Prophet Secretary for Combinatorial Auctions and Matroids

The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items because of their applications in mechanism design. The most important of these generalizations are to matroids and to combinatorial auctions (extends bipartite matching). Kleinberg-Weinberg \cite{KW-STOC12} and Feldman et al. \cite{feldman2015combinatorial} show that for adversarial arrival order of random variables the optimal prophet inequalities give a $1/2$-approximation. For many settings, however, it's conceivable that the arrival order is chosen uniformly at random, akin to the secretary problem. For such a random arrival model, we improve upon the $1/2$-approximation and obtain $(1-1/e)$-approximation prophet inequalities for both matroids and combinatorial auctions. This also gives improvements to the results of Yan \cite{yan2011mechanism} and Esfandiari et al. \cite{esfandiari2015prophet} who worked in the special cases where we can fully control the arrival order or when there is only a single item. Our techniques are threshold based. We convert our discrete problem into a continuous setting and then give a generic template on how to dynamically adjust these thresholds to lower bound the expected total welfare.Comment: Preliminary version appeared in SODA 2018. This version improves the writeup on Fixed-Threshold algorithm

On the best-choice prophet secretary problem

We study a variant of the secretary problem where candidates come from independent, not necessarily identical distributions known to us, and show that we can do at least as well as in the IID setting. This resolves a conjecture of Esfandiari et al.Comment: 7 page

Fishing For Better Constants: The Prophet Secretary Via Poissonization

Given n random variables $X_1, \ldots , X_n$ taken from known distributions, a gambler observes their realizations in this order, and needs to select one of them, immediately after it is being observed, so that its value is as high as possible. The classical prophet inequality shows a strategy that guarantees a value at least half (in expectation) of that an omniscience prophet that picks the maximum, and this ratio is tight. Esfandiari, Hajiaghayi, Liaghat, and Monemizadeh introduced a variant of the prophet inequality, the prophet secretary problem in [1]. The difference being that that the realizations arrive at a random permutation order, and not an adversarial order. Esfandiari et al. gave a simple $1-1/e \approx 0.632$ competitive algorithm for the problem. This was later improved in a surprising result by Azar, Chiplunkar and Kaplan [2] into a $1-1/e + 1/400 \approx 0.634$ competitive algorithm. In a subsequent result, Correa, Saona, and Ziliotto [3] took a systematic approach, introducing blind strategies, and gave an improved $0.669$ competitive algorithm. Since then, there has been no improvements on the lower bounds. Meanwhile, current upper bounds show that no algorithm can achieve a competitive ratio better than $0.7235$ [4]. In this paper, we give a $0.6724$-competitive algorithm for the prophet secretary problem. The algorithm follows blind strategies introduced by [3] but has a technical difference. We do this by re-interpretting the blind strategies, framing them as Poissonization strategies. We break the non-iid random variables into iid shards and argue about the competitive ratio in terms of events on shards. This gives significantly simpler and direct proofs, in addition to a tighter analysis on the competitive ratio. The analysis might be of independent interest for similar problems such as the prophet inequality with order-selectionComment: 15 page

Optimal Single-Choice Prophet Inequalities from Samples

We study the single-choice Prophet Inequality problem when the gambler is given access to samples. We show that the optimal competitive ratio of $1/2$ can be achieved with a single sample from each distribution. When the distributions are identical, we show that for any constant $\varepsilon > 0$, $O(n)$ samples from the distribution suffice to achieve the optimal competitive ratio ($\approx 0.745$) within $(1+\varepsilon)$, resolving an open problem of Correa, D\"utting, Fischer, and Schewior.Comment: Appears in Innovations in Theoretical Computer Science (ITCS) 202

Improved Revenue Bounds for Posted-Price and Second-Price Mechanisms

We study revenue maximization through sequential posted-price (SPP) mechanisms in single-dimensional settings with $n$ buyers and independent but not necessarily identical value distributions. We construct the SPP mechanisms by considering the best of two simple pricing rules: one that imitates the revenue optimal mchanism, namely the Myersonian mechanism, via the taxation principle and the other that posts a uniform price. Our pricing rules are rather generalizable and yield the first improvement over long-established approximation factors in several settings. We design factor-revealing mathematical programs that crisply capture the approximation factor of our SPP mechanism. In the single-unit setting, our SPP mechanism yields a better approximation factor than the state of the art prior to our work (Azar, Chiplunkar & Kaplan, 2018). In the multi-unit setting, our SPP mechanism yields the first improved approximation factor over the state of the art after over nine years (Yan, 2011 and Chakraborty et al., 2010). Our results on SPP mechanisms immediately imply improved performance guarantees for the equivalent free-order prophet inequality problem. In the position auction setting, our SPP mechanism yields the first higher-than $1-1/e$ approximation factor. In eager second-price (ESP) auctions, our two simple pricing rules lead to the first improved approximation factor that is strictly greater than what is obtained by the SPP mechanism in the single-unit setting.Comment: Accepted to Operations Researc