Online Social Welfare Function-based Resource Allocation

TL;DR

The paper addresses online resource allocation to a fixed population where each individual's ex-ante utility is learned over time and aggregated via a social welfare function (SWF). It introduces a confidence-sequence lifting principle: under monotonicity of the SWF, time-uniform confidence sequences on individual means lift to anytime-valid bounds on welfare, enabling both learning and inference. The authors propose SWF-UCB, a general algorithm with exact oracles for three SWF families (Weighted Power Mean, Kolm, Gini) and prove near-optimal regret $\tilde{O}(n+\sqrt{nkT})$, along with experimental validation of $\sqrt{T}$ scaling and interesting interactions between $k$ and SWF parameters. The framework also supports inference tasks such as sequential testing, optimal stopping, and policy evaluation, offering a principled approach to fairness-aware online allocation with broad theoretical and practical significance. The work thus unifies online learning and time-uniform welfare inference across a spectrum of welfare objectives, providing practical algorithms, theoretical guarantees, and insights into how different fairness notions impact learning in resource-constrained environments.

Abstract

In many real-world settings, a centralized decision-maker must repeatedly allocate finite resources to a population over multiple time steps. Individuals who receive a resource derive some stochastic utility; to characterize the population-level effects of an allocation, the expected individual utilities are then aggregated using a social welfare function (SWF). We formalize this setting and present a general confidence sequence framework for SWF-based online learning and inference, valid for any monotonic, concave, and Lipschitz-continuous SWF. Our key insight is that monotonicity alone suffices to lift confidence sequences from individual utilities to anytime-valid bounds on optimal welfare. Building on this foundation, we propose SWF-UCB, a SWF-agnostic online learning algorithm that achieves near-optimal $\tilde{O}(n+\sqrt{nkT})$ regret (for $k$ resources distributed among $n$ individuals at each of $T$ time steps). We instantiate our framework on three normatively distinct SWF families: Weighted Power Mean, Kolm, and Gini, providing bespoke oracle algorithms for each. Experiments confirm $\sqrt{T}$ scaling and reveal rich interactions between $k$ and SWF parameters. This framework naturally supports inference applications such as sequential hypothesis testing, optimal stopping, and policy evaluation.

Figures (6)

• Figure 1: Normalized regret $R(T) / \sqrt{T}$ versus time horizon $T$ for WPM, Kolm, and Gini SWFs. The normalized regret remains bounded across two orders of magnitude in $T$, consistent with our theoretical $\tilde{\mathcal{O}}(\sqrt{T})$ guarantee.
• Figure 2: Trends with varying power value $q$ for WPM and Kolm SWFs. We consider the range between egalitarian ($q=-\infty$ for WPM and Kolm) and utilitarian ($q=1$ for WPM and $q=0$ for Kolm). While there is a smooth change in the observed regret, there is significant variability with changing $k$.
• Figure 3: Trends with varying number of allocated resources $k$ for all three SWFs. We observe that there is a sharp decrease after $k=20$ for egalitarian welfare. The changes with increasing $q$ becomes less gradual for both WPM and Kolm SWFs. Geometric weights have some similarity with egalitarian welfare, and we see a similar pattern with varying $k$ for Gini SWF, although the curve is much smoother.
• Figure 4: Normalized regret $R(T) / \sqrt{T}$ versus time horizon $T$ for WPM, Kolm, and Gini SWFs with linear decay in weights. The normalized regret remains bounded across two orders of magnitude in $T$, consistent with our theoretical $\tilde{\mathcal{O}}(\sqrt{T})$ guarantee.
• Figure 5: Trends with varying power value $q$ for WPM and Kolm SWFs with linear decay in weights. $q=-\infty$ corresponds to egalitarian welfare for both, whereas $q=1$ for SWF and $q=0$ for Kolm correspond to utilitarian welfare. We observe that while there is a smooth change in the observed regret, there is significant variability with changing $k$.

Theorems & Definitions (14)

• Proposition 3.1
• Remark 3.2
• Theorem 4.1
• Theorem 5.1
• Theorem 5.2
• Proposition 5.3
• Corollary 1.1
• proof
• proof
• proof