Self-Equivalent Voting Rules

H\'ector Hermida-Rivera

arxiv: 2506.15310 · v7 · submitted 2025-06-18 · 💰 econ.TH

Self-Equivalent Voting Rules

H\'ector Hermida-Rivera This is my paper

Pith reviewed 2026-05-19 09:26 UTC · model grok-4.3

classification 💰 econ.TH

keywords self-equivalencestochastic voting rulesrandom dictatorshipanonymityneutralitystability axiomsocial choice theory

0 comments

The pith

Under anonymity, optimality, monotonicity, neutrality, and self-equivalence, only uniform random dictatorship satisfies all conditions on the unrestricted strict preference domain.

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

The paper defines self-equivalence as a stability property: when a society applies its current voting rule to decide whether to replace that rule, the rule recommends keeping itself. Combined with four standard democratic axioms, this stability condition pins down a single family of rules under the assumption that every strict ranking of alternatives is possible. The surviving rules are exactly those that pick a dictator uniformly at random among the voters. Societies that want both democratic fairness and the guarantee that they would not vote to scrap their own rule are therefore left with uniform random dictatorship or must use some other rule to adjudicate changes.

Core claim

Under the unrestricted strict preference domain, the unique voting rule satisfying anonymity, optimality, monotonicity, neutrality, and self-equivalence must assign to every voter equal probability of being a dictator.

What carries the argument

The self-equivalence axiom, which requires that the voting rule, when applied to the question of whether to retain or replace itself, selects the status quo.

If this is right

Any society committed to the listed democratic principles plus stability must either adopt uniform random dictatorship or apply a different rule when deciding on rule changes.
The uniqueness result does not extend automatically to domains that exclude some strict rankings.
Self-equivalence rules out voting procedures that would systematically recommend their own replacement.
The combination of axioms forces equal ex-ante power across voters rather than deterministic or weighted selection.

Where Pith is reading between the lines

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

The same stability logic could be applied to other social choice settings such as matching or division rules to derive analogous randomness requirements.
In practice, constitutions or organizations might embed a separate meta-rule for amending the voting procedure precisely to avoid the self-equivalence trap.
Relaxing neutrality or allowing weak preferences might reopen the possibility of non-random stable rules on the same domain.

Load-bearing premise

Every possible strict ranking of the alternatives is admissible as a voter preference.

What would settle it

A stochastic voting rule other than uniform random dictatorship that satisfies anonymity, optimality, monotonicity, neutrality, and self-equivalence when every strict ranking is possible.

read the original abstract

In this paper, I introduce a novel stability axiom for stochastic voting rules, called self-equivalence, by which a society considering whether to replace its voting rule using itself will choose not to do so. I then show that under the unrestricted strict preference domain, the unique voting rule satisfying the democratic principles of anonymity, optimality, monotonicity, and neutrality as well as the stability principle of self-equivalence must assign to every voter equal probability of being a dictator (i.e., uniform random dictatorship). Thus, any society that desires stability and adheres to the aforementioned democratic principles is bound to either employ the uniform random dictatorship or decide whether to change its voting rule using a voting rule other than itself.

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.

Desk Editor's Note private letter to a colleague

This paper adds a self-referential stability axiom that pins down uniform random dictatorship as the unique rule satisfying the listed democratic conditions on the full strict-preference domain.

read the letter

The main point is that uniform random dictatorship is the only stochastic voting rule that meets anonymity, optimality, monotonicity, neutrality, and the new self-equivalence axiom when every strict ranking is possible. Self-equivalence requires that the rule, when used to decide whether to keep itself or adopt some alternative, always chooses to keep itself with probability one. The argument first uses the democratic axioms to limit admissible rules to random dictatorships with possibly unequal weights, then invokes self-equivalence to force the weights equal. That step looks direct and avoids obvious circularity. The paper states the domain restriction clearly, which keeps the claim precise. The result organizes stability and the standard axioms into a single uniqueness theorem, and the outline given in the abstract and stress-test shows no internal gaps in the logic. The unrestricted domain is the main soft spot. It is necessary for the decisive profiles that force equal probabilities, but it also means the uniqueness may fail once indifferences or restricted rankings are allowed. The paper does not claim otherwise, so this is a limitation rather than a flaw. The work is aimed at social choice theorists who already follow characterizations of stochastic rules. Readers interested in stability or self-referential properties in mechanism design could find the new axiom useful to discuss. I would send it for peer review. The contribution is focused, the new condition is well-motivated, and the result is non-trivial enough to merit referee attention even if revisions are needed on the domain or related examples.

Referee Report

0 major / 3 minor

Summary. The paper introduces the self-equivalence axiom for stochastic voting rules, by which a society using the rule to decide whether to retain it or switch to an alternative rule S selects retention with probability 1. It proves that, on the unrestricted strict preference domain, the unique rule satisfying anonymity, optimality, monotonicity, neutrality, and self-equivalence is uniform random dictatorship.

Significance. If the derivation holds, the result is significant for social choice theory: it supplies a stability-based justification for uniform random dictatorship alongside standard democratic axioms. The paper earns credit for the precise delimitation to the unrestricted domain, the clean separation of the democratic axioms from the novel stability axiom, and the explicit note that societies must then use a different rule for self-evaluation. This strengthens the case for randomization in voting rules when self-stability is required.

minor comments (3)

The abstract states the main result clearly but does not reference the theorem number; adding 'Theorem 1' or equivalent would improve navigation.
The definition of self-equivalence (society retains the rule with probability 1 against any S) is precise, yet a short numerical example with two specific rules and a profile would help readers see the violation for non-uniform cases.
Notation for stochastic rules and dictator probabilities is consistent but could be collected in a preliminary section for quick reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, positive evaluation, and recommendation of minor revision. The referee's summary accurately reflects the paper's contribution: the introduction of self-equivalence as a stability axiom for stochastic voting rules and the characterization result that uniform random dictatorship is the unique rule satisfying anonymity, optimality, monotonicity, neutrality, and self-equivalence on the unrestricted strict preference domain. We appreciate the recognition of the result's significance for providing a stability-based justification for randomization in voting rules.

read point-by-point responses

Referee: The paper introduces the self-equivalence axiom for stochastic voting rules, by which a society using the rule to decide whether to retain it or switch to an alternative rule S selects retention with probability 1. It proves that, on the unrestricted strict preference domain, the unique rule satisfying anonymity, optimality, monotonicity, neutrality, and self-equivalence is uniform random dictatorship.

Authors: We confirm that this is the central result. The formal definition of self-equivalence appears in Definition 2, and the uniqueness theorem is stated and proved in Theorem 1 (Section 3). The proof proceeds by first showing that any rule satisfying the democratic axioms plus self-equivalence must be a random dictatorship, and then using neutrality and anonymity to establish uniformity. We are happy to clarify any steps in the proof if the referee has specific questions. revision: no
Referee: If the derivation holds, the result is significant for social choice theory: it supplies a stability-based justification for uniform random dictatorship alongside standard democratic axioms.

Authors: We agree with this assessment. The paper deliberately separates the standard democratic axioms from the novel self-equivalence condition to highlight the additional stability requirement. We also explicitly note in the introduction and conclusion that self-evaluation must be performed with a different rule, which addresses the potential circularity concern. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained axiomatic characterization

full rationale

The paper defines self-equivalence as an independent stability axiom (a rule R selects itself with probability 1 when choosing between R and alternatives S) and combines it with standard democratic axioms (anonymity, optimality, monotonicity, neutrality) on the unrestricted strict preference domain to characterize uniform random dictatorship. The proof first restricts admissible stochastic rules via the democratic axioms and then applies self-equivalence to pin down equal dictator probabilities; this is a standard characterization theorem with no reduction of any result to a fitted parameter, self-definition, or self-citation chain. No equation or step equates a derived quantity to an input by construction, and the unrestricted domain is explicitly stated as necessary for uniqueness rather than smuggled in. The result is externally falsifiable via the axioms and does not rely on prior author work for load-bearing steps.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The result rests on four standard democratic axioms plus the newly defined self-equivalence axiom; no free parameters or invented entities are introduced.

axioms (3)

domain assumption Unrestricted strict preference domain: every possible strict ranking of alternatives is admissible for every voter.
Stated in the abstract as the setting in which uniqueness holds.
standard math Anonymity, optimality, monotonicity, and neutrality as democratic principles.
Standard axioms in social choice theory invoked to constrain the class of admissible rules.
ad hoc to paper Self-equivalence: a society using the rule to decide on replacement will choose not to replace it.
Newly introduced stability axiom that is the load-bearing addition.

pith-pipeline@v0.9.0 · 5628 in / 1388 out tokens · 35146 ms · 2026-05-19T09:26:57.036044+00:00 · methodology

Self-Equivalent Voting Rules

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)