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arxiv: 2606.04959 · v1 · pith:6RFU5SL4 · submitted 2026-06-03 · cs.GT · econ.TH· q-fin.TR

Fairness and Strategy-Proofness in Automated Market Makers

Frank M. V. Feys This is my paper

pith:6RFU5SL4reviewed 2026-06-28 03:48 UTCmodel grok-4.3open to challenge →

classification cs.GT econ.THq-fin.TR
keywords automated market makersliquidity provider votingfairnessstrategy-proofnessAitchison centroidsocial choice theoryopinion pooling
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The pith

For weighted-product automated market makers with three or more assets, no voting rule on the trading function satisfies both fairness and strategy-proofness.

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

The paper establishes that liquidity providers cannot jointly decide on the trading function of an automated market maker in a way that is fair to all and immune to strategic voting when there are three or more assets. Arrovian fairness axioms lead uniquely to the weighted Aitchison centroid as the aggregator, which is a form of mean, but strategy-proofness requires a median-type rule, so only a dictator rule satisfies both. This structural obstruction explains the absence of voting mechanisms in deployed AMMs. The impossibility is specific to the weighted-product family and does not hold for two-asset cases.

Core claim

On the weighted-product family with n ≥ 3 assets, no aggregation rule is at once fair and strategy-proof. Arrovian fairness forces a unique form, the weighted Aitchison centroid, the weighted geometric mean of the providers' preferred pools. But fairness forces mean-type aggregation and strategy-proofness forces median-type, and the only rule that is both is a single-provider dictator. The obstruction is sharp: it vanishes at n = 2, where a fair strategy-proof rule exists. Under the Frongillo--Papireddygari--Waggoner equivalence, the centroid is Genest's logarithmic opinion pool, and the impossibility transfers to externally Bayesian pooling.

What carries the argument

The weighted Aitchison centroid as the unique fair aggregation rule under Arrovian axioms, which acts as the mean-type aggregator that conflicts with the median-type required by strategy-proofness.

If this is right

  • Only a single-provider dictator rule satisfies both fairness and strategy-proofness for n ≥ 3.
  • A fair and strategy-proof aggregation rule exists when there are only two assets.
  • The impossibility result carries over to externally Bayesian pooling of opinions via the established equivalence.
  • The result applies specifically within the weighted-product family of trading functions.

Where Pith is reading between the lines

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

  • This suggests that parameter setting in multi-asset AMMs may need to use fixed rules or centralized decisions rather than provider votes.
  • Designers might explore alternative notions of fairness or strategy-proofness that permit more flexible aggregation.
  • Similar impossibilities could arise in other families of trading functions if similar mean-median conflicts occur.

Load-bearing premise

The analysis is restricted to trading functions in the weighted-product family and uses standard Arrovian fairness and strategy-proofness axioms from social choice theory for the aggregation.

What would settle it

Constructing an explicit non-dictatorial aggregation rule for three-asset weighted-product AMMs that meets both the fairness axioms and strategy-proofness would falsify the central claim.

read the original abstract

No deployed automated market maker lets its liquidity providers vote on the trading function. We show this is structural, not an oversight. On the weighted-product family with $n \geq 3$ assets, no aggregation rule is at once fair and strategy-proof. Arrovian fairness forces a unique form, the weighted Aitchison centroid, the weighted geometric mean of the providers' preferred pools. But fairness forces mean-type aggregation and strategy-proofness forces median-type, and the only rule that is both is a single-provider dictator. The obstruction is sharp: it vanishes at $n = 2$, where a fair strategy-proof rule exists. Under the Frongillo--Papireddygari--Waggoner equivalence, the centroid is Genest's logarithmic opinion pool, and the impossibility transfers to externally Bayesian pooling.

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 / 3 minor

Summary. The manuscript proves an impossibility theorem for automated market makers: within the weighted-product family with n ≥ 3 assets, no aggregation rule over liquidity providers' preferred trading functions can be simultaneously Arrovian-fair and strategy-proof. Fairness uniquely forces the weighted Aitchison centroid (a mean-type aggregator), while strategy-proofness forces median-type aggregation; the only rule satisfying both is a single-provider dictator. The obstruction is sharp—it vanishes for n=2—and transfers, via the Frongillo–Papireddygari–Waggoner equivalence, to the impossibility of non-dictatorial externally Bayesian pooling.

Significance. If the result holds, it supplies a structural explanation for the absence of liquidity-provider voting on trading functions in deployed AMMs. The paper cleanly imports Arrovian axioms and strategy-proofness into AMM design, characterizes the unique fair aggregator, and leverages an existing equivalence to obtain a transfer result to logarithmic opinion pooling. These elements constitute a precise, falsifiable contribution at the intersection of mechanism design and social choice.

minor comments (3)
  1. [§2] §2 (or wherever the weighted-product family is formalized): the precise functional form of the trading function and the domain of admissible weight vectors should be stated explicitly before the characterization of the Aitchison centroid is invoked.
  2. The proof sketch in the abstract invokes the Frongillo–Papireddygari–Waggoner equivalence; a self-contained one-paragraph recap of the relevant direction of that equivalence would make the transfer step easier to verify without consulting the cited paper.
  3. Table or figure (if present) comparing the n=2 case to the n≥3 case would help readers see the boundary result at a glance.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of the manuscript, as well as the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation characterizes the unique fair aggregator (weighted Aitchison centroid) directly from Arrovian axioms on the weighted-product family, then contrasts mean-type vs. median-type behavior under strategy-proofness to reach the dictator conclusion. This proceeds from the stated axioms without any parameter fitting, self-definition of the target quantity, or load-bearing self-citation; the Frongillo–Papireddygari–Waggoner equivalence is invoked as an external transfer to logarithmic pooling and does not reduce the impossibility to prior work by the same author. The result is self-contained within the given axiom set and family restriction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on applying standard social-choice axioms to the AMM domain; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Standard Arrovian fairness and strategy-proofness axioms from social choice theory apply directly to aggregation of liquidity providers' preferred pools.
    The impossibility is derived by showing that these axioms force conflicting functional forms (mean-type vs median-type) for n≥3.

pith-pipeline@v0.9.1-grok · 5665 in / 1336 out tokens · 52327 ms · 2026-06-28T03:48:30.090994+00:00 · methodology

discussion (0)

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