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Strategic Analysis of Just-In-Time Liquidity Provision in Concentrated Liquidity Market Makers

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

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Pith's one-line read A transaction-level model of just-in-time liquidity provision in concentrated-liquidity AMMs proves an optimal single-swap strategy exists, and fitting it to real Uniswap V3 data shows JIT LPs could earn up to 69% more by accounting for…

desk verdict Solid first model of JIT LP optimization in CLMMs, with an existence theorem worth taking seriously, but the empirical headline numbers are upper bounds that ignore inclusion auctions and the paper has one genuinely sloppy lemma. read the letter →

arxiv 2509.16157 v2 pith:NGMFLC34 submitted 2025-09-19 cs.GT cs.CR

Bruno Llacer Trotti , Weizhao Tang , Rachid El-Azouzi , Giulia Fanti , Daniel Sadoc Menasche This is my paper · ORCID
classification cs.GTcs.CR
keywords just-in-timeliquidityconcentratedmarketmakersUniswapV3automatedpriceimpactimpermanentlossMEVprovisionoptimization
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 builds the first transaction-level theory of just-in-time (JIT) liquidity provision in concentrated-liquidity AMMs such as Uniswap V3. A JIT LP watches a pending swap, mints a concentrated position just before it, and burns it immediately after, earning fees while also gaining or losing from the price move the swap causes. The paper proves that for every swap there is a global profit-maximizing choice of liquidity amount and price range, and gives an algorithm that computes it. Applied to six months of real Uniswap V3 JIT transactions, the optimized strategy would have increased profits by up to 69% relative to what the JIT LPs actually earned, because most real JIT LPs ignore price impact and even take trades with negative net utility. The authors also find that if JIT LPs optimized, traders would face lower slippage while passive LPs' fee income would drop by up to 44% per trade.

What carries the argument

The load-bearing object is the tick-decomposed utility identity $U=\sum_m (F_m-C_m)$, where $F_m=\delta_m L/(L+P_m)$ is the JIT LP's pro-rata share of the fee accrued in tick $m$, and $C_m$ is the corresponding price-impact term derived as a function of the JIT liquidity $L$, the passive liquidity $P_m$, the swap size, and the market and tick prices. The key structural fact is the price-impact threshold of Theorem 3.1: for a move $q\to q'$, with $\hat q$ the projection of the current price onto the chosen range, $C\le 0$ holds exactly when $\hat q' \le (p_x/p_y)^2/\hat q$ for a price decrease, and the reverse inequality for a price increase. This turns the JIT decision into a finite search over tick ranges that are actually touched as the price travels from $q$ to the no-JIT post-swap price $q^*$, plus a one-dimensional optimization in $L$; Algorithm 1 performs that search, and Theorem 4.1 guarantees the search has a finite global winner.

What would settle it

Obtain the complete set of bids from the block-builder auctions for the same six months of USDC/WETH JIT bundles, and recompute the profit gap as $U=F-C-v(\text{bid})$ with each JIT LP charged its actual inclusion cost; if the gap between observed and optimized profit shrinks toward zero as $v$ rises with position profitability, the 69% estimate is an artefact of the constant-$v$ assumption rather than a real inefficiency.

Watch

Extended reading notes

Core claim

On the paper's model, a JIT LP's per-swap utility is $U(s;\theta)=F(s;\theta)-C(s;\theta)$, where $F$ is the trading fee it captures from the swap and $C$ is price impact (the change in dollar value of its position between mint and burn, defined as positive when the position loses value). Theorem 4.1 states that this utility always attains a global maximum over the strategy space of single positions $(L,a,b)$ subject to the budget; the argument is that tick ranges are finite, and on each range $F$ is continuous and bounded while $C$ is continuous in $L$, so the feasible interval for $L$ is compact. Algorithm 1 finds the optimum by enumerating ranges in $\mathcal{R}(q,q^*)$ and solving a one-dimensional non-concave problem for $L^*$. The paper classifies the resulting optima into three strategic archetypes: overpriced trades (pool price moves away from market price, price impact helps the LP), arbitrageur trades (price moves toward market, price impact hurts), and overshoot trades (price crosses the market price, so the LP should concentrate liquidity only in the segment where $C<0$). Fitting the algorithm to 6,829 real JIT swaps in a USDC/WETH Uniswap V3 pool, the authors find that observed JIT behavior underperforms the optimum by up to 69%, that price impact accounts for roughly 75% of total returns on average, and that optimized JIT participation could cut passive-LP fee income by up to 44% per trade while lowering trader slippage.

Load-bearing premise

The central empirical claim assumes the JIT LP's cost of winning the right to include its sandwich bundle is the same no matter which liquidity position it chooses; if that cost is actually a competitive bid that rises with the profit the position can extract, the measured 69% gain would be eaten by the auction.

Editorial extensions

If this is right

  • Overpriced trades, which push the pool price away from the market price, are unambiguously favorable: price impact is non-positive and the optimal strategy can deploy the full JIT budget.
  • Arbitrageur trades, which push the pool price toward the market price, are usually loss-making unless fees cover the price impact, so the optimal strategy is often to skip the trade.
  • If JIT LPs optimized their positions, passive LPs' fee income would fall by up to 44% per trade, while traders would enjoy lower slippage.
  • JIT liquidity acts as a preemptive arbitrage force: it dampens price dislocations and reduces the profit available to arbitrageurs who would correct the same mispricing later.
  • The three-way classification of trades (overpriced, arbitrageur, overshoot) gives a practical participation rule derived from the price-impact threshold, not just a numerical recipe.

Reading between the lines

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

  • The 69% figure likely overstates what JIT LPs could retain if inclusion is a competitive auction: the paper fixes the winning-bid cost $v$ independent of the position, so part of the measured gap may be a transfer to block builders rather than profit.
  • The threshold condition in Theorem 3.1 could be turned into a real-time classifier: label a pending swap by whether $q'$ crosses the boundary, and only dispatch capital when the label is 'overpriced' or the post-crossing segment of an overshoot trade.
  • Because the strategy space is restricted to single positions, a natural extension is to allow two or more simultaneous positions; the proof that $U$ attains a maximum would need revisiting, but the tick-decomposed utility carries over.
  • The empirical study uses one pool, and the authors say volatile pools show wider gaps, so the same counterfactual should be re-run on a high-volatility pool to see whether the 69% gap widens.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. This paper studies just-in-time (JIT) liquidity provision in concentrated liquidity market makers (CLMMs). It develops a transaction-level model in which a JIT LP chooses a position (L,a,b) immediately before a swap and earns fees minus price impact, subject to a budget constraint. The paper characterizes when price impact is favorable, analyzes how post-swap price depends on injected liquidity, proves existence of an optimal strategy (Theorem 4.1), and proposes Algorithm 1 to compute it. The empirical evaluation on Uniswap V3 USDC/WETH data (January-June 2024) claims that real JIT LPs are suboptimal, could earn up to 69% more with the optimized strategy, that JIT returns are driven mostly by price impact, and that optimized JIT activity reduces passive LP fees and trader slippage.

Significance. If the results hold, this would be a valuable first formal, transaction-level treatment of JIT LP optimization in CLMMs. The theoretical core is largely self-contained: the price-impact characterization and the existence theorem do not rely on fitted parameters, and the empirical study uses real on-chain data. The paper also articulates falsifiable predictions through the three strategic archetypes and the fee-sufficiency condition. However, the headline empirical claims are currently overstated because inclusion-auction costs are assumed away and the numerical optimizer lacks a global-optimality certificate; these issues do not invalidate the existence theorem but do affect the practical contribution.

major comments (4)
  1. [Section 3.2 / Appendix D (Lemma 3.2)] Lemma 3.2 states that q' is 'continuous, strictly decreasing in L' with lim_{L→∞} q'=q, but with q'(L)<q for finite L, convergence to q from below requires q' to be increasing in L. The proof sketch in Section 3.2 even says 'As L increases, the same ∆x has less price impact, so q'↑,' and the appendix proof does not establish the claimed direction. This is not a cosmetic typo: the sign of the monotonicity is used implicitly in Lemma 3.1 and in the continuity/boundedness argument of Theorem 4.1. Please correct the statement and make the proof consistent.
  2. [Section 4.2.2 and Section 6 (Data Collection)] The model assumes the auction/bid cost v is 'a constant that does not relate to the choice of liquidity position' (Section 4.2.2), while Section 4.1 describes inclusion as a sealed-bid auction in which the winning JIT bundle pays a tip bounded below by the second-highest bid. Section 6 explicitly does not simulate bidding auctions among JIT LPs. Consequently, the reported 'up to 69% more than current profit' and 'never loses money' (Section 6.1.1) are gross-of-auction upper bounds, not achievable net profits; under competitive bidding the surplus could be consumed by tips. The existence theorem is unaffected, but the practical headline should be either re-derived with an auction model or explicitly labeled as an upper bound.
  3. [Section 5 (Algorithm 1)] Algorithm 1's optimality guarantee is conditional on solving a non-concave, non-linear univariate problem (Eq. 19) to global optimality. The implementation uses Particle Swarm Optimization and binary search, neither of which provides a global optimality certificate. The empirical 'optimized' profits, including the 69% figure and the 'never loses money' statement in Section 6.1.1, are therefore only as strong as the heuristic solver. Please either provide a certified global method (e.g., branch-and-bound over the single variable) or report bounds on the suboptimality gap.
  4. [Appendix E (Proposition 5.1)] The proof of Proposition 5.1 contains sign errors: it states 'F_m < C_m for all m ⇒ U>0' when U=F−C would give U<0, and it concludes that the derived inequality is 'a sufficient condition for U>0' although the proposition asserts U<0. The proposition itself appears correct after rearrangement, but the appendix proof needs to be fixed to match the statement.
minor comments (4)
  1. [Appendix B, proof of Theorem 3.1] The proof divides by sqrt(q)−sqrt(q') and calls it positive, but for q<q' it is negative; the final inequality is correct only after a sign flip. Please correct the sign discussion.
  2. [Section 6] The data-collection paragraph reports 6,829 JIT swaps, while Section 6.1.1 states 6,884 JIT transactions; please reconcile these numbers.
  3. [Section 4.2.2 and Table 1] The bid/inclusion cost is denoted v in Section 4.2.2, v* in Appendix D's proof of Theorem 4.1, and 'v*_j' in Table 1; please unify the notation.
  4. [Figure 3(a)] The axis labels appear garbled ('102 101 100 0 100 101 102 103'); please format the log-scale axis clearly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the optimization theorem and empirical counterfactual are self-contained and do not reduce to their inputs.

full rationale

The paper's central derivation chain is analytical. Theorem 4.1 takes U(s;θ)=F(s;θ)-C(s;θ) from the CLMM fee formula (Eq. 14) and the tick-wise price-impact decomposition (Eq. 17), then proves existence of a maximizer by finiteness of tick intervals and continuity/compactness; no parameter is fitted to obtain the theorem. The empirical 69% claim (Section 6.1.1) is a counterfactual evaluation: real JIT transactions are replayed under Algorithm 1 using the same realized transaction set and the same budget, and the paper explicitly states it does not simulate auction outcomes (Section 6, Data Collection), so the figure is a stated upper-bound comparison rather than a fitted prediction. The single-position restriction cites the authors' own [24], but it is a conservative modeling assumption (also attributed to [27]) and does not imply the measured profit gap; it is not load-bearing for Theorem 4.1. The 'never loses money' sentence is an analytic consequence of including non-participation in the strategy space rather than a separate empirical discovery, but this is a presentation nuance, not a circular derivation. The unmodeled inclusion-auction cost is an explicitly acknowledged limitation rather than a circular step. No equation in the paper is defined in terms of its own output, and no fitted constant is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data. The central derivations rest on standard CLMM and constant-product formulas plus the stated constancy of market prices. The main abstractions are the single-position restriction, the exogenous passive-LP state, and a constant inclusion cost v, which is effectively set to zero in the empirical section. These simplifications directly affect the size of the reported empirical gains.

assumptions (5)
  • domain assumption External market prices p_x and p_y are constant throughout the trade (Section 3, Assumptions).
    Justified by single-block atomic execution; the central price-impact characterizations in Theorem 3.1 and Corollary 3.1 depend on this simplification.
  • domain assumption A JIT LP can atomically mint before and burn after the target swap and is guaranteed inclusion (Section 4.1, modeling implications).
    The utility model assumes bundle atomicity and successful inclusion; competition and auction outcomes are not modeled.
  • domain assumption The JIT LP commits to a single position (L,a,b) rather than multiple positions (Section 4.2.1).
    Justified by prior literature and data from [24], including self-cited work; restricts the strategy space and is needed for the algorithm.
  • ad hoc to paper The auction or bid cost v is a constant independent of the chosen liquidity position (Section 4.2.2).
    Adopted because losing-bid data are unavailable; this simplification is carried into the empirical section, where costs are effectively ignored.
  • domain assumption Passive LP positions s are fixed a priori and do not respond to the JIT LP's intervention (Section 4.2).
    The optimization treats passive liquidity distribution P as exogenous; strategic responses by passive LPs or other JIT LPs could alter the optimum.

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Pith. "Pith review of Strategic Analysis of Just-In-Time Liquidity Provision in Concentrated Liquidity Market Makers." pith.science (2026). https://pith.science/paper/NGMFLC34

@misc{pith2026250916157,
  author       = {Pith},
  title        = {Pith review of: Strategic Analysis of Just-In-Time Liquidity Provision in Concentrated Liquidity Market Makers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NGMFLC34}},
  note         = {Machine review of arXiv:2509.16157}
}
read the original abstract

Liquidity providers (LPs) are essential figures in the operation of automated market makers (AMMs); in exchange for transaction fees, LPs lend the liquidity that allows AMMs to operate. While many prior works have studied the incentive structures of LPs in general, we currently lack a principled understanding of a special class of LPs known as Just-In-Time (JIT) LPs. These are strategic agents who momentarily supply liquidity for a single swap, in an attempt to extract disproportionately high fees relative to the remaining passive LPs. This paper provides the first formal, transaction-level model of JIT liquidity provision for a widespread class of AMMs known as Concentrated Liquidity Market Makers (CLMMs), as seen in Uniswap V3, for instance. We characterize the landscape of price impact and fee allocation in these systems, formulate and analyze a non-linear optimization problem faced by JIT LPs, and prove the existence of an optimal strategy. By fitting our optimal solution for JIT LPs to real-world CLMMs, we observe that in liquidity pools (particularly those with risky assets), there is a significant gap between observed and optimal JIT behavior. Existing JIT LPs often fail to account for price impact; doing so, we estimate they could increase earnings by up to 69% on average over small time windows. We also show that JIT liquidity, when deployed strategically, can improve market efficiency by reducing slippage for traders, albeit at the cost of eroding average passive LP profits by up to 44% per trade.

Figures

Figures reproduced from arXiv: 2509.16157 by the authors.

Figure 1
Figure 1. Partition of the (q, q′ )-space with px/py = 1; black curve corresponds to q ′ = 1/q. Red and green regions correspond to PI ≥ 0 and PI ≤ 0, respec￾tively. 3.2 The Effects of Liquidity We analyze the limiting behavior of price impact as available liquidity changes. Suppose we have a price range (a, b) which is currently covered by total liquidity L by passive LPs. A JIT LP is considering adding liquidity to this ran… view at source ↗
Figure 2
Figure 2. (a) Illustration of the three kind of trades. Each vertical line represents [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Comparison between actual and optimized JIT LP behavior. (a) [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: CDF of the proportion of JIT returns resulting from price impact as [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]
Figure 5
Figure 5. Figure 5: Both plots compare the effects of higher JIT budget with a baseline [PITH_FULL_IMAGE:figures/full_fig_p023_5.png]
Figure 6
Figure 6. Figure 6: Arbitrageur trade: Utility curves for different choices of (a, b). The legend indicates how many ticks away the position endpoints are relative to ⌊q⌋. For example, the blue line labeled −2, −1 corresponds to the position (t⌊q⌋−2, t⌊q⌋−1). Despite the presence of fees,…
Figure 7
Figure 7. Figure 7: Overpriced trade: Utility curves for different choices of (a, b) on an overpriced trade. The legend indicates how many ticks away the positions are relative to ⌊q⌋. For example, the blue line labeled −1, 0 corresponds to the position (t⌊q⌋−1, t⌊q⌋). All strategies yiel…
Figure 8
Figure 8. Figure 8: Overshoot trade: Utility curves for different choices of (a, b) on an overshoot trade. The legend indicates how many ticks away the positions are relative to ⌊q⌋. For example, the blue line labeled −2, −1 corresponds to the position (t⌊q⌋−2, t⌊q⌋−1). Notice that some s…

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    Therefore, asLincreases,q ′ must not increase by 2 and 4.q ′ must not stay constant, either, by 1 and 3, which implies the strict monotonicity ofq ′ inL

    Pm k=1ϕk(L)ψk(q′) = ∆xis constant. Therefore, asLincreases,q ′ must not increase by 2 and 4.q ′ must not stay constant, either, by 1 and 3, which implies the strict monotonicity ofq ′ inL. The continuity ofq ′ inLoriginates from the continuity of eachϕ k andψ k. Finally, asL→∞...

  24. [2024]

    Accessed: 2025-08-07

    2025

Pith tools

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