REVIEW 4 major objections 5 minor 2 cited by
Loss-Versus-Rebalancing under Deterministic and Generalized block-times
T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash
Pith's one-line read For automated market makers on fixed-interval chains, per-block arbitrage loss is given by a closed-form formula, and constant spacing is asymptotically the best possible block-time law.
desk verdict Genuinely new and likely correct, but the proof of the key overshoot lemma is incomplete and the definition of O is typo'd; worth refereeing after a fix. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The engine is a discrete-time Markov chain whose state is the log-price inside the no-arbitrage interval $[0,\rho_b]$ with $\rho_b=\gamma/\sigma_b$, reset to 0 whenever an arbitrage occurs. Stopping the underlying random walk at its first boundary crossing, the expected time to an arbitrage and the conditional loss are expressed through the ladder height $H$ (the size of the walk's first descent below zero) and the overshoot variables $R$ and $O$. The central identities are the stopped-random-walk relations $E[\tau]=h_2+h_1(\rho_b^2+2\rho_b r_1+r_2-o_2)/(\rho_b+r_1+o_1)$ and $\mathrm{LVR}=\ell\sigma_b^{2}[h_2/2+h_1(r_2-o_2)/(2(\rho_b+r_1+o_1))]$. For Gaussian increments $h_1=1/\sqrt{2}$ and $h_2=|\zeta(1/2)|/\sqrt{\pi}$, and the boundary-crossing estimate supplies $r_1,o_1\to\kappa=|\zeta(1/2)|/\sqrt{2\pi}$ and $r_2,o_2\to\omega=1/4+\kappa^2$ exponentially fast; the collapse of this ratio to $\rho_b$ is what produces the explicit coefficient $1.7164$ in the headline formula.
What would settle it
A direct numerical check of the overshoot identities: simulate the Gaussian random walk on $[0,\rho_b]$ and estimate the right overshoot moments $r_1,r_2$ and the auxiliary 'leftover' overshoot moments $o_1,o_2$ as defined in Section 2; for $\rho_b$ from 1 to 8, test whether $r_1-o_1$ and $r_2-o_2$ decay like $e^{-c\rho_b}$. Alternatively, rerun the paper's Monte-Carlo protocol at $\rho_b=\gamma/\sigma_b\in\{0.5,1,2,3,4,5\}$ and check that measured $\overline{\mathrm{ARB}}$ tracks $\ell\sigma_b^{2}/(2+1.7164\,\gamma/\sigma_b)$ to the reported sub-0.01% accuracy; a fixed offset in the overshoot moments would appear as a wrong $O(1)$ constant.
Extended reading notes
Core claim
In the deterministic block-time model, with log-price increments $N(0,\sigma_b^2)$ per block, the long-run per-block arbitrage loss is $\overline{\mathrm{ARB}} = \ell\sigma_b^{2}/(2+\sqrt{2\pi}\,\gamma/(|\zeta(1/2)|\,\sigma_b)) + O(e^{-c\gamma/\sigma_b})$, numerically $\approx \ell\sigma_b^{2}/(2+1.7164\,\gamma/\sigma_b)$. The companion statements are $P_{\mathrm{trade}} = 1/(\gamma/(\sqrt{2}\sigma_b)+|\zeta(1/2)|/\sqrt{\pi}) + O(e^{-c\gamma/\sigma_b})$ and $\mathrm{LVR} = \ell\sigma_b^{2}|\zeta(1/2)|/(2\sqrt{\pi}) + O(e^{-c\gamma/\sigma_b})$. Extended to arbitrary block-time laws, the paper shows that the asymptotic arbitrage probability $P_{\mathrm{trade}} = \sqrt{2}\sigma_b/\gamma + O(\sigma_b/\gamma)$ is universal, while the asymptotic LVR is minimized only by constant block spacing. Hence, among all distributions with the same mean block time, the Dirac law, that is constant block spacing, gives liquidity providers the strongest protection against arbitrage.
Load-bearing premise
The load-bearing premise is Lemma 3.1: the auxiliary 'leftover' overshoot moments $o_1,o_2$ converge exponentially fast to the same limits $\kappa,\omega$ as the right overshoot moments $r_1,r_2$; the appendix derives the $r$ convergence from a cited boundary-crossing theorem and states the $o$ convergence 'likewise', but if an $O(1)$ difference between them survives, the $1.7164$ constant changes.
Editorial extensions
If this is right
- On proof-of-stake chains with fixed slot times, the formula gives LPs a closed-form adverse-selection cost per block from three observable inputs: per-block volatility, spread, and liquidity density.
- Holding average block time fixed, replacing Poisson arrivals with constant spacing reduces per-block LVR by up to roughly 17.4% in the fast-block regime.
- The asymptotic probability that a block contains an arbitrage trade is universal, so changing the block-time distribution changes the size of the loss per trade, not the frequency of trades.
- Constant block spacing uniquely minimizes both asymptotic LVR and total arbitrage among all block-time distributions with a fixed mean, giving a scheduling principle for consensus protocols.
Reading between the lines
- The same stopped-random-walk machinery should extend to fee-paying AMMs: fees shift or widen the no-arbitrage interval, and the overshoot identities should quantify how fees trade off against arbitrage loss.
- A testable prediction is that blockchains with jittered or variable slot times impose higher adverse selection on liquidity providers than fixed slots with the same average; comparing measured LVR across chains that changed their slot schedule would isolate this effect.
- Since asymptotic trade frequency is distribution-free, any reduction in LVR from changing block timing must come from shrinking the conditional loss per arbitrage; protocol designers could therefore optimize that second factor directly.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript studies loss-versus-rebalancing (LVR) for automated market makers under general block-time distributions, focusing on deterministic (constant) block times. The authors model the log-price as a resetting random walk on the no-arbitrage interval, decompose long-run average arbitrage loss as the product of the probability of an arbitrage trade and the expected loss conditional on a trade, and use Wald identities and ladder-height / overshoot results to derive closed-form asymptotic expressions. For constant block times the central formula is ARB = ℓσ_b² / (2 + √(2π) γ/(|ζ(1/2)|σ_b)) + exponentially small error, with numerical Monte Carlo support for ρ_b = γ/σ_b ≥ 1. For general block-time distributions the paper claims a universal first-order trade probability and that the Dirac (constant) block-time distribution asymptotically minimizes LVR among distributions with fixed mean.
Significance. If the mathematical derivation can be made fully rigorous, the contribution is significant for the DeFi literature: it provides the first analytical LVR formula for fixed-interval blockchains, gives a closed-form coefficient (1.7164) that can be used in practice, and identifies a design principle (constant block times minimize asymptotic arbitrage losses). The paper is commendable for being parameter-free: the constants come from cited external results (Lai, Chang–Peres, Lotov, Spitzer, Wald), not from fitting, and the Monte Carlo experiments are used only as validation. The decomposition Ptrade × LVR and the universality result for Ptrade are appealing and potentially useful beyond the specific constant-time case. However, the correctness of the headline coefficient currently rests on a proof gap and on an algebraically inconsistent definition of an auxiliary overshoot, so the central claim cannot yet be considered established.
major comments (4)
- [Section 2.3, definition of O and equations (2.3)–(2.4)] The definition O = -S_τ 1{τ=τρ_b}/p is algebraically inconsistent with the surrounding identities. On the event {τ=τρ_b}, S_τ > ρ_b, so O is negative and of order ρ_b, not a positive O(1) overshoot. Substituting this definition into (2.3) and using (2.5) forces h1 = 0, which contradicts h1 > 0. The intended object is presumably the eventual ladder height on paths that first hit the upper barrier, i.e. O = -S_{τ0} 1{τ=τρ_b}/p, where τ0 is the first time the walk is negative. This correction must be made and the identities re-derived with the corrected definition.
- [Section 6, proof of Lemma 3.1 (convergence of o1, o2)] The proof does not establish the claimed convergence for the auxiliary overshoot O. The stated consequence of Lotov's theorem, (6.1), gives the limiting distribution of the upper overshoot R and of the left overshoot L = -S_τ conditional on {τ=τ0}. It does not give the distribution of -S_{τ0} conditional on {τρ_b < τ0}, which is what the corrected O requires. The sentence 'likewise o1 = κ + O(e^{-cρ_b})' is asserted without a theorem. This gap is load-bearing: in (2.9) the denominator contains r1 + o1 and the numerator contains r2 - o2, and in (2.10) the correction term is proportional to h1(r2 - o2)/(ρ_b + r1 + o1). If o1 differs from r1 by an O(1) amount, E[τ] acquires an O(1) shift and the constant in the headline ARB formula changes; if o2 differs from r2, the leading LVR constant is unchanged but the stated exponentially small error is not justified. A proof of the convergence of O to the same limit F (or an explicit argument showing the difference is exponentially small) is required.
- [Section 6, equations (6.2)–(6.3) and the events B and A] The 'good event' argument is not coherent as written. Event B is defined as {τρ_b < τ0 and ∃ n0 ≥ τρ_b : S_{n0} ∈ [ρ_b/4, 3ρ_b/4]}; the overshoot S_{τρ_b} occurs at or before n0, and restarting the walk at n0 via the strong Markov property cannot control the distribution of S_{τρ_b} conditional on B. Similarly, the use of A for the left overshoot is unclear. This part of the proof therefore does not achieve the uniform-in-x convergence needed for the moment asymptotics. The proof of Lemma 3.1 must be reworked so that the conditioning events actually determine the overshoot variable whose distribution is being bounded.
- [Section 4, equations (4.1), (4.2), (4.4), and Table 1] There is an internal inconsistency in the constant relating h2 to the Riemann zeta function. Corollary 3.1 and Table 1 use h2 = |ζ(1/2)|/√π, leading to LVR = ℓσ_b²|ζ(1/2)|/(2√π) and to the headline denominator with √(2π)γ/(|ζ(1/2)|σ_b). In contrast, equation (4.4) yields h2 = |ζ(1/2)|/√(2π) for the constant block-time case, and equations (4.1)–(4.2) use |ζ(1/2)|/√(2π), while Table 1's 'General' row uses |ζ(1/2)|/√π. These differ by a factor of √2. The authors must correct the coefficient in (4.4) and align (4.1)–(4.2) with Table 1, and then verify that the definition of Cµ is consistent. This affects the general-distribution LVR formula and the Dirac-optimality statement. Relatedly, in Corollary 4.1 the text states that 'by Lemma 4.1 ... h2 ... are all bounded constants,' but Lemma 4.1 only bounds r1, r2, o1, o2; a separate argument for the finiteness of h2 is needed.
minor comments (5)
- [Throughout] There are several typos: 'bl ocks', 'Los s-versus-rebalancing', 'theoretical expression for ρb = γ σb smaller than 5' in the Table 2 caption, and missing labels for the panels in Figure 1.
- [Section 2.3, Wald identities] The displayed second Wald identity writes Var(X1)E[τ] + (E[X1])²E[τ²] = E[Sτ²]; with E[X1]=0 this is fine, but the E[τ²] term is unnecessary and may confuse readers. The simplified identity Var(X1)E[τ] = E[Sτ²] is what is used.
- [References] Reference [3] is duplicated in the bibliography (the Ethereum Foundation documentation appears twice), and references [5] and [19] have the same title; please check whether they are distinct works or a duplicate entry.
- [Section 5, item (3)] The SDE in the final remark has a missing parenthesis or an ambiguous grouping; please rewrite the expression so that the denominator and the drift term are unambiguous.
- [Appendix, proof of Lemma 4.1] The proof uses notation such as P(R1 ≥ x | τ0 ≥ τρ_b) without defining R1, and the displayed decomposition of the conditional probability is difficult to follow. A cleaner conditioning argument would help, especially since the same domination is needed for O under the corrected definition.
Circularity Check
No circularity: the derivation is parameter-free and relies on external random-walk theorems; the flagged O-overshoot issue is a correctness gap, not a circular reduction.
full rationale
The paper's derivation is self-contained and does not fit any parameter from the quantity it predicts. The headline ARB formula follows from the renewal decomposition ARB = Ptrade × LVR, Wald's identities, and external ladder-height moment results (Spitzer, Lai, Lotov, Chang–Peres). The constants h1, h2, κ, and ω are obtained from cited theorems, not from the target loss; Monte Carlo is used only for validation, as in Table 2. No load-bearing self-citation appears: the closely related prior work by Milionis et al. is by other authors and supplies the LVR definition and the Poisson benchmark, not the constant-block-time result. The reader's flagged concern about the auxiliary overshoot O in Section 2.3 is a real internal-consistency and proof gap: as printed, O = −Sτ 1{τ=τρb}/p would imply o1 = −(ρb+r1), contradicting Lemma 3.1's assertion o1 = κ + O(e^{-cρb}), and the appendix supplies only 'likewise' rather than a theorem for O. That is an omitted proof or typographical error affecting correctness, but it is not circularity: the claimed constants are not defined in terms of the final formula, and no fitted input is relabeled as a prediction. The Dirac optimality result is proved from Lai's ladder-height formula and Jensen's inequality, so it is also not circular. Score 0.
Assumptions & free parameters
assumptions (7)
- domain assumption Asset log-price follows a zero-drift geometric Brownian motion between blocks.
- domain assumption Block times are i.i.d. draws from a distribution µ with finite first and second moments, mean normalized to 1.
- domain assumption AMM has constant spread γ and constant liquidity density ℓ; arbitrage loss for deviation d is ℓd²/2.
- domain assumption After each arbitrage the relative log-price is reset to the lower boundary 0.
- standard math Existing theorems: Wald identities, Spitzer's ladder-height mean, Chang-Peres/Lai moments for Gaussian random walks, and Lotov's boundary-crossing theorem for Gaussian random walks with two barriers.
- standard math Strong law of large numbers applies to the regenerative arbitrage process so ARB = P_trade × LVR.
- standard math Jensen's inequality gives C_μ ≥ 0 with equality iff µ is Dirac.
Cite this review
Pith. "Pith review of Loss-Versus-Rebalancing under Deterministic and Generalized block-times." pith.science (2026). https://pith.science/paper/DOK2EVRJ
@misc{pith2026250505113,
author = {Pith},
title = {Pith review of: Loss-Versus-Rebalancing under Deterministic and Generalized block-times},
year = {2026},
howpublished = {\url{https://pith.science/paper/DOK2EVRJ}},
note = {Machine review of arXiv:2505.05113}
}
abstract
Although modern blockchains almost universally produce blocks at fixed intervals, existing models still lack an analytical formula for the loss-versus-rebalancing (LVR) incurred by Automated Market Makers (AMMs) liquidity providers in this setting. Leveraging tools from random walk theory, we derive the following closed-form approximation for the per block per unit of liquidity expected LVR under constant block time: \[ \overline{\mathrm{ARB}}= \frac{\,\sigma_b^{2}} {\,2+\sqrt{2\pi}\,\gamma/(|\zeta(1/2)|\,\sigma_b)\,}+O\!\bigl(e^{-\mathrm{const}\tfrac{\gamma}{\sigma_b}}\bigr)\;\approx\; \frac{\sigma_b^{2}}{\,2 + 1.7164\,\gamma/\sigma_b}, \] where $\sigma_b$ is the intra-block asset volatility, $\gamma$ the AMM spread and $\zeta$ the Riemann Zeta function. Our large Monte Carlo simulations show that this formula is in fact quasi-exact across practical parameter ranges. Extending our analysis to arbitrary block-time distributions as well, we demonstrate both that--under every admissible inter-block law--the probability that a block carries an arbitrage trade converges to a universal limit, and that only constant block spacing attains the asymptotically minimal LVR. This shows that constant block intervals provide the best possible protection against arbitrage for liquidity providers.
Figures
Forward citations
Cited by 2 Pith papers
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Measuring CEX-DEX Extracted Value and Searcher Profitability: The Darkest of the MEV Dark Forest
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Split the Yield, Share the Risk: Pricing, Hedging and Fixed rates in DeFi
A formal model prices DeFi yield tokens as discounted expected future yield and proposes utility-based market makers and a fixed-rate lending design on top.
Reference graph
Works this paper leans on
-
[8]
C., Roughgarden, T., & Zhang, A
Milionis, J., Moallemi, C. C., Roughgarden, T., & Zhang, A. L. (2023). Loss-versus-rebalancing under fee models with Poisson block times. Preprint, arXiv:2305.146 04 [q-fin.MF]
work page 2023
-
[1]
Adams, H., Zinsmeister, N., & Robinson, D. (2020). Unisw ap v2 Core (Whitepaper). Uniswap.org. Retrieved from https://uniswap.org/whitepaper.pdf
work page 2020
-
[2]
Angeris, G., Kao, H.-T., Chiang, R., Noyes, C., & Chitra, T. (2019). An analysis of Uniswap markets. Cryptoeconomic Systems, 1(1), Article 3. arXiv:1911.03380
arXiv 2019
-
[3]
Ethereum Foundation. (2025). Proof-of-Stake consensu s mechanism – block time documentation. Ethereum Developer Docs*. Retrieved Apr. 3 0, 2025, from https://ethereum.org/en/developers/docs/consensus-mechanisms/pos/
work page 2025
-
[4]
Hanson, R. (2007). Logarithmic Market Scoring Rules for Modular Combinatorial Information Aggregation. Journal of Prediction Markets , 1(1), 3–15
work page 2007
-
[5]
Khaniyev, T., & Kucuk, Z. (2022). Asymptotic expansions for the moments of the Gaussian random walk with two barriers. Communications in Statistics-Theory and Methods , 51(2), 376–391
work page 2022
-
[6]
Lai, T. L. (1976). Asymptotic moments of random walks wit h applications to ladder variables and renewal theory. The Annals of Probability , 4(1), 51–66
work page 1976
-
[7]
C., Roughgarden, T., & Zhang, A
Milionis, J., Moallemi, C. C., Roughgarden, T., & Zhang, A. L. (2022). Automated market making and loss- versus-rebalancing. In Proceedings of the 2nd ACM Conference on DeFi and Digital Ass ets (DeFi ’22) , Los Angeles, CA. arXiv:2208.06046. 16 A. NEZLOBIN AND M. TASSY
arXiv 2022
Show all 20 references
-
[9]
M., Reeves, D
Othman, A., Pennock, D. M., Reeves, D. M., & Sandholm, T. ( 2013). Automated Market-Making in the Large: The Gates Hillman Prediction Market. Proceedings of the ACM Conference on Electronic Commerce , 721–738
2013
-
[10]
Spitzer, F. (1956). A combinatorial lemma and its appli cation to probability theory. Transactions of the American Mathematical Society , 82(2), 323–339
1956
-
[11]
Wald, A. (1944). On cumulative sums of random variables . The Annals of Mathematical Statistics , 15(3), 283–296. Ethereum Foundation Documentation . Retrieved 30 Apr 2025 from https://ethereum.org/en/developers/docs/consensus-mechanisms/pos/
1944
-
[12]
Kendall, D. G. (1951). Some problems in the theory of que ues. Journal of the Royal Statistical Society, Series B, 13(2), 151–173
1951
-
[13]
Kingman, J. F. C. (1961). The single-server queue in hea vy traffic. Mathematical Proceedings of the Cambridge Philosophical Society, 57(4), 902–904
1961
-
[14]
Lotov, V. I. (1996). On some boundary-crossing problem s for Gaussian random walks. Annals of Probability , 24(4), 2154–2171
1996
-
[15]
Capponi, A., & Jia, Y. (2025). Order-flow toxicity and ar bitrage rents in automated market makers. Review of Financial Studies , forthcoming
2025
-
[16]
T., & Peres, Y
Chang, J. T., & Peres, Y. (1997). Ladder heights, Gaussi an random walks and the Riemann zeta function. Annals of Probability , 25(2), 787–802
1997
-
[17]
Nagaev, S. V. (2010). Exact expressions for the moments of ladder heights. Siberian Mathematical Journal , 51(4), 675–695
2010
-
[18]
Fuh, C.-D. (2007). Asymptotic expansions on moments of the first ladder height in Markov random walks with small drift. Advances in Applied Probability , 39(3), 826–852
2007
-
[19]
A., & Kucuk, Z
Khaniyev, T. A., & Kucuk, Z. (2004). Asymptotic expansi ons for the moments of the Gaussian random walk with two barriers. Statistics & Probability Letters , 69(1), 91–103
2004
-
[20]
Lotov, V. I. (2023). Exact formulas in some boundary cro ssing problems for integer-valued random walks. Izvestiya: Mathematics , 87(1), 45–60
2023
Reviewed August 15, 2026 · model on record in the stance chip above.
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