Uniform-Loss Automated Market Making for Prediction Markets

Brian Zhu; Ciamac C. Moallemi; Dan Robinson

REVIEW 1 major objections 5 minor 30 references

Uniform-Loss Automated Market Making for Prediction Markets

T0 review · 1 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A second-order boundary value problem equates prediction-market AMM design with the choice of a belief process, so that loss is proportional to pool value at every price.

desk verdict The main win-martingale/AMM correspondence is solid and worth engaging with, but Theorem 10's minimax claim is false as stated and needs a straightforward fix. read the letter →

arxiv 2607.17428 v2 pith:TXXZPNPK submitted 2026-07-19 q-fin.TR

Ciamac C. Moallemi , Dan Robinson , Brian Zhu This is my paper · ORCID

classification q-fin.TR MSC 91B2660G4460J6091G80

keywords uniformLVRpredictionmarketsautomatedmarketmakerwin-martingaleloss-versus-rebalancingboundaryvalueproblemliquidityschedulepool-valuefunction

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

The paper introduces uniform AMMs for binary prediction markets: mechanisms whose instantaneous loss-versus-rebalancing is proportional to pool value and independent of the current token price, after normalizing for the rate of information arrival. It proves a two-way equivalence: every separable win-martingale satisfying mild regularity admits a concave pool-value function that solves a boundary value problem, and every sufficiently regular concave pool-value function defines a win-martingale under which it is uniform. A deterministic liquidity schedule then implements any prescribed target expected cumulative loss over time, separating statewise from timewise subsidy. If correct, AMM designers and subsidizers can shape where and when losses occur instead of only bounding the total.

What carries the argument

The uniformity BVP G(p)^2 V''(p)+βV(p)=0 with V(0)=V(1)=0 and V>0 concave is the load-bearing object. It enforces that the state-dependent volatility G(p)^2 and the AMM curvature V''(p) cancel so that -½V''G^2 = (β/2)V at every price; β acts as the uniform loss rate per unit informational time. The singular Sturm-Liouville formulation provides existence of the eigenpair via Hardy-type bounds and the ground-state alternative, and its homogeneity allows arbitrary liquidity scaling and the closed-form dynamic liquidity schedules.

What would settle it

Take a win-martingale with jumps, e.g., a compensated Poisson-driven belief process converging to 0 or 1, and compute the realized ratio LVR/V along a simulated path; if the ratio varies with price despite a pool-value function solving the continuity-based BVP for the diffusion part, the no-jump assumption is falsified. Alternatively, on real high-frequency prediction-market data, a uniform AMM fitted from the continuous model should show a flat LVR/V across price buckets; systematic elevation near p≈0 or p≈1 would indicate missing jump risk.

Watch

Extended reading notes

Core claim

The central result is the uniformity BVP, G(p)^2 V''(p)+βV(p)=0 on (0,1) with V(0)=V(1)=0, V>0, viewed as a singular Sturm-Liouville eigenpair. Theorem 11 constructs a concave unimodal V for any separable win-martingale with volatility profile G and information clock h(t) satisfying Assumption 3; Theorem 13 constructs G from V by G^2 = -βV/V''. Theorem 14 then shows that with a uniform V fixed, scaling liquidity by the schedule L_t ∝ h(t)^2 D'(t) exp(∫ β/(2h^2)) implements the target expected loss D(t). Examples: Wright-Fisher G=√(p(1-p)) yields constant-elasticity pool value p(1-p); logistic G=p(1-p) yields the constant product market maker; Gaussian score dynamics yield V=φ(Φ^{-1}(p)); the

Load-bearing premise

The fair price is assumed to move continuously via an Itô diffusion with no jumps; if real belief updates arrive as discrete jumps, the LVR decomposition and the uniformity BVP would need a different framework, and the stated correspondence would not hold as written.

Editorial extensions

If this is right

For any separable win-martingale in the class, a concave pool-value function exists that makes instantaneous LVR a constant fraction of pool value across all price states.
Conversely, every sufficiently regular concave pool-value function embeds a volatility profile under which it is uniform — AMM geometry encodes belief dynamics.
With a uniform value function, a deterministic liquidity schedule implements any target expected cumulative loss schedule in closed form, so designers can front-load or back-load losses without changing the statewise distribution.
Under uniform LVR, the subsidizer's loss rate per dollar deployed is the same in every price state, removing the incentive for sophisticated LPs to time entry and exit.
Uniform LVR is the minimax design: among all admissible pool-value functions it minimizes the worst-case relative loss rate across prices.

Reading between the lines

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

My inference: the same BVP machinery extends naturally to non-uniform objectives by replacing the constant β with a price-dependent weight, which the paper gestures at; this would let a designer concentrate subsidy at, say, p=1/2.
My inference: empirical calibration of G and h from tick-level prediction-market data could test which canonical profile (logistic, Wright-Fisher, Gaussian score) real markets approximate; the uniform AMM would then be directly implementable.
My inference: in multi-outcome or combinatorial markets the scalar BVP would generalize to an elliptic eigenvalue problem on the simplex, with Hardy weights on the boundary; the 'uniform loss' criterion would become a normal-flux matching condition.

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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

Here's what you should know: the core construction — every separable win-martingale in their class admits a pool-value function making LVR uniform, and conversely any sufficiently regular pool-value function induces a compatible win-martingale — is real and well argued. Theorems 11 and 13 hold up on inspection. That is a genuine generalization of Moallemi and Robinson's Gaussian-only result, and the dynamic liquidity schedule in Theorem 14 is a clean, useful contribution. The examples (Wright-Fisher/sqrt invariant, logistic/CPMM, Gaussian/normal density) are informative, and the paper is honest about its scope.

But the minimax optimality claim, Theorem 10, is false as stated. The stress-test example is correct: for G(p)=min(p,1-p), which satisfies Assumption 3, the uniformity BVP G(p)^2 V'' + beta V = 0 admits positive concave classical solutions for every beta in (0,1/4) (power-law near each endpoint, matched at 1/2). Each gives uniform relative LVR equal to beta, and the functions are in C([0,1]) ∩ C^2((0,1)) with V(0)=V(1)=0. So the achievable uniform rates include all beta below the Hardy critical value 1/4; the infimum is 0 and no minimizer exists. The proof uses a Rayleigh-quotient argument over H^1_0, which implicitly selects the principal eigenvalue, but the theorem statement does not restrict to that. The fix is easy: state Theorem 10 for the maximal eigenvalue (or for the specific beta constructed in Theorem 11) and handle the non-attainment case separately. As written, it is a wrong side theorem. The main correspondence and dynamic results do not rely on it.

The other flagged concerns are less serious. The jump-diffusion limitation is a scope choice, not an error; the authors are explicit about continuous Itô paths. The omitted 'most exciting game' invariant is an example-level gap — minor. The reliance on cited Sturm-Liouville criticality results means the existence proof is as strong as those citations, which is acceptable for a theory paper, but it makes full independent verification more work. No code or data, but the theoretical contribution stands alone.

Bottom line: this is a serious paper that deserves a referee. I would not desk-reject it. But I would require the authors to fix Theorem 10 before acceptance — it is an embarrassing counterexample to a stated theorem, even if the paper's central claims survive. Anyone working on prediction-market mechanism design or AMM theory should read it, and cite it once Theorem 10 is corrected.

Referee Report

1 major / 5 minor

Summary. The paper introduces uniform automated market makers for binary prediction markets, defined by the requirement that instantaneous loss-versus-rebalancing (LVR) is proportional to pool value and independent of price, after normalizing by an information clock. The central object is the uniformity boundary value problem G(p)^2 V''(p) + β V(p) = 0 with V(0)=V(1)=0, V>0. For separable win-martingales satisfying Assumption 3, the paper claims a bidirectional correspondence: Theorem 11 constructs a concave pool-value function from a volatility profile, and Theorem 13 constructs a win-martingale from a concave pool-value function. Theorem 14 extends the framework to dynamic liquidity, giving an explicit deterministic liquidity schedule that implements any prescribed target expected cumulative loss schedule. The paper also claims a minimax optimality property for uniform LVR (Theorem 10) and illustrates the constructions on canonical volatility profiles and AMMs.

Significance. If the main existence and correspondence results hold, this is a valuable framework: it gives a constructive link between belief-process dynamics and AMM geometry, yields explicit closed-form liquidity schedules for dynamic loss targeting, and recovers canonical mechanisms such as CPMM and LMSR as uniform for specific win-martingales. The paper is generally transparent about its modeling scope (continuous Itô price processes, risk-neutral zero-rate setting) and it explicitly acknowledges non-uniqueness of admissible eigenpairs. The dynamic implementation result is a genuinely useful design tool. However, the claimed minimax optimality is false as stated, and because this claim is used in the introduction and Section 4 as a normative justification for uniform LVR, the paper requires substantive revision before it can be accepted.

major comments (1)

[Section 4, Theorem 10 / Appendix B.6] The minimax claim is false as stated. For G(p)=min(p,1-p), which satisfies Assumption 3, every β∈(0,1/4) admits a positive solution Vβ of G²V''+βV=0 with V(0)=V(1)=0. With r±=(1±√(1−4β))/2, take Vβ=A p^{r+} on [0,1/2] and match to C(1−p)^{r+}+D(1−p)^{r−} on [1/2,1]; the matching constants give a positive concave C² interior function whose relative-LVR ratio is identically β. Since β can be arbitrarily small, the theorem's lower bound fails. The proof's Rayleigh-quotient integration by parts is over H^1_0, but Vβ∉H^1_0 (near 1, V∼D(1−p)^{r−}, r−<1/2, so ∫(V')²=∞). Section 4.1 itself acknowledges a family of admissible eigenpairs. The theorem must specify the principal eigenpair and restrict the admissible class, or the minimax claim should be withdrawn.

minor comments (5)

[Section 2.2 / Section 4.2] The absorbed Brownian motion profile G(p)=1_{0<p<1} does not satisfy Assumption 3(i), which requires continuity on [0,1] with G(0)=G(1)=0. The paper calls this example 'formal,' but since it is used to derive an explicit invariant in §4.2, the mismatch should be acknowledged explicitly or the example should be presented as a limiting case.
[Section 2.3] The statement that the square-root clock h(t)=√(T−t) has 'remaining quadratic variation of the process is (T−t)' is misleading for the price process: by Proposition 4(ii) the conditional remaining quadratic variation equals P_t(1−P_t), which is stochastic. The claim is only true for the latent driving Brownian motion or for a specific underlying process, and should be clarified.
[Section 4.2] For the absorbed Brownian motion invariant, the text says 'After a series of substitutions, we can show' and gives F(x,y;L). The derivation is entirely omitted. Since this is a claimed explicit construction, a sketch or a reference to an appendix is needed.
[Section 4.4] The phrase 'The ratio −V(p)/V''(p) yields the dynamics' omits the role of β. For the CPMM the displayed dynamics dP_t=P_t(1−P_t)/h(t)dW_t is correct only after fixing β=1/4; for the LMSR the displayed dynamics uses β=1. Please state the chosen β explicitly in these examples.
[Throughout] There are several typos and small notational inconsistencies: 'ahve' (§2.3), 'fiar' (§4.1), 'It¯o' (§3.2), 'is has' / 'has is' (§4.4), 'loses' for 'losses' (§5). Also, LVR_t is defined as an instantaneous rate in Proposition 8 but used as both a rate and an increment in the proof of Theorem 14; the notation dLVR_t would remove the ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the uniformity BVP is a definitional reformulation, and the main existence/converse theorems are constructive rather than fitted; examples are validated against independent AMM benchmarks.

full rationale

The central derivation chain is self-contained. The uniformity condition LVR_t/V(P_t) constant in price is translated exactly into the BVP G(p)^2 V''(p)+βV(p)=0; this is a definitional reformulation of the stated design goal, not a hidden input. Theorem 11 proves existence of a positive concave solution by an external singular Sturm-Liouville/Hardy criticality argument (Lemma 16, citing Zettl, Pinchover, Devyver et al.), not by assuming the conclusion. Theorem 13 constructs G from V and then verifies the induced process is a genuine win-martingale via Proposition 4; the BVP is satisfied by the defining choice G^2=βV/(-V''), but the theorem proves that the constructed diffusion has the binary-resolution martingale property, so the equivalence is contentful rather than tautological. The dynamic result (Theorem 14) chooses a liquidity schedule as a control to match an externally prescribed loss schedule; showing that expected wealth solves the target ODE is an implementation theorem, not a fitted parameter renamed as a prediction. The examples are obtained by direct eigenfunction computation and are matched to known CPMM and LMSR formulas, which are independent benchmarks outside the paper's definitions. The only self-citation (Moallemi and Robinson 2024) appears as background and is rederived in Appendix A; it is not load-bearing. The paper explicitly flags non-uniqueness of admissible eigenpairs and the limited regularity of the reconstructed invariant (Prop. 7 remark), which are honest limitations rather than circular moves. A possible issue with Theorem 10's endpoint regularity or functional-analytic assumptions is a correctness concern, not a circularity, and therefore does not affect this score.

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

No data are fitted; β is the principal eigenvalue determined by G and V, and L, w0, h, D are designer-supplied inputs rather than fitted parameters. No new physical or economic entities are postulated; 'uniform AMM' is a property, not an entity.

assumptions (5)

domain assumption Risk-neutral zero-interest pricing: the YES-token fair price is a martingale, P_t = E[P_T | F_t].
Invoked in Section 2 before Definition 1; if real prices carry risk premia, the win-martingale model is not the fair-price process.
domain assumption The fair price is a continuous Itô diffusion with no drift and no jumps, dP_t = σ(P_t,t) dW_t.
Section 2; the entire LVR/Itô decomposition (Prop. 8) and uniformity BVP depend on continuity and quadratic variation.
domain assumption Separability of volatility: dP_t = G(P_t)/h(t) dW_t with G,h satisfying Assumption 3.
Definition 2 and Assumption 3; the examples and theorems are restricted to separable win-martingales.
domain assumption Arbitrageurs costlessly and continuously rebalance the pool to the fair price.
Used in Section 3.2 and Proposition 8; without this, LVR as defined is not the realized loss.
standard math Criticality/ground-state alternative for singular Sturm-Liouville operators (Hardy inequality, Pinchover 1990, Devyver et al. 2014).
Lemma 16/Theorem 11 rely on these external results for existence and endpoint behavior of the eigenfunction.

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Cite this review

Pith. "Pith review of Uniform-Loss Automated Market Making for Prediction Markets." pith.science (2026). https://pith.science/paper/TXXZPNPK

@misc{pith2026260717428,
  author       = {Pith},
  title        = {Pith review of: Uniform-Loss Automated Market Making for Prediction Markets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TXXZPNPK}},
  note         = {Machine review of arXiv:2607.17428}
}

read the original abstract

Automated market makers (AMMs) for prediction markets descend from market scoring rules, where a mechanism operator subsidizes a market to aggregate beliefs about uncertain events. The existing literature has focused on bounding the total worst-case loss to the subsidizer, but has not addressed how that loss is distributed across price states or over time. We use the framework of loss-versus-rebalancing (LVR) to study this distribution and introduce \textit{uniform AMMs}, defined by the property that instantaneous LVR is proportional to pool value and independent of the current token price. In a static setting, we show that for a broad class of \textit{win-martingales} -- processes that converge to 0 or 1 at a fixed resolution time -- there exists a pricing function that achieves uniform LVR under that process, and conversely, that any sufficiently regular pricing function induces a win-martingale under which it is uniform. We then extend the framework to dynamic liquidity management, showing that liquidity levels can be adjusted over time to implement a prescribed target expected cumulative loss schedule. This theory is illustrated with canonical examples of win-martingales and pricing functions. Our results can inform AMM designers and liquidity providers on how the inevitable cost of subsidizing price discovery can be shaped and controlled across both price and time.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

30 extracted references · 1 linked inside Pith

[1]

Journal of Prediction Markets , volume =

Hanson, Robin , title =. Journal of Prediction Markets , volume =. 2007 , publisher =

2007
[2]

2024 , journal =

pm-AMM: A Uniform AMM for Prediction Markets , author =. 2024 , journal =

2024
[3]

Proceedings of the 25th ACM Conference on Economics and Computation , series =

Angeris, Guillermo and Chitra, Tarun and Diamandis, Theo and Kulkarni, Kshitij and Evans, Alex , title =. Proceedings of the 25th ACM Conference on Economics and Computation , series =. 2024 , publisher =

2024
[4]

Zettl, Anton , title =
[5]

Journal of Differential Equations , year =

Pinchover, Yehuda , title =. Journal of Differential Equations , year =
[6]

Journal of Functional Analysis , year =

Devyver, Baptiste and Fraas, Martin and Pinchover, Yehuda , title =. Journal of Functional Analysis , year =
[7]

, title =

Archak, Nikolay and Ipeirotis, Panagiotis G. , title =. Proceedings of the 10th ACM Conference on Electronic Commerce , series =. 2009 , doi =

2009
[8]

and Kurtz, Thomas G

Ethier, Stewart N. and Kurtz, Thomas G. , title =. 1986 , series =

1986

Show all 30 references

[9]

15th Innovations in Theoretical Computer Science Conference (ITCS 2024) , series =

Frongillo, Rafael and Papireddygari, Maneesha and Waggoner, Bo , title =. 15th Innovations in Theoretical Computer Science Conference (ITCS 2024) , series =. 2024 , publisher =

2024
[10]

2023 , journal =

The Most Exciting Game , author =. 2023 , journal =

2023
[11]

2022 , journal =

Automated Market Making and Loss-Versus-Rebalancing , author =. 2022 , journal =

2022
[12]

The Review of Financial Studies , year =

Capponi, Agostino and Jia, Ruizhe , title =. The Review of Financial Studies , year =
[13]

2012 , publisher =

Ordinary Differential Equations and Dynamical Systems , author =. 2012 , publisher =

2012
[14]

1991 , type =

Gantert, Nina , title =. 1991 , type =

1991
[15]

Journal of Economic Perspectives , year =

Wolfers, Justin and Zitzewitz, Eric , title =. Journal of Economic Perspectives , year =
[16]

2006 , doi =

Wolfers, Justin and Zitzewitz, Eric , title =. 2006 , doi =

2006
[17]

and Nelson, Forrest D

Berg, Joyce E. and Nelson, Forrest D. and Rietz, Thomas A. , title =. International Journal of Forecasting , year =
[18]

The Review of Economic Studies , volume =

Corporate Prediction Markets: Evidence from Google, Ford, and Firm X , author =. The Review of Economic Studies , volume =. 2015 , doi =

2015
[19]

The Journal of Prediction Markets , volume =

Prediction Markets: An Extended Literature Review , author =. The Journal of Prediction Markets , volume =. 2007 , doi =

2007
[20]

Journal of Financial Markets , volume =

Bossaerts, Frederik and Yadav, Nitin and Bossaerts, Peter and Nash, Chad and Todd, Torquil and Rudolf, Torsten and Hutchins, Rowena and Ponsonby, Anne-Louise and Mattingly, Karl , title =. Journal of Financial Markets , volume =. 2024 , doi =

2024
[21]

and Debreu, G

Arrow, Kenneth J. and Debreu, G. Existence of an Equilibrium for a Competitive Economy , journal =. 1954 , publisher =. doi:10.2307/1907353 , note =

1954 doi
[22]

2021 , month =

Uniswap v3 Core , author =. 2021 , month =

2021
[23]

Information Systems Frontiers , volume =

Hanson, Robin , title =. Information Systems Frontiers , volume =. 2003 , month =. doi:10.1023/A:1022058209073 , url =

2003 doi
[24]

1953 , publisher =

Stochastic Processes , author =. 1953 , publisher =

1953
[25]

and Hughston, Lane P

Brody, Dorje C. and Hughston, Lane P. and Macrina, Andrea , title =. International Journal of Theoretical and Applied Finance , volume =
[26]

arXiv preprint arXiv:2404.19672 , year =

Specific Wasserstein Divergence Between Continuous Martingales , author =. arXiv preprint arXiv:2404.19672 , year =

arXiv
[27]

Exciting Games and Monge-Amp

Backhoff, Julio and Wang, Zhizhang and Zhang, Xin , journal =. Exciting Games and Monge-Amp. 2024 , note =

2024
[28]

, title =

Chen, Yiling and Pennock, David M. , title =. Proceedings of the 23rd Conference on Uncertainty in Artificial Intelligence , series =. 2007 , publisher =

2007
[29]

2025 , eprint =

Dalen, Shaw , title =. 2025 , eprint =

2025
[30]

Michael and Kreps, David M

Harrison, J. Michael and Kreps, David M. , title =. Journal of Economic Theory , volume =. 1979 , doi =

1979

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Journal of Prediction Markets , volume =

Hanson, Robin , title =. Journal of Prediction Markets , volume =. 2007 , publisher =

2007

[2] [2]

2024 , journal =

pm-AMM: A Uniform AMM for Prediction Markets , author =. 2024 , journal =

2024

[3] [3]

Proceedings of the 25th ACM Conference on Economics and Computation , series =

Angeris, Guillermo and Chitra, Tarun and Diamandis, Theo and Kulkarni, Kshitij and Evans, Alex , title =. Proceedings of the 25th ACM Conference on Economics and Computation , series =. 2024 , publisher =

2024

[4] [4]

Zettl, Anton , title =

[5] [5]

Journal of Differential Equations , year =

Pinchover, Yehuda , title =. Journal of Differential Equations , year =

[6] [6]

Journal of Functional Analysis , year =

Devyver, Baptiste and Fraas, Martin and Pinchover, Yehuda , title =. Journal of Functional Analysis , year =

[7] [7]

, title =

Archak, Nikolay and Ipeirotis, Panagiotis G. , title =. Proceedings of the 10th ACM Conference on Electronic Commerce , series =. 2009 , doi =

2009

[8] [8]

and Kurtz, Thomas G

Ethier, Stewart N. and Kurtz, Thomas G. , title =. 1986 , series =

1986