Water-Filling is Universally Minimax Optimal

We study the problem of fair online resource allocation, motivated by applications such as refugee resettlement and airline scheduling, where agents arrive sequentially and must be assigned to facilities with limited capacities. We introduce a model that maximizes the overall welfare subject to resource constraints and a Lipschitz fairness requirement, which ensures that similar agents arriving in the same batch receive similar expected outcomes. We first analyze the offline problem, proving that the value of the optimal fair allocation is at least an $Ω(1/γ)$ fraction of the optimal unfair allocation, where $γ$ is the fairness coefficient, thereby bounding the price of fairness. For the online setting, we propose an algorithm based on dual mirror descent that enforces fairness constraints within batches while estimating optimal dual variables. We prove that this algorithm achieves sublinear regret relative to the optimal offline fluid benchmark. Finally, we validate our theoretical results using real-world data from the Refugee Economies Programme, demonstrating the algorithm's performance and examining the trade-offs between welfare maximization and fairness enforcement.

Competitive analysis is central to the study of online algorithms, but upper bounds are often highly problem-specific. We develop a more unifying methodology via the minimax viewpoint. Guided by Yao's principle, we reduce worst-case competitive analysis to Bayesian online design under an arbitrary correlated prior over arrival sequences. For such a prior, let $X^*$ be the hindsight-optimal fractional solution for the realized instance, and let $X^{(t)}=\mathbb E[X^*\mid \mathcal F_t]$ be its posterior process. Our guiding rule is posterior matching: at each time $t$, choose the feasible online action that tracks the current posterior $X^{(t)}$ as closely as the online constraints permit. We show that this single principle yields optimal or near-optimal guarantees for several classical online fractional problems, including set cover, load balancing, matching and more general resource-allocation problems, recovering or improving state-of-the-art bounds in these settings with norm/concave objectives. Via known rounding reductions, it also yields randomized integral guarantees for weighted paging, MTS on star metrics, and ski-rental. At a technical level, our analysis reduces competitive guarantees to key probabilistic inequalities for the vector martingales generated by the posterior of the offline optimum. The resulting framework gives a reusable route from Bayesian online design under arbitrary correlated priors to information-theoretic worst-case competitive guarantees.

We study online fair allocation of $T$ sequentially arriving items among $n$ agents with heterogeneous preferences, with the objective of maximizing generalized-mean welfare, defined as the $p$-mean of agents' time-averaged utilities, with $p\in (-\infty, 1)$. We first consider the i.i.d. arrival model and show that the pure greedy algorithm -- which myopically chooses the welfare-maximizing integral allocation -- achieves $\widetilde{O}(1/T)$ average regret. Importantly, in contrast to prior work, our algorithm does not require distributional knowledge and achieves the optimal regret rate using only the online samples. We then go beyond i.i.d. arrivals and investigate a nonstationary model with time-varying independent distributions. In the absence of additional data about the distributions, it is known that every online algorithm must suffer $Ω(1)$ average regret. We show that only a single historical sample from each distribution is sufficient to recover the optimal $\widetilde{O}(1/T)$ average regret rate, even in the face of arbitrary non-stationarity. Our algorithms are based on the re-solving paradigm: they assume that the remaining items will be the ones seen historically in those periods and solve the resulting welfare-maximization problem to determine the decision in every period. Finally, we also account for distribution shifts that may distort the fidelity of historical samples and show that the performance of our re-solving algorithms is robust to such shifts.