Motivated by applications where a system must remain operational via continual procurement of contracts, we study two online contract selection problems under uncertain prices. At each time step, a price drawn from a known distribution is revealed online, and the decision-maker may initiate a contract of arbitrary duration, incurring a cost equal to the product of the price and the contract length; moreover, every time period must be covered by at least one active contract. We consider two models depending on how contracts cover time: a \emph{deferred model}, in which contracts are queued back-to-back, and a \emph{concurrent model}, in which contracts become active immediately and may overlap. In both settings, we seek online algorithms that minimize their competitive ratio, i.e., the ratio between the expected cost incurred by the online algorithm and the expected offline optimal cost when all prices are known in advance. We first focus on the case where prices are independent and identically distributed (i.i.d.). For the deferred model, we characterize exactly the worst-case optimal competitive ratio, which is asymptotically $ζ^* \approx 2.472$ as the time horizon grows. For the concurrent model, we prove a lower bound of $ζ^*$ on the optimal competitive ratio and an asymptotic competitive ratio of at most $4.179$. These bounds improve upon the current lower bound of $2.148$ and upper bound of $6.052$ on the optimal competitive ratio. For both models, our algorithms are quantile-based that can be easily translated into practical threshold-based algorithms for any distribution. Our proofs follow from linear programs and duality arguments in quantile spaces. Lastly, we show that, in both models, no finite competitive ratio exists when the prices are still independent but not necessarily identically distributed, proving a striking division in the two price settings.

翻译：在系统必须通过持续采购合同来维持运行的现实应用推动下，我们研究了价格不确定下的两个在线合同选择问题。在每个时间步，一个来自已知分布的价格在线展示，决策者可以启动一个任意时长的合同，产生的成本等于价格与合同长度的乘积；此外，每个时间段必须至少有一个活跃合同覆盖。我们根据合同覆盖时间的方式考虑了两个模型：一种称为\emph{延迟模型}，其中合同按先后顺序排队；另一种称为\emph{并发模型}，其中合同立即生效且可能重叠。在这两种设置中，我们寻求最小化竞争比的在线算法，即在线算法产生的期望成本与所有价格预先已知时的期望离线最优成本之比。我们首先关注价格独立同分布（i.i.d.）的情况。对于延迟模型，我们精确刻画了最坏情况下的最优竞争比，该竞争比随时间跨度增长而渐近趋于$ζ^* \approx 2.472$。对于并发模型，我们证明了最优竞争比的下界为$ζ^*$，且渐近竞争比至多为$4.179$。这些界改进了当前最优竞争比的下界$2.148$和上界$6.052$。对于这两种模型，我们的算法均基于分位数，可轻松转化为适用于任意分布的实用阈值算法。我们的证明依赖于分位数空间中的线性规划和对偶论证。最后，我们证明了当价格仍然独立但非同分布时，两种模型均不存在有限竞争比，这揭示了两种价格设置之间的显著差异。