Title: Smooth Quadratic Prediction Markets
Venue: NeurIPS 2025
Accepted: yes
Keywords: Prediction Markets
Abstract: When agents trade in a Duality-based Cost Function prediction market, they collectively implement the learning algorithm Follow-The-Regularized-Leader [Abernethy et al., 2013]. We ask whether other learning algorithms could be used to inspire the design of prediction markets. By decomposing and modifying the Duality-based Cost Function Market Maker's (DCFMM) pricing mechanism, we propose a new prediction market, called the Smooth Quadratic Prediction Market, the incentivizes agents to collectively implement general steepest gradient descent. Relative to the DCFMM, the Smooth Quadratic Prediction Market has a better worst-case monetary loss for AD securities while preserving axiom guarantees such as the existence of instantaneous price, information incorporation, expressiveness, no arbitrage, and a form of incentive compatibility.  To motivate the application of the Smooth Quadratic Prediction Market, we independently examine agents' trading behavior under two realistic constraints: bounded budgets and buy-only securities.  Finally, we provide an introductory analysis of an approach to facilitate adaptive liquidity using the Smooth Quadratic Prediction Market. Our results suggest future designs where the price update rule is separate from the fee structure, yet guarantees are preserved.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\n- The authors propose a novel prediction market model by replacing the fee of the traditional Duality-based Cost Function Market Maker (DCFMM) model (that is, the Bregman divergence between two consecutive states of the market) with a fee that is quadratic in the norm of the requested securities bundle *(Def. 5)*.\n- The proposed novel model facilitates higher profits for the market maker, since the novel payment function upper bounds the traditional one *(Th. 4)*.\n- The proposed model satisfies several of the desiderata for a prediction market, except incentive compatibility (which means that, given a market state and a trader's belief, the bundle requested by the trader in order to maximise the expected return shifts the market state to match the trader's belief). However, by showing that the update of the market state follows a gradient descent dynamic, it turns out that the novel model satisfies an *incremental* version of incentive compatibility (which means that an infinite sequence of return-maximiser trades with the same belief will shift the market state to asymptotically match the belief) *(Th. 8)*.\n- Finally, incremental incentive compatibility (the convergence of the market state to the belief of a sequence of traders) is tested in two constrained scenarios (buy-only markets and markets with budget).\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n- introduction and preliminaries are fairly clear and concise even for a reader not familiar with the literature\n- the work is technically sound and supporting lemmas for each result are clearly stated, making the proofs self-contained and easier to follow\n- the work is well written and organised into preliminaries, new model, its properties and limitations, experiments\n\n### Weaknesses\n- I am not familiar with the literature on prediction markets, but the generalisation over the traditional DCFMM model seems fairly mild, possibly hindering the impact and originality of the work\n\nQUESTIONS:\n### Major\n- Incentive compatibility and its incremental version would benefit from a more extensive discussion and comparison. In particular, under incremental IC, the market state matches the traders' belief only asymptotically. What is the consequence of this in the short-run? What if traders with different beliefs trade one after the other? Can this lead to oscillatory market states?\n- Beyond the definition of L-smoothness which allows for the bound of Theorem 4, what is the meaning / intuition of replacing the DCFMM Bregman fee with the quadratic one of Definition 5? Would other notions of distance be suitable - for example, the Bregman divergence of the strongly convex conjugate of $C$ between $C(q_t)$ and  $C(q_{t+1})$? \n-  Can the bound of Theorem 4 be made tighter? How much better is the worst-case loss of SQPM over DCFMM?\n- Given the relation between  DCFMM and FTRL, the relation between Smooth Quadratic Prediction Market and gradient descent upon the replacement of a Bregman term with a quadratic one is somewhat expected. What is the precise interplay between these elements? \n\n### Minor\n- Section 2.3 would benefit from a more explicit statement along the lines \"If $\\hat{C}$ (some property) then $C = \\hat{C}^{\\ast}$ is CIIP\"\n- L. 171: regrate is a typo\n- Caption of Figure 1: the tiles in the plots do not seem to convey any information about the different norms\n- Appendix F discusses Buy-Only Markets, but not Budget-Bounded Traders\n- Line 411: $Q$ is the gradient of the convex conjugate of $R$, not the convex conjugate of $R$. Similarly, there is something wrong with the **Given** line of **Protocol 1** in Appendix D\n\nLIMITATIONS:\n- the authors acknowledge that their model does not fulfil one of the standard desirable axioms for prediction markets, and propose a sound alternative axiom which their model do fulfil\n- an important  limitation might be that of agents with varying beliefs. It is not clear what happens of incremental incentive compatibility in this case. The authors acknowledge this limitation and propose it as a future direction.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the prediction market design problem. The authors first introduced the classical Duality-based Cost Function Market Maker (DCFMM) and its connection to Follow-The-Regularized-Leader (FTRL). By decomposing and modifying the DCFMM's pricing mechanism, this paper introduces the Smooth Quadratic Prediction Market, a novel prediction market design inspired by general steepest gradient descent. Specifically for Arrow-Debreu (AD) securities, the authors show Smooth Quadratic Prediction Market offers a better worst-case loss while preserving key axiomatic guarantees. The paper further explores agent behavior under realistic constraints (bounded budgets and buy-only securities) and provides an introductory analysis of adaptive liquidity.\n\nSTRENGTHS AND WEAKNESSES:\nOn the positive side, this paper considers a different approach for prediction market design, exploring a new direction for prediction market design beyond the well-established DCFMM framework. I found the problem presented by the paper interesting and relevant. In general, this paper proposed an innovative research topic.\n\nOn the other hand, while the paper acknowledges that the Smooth Quadratic Prediction Market does not satisfy the strong axiom of incentive compatibility, instead relying on an \"incremental incentive compatibility,\" the implications of this weaker form for real-world market behavior and equilibrium outcomes could be discussed in more detail. This might be a critical point for practical adoption. For example, the definition requires “a sequence of agents with the same belief”; is it reasonable to assume that there exists a sequence of agents that will have exactly the same belief in practice?\n\nWhile the paper provides an analytical proof (Theorem 4) that the Smooth Quadratic Prediction Market has a better worst-case loss than the DCFMM, *empirical validation* would significantly strengthen this claim. \n\nIn Section 5, the acknowledgment that proving other important properties (bounded-worst case loss, information incorporation, incentive compatibility) for the adaptive liquidity approach becomes challenging due to non-convexity is honest but highlights a significant limitation for this promising extension. The section feels somewhat incomplete due to these open problems.\n\nQUESTIONS:\nCould you elaborate on the practical implications of only satisfying \"incremental incentive compatibility\"? Are there scenarios where this weaker guarantee could lead to undesirable market behaviors?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

comments: SUMMARY:
The paper proposes Smooth Quadratic Prediction Markets (SQPM), a variant of Duality-based Cost Function Market Maker (DCFMM) for Arrow-Debereu securities. By replacing the DCFMM’s Bregman “curvature fee” with a simple quadratic fee derived from an L-smooth upper bound, the SQPM retains axioms of automated market makers such as instantaneous price, information incorporation and no arbitrage. Moreover, their SQPM incentivizes traders to implement general steepest descent paths toward their beliefs, satisfying incremental incentive compatibility. Lastly, they suggest an extension to adaptive liquidity by allowing smoothness to decrease as trading volume grows.

STRENGTHS AND WEAKNESSES:
### Strenghts
- The paper seems to make novel conncection to optimization by extending the duality between FTRL to general L-smooth cost functions. 
- The paper is well written; the theorems are clearly stated and proved, to the degree that I can understand.

QUESTIONS:
- In the absence of one‐step incentive compatibility, could a trader exploit incremental steps (e.g., splitting a large trade)? 
- How should one choose the norm ∥·∥ and smoothness parameter L in real‐world settings?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 4
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies automated market makers for Arrow–Debreu securities.  \nThe market-maker maintains a state vector $q_t\\in\\mathbb{R}^d$ recording the total shares held by traders.  \nWhen a trader requests a bundle $r_t\\in\\mathbb{R}^d$ (positive to buy, negative to sell), the market maker charges $\\textrm{pay}(q_t,r_t)$ and updates the state to $q_{t+1}:=q_t+r_t$. After trading ends a single outcome $Y\\in[d]$ is revealed and, for any $t$, the $t$-th trader earns $r_t(Y)$.\nIn the classical dual-cost function market maker (DCFMM) one assumes a convex, increasing, 1-invariant, probability-mapping  (CIIP) potential $C:\\mathbb{R}^d\\to\\mathbb{R}$ that is the Fenchel conjugate of some strictly convex and continuous $\\hat{C}$ defined in the relative interior of the probability simplex $\\Delta_{d}$ and sets  $\\text{pay}(q,r):=C(q+r)-C(q)$.  \nThis choice satisfies axioms 1–5 (price existence, information incorporation, no-arbitrage, expressiveness, one-shot incentive compatibility) and gives a finite worst-case loss.\nThe paper adds the assumption that $C$ is $L$-smooth, so that $C(q+r)\\le C(q)+\\langle\\nabla C(q),r\\rangle+\\tfrac{L}{2}\\|r\\|^{2}$ and proposes charging this upper bound instead, i.e., $\\textrm{pay}(q,r):=\\langle\\nabla C(q),r\\rangle+\\tfrac{L}{2}\\|r\\|^{2}.$\nThe new rule (SQPM) preserves axioms 1–4, lowers the market maker’s worst-case loss, and replaces Axiom 5 with a weaker Axiom 6: incremental incentive compatibility.  \nA risk-neutral trader’s best response is a steepest-descent step toward her belief, and repeated budget-limited trades converge to that belief at rate $O(1/t)$.  \nThe authors also discuss budget constraints, buy-only settings, and a sketch for adaptive liquidity.\n\nSTRENGTHS AND WEAKNESSES:\nThe upper bound idea is nice, the paper seems well-written, and the statements well-supported by theorems and proofs.\n\nQUESTIONS:\nTypos:\n- line 67 (missing f)\n- line 101 (instsantaneous)\n- line 127 (missing full stop)\n- line 171 (regrate)\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 1, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}