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Smooth Quadratic Prediction Markets

Enrique Nueve, Bo Waggoner

TL;DR

The paper proposes the Smooth Quadratic Prediction Market (SQPM), a novel market maker design for Arrow-Debreu securities that reinterprets the cost-function framework through a simple quadratic fee to induce gradient-descent-like trading. SQPM preserves core market guarantees such as existence of instantaneous prices, information incorporation, expressiveness, no arbitrage, and a form of incentive compatibility, while achieving better worst-case outcomes than the traditional DCFMM. The authors demonstrate that, under $L$-smooth CIIP cost functions, SQPM traders implement general steepest descent steps, with $\ell_2$-based and $\ell_p$-based variants providing incremental incentive compatibility and convergence of prices to traders’ beliefs. They further analyze SQPM under budget-bounded and buy-only constraints and outline adaptive liquidity mechanisms that adjust market stiffness with volume, highlighting practical implications for scalable, robust prediction markets and future extensions to broader security spaces.

Abstract

When agents trade in a Duality-based Cost Function prediction market, they collectively implement the learning algorithm Follow-The-Regularized-Leader. 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

TL;DR

The paper proposes the Smooth Quadratic Prediction Market (SQPM), a novel market maker design for Arrow-Debreu securities that reinterprets the cost-function framework through a simple quadratic fee to induce gradient-descent-like trading. SQPM preserves core market guarantees such as existence of instantaneous prices, information incorporation, expressiveness, no arbitrage, and a form of incentive compatibility, while achieving better worst-case outcomes than the traditional DCFMM. The authors demonstrate that, under -smooth CIIP cost functions, SQPM traders implement general steepest descent steps, with -based and -based variants providing incremental incentive compatibility and convergence of prices to traders’ beliefs. They further analyze SQPM under budget-bounded and buy-only constraints and outline adaptive liquidity mechanisms that adjust market stiffness with volume, highlighting practical implications for scalable, robust prediction markets and future extensions to broader security spaces.

Abstract

When agents trade in a Duality-based Cost Function prediction market, they collectively implement the learning algorithm Follow-The-Regularized-Leader. 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.
Paper Structure (30 sections, 27 theorems, 38 equations, 3 figures)

This paper contains 30 sections, 27 theorems, 38 equations, 3 figures.

Table of Contents

  1. Introduction
  2. What are Prediction Markets?
  3. Duality-based CFMM
  4. Smooth Quadratic Prediction Markets
  5. Background and Notation
  6. Bregman Divergences, Smoothness, and Strong Convexity
  7. Automated Market Makers for AD Securities
  8. Intuition of Axioms
  9. CIIP Construction
  10. Softmax
  11. Sparsemax
  12. Duality-based CFMM
  13. Smooth Quadratic Prediction Markets
  14. Smooth Quadratic Prediction Market Design
  15. $\ell_2$-based Smooth Quadratic Prediction Markets
  16. ...and 15 more sections

Key Result

Lemma 1

For a convex differentiable function $f:\mathbb{R}^{d} \to (-\infty ,+\infty ]$, it holds that $\langle \nabla f(x)-\nabla f(y), x-y \rangle \geq 0$$\forall$$x,y\in\mathbb{R}^{d}$.

Figures (3)

  • Figure 1: Let $q_{0}=(10,20,10)$, $C$ is softmax with smoothness of $L=1$, and $\mu = (1/6,1/6,2/3)$. The blue square expresses $\mu$ and the orange path towards the blue square demonstrates the updating market distribution states. 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 $\ell_{1}$-smooth, we use said norm experimentally for the sake of comparison.
  • Figure 2: Let $q_{0}=(10,20,10)$, $C$ is softmax with smoothness of $L=1$, and $\mu = (1/6,1/6,2/3)$. The agents had a budget of $B=.01$. The blue square expresses $\mu$ and the orange path towards the blue square demonstrates the updating market distribution states. 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 $\ell_{1}$-smooth, we use said norm experimentally for the sake of comparison.
  • Figure 3: Let $q_{0}=(10,20,10)$, $C$ is softmax with smoothness of $L=1$, and $\mu = (1/6,1/6,2/3)$. The blue square expresses $\mu$ and the orange path towards the blue square demonstrates the updating market distribution states in a 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 $\ell_{1}$-smooth, we use said norm experimentally for the sake of comparison.

Theorems & Definitions (52)

  • Lemma 1: hiriart2004fundamentals, Proposition 6.1.1
  • Definition 1: Bregman divergence, L-smoothness, & K-strongly convex
  • Theorem 1: kakade2009duality, Theorem 6
  • Definition 2: Automated Market Maker for AD Securities
  • Definition 3: Price-Plus-Fee Market
  • Lemma 2: abernethy2013efficient, Theorem 4.2, Lemma 4.3
  • Lemma 3
  • proof
  • Definition 4: abernethy2013efficient, DCFMM for AD Securities
  • Theorem 2: abernethy2013efficient, Theorem 3.2
  • ...and 42 more