We study the design of optimal Bayesian data acquisition mechanisms for a platform interested in estimating the mean of a distribution by collecting data from privacy-conscious users. In our setting, users have heterogeneous sensitivities for two types of privacy losses corresponding to local and central differential privacy measures. The local privacy loss is due to the leakage of a user's information when she shares her data with the platform, and the central privacy loss is due to the released estimate by the platform to the public. The users share their data in exchange for a payment (e.g., through monetary transfers or services) that compensates for their privacy losses. The platform does not know the privacy sensitivity of users and must design a mechanism to solicit their preferences and then deliver both local and central privacy guarantees while minimizing the estimation error plus the expected payment to users. We first establish minimax lower bounds for the estimation error, given a vector of privacy guarantees, and show that a linear estimator is (near) optimal. We then turn to our main goal: designing an optimal data acquisition mechanism. We establish that the design of such mechanisms in a Bayesian setting (where the platform knows the distribution of users' sensitivities and not their realizations) can be cast as a nonconvex optimization problem. Additionally, for the class of linear estimators, we prove that finding the optimal mechanism admits a Polynomial Time Approximation Scheme.

We consider a platform's problem of collecting data from privacy sensitive users to estimate an underlying parameter of interest. We formulate this question as a Bayesian-optimal mechanism design problem, in which an individual can share her (verifiable) data in exchange for a monetary reward or services, but at the same time has a (private) heterogeneous privacy cost which we quantify using differential privacy. We consider two popular differential privacy settings for providing privacy guarantees for the users: central and local. In both settings, we establish minimax lower bounds for the estimation error and derive (near) optimal estimators for given heterogeneous privacy loss levels for users. Building on this characterization, we pose the mechanism design problem as the optimal selection of an estimator and payments that will elicit truthful reporting of users' privacy sensitivities. Under a regularity condition on the distribution of privacy sensitivities we develop efficient algorithmic mechanisms to solve this problem in both privacy settings. Our mechanism in the central setting can be implemented in time O (n log n) where n is the number of users and our mechanism in the local setting admits a Polynomial Time Approximation Scheme (PTAS).

We consider descending price auctions for selling m units of a good to unit demand i.i.d. buyers where there is an exogenous bound of k on the number of price levels the auction clock can take. The auctioneer's problem is to choose price levels p1 > p2 > ․․․ > pk for the auction clock such that auction expected revenue is maximized. The price levels are announced prior to the auction. We reduce this problem to a new variant of prophet inequality, which we call batched prophet inequality, where a decision-maker chooses k (decreasing) thresholds and then sequentially collects rewards (up to m) that are above the thresholds with ties broken uniformly at random. For the special case of m=1 (i.e., selling a single item), we show that the resulting descending auction with k price levels achieves 1- 1/ek of the unrestricted (without the bound of k) optimal revenue. That means a descending auction with just 4 price levels can achieve more than 98% of the optimal revenue. We then extend our results for m>1 and provide a closed-form bound on the competitive ratio of our auction as a function of the number of units m and the number of price levels k.

We consider the problem of selling a single item to n unit-demand buyers to maximize revenue, where the buyers' values are independently distributed (not necessarily identical) according to publicly known distributions but unknown to the buyers themselves, with the option of allowing buyers to inspect the item at a cost. This problem can be interpreted as a revenue maximizing variant of Weitzman's Pandora's problem with non-obligatory inspection. We present an approximation mechanism that achieves 1/2. The proposed mechanism generalizes to the case of selling k units of an item to unit-demand buyers, obtaining 1-1/√k+3 of the optimal revenue in expectation. The mechanism is sequential and has a simple implementation that works in an online setting where buyers arrive in an arbitrary unknown order, yet achieving the aforementioned approximation with respect to the optimal offline mechanism.

Crowdfunding is quickly emerging as an alternative to traditional methods of funding new products. In a crowdfunding campaign, a seller solicits financial contributions from a crowd, usually in the form of pre-buying an unrealized product, and commits to producing the product if the total amount pledged is above a certain threshold. We provide a model of crowdfunding in which consumers arrive sequentially and make decisions about whether to pledge or not. Pledging is not costless, and hence consumers would prefer not to pledge if they think the campaign will not succeed. This can lead to cascades where a campaign fails to raise the required amount even though there are enough consumers who want the product. The paper introduces a novel stochastic process --- anticipating random walks --- to analyze this problem. The analysis helps explain why some campaigns fail and some do not, and provides guidelines about how sellers should design their campaigns in order to maximize their chances of success. More broadly, Anticipating Random Walks can also find application in settings where agents make decisions sequentially and these decisions are not just affected by past actions of others, but also by how they will impact the decisions of future actors as well.

We study an abstract optimal auction problem for selecting a subset of self-interested agents to whom to provide a service. A feasibility constraint governs which subsets can be simultaneously served; however, the mechanism may additionally choose to bundle unconstrained attributes such as payments or add-ons with the service. An agent's preference over service and attributes is given by her private type and may be multi-dimensional and non-linear. A single-agent problem is to optimizes a menu to offer an agent subject to constraints on the probabilities with which each of the agent's types is served. We give computationally tractable reductions from multi-agent auction problems to these single-agent problems. Our discussion focuses on maximizing revenue, but our results can be applied to other objectives (e.g., welfare).

We study Bayesian mechanism design problems in settings where agents have budgets. Specifically, an agent's utility for an outcome is given by his value for the outcome minus any payment he makes to the mechanism, as long as the payment is below his budget, and is negative infinity otherwise. This discontinuity in the utility function presents a significant challenge in the design of good mechanisms, and classical mechanisms fail to work in settings with budgets. The goal of this paper is to develop general reductions from budget-constrained Bayesian MD to unconstrained Bayesian MD with small loss in performance. We consider this question in the context of the two most well-studied objectives in mechanism design---social welfare and revenue---and present constant factor approximations in a number of settings. Some of our results extend to settings where budgets are private and agents need to be incentivized to reveal them truthfully.

We give a simple reduction from Bayesian incentive compatible mechanism design to algorithm design in settings where the agents’ private types are multidimensional. The reduction preserves performance up to an additive loss that can be made arbitrarily small in polynomial time in the number of agents and the size of the agents’ type spaces.

In this article we study several routing problems that generalize shortest paths and the traveling salesman problem. We consider a more general model that incorporates the actual cost in terms of gas prices. We have a vehicle with a given tank capacity. We assume that at each vertex gas may be purchased at a certain price. The objective is to find the cheapest route to go from s to t , or the cheapest tour visiting a given set of locations. We show that the problem of finding a cheapest plan to go from s to t can be solved in polynomial time. For most other versions, however, the problem is NP-complete and we develop polynomial-time approximation algorithms for these versions.

Motivated by applications in guaranteed delivery in computational advertising, we consider the general problem of balanced allocation in a bipartite supply-demand setting. Our formulation captures the notion of deviation from being balanced by a convex penalty function. While this formulation admits a convex programming solution, we strive for more robust and scalable algorithms.

In the context of auctions for digital goods, an interesting Random Sampling Optimal Price auction (RSOP) has been proposed by Goldberg, Hartline and Wright; this leads to a truthful mechanism. Since random sampling is a popular approach for auctions that aims to maximize the seller's revenue, this method has been analyzed further by Feige, Flaxman, Hartline and Kleinberg, who have shown that it is 15-competitive in the worst case -- which is substantially better than the previously proved bounds but still far from the conjectured competitive ratio of 4. In this paper, we prove that RSOP is indeed 4-competitive for a large class of instances in which the number λ of bidders receiving the item at the optimal uniform price, is at least 6. We also show that it is 4.68 competitive for the small class of remaining instances thus leaving a negligible gap between the lower and upper bound. Furthermore, we develop a robust version of RSOP -- one in which the seller's revenue is, with high probability, not much below its mean -- when the above parameter λ grows large. We employ a mix of probabilistic techniques and dynamic programming to compute these bounds.

We consider the problem of query rewrites in the context of pay-per-click search advertising. Given a three-layer graph consisting of queries, query rewrites, and the corresponding ads that can be served for the rewrites, we formulate a family of graph covering problems whose goals are to suggest a subset of ads with the maximum benefit by suggesting rewrites for a given query. We obtain constant-factor approximation algorithms for these covering problems, under two versions of constraints and a realistic notion of ad benefit. We perform experiments on real data and show that our algorithms are capable of outperforming a competitive baseline algorithm in terms of the benefit of the rewrites.