Online Fair Allocation of Perishable Resources

TL;DR

This work tackles online fair allocation with perishables by formalizing a budgeted, multi-round setting where items spoil endogenously. It introduces a perishing-aware guardrail algorithm, Perishing-Guardrail, which constructs a conservative baseline $\underline{X}$ and an adaptive guardrail $\overline{X}=\underline{X}+L_T$, guided by forecasts of future demand and spoilage; a threshold rule determines when to allocate aggressively. The authors establish lower bounds that reveal fundamental envy-efficiency limits under perishability, and show their algorithm achieves near-optimal performance in a δ-offset-expiry regime via a slow-consumption analysis and a carefully defined good event. They validate the approach through extensive simulations and a real-world ginger perishability dataset, demonstrating substantial gains in efficiency and fairness over perishing-agnostic methods, especially when spoilage is non-negligible. The results highlight offset-expiry as a critical regime and offer practical guidance for managing perishable inventories in fair and efficient ways across domains such as food distribution and vaccine logistics.

Abstract

We consider a practically motivated variant of the canonical online fair allocation problem: a decision-maker has a budget of perishable resources to allocate over a fixed number of rounds. Each round sees a random number of arrivals, and the decision-maker must commit to an allocation for these individuals before moving on to the next round. The goal is to construct a sequence of allocations that is envy-free and efficient. Our work makes two important contributions toward this problem: we first derive strong lower bounds on the optimal envy-efficiency trade-off that demonstrate that a decision-maker is fundamentally limited in what she can hope to achieve relative to the no-perishing setting; we then design an algorithm achieving these lower bounds which takes as input $(i)$ a prediction of the perishing order, and $(ii)$ a desired bound on envy. Given the remaining budget in each period, the algorithm uses forecasts of future demand and perishing to adaptively choose one of two carefully constructed guardrail quantities. We demonstrate our algorithm's strong numerical performance - and state-of-the-art, perishing-agnostic algorithms' inefficacy - on simulations calibrated to a real-world dataset.

Figures (8)

• Figure 1: Graphical representation of \ref{['thm:upper_bound_perishing', 'thm:lower_bound']}. \ref{['fig:diagram_a']} illustrates the envy-efficiency trade-off ($\Delta_{\text{\it efficiency}}$ vs. $\Delta_{\text{\it EF}}$) achieved by Perishing-Guardrail (\ref{['alg:brief-perishing']}). The dotted lines represent the impossibility results due to either demand or perishing uncertainty. The region below the solid line represents the impossibility due to the envy-efficiency trade-off; the green region is the achievable region for Perishing-Guardrail. \ref{['fig:diagram_b']} illustrates the phase transition between the performance of Perishing-Guardrail depending on the spoilage loss $\mathcal{L}^{\textsf{perish}}$ ($x$-axis) and envy parameter $L_T$ ($y$-axis).
• Figure 2: Illustrating the $\underline{X}$ construction \ref{['eq:xlower']} for the toy instance in \ref{['ex:xlower']}. The dashed line corresponds to the line $Y = X$, and the solid line to $(B - \overline{\Delta}(X)) / \overline{N}$. Here, $\underline{X}$ is represented by the red star, the point at which the solid and dashed lines intersect.
• Figure 3: Maximum feasible allocation $\underline{X}$ vs. $\alpha$, for $T_b \sim \emph{Geometric}(T^{-(1 + \alpha)})$, $B = 200$, $T = 150$, $N_t$ drawn from a truncated normal distribution $\mathcal{N}(2, 0.25)$, and $\delta = 1/T$. Here, $\underline{X}$ was calculated via line search, with Monte Carlo simulation used to estimate $\overline{\Delta}(X)$ for each value of $X$. The dashed line represents the "naive" allocation $B / \overline{N} = 0.89$ which ignores possible perishing, and the green line is the curve of best fit to $\underline{X}$.
• Figure 4: Empirical trade-off between $\mathbb E \mathopen{}\mathclose{\left[ \Delta_{\text{\it efficiency}} \right]$ and $\mathbb E \mathopen{}\mathclose{\left[ \Delta_{\text{\it EF}}} \right]$. The points on the trade-off curve correspond to increasing values of $L_T$, from left to right. Static-$\frac{B}{\overline{N}}$ and Static-$\underline{X}$ respectively correspond to Vanilla-Guardrail and Perishing-Guardrail for $L_T = 0$.
• Figure 5: Algorithm comparison across $\mathbb E \mathopen{}\mathclose{\left[ \Delta_{\text{\it EF}} \right], \mathbb E \mathopen{}\mathclose{\left[ \Delta_{\text{\it efficiency}}} \right], \mathbb E \mathopen{}\mathclose{\left[ \textsc{Envy}} \right]$, and $\mathbb E \mathopen{}\mathclose{\left[ \text{Spoilage}} \right]$, for $\alpha \in \{0.1,0.2,0.25,0.3\}$.

Theorems & Definitions (47)

• Remark 2.1
• Definition 2.2: Counterfactual Envy, Hindsight Envy, and Efficiency
• Theorem 2.3: Theorems 1 and 2, sinclair2021sequential
• Definition 3.1: Offset-expiring process
• Theorem 3.2
• Definition 3.3: $\sigma$-induced loss
• Remark 3.4
• Remark 3.5
• Example 3.6
• Theorem 3.7