Mechanism Design For Covering Problems

Algorithmic mechanism design deals with efficiently-computable algorithmic constructions in the presence of strategic players who hold the inputs to the problem and may misreport their input if doing so benefits them. Algorithmic mechanism design finds applications in a variety of internet settings such as resource allocation, facility location and e-commerce, such as sponsored search auctions. There is an extensive amount of work in algorithmic mechanism design on packing problems such as single-item auctions, multi-unit auctions and combinatorial auctions. But, surprisingly, covering problems, also called procurement auctions, have almost been completely unexplored, especially in the multidimensional setting. In this thesis, we systematically investigate multidimensional covering mechanism- design problems, wherein there are m items that need to be covered and n players who provide covering objects, with each player i having a private cost for the covering objects he provides. A feasible solution to the covering problem is a collection of covering objects (obtained from the various players) that together cover all items. Two widely considered objectives in mechanism design are: (i) cost-minimization (CM) which aims to minimize the total cost incurred by the players and the mechanism designer; and (ii) payment minimization (PayM), which aims to minimize the payment to players. Covering mechanism design problems turn out to behave quite differently from packing mechanism design problems. In particular, various techniques utilized successfully for packing problems do not perform well for covering mechanism design problems, and this necessitates new approaches and solution concepts. In this thesis we devise various techniques for handling covering mechanism design problems, which yield a variety of results for both the CM and PayM objectives. In our investigation of the CM objective, we focus on two representative covering problems: uncapacitated facility location (UFL) and vertex cover. For multi-dimensional UFL, we give a black-box method to transform any Lagrangian-multiplier-preserving ρ-approximation algorithm for UFL into a truthful-in-expectation, ρ-approximation mechanism. This yields the first result for multi-dimensional UFL, namely a truthful-in-expectation 2-approximation mechanism. For multi-dimensional VCP (Multi-VCP), we develop a decomposition method that reduces the mechanism-design problem into the simpler task of constructing threshold mechanisms, which are a restricted class of truthful mechanisms, for simpler (in terms of graph structure or problem dimension) instances of Multi-VCP. By suitably designing the decomposition and the threshold mechanisms it uses as building blocks, we obtain truthful mechanisms with approximation ratios (n is the number of nodes): (1) O(r2 log n) for r-dimensional VCP; and (2) O(r log n) for r-dimensional VCP on any proper minor-closed family of graphs (which improves to O(log n) if no two neighbors of a node belong to the same player). These are the first truthful mechanisms for Multi-VCP with non-trivial approximation guarantees. For the PayM objective, we work in the oft-used Bayesian setting, where players’ types are drawn from an underlying distribution and may be correlated, and the goal is to minimize the expected total payment made by the mechanism. We consider the problem of designing incentive compatible, ex-post individually rational (IR) mechanisms for covering problems in the above model. The standard notion of incentive compatibility (IC) in such settings is Bayesian incentive compatibility (BIC), but this notion is over-reliant on having precise knowledge of the underlying distribution, which makes it a rather non- robust notion. We formulate a notion of IC that we call robust Bayesian IC (robust BIC) that is substantially more robust than BIC, and develop black-box reductions from robust BIC-mechanism design to algorithm design. This black-box reduction applies to single- dimensional settings even when we only have an LP-relative approximation algorithm for the algorithmic problem. We obtain near-optimal mechanisms for various covering settings including single- and multi-item procurement auctions, various single-dimensional covering problems, and multidimensional facility location problems. Finally, we study the notion of frugality, which considers the PayM objective but in a worst-case setting, where one does not have prior information about the players’ types. We show that some of our mechanisms developed for the CM objective are also good with respect to certain oft-used frugality benchmarks proposed in the literature. We also introduce an alternate benchmark for frugality, which more directly reflects the goal that the mechanism’s payment be close to the best possible payment, and obtain some preliminary results with respect to this benchmark

Near-Optimal and Robust Mechanism Design for Covering Problems with Correlated Players

We consider the problem of designing incentive-compatible, ex-post individually rational (IR) mechanisms for covering problems in the Bayesian setting, where players' types are drawn from an underlying distribution and may be correlated, and the goal is to minimize the expected total payment made by the mechanism. We formulate a notion of incentive compatibility (IC) that we call {\em support-based IC} that is substantially more robust than Bayesian IC, and develop black-box reductions from support-based-IC mechanism design to algorithm design. For single-dimensional settings, this black-box reduction applies even when we only have an LP-relative {\em approximation algorithm} for the algorithmic problem. Thus, we obtain near-optimal mechanisms for various covering settings including single-dimensional covering problems, multi-item procurement auctions, and multidimensional facility location.Comment: Major changes compared to the previous version. Please consult this versio

Makespan Minimization via Posted Prices

We consider job scheduling settings, with multiple machines, where jobs arrive online and choose a machine selfishly so as to minimize their cost. Our objective is the classic makespan minimization objective, which corresponds to the completion time of the last job to complete. The incentives of the selfish jobs may lead to poor performance. To reconcile the differing objectives, we introduce posted machine prices. The selfish job seeks to minimize the sum of its completion time on the machine and the posted price for the machine. Prices may be static (i.e., set once and for all before any arrival) or dynamic (i.e., change over time), but they are determined only by the past, assuming nothing about upcoming events. Obviously, such schemes are inherently truthful. We consider the competitive ratio: the ratio between the makespan achievable by the pricing scheme and that of the optimal algorithm. We give tight bounds on the competitive ratio for both dynamic and static pricing schemes for identical, restricted, related, and unrelated machine settings. Our main result is a dynamic pricing scheme for related machines that gives a constant competitive ratio, essentially matching the competitive ratio of online algorithms for this setting. In contrast, dynamic pricing gives poor performance for unrelated machines. This lower bound also exhibits a gap between what can be achieved by pricing versus what can be achieved by online algorithms

Truthful Allocation in Graphs and Hypergraphs

We study truthful mechanisms for allocation problems in graphs, both for the minimization (i.e., scheduling) and maximization (i.e., auctions) setting. The minimization problem is a special case of the well-studied unrelated machines scheduling problem, in which every given task can be executed only by two pre-specified machines in the case of graphs or a given subset of machines in the case of hypergraphs. This corresponds to a multigraph whose nodes are the machines and its hyperedges are the tasks. This class of problems belongs to multidimensional mechanism design, for which there are no known general mechanisms other than the VCG and its generalization to affine minimizers. We propose a new class of mechanisms that are truthful and have significantly better performance than affine minimizers in many settings. Specifically, we provide upper and lower bounds for truthful mechanisms for general multigraphs, as well as special classes of graphs such as stars, trees, planar graphs, k-degenerate graphs, and graphs of a given treewidth. We also consider the objective of minimizing or maximizing the L^p-norm of the values of the players, a generalization of the makespan minimization that corresponds to p = ?, and extend the results to any p > 0

Truthful Mechanisms For Resource Allocation And Pricing In Clouds

A major challenging problem for cloud providers is designing efficient mechanisms for Virtual Machine (VM) provisioning and allocation. Such mechanisms enable the cloud providers to effectively utilize their available resources and obtain higher profits. Recently, cloud providers have introduced auction-based models for VM provisioning and allocation which allow users to submit bids for their requested VMs. We formulate the dynamic VM provisioning and allocation problem for the auction-based model as an integer program considering multiple types of resources. We then design truthful greedy and optimal mechanisms for the problem such that the cloud provider provisions VMs based on the requests of the winning users and determines their payments. We show that the proposed mechanisms are truthful, that is, the users do not have incentives to manipulate the system by lying about their requested bundles of VM instances and their valuations. We perform extensive experiments using real workload traces in order to investigate the performance of the proposed mechanisms. Our proposed mechanisms achieve promising results in terms of revenue for the cloud provider

Modern public finances as a proposal for an emerging country: The social approach in the fight against poverty in Mexico

In Mexico, the management of public resources has been questioned by the State, and mainly the results that the public administration at its three levels (federal, state and municipal), by the lack of transparency in the application and verification of public resources. The experience that gives us the operation of different emerging programs that focused on reducing social and economic inequality in the country, we can locate them as the first attempts in the search for a solution that is complex. Moving from the role of the benefactor and welfare state to the promoter of the regions and in the recognition of the focalization of priority attention areas, the path that has been taken is not only the beginning. Recognizing the public nature of public finances as a promoter of social development, we must understand it as the one assumed by the State through social, political and economic co-responsibility to solve poverty and marginalization of its own public policy orientation and vision of solution has been made since the eighties. From the above, we can point out some preliminary conclusions including the study of the indigenous language-speaking population with a high level of social exclusion in the methodology for the definition of multidimensional poverty in Mexico, will allow the allocation of public resources in the fight against poverty to be effective since it will make it possible to identify to the target population that is subject to social exclusion and marginalization. This invites us to a final reflection: What to do to address the just social demands of the indigenous population that is immersed in poverty, marginalization and exclusion? What to do so that they do not leave their communities, and if they have already done so, how to attend to the needs of family groups that are due to the expense of a remittance that may never arrive

Trust-Based Mechanisms for Robust and Efficient Task Allocation in the Presence of Execution Uncertainty

Vickrey-Clarke-Groves (VCG) mechanisms are often used to allocate tasks to selfish and rational agents. VCG mechanisms are incentive-compatible, direct mechanisms that are efficient (i.e. maximise social utility) and individually rational (i.e. agents prefer to join rather than opt out). However, an important assumption of these mechanisms is that the agents will always successfully complete their allocated tasks. Clearly, this assumption is unrealistic in many real-world applications where agents can, and often do, fail in their endeavours. Moreover, whether an agent is deemed to have failed may be perceived differently by different agents. Such subjective perceptions about an agent’s probability of succeeding at a given task are often captured and reasoned about using the notion of trust. Given this background, in this paper, we investigate the design of novel mechanisms that take into account the trust between agents when allocating tasks. Specifically, we develop a new class of mechanisms, called trust-based mechanisms, that can take into account multiple subjective measures of the probability of an agent succeeding at a given task and produce allocations that maximise social utility, whilst ensuring that no agent obtains a negative utility. We then show that such mechanisms pose a challenging new combinatorial optimisation problem (that is NP-complete), devise a novel representation for solving the problem, and develop an effective integer programming solution (that can solve instances with about 2×105 possible allocations in 40 seconds).