Mandate without Managers: Automated Market Makers as Verifiable Portfolio Products

Ionut Florescu; Sean O'Leary; Zachary Feinstein

REVIEW 3 major objections 4 minor 17 references

Mandate without Managers: Automated Market Makers as Verifiable Portfolio Products

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A fee rule turns AMM pools into verifiable portfolio funds.

desk verdict Clean fee characterization for multi-asset G3M rebalancing bands, surrounded by empirical claims that are honest but conditional; the theorem deserves peer review. read the letter →

arxiv 2608.02917 v1 pith:Q627YAO2 submitted 2026-08-03 q-fin.TR q-fin.MFq-fin.PM

Zachary Feinstein , Ionut Florescu , Sean O'Leary This is my paper · ORCID

classification q-fin.TRq-fin.MFq-fin.PM MSC 91G1091G80

keywords automatedmarketmakergeometricmeantarget-weightportfoliobandrebalancingno-arbitrageregiontrackingerrormandateloss-versus-rebalancing

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

This paper argues that an automated market maker should be read not only as an exchange but as a portfolio product: a geometric mean market maker (G3M) enforces a stated target-weight mandate by construction. The authors introduce a multi-asset fee parameter $\gamma$ and prove that competitive arbitrage keeps every realized portfolio weight at or above $(1-\gamma)$ times its target, so the pool behaves as a band-rebalanced strategy whose compliance is checkable from observable reserves. They simulate such pools against three real funds, a 60/40 balanced fund, an equal-weighted sector fund, and an equal-weighted Dow fund, using only arbitrage order flow, and they find fee ranges where the G3M beats each fund on both annualized return and tracking error. That matters because it shows a mechanical, manager-free structure can deliver a common investment product and, in principle, be verified by investors and regulators.

What carries the argument

The load-bearing object is the geometric mean market maker invariant $L = \prod_{i=1}^N x_i^{w_i}$ with target weights $w$, equipped with the paper's fee function $g^\gamma_w(\rho) = (1 + \alpha(\rho))^\gamma \prod_{i=1}^N (1+\rho_i)^{(1-\gamma)w_i}$, where $\alpha(\rho)$ is the common proportional component of any reserve update. The fee structure splits every operation into a fee-free proportional mint or burn and a fee-bearing residual swap, preserving path independence. The arbitrageur's optimization then has a convex reformulation whose KKT dual variables identify realized weights as $(1-\gamma) w_i$ plus an element of $\gamma$ times the simplex, which is what converts the fee parameter into an ex ante bound on mis-weighting.

What would settle it

Run a fee-bearing G3M with observable reserves under sustained arbitrage over a period with large price moves; if any asset's realized weight is observed below $(1-\gamma) w_i$ while arbitrageurs are actively trading, Theorem 5 is contradicted. Equivalently, repeat the backtests with real order flow rather than arbitrage-only daily closes: if no fee level yields simultaneous dominance in both CAGR and tracking error for these benchmark funds, the empirical claim fails.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 5: for a G3M with the proposed fee structure, no profitable arbitrage trade occurs if and only if every realized weight satisfies $\hat w_i \ge (1-\gamma) w_i$. Under active arbitrage, realized weights therefore live in the band $\hat w_i \in [(1-\gamma) w_i,\; \gamma + (1-\gamma) w_i]$, making the fee parameter the width of a rebalancing band rather than merely a spread. Empirically, the paper claims that for fee ranges such as $\gamma \in [2.73\%, 3.90\%]$ for the monthly-tracking 60/40 case, $\gamma \in [3.22\%, 7.09\%]$ for equal sectors under the economic mandate, and $\gamma \in [3.32\%, 9.90\%]$ for equal-weight Dow components, a simulated G3M fed only with daily arbitrage flow Pareto-dominates the incumbent fund in compound annual return and tracking error simultaneously.

Load-bearing premise

The argument holds if competitive, frictionless arbitrageurs actually trade whenever the pool leaves the fee-induced no-arbitrage band; in the simulations this is modeled as at most one arbitrage trade per day at the close, and if real arbitrage is slower, costlier, or absent, weights can drift outside the band and the claimed returns and tracking errors are not guaranteed.

Editorial extensions

If this is right

A fee-bearing G3M with fees in the identified ranges is a band-rebalanced target-weight product: its mandate is encoded in the invariant weights and its allowable drift is set entirely by $\gamma$.
Compliance becomes a public computation: from observable reserves and contemporaneous prices, anyone can check whether $\hat w_i \ge (1-\gamma) w_i$ holds, replacing manager attestation with mechanism.
The loss-versus-rebalancing paid to arbitrageurs is reconceived as the execution cost of maintaining the mandate rather than pure adverse selection, so AMM profitability should be judged against alternative rebalancing mechanisms.
Increasing $\gamma$ widens the no-arbitrage band but does not monotonically worsen tracking error, because retained fee revenue partly offsets the cost of larger weight deviations, which explains the non-monotone frontiers.
The dominant fee ranges, roughly 3% to 10%, should be read as rebalancing-band widths rather than as competitive exchange spreads.

Reading between the lines

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

If the result extends beyond the three tested mandates, any target-weight strategy whose constituents trade on liquid markets could in principle be encoded as a G3M, generalizing the ETF product structure to arbitrary custom baskets without a manager.
The verifiability property suggests a regulatory reading: the pool could serve as its own compliance report, with the no-arbitrage band acting as a precommitted risk limit; testing that would require a live deployment with price feeds and adversarial order flow, not just arbitrage-only simulation.
The once-per-day close arbitrage assumption is conservative in frequency but optimistic in frictionlessness; a natural next test is continuous intraday arbitrage with spread and inventory costs, which would shift the effective band and likely shrink the dominance regions.
Because the dominance comparison uses NAV-based tracking for the incumbent funds, part of the apparent advantage may reflect NAV smoothing; comparing ETF market prices instead could separate mechanism quality from measurement lag.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This paper is worth your time for Theorem 5 and the framing that follows; the empirical part, though, is thinner than the abstract suggests.

The genuinely new thing is the multi-asset fee-bearing G3M characterization. Theorem 5 gives a clean necessary-and-sufficient condition—no profitable arbitrage iff each realized weight is at least (1−γ)w_i—and the KKT proof in Appendix A.2 is sound. Corollary 6's weight band is a direct consequence. The fee structure (splitting each operation into proportional mint/burn plus residual swap) is a genuine design choice, and the path-independence discussion in Lemma 4 is handled carefully. This extends the two-asset fee analysis in [6] in a nontrivial way, and I see no circularity in the derivation.

The comparison with VBIAX/EQL/EDOW is a fair idea. Using arbitrage-only order flow as a conservative bound is reasonable, and the paper is honest that dominance regions depend on which benchmark you call the mandate. It is also honest that the G3M's market-making P&L is non-positive by construction.

The soft spots are all in the empirical section. First, the data and code are not shipped, and the results come from Bloomberg data only; that alone makes the Table 1 numbers uncheckable. Second, the model assumes arbitrageurs transact at most once per day at the close, frictionlessly, with no transaction costs or gas. The theorem tells you when a profitable trade exists; it does not tell you whether anyone shows up, or how often. Corollary 6 is explicitly conditional on "active arbitrage," but the practical verifiability claim—weights live in the band—needs that participation. The stress-test note has this right. The dominance regions are point estimates under an assumption, not robust bounds. Third, some samples are short (EDOW: Nov 2024–Jun 2026), and the fee range is scanned after the fact, which inflates the appearance of a wide dominance region.

None of this sinks the theory. The caveats are appropriately stated inside the paper—more than most. But the empirical claim as written ("can dominate") is conditional, and a serious referee should demand code, data, and sensitivity to arbitrage frequency and costs before accepting that part.

My recommendation: send it to peer review. The theoretical contribution deserves a serious referee; the empirical section needs revision and verification. This is a paper for math-finance and DeFi readers, and for anyone working on AMM fee design.

Referee Report

3 major / 4 minor

Summary. The paper proposes viewing a geometric mean market maker (G3M) as a programmatic portfolio product rather than only as an exchange. It introduces a one-parameter fee structure γ ∈ [0,1) under which the G3M invariant is replaced by a geometric interpolation between the fee-free invariant and proportional minting. The central theoretical result, Theorem 5, characterizes the no-arbitrage region: a profitable pure swap exists if and only if some realized asset weight falls below (1−γ)w_i, and Corollary 6 bounds the resulting mis-weighting band. The paper then backtests G3M pools with arbitrage-only order flow, at most one trade per day at the close, against VBIAX, EQL, and EDOW, reporting fee ranges in Table 1 where the simulated G3M dominates on both CAGR and tracking error.

Significance. The theoretical contribution is meaningful: Theorem 5 gives a clean, closed-form characterization of the fee-induced no-arbitrage band for a multi-asset G3M, extending prior two-asset results, and the proof via a convex program and KKT conditions is coherent. The fee construction in Definition 3 is novel and has attractive consistency properties. If the empirical claims held robustly, the paper would establish a new way to think about AMM fees as rebalancing-band parameters and would support the verifiability of portfolio mandates from observable reserves. The empirical results are suggestive but strongly conditional on the arbitrage-participation assumption and on proprietary data; they do not yet establish robust dominance over incumbent funds.

major comments (3)

[§3 (first paragraph) and Corollary 6] The empirical dominance regions in Table 1 and the band-compliance statement of Corollary 6 rest on the assumption that a profitable arbitrage trade is actually executed, modeled as at most one frictionless trade per day at the close. Theorem 5 proves only that a profitable swap exists when a weight falls below (1−γ)w_i; it does not prove that an arbitrageur appears, how often, or at what cost. If arbitrage is slower, costlier, or absent, realized weights can drift outside the no-arbitrage band, and the reported CAGR/tracking-error results are not guaranteed. The manuscript should either supply a model or evidence of arbitrage participation or explicitly frame the enforcement and performance claims as conditional on this assumption, with sensitivity analysis on trading frequency and transaction/gas costs.
[§3, Table 1] The fee parameter γ is swept over [0%,10%] on the same sample used to identify the dominance regions, with no out-of-sample validation, bootstrap, or statistical significance assessment. For EDOW the sample period is less than 20 months. The dominance regions are therefore in-sample point estimates under the stated model, not robust lower/upper bounds. I recommend adding sub-sample or bootstrap checks and tempering the abstract's 'outperform' language accordingly.
[§3 (methodology) and Figures 1–3] All data are from Bloomberg and no code or data are provided, and the text does not define the annualized tracking error formula or the exact simulation algorithm (e.g., when the daily arbitrage trade is triggered and how monthly versus daily TE is annualized). This prevents independent verification of the empirical claims. Please provide code/data or a sufficiently detailed algorithmic appendix, and define the TE metric precisely.

minor comments (4)

[Throughout] The text and figures contain numerous typographical and OCR artifacts (e.g., 'a nd', 'e conomic', 'B%)ch(ark', 'BIAX'); these should be corrected before publication.
[§2.2, Lemma 4] The term 'path independence' is used for Lemma 4, but the lemma establishes only an inequality (5), with equality under a restrictive common-minimizer condition; the terminology should either be justified or changed to something like 'super-multiplicative' to avoid confusion.
[§2.2, Remark 2] The drift bound 2γ(1−min_i w_i) is stated without derivation; a one-sentence explanation that it follows from maximizing the L1 distance over the simplex would help readers verify the extremal case.
[§3 (conventions)] The phrase 'arbitrage-only order flow' is described as providing a 'conservative bound on fees,' but this assumption is not necessarily conservative for tracking error, since the absence of uninformed order flow also changes the rebalancing dynamics; this asymmetry should be acknowledged.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 5 is proved from the fee-modified G3M invariant via KKT, and the empirical backtests are conditional simulations rather than fitted predictions.

full rationale

The paper's central theoretical claim, Theorem 5, is derived in Appendix A.2 from the arbitrageur's optimization problem min {c^T z | g^γ_w(z-1) = 1}. The KKT conditions imply the post-trade weights satisfy ŵ = (1-γ)w + μ/λ with μ ≥ 0, so the lower-bound condition ŵ_i ≥ (1-γ)w_i is a consequence of the optimization, not an input. The converse is proved by constructing λ = Σ_i c_i and μ = λ(ŵ - (1-γ)w) and verifying the KKT conditions directly. Thus the 'if and only if' claim is self-contained and does not assume the result. The fee parameter γ is an independently chosen design parameter, and the no-arbitrage band in Corollary 6 is characterized from Theorem 5, not postulated. The only citation to the authors' own work, [3], is used for the path independence of the fee-free γ = 0 G3M in Lemma 4; the main inequality of Lemma 4 is proved in Appendix A.1, and the fee-free invariant property is standard and does not carry the weight of Theorem 5 or of the empirical dominance findings. Section 3's simulations are explicitly conditioned on an external premise — arbitrage-only order flow executed at most once per day at the close — and the fee sweep over γ ∈ [0%, 10%] is an empirical search reported as backtest results, not a parameter fitted inside the theory and then relabeled as a prediction. No equation in the paper reduces to its own input by definition, no fitted parameter is renamed as a prediction, and no load-bearing claim is justified solely by a self-citation. Therefore there is no significant circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central theory rests on the standard G3M invariant, a convex KKT framework, and an author-introduced fee rule. No new physical or asset-level entities are postulated. The empirical contribution adds a scanned fee parameter and several domain assumptions about arbitrage behavior and data quality.

free parameters (1)

Fee parameter gamma = swept over [0%,10%]; dominance intervals e.g. 2.73%-3.90%, 3.22%-7.09%, 3.32%-9.90%
Design parameter of the fee structure, not fitted to data in the theory. In the empirical section it is scanned and intervals where the G3M Pareto-dominates are reported, making the headline claim conditional on a post hoc chosen parameter region.

assumptions (4)

domain assumption Competitive, risk-neutral, frictionless arbitrageurs can transact at external prices s and choose the value-maximizing trade against the pool.
Theorem 5 characterizes the no-arbitrage region only if arbitrageurs solve the stated optimization; Section 2.2 and Section 3 assume arbitrage-only order flow trading at the daily close.
domain assumption External dollar prices s exist, are observable, and are the prices at which arbitrage can be executed.
Both the weight bound and the simulations require a reliable external price source; the paper uses Bloomberg total return data without estimating bid-ask frictions. See Section 3.
ad hoc to paper The fee structure in Definition 3, a geometric interpolation between proportional minting and the fee-free G3M update, is an appropriate model of AMM fees.
This fee rule is introduced by the authors and is described as distinct from prior fee structures; the theoretical result depends on this construction, not on Balancer's implemented fee schedule.
domain assumption Path-independence properties of the fee-free G3M from Bichuch and Feinstein [3] hold as cited.
Lemma 4 and the fee design rely on the path-independence result in [3], which is accepted without reproduction here; one author of the present paper is also an author of [3].

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Cite this review

Pith. "Pith review of Mandate without Managers: Automated Market Makers as Verifiable Portfolio Products." pith.science (2026). https://pith.science/paper/Q627YAO2

@misc{pith2026260802917,
  author       = {Pith},
  title        = {Pith review of: Mandate without Managers: Automated Market Makers as Verifiable Portfolio Products},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q627YAO2}},
  note         = {Machine review of arXiv:2608.02917}
}

read the original abstract

Automated market makers (AMMs) are typically interpreted and evaluated as decentralized exchanges. Herein, we take the perspective envisioned by Balancer that an AMM can also be viewed as a portfolio technology that programmatically enforces an economic mandate. In particular, we follow the geometric mean market maker (G3M) invariant employed by that protocol in order to enforce a target-weighted portfolio. We introduce a multi-asset fee structure to the G3M under which competitive arbitrage implements a band-rebalancing strategy with mis-weighting bounded ex ante, allowing compliance with the mandate to be verified directly from the pool's observable holdings. We then compare simulated G3M portfolios against the realized performance of VBIAX, EQL, and EDOW on annualized returns and tracking error against the portfolio mandate. Across these historical case studies, and using arbitrage-only order flow, the G3M is found to outperform the incumbent funds in both metrics for certain fee ranges.

Figures

Figures reproduced from arXiv: 2608.02917 by the authors.

**Figure 2.** Performance of the G3M against EQL on tracking an equal- [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Performance of the G3M against EDOW on tracking an equa [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Review history (2 revisions) →

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Reference graph

Works this paper leans on

17 extracted references · 14 canonical work pages

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Optimal fees for geometric mean market makers

Alex Evans, Guillermo Angeris, and Tarun Chitra. Optimal fees for geometric mean market makers. In International Conference on Financial Cryptography and Da ta Security , pages 65–79. Springer, 2021

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Improved price oracles: Cons tant function market makers

Guillermo Angeris and Tarun Chitra. Improved price oracles: Cons tant function market makers. In Proceedings of the 2nd ACM Conference on Advances in Financi al Technologies, pages 80–91, 2020. 4Intriguingly, and subject to future study, within the domin ating fee ranges reported in Table 1, the realized L VR (net of fee revenue) annualizes to a cost of t...

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The geometry of constant function market makers

Guillermo Angeris, Tarun Chitra, Theo Diamandis, Alex Evans, and K shitij Kulkarni. The geometry of constant function market makers. arXiv preprint arXiv:2308.08066 , 2023

arXiv 2023
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Axioms for automated mark et makers: A mathematical frame- work in ﬁntech and decentralized ﬁnance

Maxim Bichuch and Zachary Feinstein. Axioms for automated mark et makers: A mathematical frame- work in ﬁntech and decentralized ﬁnance. Operations Research, 74(3):1187–1202, 2026

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Rebalancing with transa ction costs: Theory, simulations, and actual data

Rim El Bernoussi and Michael Rockinger. Rebalancing with transa ction costs: Theory, simulations, and actual data. Financial Markets and Portfolio Management , 37(2):121–160, 2023

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Liquidity provider returns in geometric mean markets

Alex Evans. Liquidity provider returns in geometric mean markets . 2021

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A non-custodial port folio manager, liquidity provider, and price sensor

Fernando Martinelli and Nikolai Mushegian. A non-custodial port folio manager, liquidity provider, and price sensor. https://docs.balancer.fi/whitepaper.pdf, 2019

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Automated market making and loss-versus-rebalancing

Jason Milionis, Ciamac C Moallemi, Tim Roughgarden, and Anthony Lee Zhang. Automated market making and loss-versus-rebalancing. arXiv preprint arXiv:2208.06046 , 2022

arXiv 2022

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The fundamental theorem of asset pr icing under proportional transaction costs in ﬁnite discrete time

Walter Schachermayer. The fundamental theorem of asset pr icing under proportional transaction costs in ﬁnite discrete time. Mathematical Finance: An International Journal of Mathema tics, Statistics and Financial Economics, 14(1):19–48, 2004

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Optimal rebalancing in dynamic amms

Matthew Willetts and Christian Harrington. Optimal rebalancing in dynamic amms. arXiv preprint arXiv:2403.18737, 2024

2024 arXiv
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Rebalancing-versus -rebalancing: Improving the ﬁdelity of loss-versus-rebalancing

Matthew Willetts and Christian Harrington. Rebalancing-versus -rebalancing: Improving the ﬁdelity of loss-versus-rebalancing. arXiv preprint arXiv:2410.23404 , 2024

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[12]

Pools as portfolios: O bserved arbitrage eﬃciency & lvr analysis of dynamic weight amms

Matthew Willetts and Christian Harrington. Pools as portfolios: O bserved arbitrage eﬃciency & lvr analysis of dynamic weight amms. arXiv preprint arXiv:2602.22069 , 2026. A Proofs A.1 Proof of Lemma 4 Proof. First, we refer to [3] for the path independence of the γ = 0 settin...

2026
[13]

Stationarity in ui: ci exp(ui) − βi(λ, µi) = 0 for every asset i
[14]

Stationarity in t: ∑ N i=1 µi − λγ = 0
[15]

Primal feasibility : γt + (1 − γ)∑ N i=1 wiui ≥ 0, t ≤ ui ∀i
[16]

Dual feasibility : λ ≥ 0, µ ∈ RN + ; and
[17]

Complementary slackness : λ(γt + (1 − γ)∑ N i=1 wiui) = 0 and µi(t − ui) = 0 for every asset i. Letting u∗ be the outcome of the arbitrageur’s optimization, then (following th ese KKT conditions) the post-arbitrage weights are ˆwi = ci exp(u∗ i ) c⊤ exp(u∗) = λ(1 − γ)wi + µi λ...

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [6]

Optimal fees for geometric mean market makers

Alex Evans, Guillermo Angeris, and Tarun Chitra. Optimal fees for geometric mean market makers. In International Conference on Financial Cryptography and Da ta Security , pages 65–79. Springer, 2021

work page 2021

[2] [1]

Improved price oracles: Cons tant function market makers

Guillermo Angeris and Tarun Chitra. Improved price oracles: Cons tant function market makers. In Proceedings of the 2nd ACM Conference on Advances in Financi al Technologies, pages 80–91, 2020. 4Intriguingly, and subject to future study, within the domin ating fee ranges reported in Table 1, the realized L VR (net of fee revenue) annualizes to a cost of t...

work page 2020

[3] [2]

The geometry of constant function market makers

Guillermo Angeris, Tarun Chitra, Theo Diamandis, Alex Evans, and K shitij Kulkarni. The geometry of constant function market makers. arXiv preprint arXiv:2308.08066 , 2023

arXiv 2023

[4] [3]

Axioms for automated mark et makers: A mathematical frame- work in ﬁntech and decentralized ﬁnance

Maxim Bichuch and Zachary Feinstein. Axioms for automated mark et makers: A mathematical frame- work in ﬁntech and decentralized ﬁnance. Operations Research, 74(3):1187–1202, 2026

work page 2026

[5] [4]

Rebalancing with transa ction costs: Theory, simulations, and actual data

Rim El Bernoussi and Michael Rockinger. Rebalancing with transa ction costs: Theory, simulations, and actual data. Financial Markets and Portfolio Management , 37(2):121–160, 2023

work page 2023

[6] [5]

Liquidity provider returns in geometric mean markets

Alex Evans. Liquidity provider returns in geometric mean markets . 2021

work page 2021

[7] [7]

A non-custodial port folio manager, liquidity provider, and price sensor

Fernando Martinelli and Nikolai Mushegian. A non-custodial port folio manager, liquidity provider, and price sensor. https://docs.balancer.fi/whitepaper.pdf, 2019

work page 2019

[8] [8]

Automated market making and loss-versus-rebalancing

Jason Milionis, Ciamac C Moallemi, Tim Roughgarden, and Anthony Lee Zhang. Automated market making and loss-versus-rebalancing. arXiv preprint arXiv:2208.06046 , 2022

arXiv 2022