Posted Price Mechanisms for Online Allocation with Diseconomies of Scale

TL;DR

This work addresses online k-selection with diseconomies of scale (OSDoS), where a seller with finite inventory $k$ must price units to sequential buyers with valuations in $[L,U]$ while marginal production costs rise. The authors introduce a tight lower-bound framework, defining the critical competitive ratio $oldsymbol{eta}_S^*(k)$ and developing a randomized dynamic pricing scheme, r-Dynamic, that achieves $oldsymbol{eta}_S^*(k) imes e^{oldsymbol{eta}_S^*(k)/k}$-competitiveness, and is exact for $k=2$. The paper also presents a representative-allocation-function approach to capture any randomized online algorithm on carefully constructed hard instances, enabling both tight lower bounds and a structured path to near-optimal online designs. Empirically, r-Dynamic outperforms deterministic dynamic and static randomized counterparts across varied inventory regimes, illustrating strong practical impact for online allocation with diseconomies of scale. Overall, the work advances the theory and practice of dynamic pricing under convex production costs, offering near-optimal guarantees across inventory sizes and motivating future extensions to more complex resource environments.

Abstract

This paper addresses the online $k$-selection problem with diseconomies of scale (OSDoS), where a seller seeks to maximize social welfare by optimally pricing items for sequentially arriving buyers, accounting for increasing marginal production costs. Previous studies have investigated deterministic dynamic pricing mechanisms for such settings. However, significant challenges remain, particularly in achieving optimality with small or finite inventories and developing effective randomized posted price mechanisms. To bridge this gap, we propose a novel randomized dynamic pricing mechanism for OSDoS, providing a tighter lower bound on the competitive ratio compared to prior work. Our approach ensures optimal performance in small inventory settings (i.e., when $k$ is small) and surpasses existing online mechanisms in large inventory settings (i.e., when $k$ is large), leading to the best-known posted price mechanism for optimizing online selection and allocation with diseconomies of scale across varying inventory sizes.

Figures (2)

• Figure 1: The blue curve (i.e., r-Dynamic) corresponds to the competitive ratio of Algorithm \ref{['alg:kselection-cost']} that uses randomized dynamic pricing. The red curve (i.e., d-Dynamic) and the yellow curve (i.e., r-Static) correspond to the competitive ratios of the deterministic dynamic pricing mechanism developed by Tan2023 and the static randomized pricing mechanism by sun2024static. In this figure, we set $L=1$, $U=10$, and $f(i)=\frac{i^{2}}{59}$.
• Figure 2: CDF plots of empirical competitive ratios of r-Dynamic (Algorithm \ref{['alg:kselection-cost']}), d-DynamicTan2023 and r-Staticsun2024static.

Theorems & Definitions (31)

• Theorem 1: Informal Statement of Theorem \ref{['lower-bound-main-theorem']}
• Theorem 2: Informal Statement of Theorem \ref{['upper-bound-large-inventory-cr']}
• Theorem 3: Lower Bound
• Definition 1: Instance $\mathcal{I}^{(\epsilon)}$
• Definition 2: Allocation Functions
• Lemma 1: Monotonicity
• Proposition 1: Representation based on $\boldsymbol{\psi}$
• Lemma 2: Necessary Conditions
• Lemma 3
• Lemma 4