301 research outputs found
On Simultaneous Two-player Combinatorial Auctions
We consider the following communication problem: Alice and Bob each have some
valuation functions and over subsets of items,
and their goal is to partition the items into in a way that
maximizes the welfare, . We study both the allocation
problem, which asks for a welfare-maximizing partition and the decision
problem, which asks whether or not there exists a partition guaranteeing
certain welfare, for binary XOS valuations. For interactive protocols with
communication, a tight 3/4-approximation is known for both
[Fei06,DS06].
For interactive protocols, the allocation problem is provably harder than the
decision problem: any solution to the allocation problem implies a solution to
the decision problem with one additional round and additional bits of
communication via a trivial reduction. Surprisingly, the allocation problem is
provably easier for simultaneous protocols. Specifically, we show:
1) There exists a simultaneous, randomized protocol with polynomial
communication that selects a partition whose expected welfare is at least
of the optimum. This matches the guarantee of the best interactive, randomized
protocol with polynomial communication.
2) For all , any simultaneous, randomized protocol that
decides whether the welfare of the optimal partition is or correctly with probability requires
exponential communication. This provides a separation between the attainable
approximation guarantees via interactive () versus simultaneous () protocols with polynomial communication.
In other words, this trivial reduction from decision to allocation problems
provably requires the extra round of communication
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
Bayesian Incentive Compatibility via Fractional Assignments
Very recently, Hartline and Lucier studied single-parameter mechanism design
problems in the Bayesian setting. They proposed a black-box reduction that
converted Bayesian approximation algorithms into Bayesian-Incentive-Compatible
(BIC) mechanisms while preserving social welfare. It remains a major open
question if one can find similar reduction in the more important
multi-parameter setting. In this paper, we give positive answer to this
question when the prior distribution has finite and small support. We propose a
black-box reduction for designing BIC multi-parameter mechanisms. The reduction
converts any algorithm into an eps-BIC mechanism with only marginal loss in
social welfare. As a result, for combinatorial auctions with sub-additive
agents we get an eps-BIC mechanism that achieves constant approximation.Comment: 22 pages, 1 figur
Optimal Approximation Algorithms for Multi-agent Combinatorial Problems with Discounted Price Functions
Submodular functions are an important class of functions in combinatorial
optimization which satisfy the natural properties of decreasing marginal costs.
The study of these functions has led to strong structural properties with
applications in many areas. Recently, there has been significant interest in
extending the theory of algorithms for optimizing combinatorial problems (such
as network design problem of spanning tree) over submodular functions.
Unfortunately, the lower bounds under the general class of submodular functions
are known to be very high for many of the classical problems.
In this paper, we introduce and study an important subclass of submodular
functions, which we call discounted price functions. These functions are
succinctly representable and generalize linear cost functions. In this paper we
study the following fundamental combinatorial optimization problems: Edge
Cover, Spanning Tree, Perfect Matching and Shortest Path, and obtain tight
upper and lower bounds for these problems.
The main technical contribution of this paper is designing novel adaptive
greedy algorithms for the above problems. These algorithms greedily build the
solution whist rectifying mistakes made in the previous steps
Single Parameter Combinatorial Auctions with Partially Public Valuations
We consider the problem of designing truthful auctions, when the bidders'
valuations have a public and a private component. In particular, we consider
combinatorial auctions where the valuation of an agent for a set of
items can be expressed as , where is a private single parameter
of the agent, and the function is publicly known. Our motivation behind
studying this problem is two-fold: (a) Such valuation functions arise naturally
in the case of ad-slots in broadcast media such as Television and Radio. For an
ad shown in a set of ad-slots, is, say, the number of {\em unique}
viewers reached by the ad, and is the valuation per-unique-viewer. (b)
From a theoretical point of view, this factorization of the valuation function
simplifies the bidding language, and renders the combinatorial auction more
amenable to better approximation factors. We present a general technique, based
on maximal-in-range mechanisms, that converts any -approximation
non-truthful algorithm () for this problem into
and -approximate truthful
mechanisms which run in polynomial time and quasi-polynomial time,
respectively
- …