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Pablo Daniel Azar

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3 papers

1 author row

SODA Conference 2014 Conference Paper

Prophet Inequalities with Limited Information

Pablo Daniel Azar
Robert Kleinberg
S. Matthew Weinberg

In the classical prophet inequality, a gambler observes a sequence of stochastic rewards V 1, …, V n and must decide, for each reward V i, whether to keep it and stop the game or to forfeit the reward forever and reveal the next value V i. The gambler's goal is to obtain a constant fraction of the expected reward that the optimal offline algorithm would get. Recently, prophet inequalities have been generalized to settings where the gambler can choose k items, and, more generally, where he can choose any independent set in a matroid. However, all the existing algorithms require the gambler to know the distribution from which the rewards V 1, …, V n are drawn. The assumption that the gambler knows the distribution from which V 1, …, V n are drawn is very strong. Instead, we work with the much simpler assumption that the gambler only knows a few samples from this distribution. We construct the first single-sample prophet inequalities for many settings of interest, whose guarantees all match the best possible asymptotically, even with full knowledge of the distribution. Specifically, we provide a novel single-sample algorithm when the gambler can choose any k elements whose analysis is based on random walks with limited correlation. In addition, we provide a black-box method for converting specific types of solutions to the related secretary problem to single-sample prophet inequalities, and apply it to several existing algorithms. Finally, we provide a constant-sample prophet inequality for constant-degree bipartite matchings. In addition, we apply these results to design the first posted-price and multi-dimensional auction mechanisms with limited information in settings with asymmetric bidders. Connections between prophet inequalities and posted-price mechanisms are already known, but applying the existing framework requires knowledge of the underlying distributions, as well as the so-called “virtual values” even when the underlying prophet inequalities do not. We therefore provide an extension of this framework that bypasses virtual values altogether, allowing our mechanisms to take full advantage of the limited information required by our new prophet inequalities.

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SODA Conference 2013 Conference Paper

Optimal and Efficient Parametric Auctions

Pablo Daniel Azar
Constantinos Daskalakis
Silvio Micali
S. Matthew Weinberg

Consider a seller who seeks to provide service to a collection of interested parties, subject to feasibility constraints on which parties may be simultaneously served. Assuming that a distribution is known on the value of each party for service—arguably a strong assumption—Myerson's seminal work provides revenue optimizing auctions [12]. We show instead that, for very general feasibility constraints, only knowledge of the median of each party's value distribution, or any other quantile of these distributions, or approximations thereof, suffice for designing simple auctions that simultaneously approximate both the optimal revenue and the optimal welfare. Our results apply to all downward-closed feasibility constraints under the assumption that the underlying, unknown value distributions are monotone hazard rate, and to all matroid feasibility constraints under the weaker assumption of regularity of the underlying distributions. Our results jointly generalize the single-item results obtained by Azar and Micali [2] on parametric auctions, and Daskalakis and Pierrakos [6] for simultaneously approximately optimal and efficient auctions.

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STOC Conference 2012 Conference Paper

Rational proofs

Pablo Daniel Azar
Silvio Micali

We study a new type of proof system, where an unbounded prover and a polynomial time verifier interact, on inputs a string x and a function f, so that the Verifier may learn f(x). The novelty of our setting is that there no longer are "good" or "malicious" provers, but only rational ones. In essence, the Verifier has a budget c and gives the Prover a reward r ∈ [0,c] determined by the transcript of their interaction; the prover wishes to maximize his expected reward; and his reward is maximized only if he the verifier correctly learns f(x). Rational proof systems are as powerful as their classical counterparts for polynomially many rounds of interaction, but are much more powerful when we only allow a constant number of rounds. Indeed, we prove that if f ∈ #P, then f is computable by a one-round rational Merlin-Arthur game, where, on input x, Merlin's single message actually consists of sending just the value f(x). Further, we prove that CH, the counting hierarchy, coincides with the class of languages computable by a constant-round rational Merlin-Arthur game. Our results rely on a basic and crucial connection between rational proof systems and proper scoring rules, a tool developed to elicit truthful information from experts.

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