[Paper Review] Online Social Welfare Function-based Resource Allocation

Kanad Pardeshi, Samsara Foubert+1 more|arXiv (Cornell University)|Feb 1, 2026

Advanced Bandit Algorithms Research0 citations

TL;DR

The paper formalizes online resource allocation over a fixed population using social welfare functions (SWFs) and delivers a confidence-sequence framework plus an SWF-UCB algorithm with near-optimal regret guarantees across three SWF families (Weighted Power Mean, Kolm, Gini).

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 $ ilde{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.

Motivation & Objective

• Formalize repeated resource allocation over a population with SWF-based welfare aggregation.
• Develop a time-uniform confidence sequence framework that requires only SWF monotonicity.
• Provide efficient online learning algorithms with regret guarantees for SWF-based welfare maximization.
• Instantiate the framework for three SWF families (Weighted Power Mean, Kolm, Gini) with bespoke oracles.
• Enable inference tasks such as sequential testing, optimal stopping, and policy evaluation.

Proposed method

• Define ex-ante utilities _i = [U_i] and policy p with sum(p_i)=k, aggregating via M(a mu a o p).
• Prove a confidence-sequence lifting theorem: monotonic SWFs allow turning coordinate-wise CSs for individual means into anytime-valid bounds on M(a mu a o p^*).
• Develop SWF-UCB: a general online learning algorithm that optimizes M(a mu a o p) using upper confidence bounds.
• Provide exact oracles for policy optimization per SWF family (WPM, Kolm via water-filling; Gini via greedy block-based method).
• Use dependent rounding to sample allocations meeting cardinality and marginal probability constraints.
• Prove regret bound: N= number of individuals, k resources per round, T rounds: tilde{O}(n + sqrt(n k T)).
• Demonstrate non-monotone regret in k and discuss inference applications enabled by the CS framework.

(a) WPM SWF

Experimental results

Research questions

• RQ1 How can monotonic SWFs be leveraged to obtain anytime-valid welfare guarantees in online resource allocation?
• RQ2 What is the tight regret performance of online SWF-based allocation under bounded feedback?
• RQ3 How can we efficiently optimize allocations for different SWF families (WPM, Kolm, Gini) in an online setting?
• RQ4 Can the framework support sequential inference tasks such as testing, stopping, and policy evaluation?
• RQ5 How do SWF parameters (weights, fairness parameter q) influence learning dynamics and regret?

Key findings

• SWF monotonicity suffices to lift coordinate-wise CSs to anytime-valid welfare bounds (Theorem 4.1).
• SWF-UCB achieves near-optimal tilde{O}(n + sqrt(n k T)) regret for allocating k resources among n individuals over T steps.
• Exact oracles are provided for WPM and Kolm via water-filling in O(n log n); Gini via a greedy block-based algorithm in O(k n).
• Experiments show sqrt{T} scaling of regret across WPM, Kolm, and Gini, with non-monotone dependence of regret on k (highest near intermediate k).
• Intermediate SWF parameters yield richer learning dynamics, and the framework supports sequential inference tasks (e.g., sequential testing, optimal stopping, policy evaluation).

(b) Kolm SWF