Online Social Welfare Function-based Resource Allocation

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.

💡 Research Summary

The paper tackles a fundamental problem that arises whenever a central planner must repeatedly allocate a limited number of identical, indivisible resources to a fixed population over many time steps. Each individual’s utility from receiving a resource is stochastic with an unknown mean, and the planner evaluates any allocation by aggregating the expected utilities through a social welfare function (SWF). The authors formalize this setting, impose three natural conditions on the SWF—monotonicity, concavity, and ℓ∞‑Lipschitz continuity—and develop a unified online learning and inference framework that works for any SWF satisfying these assumptions.

The key theoretical contribution is a “confidence‑sequence lifting” theorem. Starting from standard (time‑uniform) confidence sequences for each individual mean utility, monotonicity alone allows the construction of a valid confidence sequence for the optimal welfare value M(μ⊙p*). Specifically, lower and upper confidence bounds on the means are plugged into the SWF, and the resulting optimistic and pessimistic allocations (p↓t and p↑t) yield lower and upper bounds on the optimal welfare that hold uniformly over time with probability 1‑δ. This result requires no additional smoothness beyond monotonicity and thus provides a very general statistical primitive.

Building on this, the authors propose SWF‑UCB, an Upper‑Confidence‑Bound style algorithm that is SWF‑agnostic. At each round t, the algorithm updates coordinate‑wise confidence intervals for the means, computes the optimistic mean vector μ↑t, and solves a constrained maximization problem
 p​t = argmax_{p∈𝒫k} M(μ↑t ⊙ p)
where 𝒫k = {p∈