online auction designers

Abstract: Controlling tail risk is an important objective in online optimization, and recently it has been studied in the context of competitive analysis. Continuing this line of research, we investigate pure tail constraints, which capture the tradeoff between expected and worst-case competitiveness. For two fundamental search problems, online bidding and line search, we derive the Pareto-optimal frontiers of this tradeoff. We then investigate another classic problem, TCP acknowledgment, which has structure similar to the iterated ski rental problem. There, we construct an algorithm whose tradeoff coincides with the known Pareto-optimal tradeoff for ski rental. The lower bounds for this problem are substantially more involved as the problem exhibits adaptive structure: an online algorithm observes requests of the adversary (packet arrivals) and may adaptively adjust its actions (acknowledgments) on this basis. We emphasize that all previous work on tail risk in the context of competitive analysis was restricted to non-adaptive problems, where the feedback given to an algorithm was essentially limited to a binary indicator of whether the algorithm has succeeded or not. Nonetheless, we identify a set of constraints implied by tail bounds in this adaptive setting, and show that they imply a nontrivial lower bound on the TCP acknowledgment problem.

Abstract: We study the online allocation of indivisible goods among $n$ agents, where each good must be allocated immediately and irrevocably upon arrival. Against an adaptive adversary, Neoh and Teh [2026] proved that no algorithm can guarantee a positive approximation to proportionality up to one good (PROP1) that is independent of the number of goods, and the same holds for proportionality up to $k$ goods (PROP$k$) for any fixed $k$. We instead consider an oblivious adversary, which fixes the input in advance. Choo et al. [2026] showed that the uniformly random allocation returns a $Θ(1/\log(n/δ))$-PROP1 allocation with probability at least $1-δ$. We improve this to $Ω(1/\log\log(n/δ))$; our algorithm does not take $δ$ as input, so the same algorithm achieves this bound for every $δ\in(0,1)$. Moreover, with the same probability, a variant of our algorithm gives every agent almost her bundle, and even without adding any good when no single good is too valuable relative to this share. In contrast, for envy-freeness up to one good (EF1), we show that, for every $α\in(0,1]$, every randomized algorithm has an input on which its probability of returning an $α$-EF1 allocation is at most $e^{-Ω(n)}$. For envy-freeness up to any good (EFX), this probability is at most $1/n!$ with only $n+1$ goods, a bound that is optimal within a factor of $(n+1)/2$. For the maximin share (MMS), this probability is at most $5/6$, however small $α$ is. Allowing more removals gives a positive envy-freeness guarantee: allocating each good to a uniformly random agent among those with positive values achieves, with high probability, an approximation factor arbitrarily close to one for envy-freeness up to logarithmically many goods, and logarithmically many goods are necessary for this rule.

Abstract: The matroid secretary problem asks an online algorithm to select a high-weight independent set from elements arriving in uniformly random order, with immediate and irrevocable decisions. Singla (2026) recently gave a $4$-competitive algorithm for arbitrary matroids using only the number of elements and independence queries on already-arrived elements. Following his approach, we obtain an improved competitive ratio of $2+\sqrt3\approx3.7321$ in the same information model. Our algorithm accepts every element of a fixed canonical optimum with probability at least $2-\sqrt3$ and uses $O(n^2)$ independence queries. The algorithm modifies Singla's reversible reference process by retaining a randomly chosen part of the sample as a reserve whose membership in the reference greedy solution is not frozen. Balancing the remaining sample and post-sample elements preserves reversibility and allows an exact calculation of the probability that an exchange partner blocks a target element. The resulting guarantee has a direct analytic proof.