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Counting Votes with Multisets

T0 review · 1 major / 0 minor · reviewed 2026-06-28 · grok-4.3

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Pith's one-line read Multisets viewed through their free commutative monoid, functor and monad structures derive and express outcomes for instant-runoff, De Borda and single transferable vote elections.

desk verdict This paper applies the standard commutative monoid, functor, and monad structures on multisets to three voting systems but shows no evidence of non-trivial simplification or new results. read the letter →

arxiv 2606.05218 v1 pith:TPSISQNG submitted 2026-05-29 math.GM

Bart Jacobs , Michael Johnson , Richard Buckland This is my paper · ORCID
classification math.GM
keywords multisetscommutativemonoidfunctormonadinstant-runoffDeBordasingletransferablevotecounting
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that multisets, which permit repeated elements, serve as a natural datatype for recording election results such as tallies of yes and no votes. It applies this datatype to the three standard counting procedures of instant-runoff, De Borda, and single transferable vote. The argument rests on three categorical facts: multisets form a free commutative monoid, they form a functor, and they form a monad. These facts are used to rewrite the counting steps in a uniform algebraic style rather than as ad-hoc procedural lists. A reader would care because the same algebraic language might then apply to further voting rules or to proofs that different counting methods are equivalent under stated conditions.

What carries the argument

Multisets equipped with the structure of a free commutative monoid, functor and monad

What would settle it

A direct comparison in which the algebraic expressions for the three algorithms turn out to be no shorter, clearer or more uniform than the standard procedural descriptions.

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Extended reading notes

Core claim

Multisets form a free commutative monoid, a functor and a monad; these three abstract properties can be used to derive and express the election outcomes in instant-runoff voting, De Borda counting and single transferable vote.

Load-bearing premise

The monoid-functor-monad presentation supplies a non-trivial simplification or derivation advantage over ordinary procedural or set-based descriptions of the same three voting algorithms.

Editorial extensions

If this is right

  • Vote counts and transfers in the three systems become instances of monoid addition and monad operations on multisets.
  • The functorial action of multisets supplies a uniform way to lift rankings or preferences into aggregate tallies.
  • De Borda scores arise directly from the commutative monoid operation applied to ranked ballots.
  • Single transferable vote eliminations and transfers follow from the monad structure that handles redistribution of votes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same monoid and monad language could be tested on other ranked or rated voting methods not covered in the paper.
  • Formal equivalence proofs between variants of these algorithms might become shorter once both are written in the common multiset language.
  • Implementation of the counting rules in a functional programming language could directly reuse the monad operations already present in the type system.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 0 minor

Summary. The manuscript claims that multisets form a free commutative monoid, a functor, and a monad, and that these categorical structures can be applied to derive and express election outcomes in instant-runoff voting, De Borda counting, and single transferable vote (STV). The emphasis is on using these properties for vote counting algorithms rather than on the category theory itself.

Significance. A demonstration that the monoid, functor, or monad laws yield shorter derivations, new invariants, or clearer composition rules for the three voting systems would provide a unified categorical view of algorithms that already treat votes as bags. No such demonstration appears in the supplied abstract, so the significance cannot yet be assessed; the structures invoked are elementary and apply to any counting process.

major comments (1)
  1. [Abstract] Abstract: the claim that the free commutative monoid, functor, and monad properties 'can be put to good use in deriving and expressing election outcomes' is unsupported by any derivation, example, or verification. The abstract supplies no concrete illustration showing that monad laws or functoriality produce a result not immediate from direct multiset enumeration or standard procedural rules.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the report. The sole major comment concerns the abstract, which we address directly below. We agree the abstract can be strengthened and will revise it accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the free commutative monoid, functor, and monad properties 'can be put to good use in deriving and expressing election outcomes' is unsupported by any derivation, example, or verification. The abstract supplies no concrete illustration showing that monad laws or functoriality produce a result not immediate from direct multiset enumeration or standard procedural rules.

    Authors: We agree that the abstract would be improved by including a brief concrete illustration of the claimed application. The body of the manuscript already contains the derivations for instant-runoff (using the commutative monoid to aggregate first preferences and eliminate candidates), De Borda (using the functorial action on score multisets), and STV (using monad bind for vote transfers). To address the comment we will revise the abstract to incorporate one short worked example, e.g., a three-candidate instant-runoff instance showing how the monoid operation directly yields the elimination step without additional procedural machinery. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: standard monoid/functor/monad facts applied to voting without reduction to self-defined inputs

full rationale

The paper invokes externally established properties of multisets (free commutative monoid, functor, monad) to express voting procedures. These are standard category-theoretic facts independent of the present work and not derived via self-citation or internal fitting. No equations or derivations in the abstract or described claims reduce election outcomes to quantities defined inside the paper by construction. The application may or may not yield simplification, but that is a question of utility, not circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on three standard facts about multisets that are taken from category theory and not derived in the paper.

assumptions (3)
  • standard math Multisets form a free commutative monoid
    Explicitly listed in the abstract as one of the relevant abstract properties.
  • standard math Multisets form a functor
    Explicitly listed in the abstract as one of the relevant abstract properties.
  • standard math Multisets form a monad
    Explicitly listed in the abstract as one of the relevant abstract properties.

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Cite this review

Pith. "Pith review of Counting Votes with Multisets." pith.science (2026). https://pith.science/paper/TPSISQNG

@misc{pith2026260605218,
  author       = {Pith},
  title        = {Pith review of: Counting Votes with Multisets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TPSISQNG}},
  note         = {Machine review of arXiv:2606.05218}
}
read the original abstract

A multiset is a 'set' in which elements may occur multiple times. These structures are ideal for expressing the outcome of an election, for instance of the form 60 'yes' and 40 'no'. Moreover, multisets are a useful datatype in vote counting algorithms. This will be illustrated in three different forms of vote counting, known as: 'instant-runoff', 'De Borda', and 'single transferrable vote'. The relevant abstract properties of multisets are: (1) they form a (free) commutative monoid, and (2) they form a functor, and (3) also a monad. This paper illustrates how such categorical properties can be put to good use in deriving and expressing election outcomes. The emphasis is not on the (elementary) category theory involved, but on its application in voting systems.

Figures

Figures reproduced from arXiv: 2606.05218 by the authors.

Figure 1
Figure 1. An urn with four red, three blue and two green balls, described as multiset on the left, with the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Addition of multisets can be represented via a join of contents of two urns / ballot boxes. The [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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