In this work we initiate the study of buy-and-sell prophet inequalities. We start by considering what is arguably the most fundamental setting. In this setting the online algorithm observes a sequence of prices one after the other. At each time step, the online algorithm can decide to buy and pay the current price if it does not hold the item already; or it can decide to sell and collect the current price as a reward if it holds the item. We show that for i.i.d. prices a single-threshold online algorithm achieves at least $1/2$ of the expected profit of the optimal offline algorithm and we prove that this is optimal. For non-i.i.d. prices in random order, where prices are no longer independent, we give a single-threshold online algorithm that achieves at least a $1/16$ fraction of the expected profit of the optimal offline algorithm. We also show that for this setting no online algorithm can yield a better than $1/3$ approximation, and thus establish a formal separation from the i.i.d. case. On the other hand, we present a threshold-based online algorithm for this setting that yields a $1/2-o(1)$ approximation. For non-i.i.d. prices no approximation is possible. We use the results for these base cases to solve a variety of more complex settings. For instance, we show a $1/2-o(1)$ approximation for settings where prices are affiliated and the online algorithm has only access to a single sample. We also extend our upper and lower bounds for the single item case to $k$ items, and thus in particular show that it is impossible to achieve $1-o(1)$ approximations. For the budgeted version, where fractions of an item can be bought, and gains can be reinvested, we show a constant-factor approximation to the optimal offline algorithm's growth rate. In a setting with $k$ item types and price streams, we achieve a $\Omega(1/k)$ approximation for the unit-capacity case, which is optimal.

翻译：本文首次提出买卖先知不等式的研究。我们从最基本情境入手——在线算法需逐一观察价格序列。在每个时间步，若当前未持有物品，在线算法可选择以现价买入；若已持有物品，则可选择以现价卖出获利。研究表明：对于独立同分布价格，单阈值在线算法可获得最优离线算法期望收益的至少1/2，且该下界是紧的。对于随机顺序非独立同分布价格（价格不再独立），我们提出单阈值在线算法，其期望收益可达最优离线算法的1/16；同时证明该场景下任何在线算法无法获得优于1/3的近似比，从而建立与独立同分布情形之间的形式化分离。另一方面，我们为此场景设计了基于阈值的在线算法，可实现1/2-o(1)的近似比。对于非独立同分布价格，则无法获得任何近似保证。我们利用这些基础结论解决多种复杂场景：例如，当价格关联且在线算法仅能获取单个样本时，可实现1/2-o(1)的近似比。我们还将单物品情形下的上下界推广至k件物品，从而证明无法达到1-o(1)的近似比。对于允许购买物品分数份额且收益可再投资的预算约束版本，我们证明了与最优离线算法增长率相差常数因子的近似比。在含k种物品类型与价格流的多物品场景中，我们针对单位容量情形实现了紧的Ω(1/k)近似比。