument that the subdomain is the natural setting for the Borda rule in social rankings. Without such clarification the claim that the solution is the canonical Borda-type SRS for general social rankings is not fully supported.
minor comments (2)
  1. [Abstract] The abstract refers to 'several possible definitions of the Borda rule as an SRS' but does not enumerate them; a brief list or reference to the relevant definitions in §2 would improve readability.
  2. [§3] Notation for the weak consistency axiom and the perfunctory-participation independence condition could be introduced earlier and used consistently to aid comparison with the classical Borda properties.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comment point by point below and will incorporate revisions to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Main characterization result (§4)] The central characterization (main result, likely Theorem in §4) establishes uniqueness only after restricting closeness to unanimity to the linear and symmetric subdomain. This restriction appears load-bearing: the proof strategy uses it to exclude other candidate Borda-type SRSs, yet the manuscript does not supply either a counterexample showing that uniqueness fails on the full domain or a substantive argument that the subdomain is the natural setting for the Borda rule in social rankings. Without such clarification the claim that the solution is the canonical Borda-type SRS for general social rankings is not fully supported.

    Authors: We agree that the restriction to the linear and symmetric domain is load-bearing for the uniqueness result in our characterization. This domain is the natural setting for extending the Borda rule because it mirrors the classical social-choice environment of strict linear orders with symmetric coalition treatment, which is particularly relevant for ordinal evaluations of individual competence in coalitional settings. To address the concern directly, we will revise the manuscript to include an explicit counterexample demonstrating that uniqueness fails on the full domain, along with a substantive discussion justifying why the linear and symmetric subdomain is the appropriate context for the Borda-type SRS. This addition will clarify the scope of the result without altering the main theorem. revision: yes

Circularity Check

0 steps flagged

Axiomatic characterization relies on external normative axioms with no definitional reduction

full rationale

The paper characterizes a Borda-type social ranking solution (SRS) via weak consistency, closeness to unanimity restricted to the linear symmetric domain, neutrality, and independence of perfunctory participation. These axioms are presented as independent normative requirements drawn from social choice theory rather than quantities defined in terms of the target Borda SRS itself. The derivation proceeds by showing that any SRS satisfying the listed axioms must coincide with the chosen Borda-type rule inside the stated domain; no equation or parameter is fitted to data and then relabeled as a prediction, and no load-bearing step reduces to a self-citation whose content is merely the present result restated. The domain restriction is a scope limitation that narrows the uniqueness claim but does not create a self-definitional loop. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The characterization rests on standard axioms of social choice (neutrality, consistency variants) plus two domain-specific assumptions (linear and symmetric domain for closeness to unanimity) and the modeling choice of how to embed the Borda rule into the social ranking framework. No free parameters or invented entities are introduced.

axioms (4)
  • domain assumption Weak consistency for social ranking solutions
    Invoked as one of the characterizing properties in the abstract.
  • domain assumption Closeness to unanimity restricted to linear and symmetric domains
    Explicitly stated as the setting under which the property is applied.
  • standard math Neutrality (names of individuals do not matter)
    Standard anonymity-type axiom in social choice.
  • domain assumption Independence of perfunctory participation
    New modeling assumption introduced for the characterization.

pith-pipeline@v0.9.0 · 5712 in / 1589 out tokens · 39318 ms · 2026-05-19T23:56:05.353813+00:00 · methodology

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