Smooth Quadratic Prediction Markets
Authors: Enrique Nueve
Bo Waggoner
Venue: NeurIPS 2025 poster ()
Keywords: Prediction Markets
Primary area: theory
TL;DR: We propose a new prediction market framework with close ties to general steepest descent.

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.

Smooth Quadratic Prediction Markets
Enrique Nueve Department of Computer Science University of Colorado Boulder edu
Bo Waggoner Department of Computer Science University of Colorado Boulder bwage wooorddo........ edu
enrique. www.e
Abstract
When agents trade in a Duality-based Cost Function prediction market, they collec- tively implement the learning algorithm Seeoo-Th--Reglariieee nethy 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 better worst-case monetary loss for AD securities while preserving axiom guarantees such as the existence of instantaneous price, information incorporation, expressiveness, 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.
[Aber-
Introduction
What are Prediction Markets? Prediction markets allow traders to buy and sell securities whose payoffs depend on the realization future events [Hanson, 2003]. Prediction market platforms such Kalshi and Polymarket average work, we examine prediction markets that trade Arrow-Debreu (AD) securities, which pay $1 particular state of the world is reached and SO otherwise [Arrow, 1964]. Given a finite random variable y = fyn,...,Wi yd over mutually exclusive and exhaustive outcomes, AD securities are designed to elicit full probability distribution with respect to V, reflected by the current prices of the securities.
millions of dollars per event-based market. In this
-based CFMM The work of Abernethy et al. [2013] introduced general framework for designing automated prediction markets over combinatorial or infinite state outcome spaces named the bbsdd Cost Function Market Maker (DCFMM), As stated by Abermethy et al. [2013], "An automated market maker a market institution that adaptively sets prices for each security and is always willing accept trades these prices." " The DCFMM satisfies the desirable market axioms of no arbitrage, bounded worst-case loss, information incorporation, expressiveness, incentive compatibility, and efficient computability. The DCFMM over the last decade has been examined in numerous works such as Abernethy et al. [2014], Devanur et al. [2015]. Frongillo and Waggoner [2017]. Frongillo et al. [2023]. A core observation of Abernethy et al. [2013] that the DCFMM uses Follow- Thee--hheeeaa Rethe-Rgggaaiizd-LL.e. as loss vectors. Although not our main contribution, as has been hinted the literature before by Devanur et al. [2015], to our knowledge, we are the first to formally show (in Appendix C) a tight
(FTRL) to set prices for securities, by interpreting past trades
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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equivalence between Continuous FTRL and DCFMM in terms of regret and worst monetary loss of the mechanism. Inspired by this relation, we ask other machine learning algorithms could be used to design new prediction market with at least the same or better properties than the DCFMM. This work proposes new market maker payment mechanism called the Smooth Quadratic Prediction Market, which is analogous to the learning algorithm general steepest descent in terms of trader's s incentives.
Smooth Quadratic Prediction Markets We propose the Smooth Quadratic Prediction Market, which increases profitability for the market maker relative the DCFMM but still has the performance guarantees of the DCFMM, such as no arbitrage, bounded worst-case loss, information incorporation, expressiveness, computational efficiency, and form of incentive compatibility. We show that the Smooth Quadratic Market can be interpreted as replacing a Bregman divergence term in the DCFMM payments by a simpler quadratic "fee". In the body of this work, we show that the Smooth Quadratic Prediction Market has many of the same axiom guarantees as that of the DCFMM, demonstrate how agents are incentivized follow general steepest descent when trading, examine how agents trade under either bounded budgets buy-only market, and provide approach to facilitate adaptive liquidity.
Background and Notation
Let us denote the all ones vector by (I....,1) 1) E Rd. Let us denote by the vector d; = L....... ................... 0). i.c. all-zero except for a one in the i-th position. Comparison between vectors is pointwise, e.g. > q' if q; > of for all = I....,d d and similarly for 2. We say q zे d' when qi 2 gf for all and y * q'. We denote ((u)+): = max(u;, 0). For a set Y C IRd, we denote by closure(Y) as the smallest closed set containing all of the limit points of Y. Define R2C be the nonnegative
orthant. Let As = {p E R4 ||l| = 1 be the set of probability distributions over outcomes, represented as vectors. Let relint( (aa) = A. \ {p E Aa/3 E [d], = 0). We let || || RR + R. denote general p-norm. Given norm || |]. we define it's dual norm ||y||= supl|=/|sı (z. v). Let f: _dd + (-o0, +oo| be a function. We define the Fenchel conjugate of f by /* R - [-00, such that (") sUPzedom(J) (y. [(x). We will use the following definitions for function f.
convex: 4 x, R AE 10.1). + (1 -)v) Sff(z) + +(1+++11---0110 A)f(v)
increasing: f(g) > (d') V g.9g E Rd with q ž2 d'
l-invariant: f(g + a1) [() + for ER E R.
probability mapping: f is twice-differentiable Vf Rd + As. and closure ((uure(([f()) (g) As.
q E [R)])
We shall refer to function which convex, increasing, 1-invariant, and probability mapping as CHIP.
2.1 Bregman Divergences, Smoothness, and Strong Convexity
This section presents core concepts used throughout this work. We emphasize that we define both L-smoothness and K-strong convexity through general norm |]. not just via the 2-norm, as is common in some literature.
Lemma 1 (Hiriart-Urrur and Lemaréchal [2004], Proposition 6.1.1). For convex differentiable function f Rd + 1-,0,,00) it holds that (V Vf(y), x v) 2 O. x, Rd.
Definition (Bregman divergence, L-smoothness, & K-strongly convex). For a differentiable function f Rd + -00, we define the Bregman divergence as Dy(x, = f(x) (f(v)
(v), ar y) If Rd (-00, + is convex und differentiable, we say that f is L-smooth w.E.f ||- if x, E Rdd it holds that Dy v) S 2 1| yll2 where I. 2 0. If .Rd - (-oo, + is convex and differentiable,
say that f is K-strongly conver w.t. (2(x, (2,)) y) where K 2 0.
if x, Rd it holds that
K | 2 <

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Conceptually, K-strongly convex acts quadratic lower bound of convex function, while L- smoothness serves as quadratic upper bound of convex function. Finally. we present a theorem demonstrating the dual relationship between strong convexity and smoothness.
Theorem 1 (Shalev-Shwartz [2009]). Assume that f is closed and convex function. Then f is K-strongly convex w.r.t. a norm || iff. I is * -smooth w.r.l. the dual norm 1|
2.2 Automated Market Makers for AD Securities
We first introduce the general framework for an automated market maker.
Definition 2 (Automated Market Maker for AD Securities). Say we have a finite random variable (vi,..., yd) over d mutually exclusive and exhaustive outcomes. An automated market maker for AD securities, with initial state qo R operates follows, At round E No.
1. A trader can request any bundle of securities E Rd.
2. The trader pays the market maker some amount Pay(ge, E R in cash.
3. The market state updates to qt+1 9t Tt-
After an outcome of the form Y === 1/4 occurs, for each round the trader responsible for the trade Te paid (rt); in cash, i.e. the number of shares purchased in outcome yc. The market payout for the bundle Tn and the outcome Y = y is expressed via (Tt, p(u)) where p y - Sy. At any state qt, the market maker can infer the belief of the market via the instsantaneous price InstP (gt))(((g))))) ( = f(g,
where f(g,,)) r2) V., Pay(q,, rt)-
Intuitively, the instantaneous price of security i is the amount that would be paid for an arbitrarily small amount of security We now define particular family of automated market makers, which captures the DCFMM and the Smooth Quadratic Prediction Market.
Definition 3 (Price-Plus- Plus-Fee Market). A Price-Plus-Fee Market is automated market maker of the following form. Let C Rd > be CHIP. Then
Pay(gr, Tt) = (VC(a)) + Feely,, Tr) = (pr, r2) + Fee(gr, rT)
where Feelg, o( [rel)). We note that in this case, InstPrice(q:) pt VC(gt).
Given that both markets we examine in this work are Price-Plus-Fee markets, we shall use VC(()
and InstPrice (G) interchangeably.
Generally, is common practice to expect prediction markets to satisfy some form of the following axioms.
Axiom (Existence of Instantaneous Price): C is continuous and differentiable everywhere Rdd
Axiom 2. (Information Incorporation); for any qr Rd. Pay(q + r,r) 2 Pay(g,r).
Axiom 3. (No Arbitrage) For Price-Plus Fee market, no arbitrage defined such that for any given trade sequence = .o. E (IR)) and initial state go E R. it holds that
min min vEy
(yn() go +
Σ rd) 1=0
Paylge rt).
1=0
For the case where = -6, equality must hold in the previous inequality.
Axiom 4. (Expressivennsss): For any p Da and 0, 3 E Rd such that ||Insprice(g)) pll
Axiom 5. (Incentive Compatibility): Assume that the market is at state qu and that the agent has
belief E Ag. To maximize expected return
arg max PLERd
(1,TT))
re)
Expected Payout
Payment Market
the agent will purchase a bundle Tr such that for q++1 ge + Tt it holds that InstPrice p.
3

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Intuition of Axioms Axiom states that any market state can mapped to a distribution. Axiom states that if trader were to purchase the same bundle twice, the price would be larger the second time. Axiom 3 states that for any bundle purchase there exists an outcome where the trader losses money, Axiom 4 states that the state of the market can mapped a distribution that expresses belief arbitrarily close. Axiom 5 states that trader desires to move the market to a state such that VC maps the market state to their belief of the distribution y. See Abernethy et al. [2013] for further discussion regarding the intuition of axioms.
2.3 CIIP Construction
may seem unclear how one could construct a C. which is CIIP. We later show that via a C being CHIP, Price-Plus-Fee Market is able automatically satisfy many of our desired axioms In this subsection we provide supporting lemmas for constructing CII functions and provide some examples.
Lemma 2 (Abernethy et al. [2013], Theorem 4.2, Lemma 4.3). Let C : Rd + IR be defined over relint( Aa). C be strictly convex over its domain, and define C = Ó As C is strictly convex, (C(g) = arg paxpeedm(C)) (q.p) p) - Cp)) Furthermore, assuming dom(C) is restricted to either relint( Aa) or Ad, we also have closure (losur((VVC(g)) 1 (R4)) = Aa-
Note, if C: Rd + was - -strongly convex w.r.t. || then C' would be L-smooth w.r.t. || 1/1 via Theorem 1.
C which is one-invariant allows for InstPrice(ge) = InstPrice(ge+1) where (++1 = + al such that a R. This is desirable property since the purchase of qr+1 = 9t + al incorporates information uniformly.
Lemma 3. W.r.t. a function f restricted to the Aa (or relint(Aa)) the Fenchel conjugate f" is one-invariant.
Proof. Let q E IRd and R. Observe the following f' al) suppe al, f(p) suppes. (q.P) f(p) + a(1,p) f() +
We proceed to give examples of Ò' and the corresponding C' which satisfies CIIP. For the case of C(p) = L- >i= pt log the derived C is softmax.
Softmax Let C(p) L- - pt logps Then C(g) supperelim(AA)) (g.p) O(p) c/a "E:- coo where
L-1 log(Ei= e. Furthermore, VC(g) arg maxperelim( (1- p) ((p)
C(q) is L-smooth w.r.t. || ||2 and ||. ll Also, it holds that (VC(g) E RRd] = int(Aa) and thus closure((VC(g)) g E [Rd])= = Ag.
For the case of C(p) 4 lpl.. the derived C is referred in the machine learning literature as sparsemax [Martins and Astudillo, 2016, Niculae and Blondel, 2017].
lellss for L > 0. Then C(q) suppe (g.P) C(p) 2 || Furthermore, VC(g) = arg maxpes, (q.p) C(p) = arg min A. l|g 2.12 where C(g) is 1 smooth w.r.t ||- l12. Also, it holds that (VC((g) | q E Rd- = As and thus closure (VC()) g E RR) Aa.
Sparsemax Let ((p)
A p) - 1
2.4 based CFMM
We now define the DCFMM which provides a construction scheme for a cost function C' respectively and particular Pay(-.-) scheme.
Definition 4 (Abernethy et al. [2013]. DCFMM for AD Securities). Let C. IRd - R be CIIP such that C= E where Ò strictly convex and continuous over all of relint( (Aa). At state qs for bundle To define the payment of a DCFMM by Pay Payp(g,,7 TT) Clg. + rt) C(gt).

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With respect to the Price-Plus- Plus-Fee Market (Definition 3), one could think of the DCFMM market payment linear term plus an implicit fee based on the Bregman divergence by
Payn (gr, rr) C(g++) C(q.) (pr, rx) + Do (e(e
Breg-
rx) and at+1 9: + T₁. Intuitively, the DCFMM charges the instantancous price per share a bundle r, via (pc, and then charges a a "curvature fee" via the Bregman Fee to hedge the potential loss for large trades. The DCFMM satisfies Axioms 1-5, which we now show.
since Dc(9(+1.9) C(q(+1)
- C(q.)
Theorem (Abernethy et al. [2013], Theorem 3.2). The DCFMM satisfies Axioms 1-4.
Trivially, with Pay D plugged into Eq (1) and taking the gradient with respect Q++1 and setting it equal to zero, we can see that Axiom is also satisfied for the DCFMM. Furthermore, the DCFMM has the nice property that the worst-case loss of the market is bounded.
Theorem 3 3 (Abernethy al. [2013], Theorem 4.4). The DCFMM has a worst-case loss more than SuPrer()) C(p) minpee, (p)).
As noted by Abernethy al. [2013], FTRL's regrate rate and DCFMM's worst-case loss similar. We show in Appendix C the equivalence to Continuous FTRL's regret rate and DCFMM's worst-case loss.
3 Smooth Quadratic Prediction Markets
3.1 Smooth Quadratic Prediction Market Design
We now introduce the Smooth Quadratic Prediction Market. Given a smooth CII function C' w.r.t. a general norm || Il. propose charging fee based the upper quadratic bound obtained via the L-smoothness condition.
Definition 5 (Smooth Quadratic Prediction Market), Let Idd + R. CHIP. Assume that C is L-smooth w.e.. || ||. At state q: for bundle Tı, we define the payment of Smooth Quadratic Prediction Market by
L 11
Payı(u, Ti) (pp,re) +
Q-Pee
In the remainder of this section, we show that the Smooth Quadratic Prediction Market satisfies Axioms 1-4 and has a better worst-case loss than the DCFMM. Note, we don't claim that the Smooth Quadratic Prediction Market satisfies Axiom 5 Incentive Compatibility. Later in Section 3.2 and 3.3, we show that traders within the Smooth Quadratic Prediction Market satisfy instead a form of incremental incentive compatibility, and interestingly, the traders mimic the update steps of gradient (general) steepest descent while trading.
Lemma By the assumption that C is CIIP, the Smooth Quadratic Prediction Market satisfies Axiom 1 Existence of Instantaneous Price.
Lemma 5. The Smooth Quadratic Prediction Market satisfies Axiom 2 Information Incorporation.
Proof. Since C' is convex and differentiable, by the monotonicity of gradients (Lemma 1), we have that
(VC(q++1) VC(g)). -q) 2 0 (VC(g + rr), 2 (VC(g).
where gr+1 = q: Tt. Then by adding 1 ||rrll2 2| to both sides we get
(VC() + +r), + ||rl (VC(g), + _ ||rel| + Payz + (.) 2 (gt,
Lemma 6. The Smooth Quadratic Prediction Market satisfies Axiom 3 No Arbitrage.
5

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Proof. By the definition of Pay D. and Pay y. we have that for any sequence $ = ....... 9i (including the empty sequence as both pay functions are equal zero) that
Σ Pay (46, rz)
Pay L (g) rz)
1=0
By (Theorem 3.2, Abernethy et al. [2013]) the DCFMM satisfies no arbitrage, we have that
min
(p())) m5)
m))
Payplairs)
yEy
1=0
A
Therefore by combing the two inequalities we have that the claim holds.
Lemma 7. The Smooth Quadratic Prediction Market satisfies Axiom 4 Expressiveness.
Proof. By the CHIP assumption on C, holds that closure every limit point of the set (VC(g) [R] in Ag. Hence, by definition of Expressiveness, the claim holds.
}) Aa which states
Theorem 4. For any fixed trade history = (ro...., Te), the Smooth Quadratic Prediction Market has better worst-case loss then the DCFMM.
Proof. Observe for any rs)) for E 0...,, that
C(a + +r)) C(a) (ps, m²) + De((1+1:0) S (p..rs) - ||rall
via Definition 1. Hence, the collected revenue of the the Smooth Quadratic Prediction Market is greater overall than that of the DCFMM implying a better worst-case loss then the DCFMM.
3.2 based Smooth Quadratic Prediction Markets
As mentioned, the core purpose of a prediction market with AD securities to elicit a distribution over finite set of outcomes. Hence, it is of essence to show that traders are incentivized to move the market state such that VC maps the market state to a trader's belief; however, recall that we stated the Smooth Quadratic Prediction Market does not satisfy the Axiom of Incentive Compatibility, Interestingly, the expectation maximizing trading behavior an agent w.r.t. the ly-based Smooth Quadratic Prediction Market is expressible via gradient descent. We show that the fa-based Smooth Quadratic Prediction Market satisfies a form of incremental incentive compatibility. Overall, the core result of this section, Theorem 6, states that trader is incentivized move the market state such that the market state maps to their belief via a sequence of bundle purchases instead via a single transaction. We formally define incremental incentive compatibility follows.
Axiom 6. (Incremental Incentive Compatibility)): Assume the market is at state qo and that a sequence of agents with the same belief μ Aa purchases bundles rt relative to maximizing their expected payout
arg 7+ERd
(1. Expected Poyout
Pay(ae re)
Payment Market
Then lim VC(g.) = p.
We now formally define gradient descent (GD) and provide a supporting Theorem and Lemma for GD which will be used in proving incremental incentive compatibility for the ba-based Smooth Quadratic Prediction Market.
Definition 6 (Gradient Descent). Let xo E Rd,, and let 7 > 0 be step size. Given differentiable function f. the gradient descent (GD) algorithm defines sequences (Ir)teNo satisfying Lt+1
Tt [xxx).
Theorem 5 (Garrigos and Gower [2023], Theorem 3.4). Assume that f is convex and ba-based L-smooth, for some L O. Let (eee be the sequence of iterates generated by the GD algorithm.
with stepsize satisfying 0 < 7 < 1
T. Then, for all x² E arg min f and for all t E Ny we have that
11x(0) x 11
f(x(1)) - inf <
2-

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Lemma 8 (Sidford [2024], Lemma 6.1.6). If Rd - R is L-smooth w.t.. || x² E arg min f and x E TR& it holds that
then for all
1
L 2 2 2--2111
2L S f(x')
Theorem 6. With respect to some CII C, define a E--based Smooth Quadratic Prediction Market. The market satisfies Axiom 6 Incremental Incentive Compatibility furthermore lim VC((g)) μ at rate of 7.
Proof. Let C(g) C(q) (M, g) and note that VC(()) (C( μ.. Note that the utility of an agent is equivalent the following
(VC(a)) 4+1 q:) + - 2 11 2rl13
min 2++1 ERd
(u, 4++1 Tt) Expected Payout
Payment Market
qt (L) is a GD step we get that VC(()) = nr, i.e. for q" E arg min C(q) holds that (g()") μs. Hence, 4++1 - q: (1) (VC()) - p)) = q: (C)))())) is GD update for C(q) = Cgg) - (n,, q) with learning rate + whose minimum point is a q" such that VC(g") = μ Also note that if (C)))) = μ then Qt+1 + ( (VC(() 14) 9: ( VC())) hence, any minimizer of C is stationary point. Observe that C is ba-based smooth and hence Theorem 5 and Lemma 8 apply.
Hence, the update of the market state 4+41 < @t 1000)-0()))) ()C())))
performed on C. Observe, setting VC
3.3
ba-da Smooth Quadratic Prediction Markets
We now generalize our results for the Smooth Quadratic Prediction Market defined w.r.t. || p E [o]. By altering the norm, the Smooth Quadratic Prediction Market gets varying behavior pertaining to the charge for Q-fee and path convergence of heN, + arg min C(q).. In this generalized setting, agents are incentivized to trade following the general steepest descent algorithm in order to maximize their expected return. We now formally define general steepest descent (SD) and provide supporting Theorem for SD which will be used in proving incremental incentive compatibility for the bp-based Smooth Quadratic Prediction Market.
|lp for
Definition (General Steepest Descent). Given a function f that is differentiable and L-smooth w.r.t. ||- I. the general steepest descent (SD) method w.r.t. the norm || iteratively defines sequences (ra)tENo via T++1 E arg min (V ((xe),, xx.) + ||0 xx.1?
Theorem (Kelner al. [2014], Theorem 1). If ʃ is convex and L-smooth w.r.f. || 1], then SD w.r.t. || satisfies f(xt) - inf f < 2LK2 (Ma))ENG is sequence of iterates generated by the SD algorithm.
t44 with K == faxx:f(s)Sf(tm) min )=int f || - x²1l where
Theorem 8. With respect to some CHI C, define an Ep bp-based Smooth Quadratic Prediction Market. The market satisfies Axiom 6 Incremental Incentive Compatibility furthermore lim VC(gg)) 4 at rate of :.
Proof. Observe that the utility of an agent equivalent to the following
L
arg min GR
((VC())) ae+1 - qe) +
2 -1 - -el1?)
(1, 4 - qo)
Expected Payout
Payment Market
(2)
L (VC(g), 4++1 q.) 2 |1uc+1 - 0el12
+ arg min Q++ERR
as 4++1 = 9: + rt. Hence, Eq. (2) a SD step for T = C(g) - (p, q). Observe that T is & based smooth and hence Theorem 7 and Lemma 8 apply.
In Figure we demonstrate the varying convergence behavior of the Smooth Quadratic Prediction Market when using different norms. There are numerous convergence results for gradient (general
7

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f2 Norm
I Norm
t. Norm
Asset 1
Asset 2
Asset 1
Asset 2
Asset 1
Asset 2
Asset 3
Asset
Asset 3
Figure I: Let qu = (10, 20, 10), C' is softmax with smoothness L = 1, and μ /1/6,166, 2/3). The blue square expresses f and the orange path towards the blue square demonstrates the updating market distribution states. As denote by the titles's s of each plot, we vary the norm used for the Smooth Quadratic Prediction Market. Note although softmax is not f-smooth, experimenallyy for the sake of comparison.
said norm
steepest) descent. To avoid redundancy with the literature, we refer the reader to Appendix D for further convergence results, which may be of interest to the application of Smooth Quadratic Prediction Markets.
4 Smooth Quadratic Prediction Markets with Constraints
In this section, examine the trading behavior of pnrectation-maximminn agents under realistic constraints (independently):
traders and buy-only markets. For mathematical sake, solely examine the ba-baeed Smooth Quadratic Prediction Market buy-only market and perform experiments for the budget-bounded trader setting. Furthermore, for both types of constraints, we perform simulations for the and &oo cases. Overall, either analytically observe that in both situations, the market state converges to the belief of sequence of traders, regardless of the used p-norm, motivating the use of the Smooth Quadratic Prediction Market in either of these realistic scenarios." We refer the reader to Appendix E for the omitted analysis within this section.
experimentally,
Budget-Bound Bounded Traders notation of natural budget constraint, which states that the loss of the agent is at most their budget, for all y y. The objective of an agent expressed by
Following the work of Fortnow and Sami [2012], we consider the
min reRd (VVC())) re) L 1mml1))
(p,re)
Expected Payout
Payment to Market
L ((VC(g)) re) 2 (1r.113)
(b
B
VyEY
Payment to Market
Pnyout
Budget
where Ry Experimentally, using VC equal to softmax, we were able to observe convergence regardless of the initial state, belicf distribution, and budget used. The rate of convergence was proportional to the size the budget. Figure in Appendix demonstrates the convergence behavior of the market under budget constraints.
The Buy-Only Market Li and Vaughan [2013] assumes that 0 A for all t No: hence, only positive bundles purchases are allowed. The objective of the trader is expressed
Buy-Only Market
"Code can be found at: http://ghhhoooonnnnneeeeeeeehhee
Markets

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via the equation below.
L (VC())) r.) + - (mrl11
min NER
(p.Ti) Expected Payout
Payment to Market
s.t.
Buy Only Constraint
From solving the Lagrange dual problem, we get that the update step is @++1 + {((VC(()
1+)+ - (V -4) and observe when (C():) = μ we have stationary point. Observe the update
a coordinate GD update for C when (VC(+:) - μ4); < 0 and gt+1,i = dr, when (VC(4)) n)x 2 0. Hence, not just experimentally but analytically, we are able to show incremental incentive compatibility.
Smooth Quadratic Prediction Markets with Adaptive Liquidity
Inspired by the paper A General Volume-Parameterio Market Making Framework [Abernethy et al., 2014], we adapt aspects of the framework to our setting to facilitate adaptive liquidity. Liquidity in this context can be thought of the sensitivity price changes with respect bundle purchases. high liquidity would indicate that large bundle purchase would not cause significant price change and vice-versa. Adaptive liquidity can be facilitated by decreasing the smoothness of C' as the volume of trades increases.
Let S (R2)* denote the history space of the market consisting of finite (and possibly empty) sequences of bundles. We define a volume update function via an asymmetric norm.
Definition 8 (VPM). Say we have a finite random variable y = ......., Va' over d mutuually exclusive and exhaustive outcomes. Let C Rdd x 1 - IR be CHP and increasing in v (it's second argument). Assume C L-smooth w.r. some general norm || 1. Furthermore, as increases, the smoothness of O should decrease. Let Rd + R. be asymmetric norm (refer to Def. 15 in Appendix F) and define w.r.t. a as- To...,Fe E the volume update function ((s) = Vu + g(T)) The VPM for AD securities defined by C with initial state qo. and initial volume th R operates as follows. At round t No.
1. A trader can request any bundle of securities E Rd.
2. The trader pays the market maker some amount Pay(g.rgg)) IR (such that Pay is dependent on C³) in cash.
3. The market state updates to 4++1 = + Tt and U++1 = V(s) such that = To.....
After an outcome of the form Y = Yt occurs, for each round the trader responsüble for the trade Tı is paid (rz); in cash, i.e. the number of shares purchased in outcome yi. The market payout for the bundle Tt and the outcome Y = is expressed via (Fe, p(y)) where y + ōy. At any state (ge, vi), the market maker can infer the belief of the market via the instsantaneous price InstPrice(g;;2) v2) f(i, 0; vx) where f(g., rV; v.) V,, rG; v1).
For the case of the VPM-DCFMM, the pay function is the following
Payp= (g,77))) (a r + g(re)) Cº (g: (g):v))

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The following inequality due to the smoothness of C motivates our proposed payment of Pay g
Pay D (qt, r2; v2) = Cat + reg + g(r:)) C0 (gt))))
C (qi + r + g(r))) - Ca (4:5 + a((r))) (C (C: uz + g(rr)()))))))>)))))) (o:
Payment Breg. Market
Liquidity Fee20
(VCC (gG + g(re)) + (n (1+ + r2, (.))
(C (C::: U2 + g(rr)) - C (1e: vx)
Payment Breg, Market
Liquidity Fee20
(VC) (gr + g(r)), r)) ( L
1mall9) + CO
(a (gei 12 + g(ri)) C (g:: 04)
Payment Smooth-Quad Market
Liquidity Fee20
Pay (gt, Tzi Ue).
Using this approach, we can easily show that Pay (96, facilitates no arbitrage (proof in Appendix F).
Lemma 9. Payzs satisfies Axiom No Arbilrage.
However, proving other important properties such as bounded-worst case loss, information incorpora- tion, and some form of incentive compatibility becomes challenging due to the non-convexity of the expected return
arg max
(p. Tt)
Pay (qt, Te:
Expected Payout
Payment to Market
We leave proving said results to future work. So, although this approach facilitates adaptive liquidity and no arbitrage, more work is required justify its use.
Conclusion
Recap In this work, proposed prediction market framework based on the traditional DCFMM framework. By doing so, the Smooth Quadratic Prediction Market satisfies many of the same axioms of the DCFMM while facilitating higher profits for the market maker. Although the Smooth Quadratic Prediction does not satisfy the axiom of incentive compatibility, we show that the axiom is satisfied in incremental sense and interestingly relate the methodology of general steepest descent with said behavior. We also examined the Smooth Quadratic Prediction Market under the constraints of budget-bounder traders and buy-only markets. Finally, we also presented introductory work on how the Smooth Quadratic Prediction Market could facilitate adaptive liquidity. Although this work provides some core insights into the properties of the Smooth Quadratic Prediction Market, there are many future directions for this work.
Future Directions One direction would be generalize this work beyond AD securities. The original work of Abernethy et al. [2013] demonstrates how the DCFMM can be used for combinatorial and infinite space outcome securities. Given the close relation between the design of the Smooth Quadratic Prediction Market and the DCFMM, we believe this generalization to more securities would be a feasible future direction. Another interesting future direction would be to analyze the convergence of the market when agents have varying beliefs. Using a random selection mechanism to select whose trades get processed, the analysis could be reduced to stochastic gradient descent. Another direction would be to further prove/disprove axiom guarantees regarding our proposed approach to adaptive liquidity. Finally, inspired by the shown equivalence of DCFMM and constant- function market makers (CFMMs) for asset exchanges by Frongillo et al. [2023], one application would be to use the Smooth Quadratic Prediction Market to run an asset exchange.
Broader Impacts: Our work informs the design of prediction markets, and thus, our work may influence the choices of prediction market makers and traders. Due to the inherent risk of losing money in prediction markets, we acknowledge the possibility of negative impacts due to this work.
10

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Acknowledgments and Disclosure of Funding
We thank the following people for their feedback throughout this project: Stephen Becker for talks on gradient descent, Maneesha Papireddygari and Rafael Frongillo for talks about liquidity provisioning. Drona Khurana for talks about convergence results, and Aaron Sidford for notes on general steepest descent.
References
Jacob Abernethy, Yiling Chen, and Jennifer Wortman Vaughan. Efficient market making via con- vex optimization, and connection online learning. ACM Transactions on Economics and Computation (TEAC), 1(2):1-39, 2013.
Jacob D Abernethy, Rafael M Frongillo. Xiaolong Li, and Jennifer Wortman Vaughan. A general Salume-parameterizes market making framework. Proceedings of the fifteenth ACM conference economics and computation, pages 413-430, 2014.
Andersen Ang. Convergence rate of gradient descent algorithm -smooth ... Jan 2025. URL https: / angms. wwwceeencc/eecccee doc/ http:/CVVVVCVVGVCCCC
pdf.
Kenneth J Arrow. The role of securities in the optimal allocation of risk-bearing, The review of economic studies, (1(2)911-9)) 1964.
Yun Kuen Cheung and Georgios Piliouras. Online optimization in games via control theory: Connect- ing regret, passivity and poincaré recurrence. In International Conference Machine Learning, pages 1855-1865. PMLR, 2021.
Nikhil Devanur, Miroslav Dudik, Zhiyi Huang, and David M Pennock. Budget constraints in prediction markets. 2015.
Lance Fortnow and Rahul Sami. Multi-outcome and multidimensional market scoring rules (manuscript). 2012.
Rafael Frongillo and Bo Waggoner. An axiomatic study of scoring rule markets. arXiv preprint UrXiv:1709.1006 2017.
Rafael Frongillo, Maneesha Papireddygari, and Bo Waggoner. An axiomatic characterization of cfmmss and equivalence to prediction markets. arXiv preprint urXiv:2302.00196 2023.
Guillaume Garrigos and Robert M Gower. Handbook of convergence theorems for (stochastic) gradient methods. arXiv preprint arXiv:2301.1 1235, 2023.
Robin Hanson. Combinatorial information market design. Information Systems Frontiers, 5:107-119, 2003.
Jean-Baptiste Hiriart-Urruty and Claude Lemaréchal. Fundamentals of convex analysis. Springer Science & Business Media, 2004.
Jonathan Kelner, Yin Tat Lee, Lorenzo Orecchia, and Aaron Sidford. An almost-linean algorithm for approximate max flow in undirected graphs, and its multicommodity generalizations. In Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms, pages 217-226. SIAM, 2014.
Joon Kwon and Panayotis Mertikopoulos. A continuous-time approach to online optimization. Journal of Dynamics and Games, 4(2):125-148, 2017.
Xiaolong Li and Jennifer Wortman Vaughan. An axiomatic characterization of adaptive-liquidity market makers. Proceedings of the fourteenth ACM Conference on Electronic Commerce, pages 657-674, 2013.
Andre Martins and Ramon Astudillo. From softmax to sparsemax: A sparse model of attention and multi-label classification. In International conference on machine learning, pages 1614-1623. PMLR, 2016.
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Panayotis Mertikopoulos. Christos Papadimitriou, and Georgios Piliouras. Cycles in adversarial regularized learning. In Proceedings of the twenty-ninth annual ACM-SIAM symposium on discrete algorithms, pages 2703-2717. SIAM, 2018.
Vlad Niculae and Mathieu Blondel. A regularized framework for sparse and structured neural attention. Advances in neural information processing systems, 30, 2017.
Shai Shalev-Shwartz. The duality of strong convexity and strong smoothness applications to machine learning, 2009. URL https: / /wwpwwww... cs htt.ahh............aaaaoaa//e/e. ShalevAOFSRO9 pdf. Accessed: 2025-08-04.
Aaron Sidford. Optimization algorithms. 2024. URL
file//
d/1-AWta = AZiwPFCJQp/viee
Stephen J. Wright and Benjamin Recht. Optimization for Data Analysis. Cambridge University Press, 2022.
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A Gradient Theorem of Line Integrals
Definition 9 (Vector field). A vector field is map F :R - Rd
Definition (Conservative Vector Field). If the vector field F is the gradient of a function f. then
F is called gradient conservative vector field. The function is called the potential scalar of F.
Definition 11 (Differentiable Multivariable Function). at the point a if there exists a linear transformation T _Rn - mm that satisfies the condition
The function f RR" - R is differentiable
[1f(x)
f(a) T a)ll
lim
2 all
A function f is said to be differentiable if f is differentiable at all points within dom( Definition 12 (Line integral of Vector Field). For a vector field F S R + Rd, the line integral along piecewise smooth curve C. U, in the direction of r, is defined
J. F(r) - dr dr= (P(r())))
where is the dot product and r [a, b] + s a bijective parametrization of the curve s such that r(a) and rb) give the endpoints of s.
Theorem 9 (Gradient Theorem of Line Integrals). For & U C CR' + R as differentiable functioooo and 8 as any continuous curve U which starts at point p and ends point q. then
[
(V(r)) dr ②(q) Q(p)
where V denotes the gradient vector field of Φ.
B Continuous FTRL
variety of works have examined FTRL in the continuous setting (CFTRL) such as Kwon and Mertikopoulos [2017], Mertikopoulos et al. [2018], Cheung and Piliouras [2021]. Let C ç Rd denote non-empty compact convex set.
Definition 13 (Regularizer). A convex function R R + R u (+20} will be called regularizer function on C if dom R) C and Rle is strictly convex and continuous.
For given regularizer function on we let Rmax = max R(x) and Rmin minggg R(x). Definition 14 (Choice Map). The choice map associated to regularizer function R on C will be the map QR gdd > C defined as
QR(v) arg argm - R(x)}, E R
aEC
Note: QR is the convex conjugate of R and we have argmax since R is strict convex and continuous hence has a well-defined unique sup.
In continuous time, instead of a sequence of payoff vectors (un)neR, in R4, the agent will be facing a measurable and locally integrable stream payoff vectors (U.)EE Rdd Consider the process:
:= = QR(n / (udss) arg mrx nt S ugds) R(x)), sec
where (N.TER, is a positive, nonincreasing and piecewise continuous parameter, while of E denotes the agent's action time given the history of payoff vectors u, < t. The dynamics of CFTRL can expressed by
y(t) = v() + fo u(x(s))ds z(t) Rn(v))
is worth noting that when R the entropy function, the dynamics reduce to replicator dynamics of evolutionary game theory.
13

---

Theorem 10 (Kwon and Mertikopoulos [2017], Theorem 4.1). If R is a regularizer function on C. and (nE) positive, non-increasing and piecewise continuous parameter, then, for every locally intergrable payoff stream (u))EER_ E ERd, we have:
max ( ZEC
(us, - S. [u,,,, 2
Rmaz
Rmin
C
Constructing DCFMM with AD securities via CFTRL
Protocol Constructing DCFMM with AD securities via Continuous FTRL
Given: regularizer R RRd + R, VC R, = Ad. and t0 = 0.
Repeat:
Market receives trade bundle 7r E IRd
2. Charge trader for bundle rz:
Clq + +r) C(qi)= = S (1)+1 + trr)
where
2 (g: srt) VC(@ + srt) arg max(p, / 9: + tridf) R(p) PEA,
which CFTRL.
3. Set a+11 = 9t + Tt and
Theorem 11. By CFTRL defined via Protocol with fixed learning rate of 1 1, the pricing of bundle at given state for DCFMM can be expressed by an action taken by CFTRL Thus, the worst-case loss of DCFMM is equivariant to the regret of CFTRL.
Proof. any round, the market receives a bundle rt Rdd and runs CFTRL
E E (gt + STt) = VC + sr;) = arg max [. rEA,
+ tridt) - R(p)
where the equality holds by Kwon and Mertikopoulos [2017] [2017][Propositioo 2]. Also, observe that
L (ol (a: ST.). rt)ds C(ge + m²) C(g.)
parametric Form
where the equality holds by the Fundamental Theorem of Line Integrals. Thus,
- Σ
C(a + rr) - C(ge)
(af + ard..)dd
Payments to Market
t=
CFTRL Lows
and that
max (p, go.)
max (p.
+
r()
max AE
(p. q: + sr:)ds.
PEAd
2=0
Worst-Case Market Payout
Worst-Case Action
Hence, via the equalities, the regret rate of CFTRL matches the worst-case loss of the DCFMM.
Corollary 1. Given a DCFMM defined via strictly convex and differentiable defined over all of Ag. then the cosst-case loss is lower bounded by Rmoz Rmin maXpep(Y) R(p) -minpes, R(p).
14

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Observe this worst-case loss matches the original derived worst-case loss in Abemnethy et al. [2013][Theorem. 4.4.] for DCFMM with AD securities (up difference of max and sup in terms of R, we leave generalizing in terms of sup to future work).
D Gradient and Steepest Descent Results
Theorem 12 (Ang [2025], Theorem 1). f convex and lz-based L-smooth, then for gradient descent where -t+1 =
f(xe), | - x2112 < 1 x= 112 if In arg min f and
| x= 1:2 I x=112 112 otherwise.
Lemma 10 (Sidford [2024], Lemma 6.1.5), For differentiable f Rd IR, norm || and x E lk"
define Q R IR
L || ul?
Q f(x) (VS(x),
where || is an arbitrary norm and L > 0. ff f is smooth w.e.t. || and y² arg min Q(v) then f(y") fz) ((=))
Theorem 13 (Wright and Recht [2022], Theorem 3.5). Suppose that f is bounded below and is I-smooth w.r.f. || ||. Then all accumulation points x of the sequence (#t} generated by a scheme that satisfies
1 [V
f(x++1) f(x))
2L
are stationary, that is, (z(z) 0. If in addition f is convex, each such V is solution of minzeR" f(x).
Lemma 11 (Sidford [2024], Lemma 6.1.7). If R. + R is differentiable and p-strong convex w.E.f. ||-|| then for all x, E arg min f and 1R" it holds that
| x. s()) (x..) S [VV[())
2/.
Lemma 12. If Rd - IR is differentiable, p-strong corvex w.rlt || -1ps and I-smooth w.r.f. || ll.. then it holds that |/y 1- x1 +|l x.11i, where x. E arg min f and x and are from Lemma 10.
Proof. By Lemma 10, and 11, we get
L " 5 ly xz 1| - f(y) f(xx) S f(x) f(x.) S |2 x.112
Hence, we get [|y =1I < + || xr,1ii. Ili..
E
Smooth Quadratic Prediction Markets with Constraints Analysis
Buy-Only Market For analysis sake, we substitute min argument of r. with g++1 +
min 1 Eli
((VC(g), a+1 q+1
a)) + " 11 20l12
(c, a++1 q.)
Expocted Payout
Payment Market
s.t. q: g++1
Let r 9++1 and VC(g:) - p. This simplifies the objective too
L
min (c,r) +
2 1r
s.t. -r
15

---

Budget-Bounded f2 Norm
Budget-Bounded 4 Norm
Budget-Boundes f., Norm
Asset 2
Asset 1
Asset 2
Asset 1
Asset 2
Asset
Asset 3
Asset
Asset 3
Figure 2: Let 9o = (10, 20, 10), C' is softmax with smoothness of L 1, and μ (1/6,166 2/3). The agents had budget of B = .01. The blue square expresses μa and the orange path towards the blue square demonstrates the updating market distribution states, As denote by the titles's 's of each plot, we vary the used for the Smooth Quadratic Prediction Market. Note although softmax is not -mothh, we use said norm experimentally for the sake of comparison.
Buy-Only & Norm
Buy-Only & Norm
Buy-Only ta Norm
Asset 1
Asset
Asset 1
Asset 2
Asset 1
Asset 2
Asset 3
Asset 3
Asset 3
Figure 3: Let qu = (10, 20, 10), C' is softmax with smoothness L = 1, and μ= (1/6, 1/6, 2/3). The blue square expresses y and the orange path towards the blue square demonstrates the updating market distribution states in buy-only market. As denote by the titles's of each plot, we vary the norm used for the Smooth Quadratic Prediction Market. Note although softmax is not fi-smooth, use said norm experimentally for the sake of comparison.
Let the Lagrangian be denoted by L(r, A) (c,r) + +|rr|2 (A, where R Solving for from min,ers C(r, A), we get r* + c). Note the dual problem is
-1 (c, c)
1
(A, A) (N.c).
max X
2L 2L
We claim that X C.. We verify
+ (c+ c) and X c via KKT conditions.
Stationarity
0c de (r", X²) = -4 E [d]): c + c - c- c+ =0
Complimentary Slackness (-X;r; = 0v [d): lf > 0 then we have (c -c;) = 0 and if S 0 we have = (c: 0) = 0.
Primal Feasibility: If c; 2 0 then we have { c;) 0 and if q we have &:
Dual Feasibility: 0 c4 by definition
16

---

Hence, we get that the update step is
a++1 = qt + L ((VC())) - 1)+ (VC(#) - -+))
and observe when VC(g)) = μl we have a stationary point.
Smooth Quadratic Prediction Markets with Adaptive Liquidity
Definition 15 (Asymmetric norm). A function g RR Ry is asymmetric norm it satisfies I, E I
Non-negativtty: g(x) 2 0
Definiteness: g(æ) = g0=x) and only if
Positive homogeneity: g(arr) ag(x) for all
Triangle inequality: g(x S g(x)) (
Lemma 9. Payz= satisfies Axiom 3 No Arbitrage.
Proof. By (Lemma 3.6, Abernethy et al. [2014]) the VPM-DCFMM satisfies no arbitrage, i.e, for all ro....,Tr E S holds that (p(y), Ei=o T²) S Ei Pay Do (4t, r4: 0y) for some V. However, it also holds that Ei= Pay no (%s Fi; U3) I Σ Pay (g,, (. Therefore by combing the two inequalities we have that (p(v), Ei=o r1) & Ei
Pay r (7, Fz; m.)
17

---

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Reviews:
[{"review_id": "pY7YBz3FEc", "reviewer": "Reviewer_gCrc", "summary": "The 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.", "questions": "Typos:\n- line 67 (missing f)\n- line 101 (instsantaneous)\n- line 127 (missing full stop)\n- line 171 (regrate)", "limitations": "yes", "rating": 5, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The upper bound idea is nice, the paper seems well-written, and the statements well-supported by theorems and proofs.", "was_revised": true, "final_justification": "I'm not familiar with the line of work this paper is about, but I enjoyed reading it and to me it's a valuable contribution. Hence, even if with low confidence, I recommend the paper for acceptance.", "rebuttal": ""}, {"review_id": "UbYg9j2kjh", "reviewer": "Reviewer_7cVQ", "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.", "questions": "- In the absence of one‐step incentive compatibility, could a trader exploit incremental steps (e.g., splitting a large trade)? \n- How should one choose the norm ∥·∥ and smoothness parameter L in real‐world settings?", "limitations": "yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strenghts\n- The paper seems to make novel conncection to optimization by extending the duality between FTRL to general L-smooth cost functions. \n- The paper is well written; the theorems are clearly stated and proved, to the degree that I can understand.", "was_revised": true, "final_justification": "Although, it is not exactly my area of expertise, I remain confident about the contributions of this paper.", "rebuttal": "> In the absence of one‐step incentive compatibility, could a trader exploit incremental steps (e.g., splitting a large trade)?\n\nOur reply: The agent will prefer to split a large trade over several trades rather than making a single large trade. This is a potential drawback, but we view it as benign for several reasons:\nThere’s no arbitrage opportunity or free profit available; the market maker’s loss is still bounded. Hence, we would not view trade-splitting as an “exploit” opportunity.\nIf the market charges a constant fee to transact (or we imagine a small cost of doing each transaction due to friction), the trader will not want to split trades below a certain level. We leave formalization of this to future work.\nIf trades are small relative to the size of the market (i.e., the liquidity), the benefit from splitting trades is extremely small. So for typical traders, this will not apply. On the other hand, for extremely large trades, it might even be preferable to the market designer if the trader splits the trades, which gives a more continuous price path.\n\n\n> How should one choose the norm ∥·∥ and smoothness parameter L in real‐world settings?\n\nOur reply:\n\nWe clarify that the designer needs to choose the convex cost function $C$ and the norm, after which L should always be chosen to equal the smooth convexity parameter of $C$ with respect to that norm.\n\nChoosing $C$ is a significant question in the prediction market literature, with some understanding but no simple answer. We can “stretch” $C$ by scaling its convex conjugate such that L is scaled up or down. In general, stretching to obtain a larger L (more curvature) will result in a better worst-case loss, but will provide less liquidity to the market because prices change more rapidly in response to trades. A smaller L will have a large worst-case loss, but it provides a lot of liquidity.\n\nRegarding the choice of norm, one impact we have observed is on the dynamics of prices. The way trades evolve for L1 versus L2 is different. L1 and L infinity have a “sparse” behavior compared to L2. A trade in one asset is more likely to affect only a few prices, rather than all prices. But it changes those few prices more dramatically, so the updates are less smooth. (See figure 1 for an illustration.)"}, {"review_id": "4dLhcksfvH", "reviewer": "Reviewer_UR8r", "summary": "This 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.", "questions": "Could 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?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "On 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.", "was_revised": true, "final_justification": "The authors addressed my questions, I raised my score.", "rebuttal": "> Could 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\nOur reply: Due to the similarity of questions, we refer our response to the reply of 7cVQ’s first question above.\n\n> Is it reasonable to assume that there exists a sequence of agents that will have exactly the same belief in practice?\n\nOur reply: We are not necessarily stating that we expect a sequence of agents with the same beliefs to come to the market; however, we primarily aim to demonstrate that the market functions as a belief aggregator in this particular scenario (a common baseline expectation in the prediction market literature). Realistically, each trader will have a unique belief. Even so, given that traders are incentivized to follow gradient descent (at least when the market uses the two norm), each trader's purchase moves the market state towards a state that maps to their particular belief (this is a byproduct of Theorem 12 in the appendix)."}, {"review_id": "hZOBWDQZdW", "reviewer": "Reviewer_dHk7", "summary": "- 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).", "questions": "### 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", "limitations": "- 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.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "### 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", "was_revised": true, "final_justification": "The authors satisfactory addressed all of my questions. My rating remains unchanged.", "rebuttal": "> 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?\n\nOur reply: This is a key question for the proposed design. In terms of theory, we show that rates of convergence are fast. Additionally, since traders are incentivized to follow gradient descent, at least when the market uses the two-norm, we know that each trade moves the market state closer to a market state that maps to their belief (a non-asymptotic guarantee).\n\nIn practice, if there is a constant fee or cost of effort to make each transaction, and/or if the liquidity of the market is high, we expect the trader to make few trades and stop quickly. So the behavior is expected to be very similar to the DCFMM in practice, but with a simpler and arguably preferable fee structure.\n\n\n> What if traders with different beliefs trade one after the other? Can this lead to oscillatory market states?\n\nOur reply: This is a bit unclear even in standard prediction markets. There are Bayesian models with information aggregation, but it’s not always clear how the theory of IC translates to actual markets with many participants. However, these markets do work well and converge in practice, and the SQPM should be no different\n\n\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\nOur reply: It’s actually quite tricky, we think experts would be surprised that it is possible at all to find a fee that threads the needle between:\nlarge enough to maintain the market maker’s worst-case loss, but\nsmall enough to retain a version of incentive compatibility: the agents aren’t charged too much, so they push the prices to match their beliefs.\nWe don’t know of another fee proposal that would work. However, the intuition for the quadratic fee can be viewed from several angles: It uses the quadratic upper-bound that is tangent to the cost function at the current point, so it is “locally” curved correctly. This technique in optimization is called majorization-minimization, which gives another intuition.\n\n> Can the bound of Theorem 4 be made tighter? How much better is the worst-case loss of SQPM over DCFMM?\n\n\nOur reply: We believe that the difference can be made arbitrarily small: take any sequence of trades, and “split” the trades to be a large number of arbitrarily small trades. In this case, the worst-case loss of the two designs should converge to be the same.\n\n\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\nOur reply: While the relationship to FTRL was noted, the literature didn’t seem to know how to exploit it previously – even in this work, our expansion to continuous FTRL in the appendix is novel. Additionally, our presentation of the DCFMM as a breakdown into a linear term plus a Bregman term is not the standard presentation, and part of our contribution is noting that this breakdown is useful and can be modified to obtain the quadratic fee market. The relation to gradient descent: One thing that is unexpected is that the market maker is still using FTRL as its algorithm to update prices. Yet the strategic process surprisingly results in gradient descent on a related function, the convex conjugate of the FTRL regularizer.\n\n> Minor questions: \n\nOur reply: thank you for the feedback, we acknowledge the presented typos and unclear aspects and will use this feedback to improve the writing."}]

Author rebuttal:

Meta-review: The reviewers all agreed that this paper provided interesting technical contributions to the field of prediction markets, with novel connections to optimizations and online learning. The reviewers also agreed that this paper was well-written, and the main results were clearly presented. The reviewers were largely positive before the rebuttal phase, and feel that the authors satisfactorily addressed any remaining concerns through their rebuttals.
Decision: Accept (poster)