Sequential Posted Price Mechanism Research Articles

Sequential Posted-Price Mechanisms with Correlated Valuations

We study the revenue performance of sequential posted-price mechanisms and some natural extensions for a setting where the valuations of the buyers are drawn from a correlated distribution. Sequential posted-price mechanisms are conceptually simple mechanisms that work by proposing a “take-it-or-leave-it” offer to each buyer. We apply sequential posted-price mechanisms to single-parameter multiunit settings in which each buyer demands only one item and the mechanism can assign the service to at most k of the buyers. For standard sequential posted-price mechanisms, we prove that with the valuation distribution having finite support, no sequential posted-price mechanism can extract a constant fraction of the optimal expected revenue, even with unlimited supply. We extend this result to the case of a continuous valuation distribution when various standard assumptions hold simultaneously (i.e., everywhere-supported, continuous, symmetric, and normalized (conditional) distributions that satisfy regularity , the MHR condition , and affiliation ). In fact, it turns out that the best fraction of the optimal revenue that is extractable by a sequential posted-price mechanism is proportional to the ratio of the highest and lowest possible valuation. We prove that a simple generalization of these mechanisms achieves a better revenue performance; namely, if the sequential posted-price mechanism has for each buyer the option of either proposing an offer or asking the buyer for its valuation, then a Ω (1/max { 1, d }) fraction of the optimal revenue can be extracted, where d denotes the degree of dependence of the valuations, ranging from complete independence ( d =0) to arbitrary dependence ( d = n -1).

Posting Prices with Unknown Distributions

We consider a dynamic auction model, where bidders sequentially arrive to the market. The values of the bidders for the item for sale are independently drawn from a distribution, but this distribution is unknown to the seller. The seller offers a personalized take-it-or-leave-it price for each arriving bidder and aims to maximize revenue. We study how well can such sequential posted-price mechanisms approximate the optimal revenue that would be achieved if the distribution was known to the seller. On the negative side, we show that sequential posted-price mechanisms cannot guarantee a constant fraction of this revenue when the class of candidate distributions is unrestricted. We show that this impossibility holds even for randomized mechanisms and even if the set of possible distributions is very small or when the seller has a prior distribution over the candidate distributions. On the positive side, we devise a simple posted-price mechanism that guarantees a constant fraction of the known-distribution revenue when all candidate distributions exhibit the monotone hazard rate property.

Sequential Peer Prediction: Learning to Elicit Effort using Posted Prices

Peer prediction mechanisms are often adopted to elicit truthful contributions from crowd workers when no ground-truth verification is available. Recently, mechanisms of this type have been developed to incentivize effort exertion, in addition to truthful elicitation. In this paper, we study a sequential peer prediction problem where a data requester wants to dynamically determine the reward level to optimize the trade-off between the quality of information elicited from workers and the total expected payment. In this problem, workers have homogeneous expertise and heterogeneous cost for exerting effort, both unknown to the requester. We propose a sequential posted-price mechanism to dynamically learn the optimal reward level from workers' contributions and to incentivize effort exertion and truthful reporting. We show that (1) in our mechanism, workers exerting effort according to a non-degenerate threshold policy and then reporting truthfully is an equilibrium that returns highest utility for every worker, and (2) The regret of our learning mechanism w.r.t. offering the optimal reward (price) is upper bounded by Õ(T{3/4) where T is the learning horizon. We further show the power of our learning approach when the reports of workers do not necessarily follow the game-theoretic equilibrium.

Matroid prophet inequalities and applications to multi-dimensional mechanism design

Consider a gambler who observes a sequence of independent random numbers and is allowed to stop at any time, claiming reward equal to the most recent observation. The famous prophet inequality of Krengel, Sucheston, and Garling asserts that a gambler who knows the distribution of each random variable can achieve half as much reward, in expectation, as a “prophet” who knows the sampled values and can choose the largest one. We generalize this result to settings in which the gambler and the prophet are allowed to make multiple selections, subject to a matroid constraint, showing that the gambler can still achieve half as much reward as the prophet.Our results imply improved bounds on the ability of sequential posted-price mechanisms to approximate optimal mechanisms in single-parameter and multi-parameter Bayesian settings. In particular, we provide the first efficiently computable constant-factor approximations to the Bayesian optimal revenue in certain multi-parameter settings.

Budget-Constrained Auctions with Heterogeneous Items

In this paper, we present the first approximation algorithms for the problem of designing revenue optimal Bayesian incentive compatible auctions when there are multiple (heterogeneous) items and when bidders can have arbitrary demand and budget constraints. Our mechanisms are surprisingly simple: We show that a sequential all-pay mechanism is a 4 approximation to the revenue of the optimal ex-interim truthful mechanism with discrete correlated type space for each bidder. We also show that a sequential posted price mechanism is a O(1) approximation to the revenue of the optimal ex-post truthful mechanism when the type space of each bidder is a product distribution that satisfies the standard hazard rate condition. We further show a logarithmic approximation when the hazard rate condition is removed, and complete the picture by showing that achieving a sub-logarithmic approximation, even for regular distributions and one bidder, requires pricing bundles of items. Our results are based on formulating novel LP relaxations for these problems, and developing generic rounding schemes from first principles. We believe this approach will be useful in other Bayesian mechanism design contexts.