Online Stochastic Allocation with Increasing Returns

2026-09-28

AI Summary

This study addresses the limitations of the traditional diminishing returns assumption in online resource allocation by tackling the failure of classical algorithms caused by increasing returns under scale and network effects. It first establishes the impossibility of achieving a constant competitive ratio under heterogeneous functions. To overcome this, the work proposes a novel deterministic algorithm for structured discretely concave functions, leveraging intermediate objective repair, prefix-robust allocation sequences, and offline objective computation. The proposed approach achieves a tight competitive ratio of 1/2 under homogeneous functions and provides a deterministic guarantee of 0.2466 for heterogeneous discretely concave functions. These results significantly outperform existing baselines while establishing theoretical upper bounds, thereby advancing the fundamental understanding of online resource allocation in the presence of increasing returns.

Abstract

Online resource allocation is a fundamental problem in revenue management, sponsored search, and platform operations. Most prior work assumes nonincreasing assignment rewards, capturing diminishing returns. We instead study increasing returns, where assigning more customers to the same product can unlock larger value through scale, visibility, or network effects. We consider capacity-limited products assigned to sequentially arriving customers, where the reward of each product depends on the total number of customers assigned to it. We focus on full compatibility, where every product can be assigned to every customer. The number of customers is unknown to the online algorithm. We show that representative classical approaches, such as online greedy algorithm and LP-based independent rounding, can perform arbitrarily bad in this setting, even when products share a common bonus function. For homogeneous bonus functions, we give a simple polynomial-time algorithm that achieves a tight $1/2$-competitive ratio. For arbitrary heterogeneous bonus functions, we show that no constant competitive ratio independent of the number of products is possible: for $m$ products, the optimal competitive can be as small as $\Theta\left({1}/{\sqrt m}\right)$. This shows that heterogeneous delayed rewards can force any online algorithm to guess the realized arrival counts. We then identify a structured heterogeneous regime that restores a constant guarantee. When each bonus function is nonnegative, nondecreasing, and discrete concave, we design an intermediate target repair algorithm with a deterministic pathwise guarantee of $0.2466$. The algorithm repeatedly computes an offline target for a larger demand level and repairs the current allocation toward it using a prefix-robust assignment order. This coordinates buildup while remaining robust to early stopping and yields a distribution-free guarantee.

Problem

• Online Stochastic Allocation
• Increasing Returns
• Resource Allocation
• Competitive Ratio
• Revenue Management

Innovation

• Online Stochastic Allocation
• Increasing Returns
• Competitive Ratio
• Target Repair Algorithm
• Discrete Concavity