Online Contract Selection for Continual Coverage

TL;DR

This paper investigates online contract selection problems under uncertain prices, providing optimal competitive ratios for specific models and demonstrating fundamental differences when price distributions are non-identical.

Contribution

It characterizes the optimal competitive ratios for two models with i.i.d. prices and introduces quantile-based algorithms, highlighting a key division in non-i.i.d. settings.

Findings

Exact worst-case competitive ratio for deferred model: ~2.472

Lower bound of 2.472 and upper bound of 4.179 for concurrent model

No finite competitive ratio when prices are independent but not identically distributed

Abstract

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 $\zeta^* \approx 2.472$ as the time horizon grows. For the concurrent model, we prove a lower bound of $\zeta^*$ 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.