2,004 research outputs found
Cost Sharing over Combinatorial Domains: Complement-Free Cost Functions and Beyond
We study mechanism design for combinatorial cost sharing models. Imagine that multiple items or services are available to be shared among a set of interested agents. The outcome of a mechanism in this setting consists of an assignment, determining for each item the set of players who are granted service, together with respective payments. Although there are several works studying specialized versions of such problems, there has been almost no progress for general combinatorial cost sharing domains until recently [S. Dobzinski and S. Ovadia, 2017]. Still, many questions about the interplay between strategyproofness, cost recovery and economic efficiency remain unanswered.
The main goal of our work is to further understand this interplay in terms of budget balance and social cost approximation. Towards this, we provide a refinement of cross-monotonicity (which we term trace-monotonicity) that is applicable to iterative mechanisms. The trace here refers to the order in which players become finalized. On top of this, we also provide two parameterizations (complementary to a certain extent) of cost functions which capture the behavior of their average cost-shares. Based on our trace-monotonicity property, we design a scheme of ascending cost sharing mechanisms which is applicable to the combinatorial cost sharing setting with symmetric submodular valuations. Using our first cost function parameterization, we identify conditions under which our mechanism is weakly group-strategyproof, O(1)-budget-balanced and O(H_n)-approximate with respect to the social cost. Further, we show that our mechanism is budget-balanced and H_n-approximate if both the valuations and the cost functions are symmetric submodular; given existing impossibility results, this is best possible. Finally, we consider general valuation functions and exploit our second parameterization to derive a more fine-grained analysis of the Sequential Mechanism introduced by Moulin. This mechanism is budget balanced by construction, but in general only guarantees a poor social cost approximation of n. We identify conditions under which the mechanism achieves improved social cost approximation guarantees. In particular, we derive improved mechanisms for fundamental cost sharing problems, including Vertex Cover and Set Cover
Dispersion for Data-Driven Algorithm Design, Online Learning, and Private Optimization
Data-driven algorithm design, that is, choosing the best algorithm for a
specific application, is a crucial problem in modern data science.
Practitioners often optimize over a parameterized algorithm family, tuning
parameters based on problems from their domain. These procedures have
historically come with no guarantees, though a recent line of work studies
algorithm selection from a theoretical perspective. We advance the foundations
of this field in several directions: we analyze online algorithm selection,
where problems arrive one-by-one and the goal is to minimize regret, and
private algorithm selection, where the goal is to find good parameters over a
set of problems without revealing sensitive information contained therein. We
study important algorithm families, including SDP-rounding schemes for problems
formulated as integer quadratic programs, and greedy techniques for canonical
subset selection problems. In these cases, the algorithm's performance is a
volatile and piecewise Lipschitz function of its parameters, since tweaking the
parameters can completely change the algorithm's behavior. We give a sufficient
and general condition, dispersion, defining a family of piecewise Lipschitz
functions that can be optimized online and privately, which includes the
functions measuring the performance of the algorithms we study. Intuitively, a
set of piecewise Lipschitz functions is dispersed if no small region contains
many of the functions' discontinuities. We present general techniques for
online and private optimization of the sum of dispersed piecewise Lipschitz
functions. We improve over the best-known regret bounds for a variety of
problems, prove regret bounds for problems not previously studied, and give
matching lower bounds. We also give matching upper and lower bounds on the
utility loss due to privacy. Moreover, we uncover dispersion in auction design
and pricing problems
Public projects, Boolean functions and the borders of Border's theorem
Border's theorem gives an intuitive linear characterization of the feasible
interim allocation rules of a Bayesian single-item environment, and it has
several applications in economic and algorithmic mechanism design. All known
generalizations of Border's theorem either restrict attention to relatively
simple settings, or resort to approximation. This paper identifies a
complexity-theoretic barrier that indicates, assuming standard complexity class
separations, that Border's theorem cannot be extended significantly beyond the
state-of-the-art. We also identify a surprisingly tight connection between
Myerson's optimal auction theory, when applied to public project settings, and
some fundamental results in the analysis of Boolean functions.Comment: Accepted to ACM EC 201
Complexity Theory, Game Theory, and Economics: The Barbados Lectures
This document collects the lecture notes from my mini-course "Complexity
Theory, Game Theory, and Economics," taught at the Bellairs Research Institute
of McGill University, Holetown, Barbados, February 19--23, 2017, as the 29th
McGill Invitational Workshop on Computational Complexity.
The goal of this mini-course is twofold: (i) to explain how complexity theory
has helped illuminate several barriers in economics and game theory; and (ii)
to illustrate how game-theoretic questions have led to new and interesting
complexity theory, including recent several breakthroughs. It consists of two
five-lecture sequences: the Solar Lectures, focusing on the communication and
computational complexity of computing equilibria; and the Lunar Lectures,
focusing on applications of complexity theory in game theory and economics. No
background in game theory is assumed.Comment: Revised v2 from December 2019 corrects some errors in and adds some
recent citations to v1 Revised v3 corrects a few typos in v
Prophet inequalities made easy: stochastic optimization by pricing nonstochastic inputs
We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold algorithms for settings beyond binary selection. Our analysis takes the form of an extension theorem: we derive sufficient conditions on prices when all weights are known in advance, then prove that the resulting approxima- tion guarantees extend directly to stochastic settings. Our framework unifies and simplifies much of the existing literature on prophet inequalities and posted price mechanisms and is used to derive new and improved results for combinatorial markets (with and without complements), multidimensional matroids, and sparse packing problems. Finally, we highlight a surprising connection between the smoothness framework for bounding the price of anarchy of mechanisms and our framework, and show that many smooth mechanisms can be recast as posted price mechanisms with comparable performance guarantees
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