Smooth Quadratic Prediction Markets

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: A New Approach to Automated Market Making

The paper "Smooth Quadratic Prediction Markets" by Enrique Nueve and Bo Waggoner proposes an innovative method for designing prediction markets by leveraging principles from machine learning. It presents a novel framework named the Smooth Quadratic Prediction Market (SQPM), which enhances profitability while maintaining essential axiomatic properties of prediction markets such as no arbitrage, expressiveness, and information incorporation.

Quadratic Fee Structure

A key feature of the SQPM is its fee structure, which employs an upper quadratic bound based on L-smoothness. This fee structure replaces the Bregman divergence term used in traditional Duality-based Cost Function Market Maker (DCFMM) approaches with a simpler quadratic term. The L-smooth quadratic fee is shown to result in better worst-case monetary loss compared to the Bregman divergence-based method in the DCFMM framework. Moreover, the paper demonstrates that this approach satisfies many essential axiomatic properties, including the existence of instantaneous price and no arbitrage, without sacrificing market expressiveness or computational efficiency.

Trading Dynamics and Algorithmic Parallels

The paper explores how traders within SQPM markets are incentivized to emulate general steepest descent—a machine learning algorithm—while trading. Specifically, it introduces the concept of incremental incentive compatibility, where traders sequentially update their positions to align the market distribution with their beliefs. This behavior mirrors gradient descent strategies, suggesting that traders are effectively using learning principles to optimize their actions in the market context.

Constraints and Adaptive Liquidity

The paper addresses practical constraints typically observed in real-world markets, such as bounded budgets and buy-only security constraints. It examines how these constraints affect trader behavior and market convergence, showing through experiments that the market reliably converges to the traders' belief distributions across varying norms and constraints.

Additionally, SQPM introduces an introductory analysis on adaptive liquidity—a mechanism that adjusts liquidity dynamically based on trading volume. This concept is inspired by previous works in variable liquidity frameworks but adapted here to work seamlessly within the quadratic prediction market setup.

Implications and Future Work

The proposed SQPM framework has significant implications for both theoretical research and practical applications in prediction market design. It aligns market operation with optimization principles, potentially offering more efficient and profitable markets. Future research directions proposed include extending the framework to more complex securities beyond AD types, analyzing market behavior with diverse trader beliefs, and exploring deeper into adaptive liquidity mechanisms.

While the paper offers a strong foundation for SQPM, it leaves several areas open for further exploration, particularly concerning adaptive liquidity properties and broader generalization beyond AD securities. Thus, the work provides both immediate enhancements for current prediction market models and profound avenues for future research in market design influenced by machine learning algorithms.