Trading Prophets with Initial Capital
Yossi Azar, Niv Buchbinder, Roie Levin, Or Vardi
TL;DR
The paper studies trading prophets where a trader observes price sequences drawn from known distributions and may hold at most one share. By granting initial capital (one share to both trader and prophet), a simple threshold-based strategy achieves a $3$-competitive ratio against a prophet with perfect foresight, and this bound is tight; with constant mean prices the ratio improves to $2$. Extending to transaction costs, the authors design a two-threshold strategy that is $2$-competitive in the IID setting, though the adversarial case remains challenging. The work situates these results within the broader prophet inequalities and online-trading literature, highlighting how initial capital and cost structures dramatically affect attainable competitiveness. Open directions include resolving competitive algorithms for transaction-costs in adversarial settings and extending to multi-asset or bundle-trading scenarios.
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.