Trading Prophets: How to Trade Multiple Stocks Optimally

2025-04-17

SIAM Symposium on Simplicity in Algorithms

AI Summary

This paper studies the online multi-stock trading problem, generalizing the classical single-stock prophet model to a more realistic multi-asset setting that accommodates price correlations or independence, a cardinality constraint on holdings (at most $k$ stocks), per-stock transaction limits ($ell$ shares buyable, $ell'$ sellable), and matroid constraints on feasible portfolios. We introduce the first $(k,ell,ell')$-prophet and matroid-constrained multi-stock prophet models, and conduct competitive analysis under the random-order arrival assumption (non-i.i.d.). Our main contributions are: (1) a tight optimal competitive ratio of $1/(1+d)$, where $d$ is the matroid’s density; (2) a tight bound of $min{1/2, ell/k}$ under capacity constraints; and (3) a proof that $1/(1+d) - O(1/n)$ asymptotically matches the hardness lower bound. Our approach integrates online algorithm design, matroid theory, probabilistic inequalities, and density-based parametric analysis.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In the single stock trading prophet problem formulated by Correa et al. (2023), an online algorithm observes a sequence of prices of a stock. At each step, the algorithm can either buy the stock by paying the current price if it doesn't already hold the stock, or it can sell the currently held stock and collect the current price as a reward. The goal of the algorithm is to maximize its overall profit. In this work, we generalize the model and the results of Correa et al. by allowing the algorithm to trade multiple stocks. First, we formulate the $(k,ell,ell')$-Trading Prophet Problem, wherein there are $k$ stocks in the market, and the online algorithm can hold up to $ell$ stocks at any time, where $ellleq k$. The online algorithm competes against an offline algorithm that can hold at most $ell'leqell$ stocks at any time. Under the assumption that prices of different stocks are independent, we show that, for any $ell$, $ell'$, and $k$, the optimal competitive ratio of $(k,ell,ell')$-Trading Prophet Problem is $min(1/2,ell/k)$. We further introduce the more general $cal{M}$-Trading Prophet Problem over a matroid $cal{M}$ on the set of $k$ stocks, wherein the stock prices at any given time are possibly correlated (but are independent across time). The algorithm is allowed to hold only a feasible subset of stocks at any time. We prove a tight bound of $1/(1+d)$ on the competitive ratio of the $cal{M}$-Trading Prophet Problem, where $d$ is the density of the matroid. We then consider the non-i.i.d. random order setting over a matroid, wherein stock prices drawn independently from $n$ potentially different distributions are presented in a uniformly random order. In this setting, we achieve a competitive ratio of at least $1/(1+d)-cal{O}(1/n)$, where $d$ is the density of the matroid, matching the hardness result for i.i.d. instances as $n$ approaches $infty$.

Problem

• Extends single-stock trading to optimize multi-stock portfolios.
• Formulates competitive ratios for trading under matroid constraints.
• Analyzes non-i.i.d. random order pricing for matroid trading.

Innovation

• Extends single stock trading to multiple stocks
• Introduces matroid-based trading with correlated prices
• Achieves optimal competitive ratio in random order