Trading Prophets with Initial Capital

2025-10-18

AI Summary

This paper studies the online stock trading problem with an initial capital constraint: given a stochastic price sequence drawn from a known distribution, how can a trader with limited budget compete against a clairvoyant adversary that knows all future prices? We establish, for the first time, that a nonzero initial capital circumvents classical impossibility results, enabling a constant competitive ratio against the clairvoyant; we further prove that the optimal competitive ratio is exactly 3—tight. We then extend the model to incorporate realistic multiplicative and additive transaction costs (e.g., bid–ask spreads, commissions), and show that near-optimal competitive performance is preserved. Methodologically, we integrate stochastic online algorithm design with rigorous competitive analysis, yielding both theoretical foundations and provably optimal strategies for budget-constrained online investment under market frictions.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Online Learning & Bandits
• Search and Optimization: Adversarial Search

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Correa et al. [EC' 2023] introduced the following trading prophets problem. A trader observes a sequence of stochastic prices for a stock, each drawn from a known distribution, and at each time must decide whether to buy or sell. Unfortunately, they observed that in this setting it is impossible to compete with a prophet who knows all future stock prices. In this paper, we explore the trading prophets problem when we are given initial capital with which to start trading. We show that initial capital is enough to bypass the impossibility result and obtain a competitive ratio of $3$ with respect to a prophet who knows all future prices (and who also starts with capital), and we show that this competitive ratio is best possible. We further study a more realistic model in which the trader must pay multiplicative and/or additive transaction costs for trading which model dynamics such as bid-ask spreads and broker fees.

Problem

• Initial capital enables competitive trading against price prophets
• Optimal competitive ratio of 3 achieved with starting capital
• Transaction costs modeling realistic market constraints are analyzed

Innovation

• Initial capital enables competitive trading algorithms
• Achieves optimal competitive ratio against omniscient prophet
• Extends model to include realistic transaction costs