103 research outputs found
Approximately Optimal Mechanism Design: Motivation, Examples, and Lessons Learned
Optimal mechanism design enjoys a beautiful and well-developed theory, and
also a number of killer applications. Rules of thumb produced by the field
influence everything from how governments sell wireless spectrum licenses to
how the major search engines auction off online advertising. There are,
however, some basic problems for which the traditional optimal mechanism design
approach is ill-suited --- either because it makes overly strong assumptions,
or because it advocates overly complex designs. The thesis of this paper is
that approximately optimal mechanisms allow us to reason about fundamental
questions that seem out of reach of the traditional theory.
This survey has three main parts. The first part describes the approximately
optimal mechanism design paradigm --- how it works, and what we aim to learn by
applying it. The second and third parts of the survey cover two case studies,
where we instantiate the general design paradigm to investigate two basic
questions. In the first example, we consider revenue maximization in a
single-item auction with heterogeneous bidders. Our goal is to understand if
complexity --- in the sense of detailed distributional knowledge --- is an
essential feature of good auctions for this problem, or alternatively if there
are simpler auctions that are near-optimal. The second example considers
welfare maximization with multiple items. Our goal here is similar in spirit:
when is complexity --- in the form of high-dimensional bid spaces --- an
essential feature of every auction that guarantees reasonable welfare? Are
there interesting cases where low-dimensional bid spaces suffice?Comment: Based on a talk given by the author at the 15th ACM Conference on
Economics and Computation (EC), June 201
A Unifying Hierarchy of Valuations with Complements and Substitutes
We introduce a new hierarchy over monotone set functions, that we refer to as
(Maximum over Positive Hypergraphs). Levels of the hierarchy
correspond to the degree of complementarity in a given function. The highest
level of the hierarchy, - (where is the total number of
items) captures all monotone functions. The lowest level, -,
captures all monotone submodular functions, and more generally, the class of
functions known as . Every monotone function that has a positive
hypergraph representation of rank (in the sense defined by Abraham,
Babaioff, Dughmi and Roughgarden [EC 2012]) is in -. Every
monotone function that has supermodular degree (in the sense defined by
Feige and Izsak [ITCS 2013]) is in -. In both cases, the
converse direction does not hold, even in an approximate sense. We present
additional results that demonstrate the expressiveness power of
-.
One can obtain good approximation ratios for some natural optimization
problems, provided that functions are required to lie in low levels of the
hierarchy. We present two such applications. One shows that the
maximum welfare problem can be approximated within a ratio of if all
players hold valuation functions in -. The other is an upper
bound of on the price of anarchy of simultaneous first price auctions.
Being in - can be shown to involve two requirements -- one
is monotonicity and the other is a certain requirement that we refer to as
(Positive Lower Envelope). Removing the monotonicity
requirement, one obtains the hierarchy over all non-negative
set functions (whether monotone or not), which can be fertile ground for
further research
Combinatorial auctions with restricted complements
Abstract Complements between goods -where one good takes on added value in the presence of another -have been a thorn in the side of algorithmic mechanism designers. On the one hand, complements are common in the standard motivating applications for combinatorial auctions, like spectrum license auctions. On the other, welfare maximization in the presence of complements is notoriously difficult, and this intractability has stymied theoretical progress in the area. For example, there are no known positive results for combinatorial auctions in which bidder valuations are multi-parameter and non-complement-free, other than the relatively weak results known for general valuations. To make inroads on the problem of combinatorial auction design in the presence of complements, we propose a model for valuations with complements that is parameterized by the "size" of the complements. The model permits a succinct representation, a variety of computationally efficient queries, and non-trivial welfare-maximization algorithms and mechanisms. Specifically, a hypergraph-r valuation v for a good set M is represented by a hypergraph H = (M, E), where every (hyper-)edge e ∈ E contains at most r vertices and has a nonnegative weight w e . Each good j ∈ M also has a nonnegative weight w j . The value v(S) for a subset S ⊆ M of goods is defined as the sum of the weights of the goods and edges entirely contained in S. We design the following polynomial-time approximation algorithms and truthful mechanisms for welfare maximization with bidders with hypergraph valuations. 1. For bidders whose valuations correspond to subgraphs of a known graph that is planar (or more generally, excludes a fixed minor), we give a truthful and (1 + ǫ)-approximate mechanism. 2. We give a polynomial-time, r-approximation algorithm for welfare maximization with hypergraph-r valuations. Our algorithm randomly rounds a compact linear programming relaxation of the problem. 3. We design a different approximation algorithm and use it to give a polynomial-time, truthful-in-expectation mechanism that has an approximation factor of O(log r m)
When Are Welfare Guarantees Robust?
Computational and economic results suggest that social welfare maximization and combinatorial auction design are much easier when bidders\u27 valuations satisfy the "gross substitutes" condition. The goal of this paper is to evaluate rigorously the folklore belief that the main take-aways from these results remain valid in settings where the gross substitutes condition holds only approximately. We show that for valuations that pointwise approximate a gross substitutes valuation (in fact even a linear valuation), optimal social welfare cannot be approximated to within a subpolynomial factor and demand oracles cannot be simulated using a subexponential number of value queries. We then provide several positive results by imposing additional structure on the valuations (beyond gross substitutes), using a more stringent notion of approximation, and/or using more powerful oracle access to the valuations. For example, we prove that the performance of the greedy algorithm degrades gracefully for near-linear valuations with approximately decreasing marginal values; that with demand queries, approximate welfare guarantees for XOS valuations degrade gracefully for valuations that are pointwise close to XOS; and that the performance of the Kelso-Crawford auction degrades gracefully for valuations that are close to various subclasses of gross substitutes valuations
Algorithms as Mechanisms: The Price of Anarchy of Relax-and-Round
Many algorithms that are originally designed without explicitly considering
incentive properties are later combined with simple pricing rules and used as
mechanisms. The resulting mechanisms are often natural and simple to
understand. But how good are these algorithms as mechanisms? Truthful reporting
of valuations is typically not a dominant strategy (certainly not with a
pay-your-bid, first-price rule, but it is likely not a good strategy even with
a critical value, or second-price style rule either). Our goal is to show that
a wide class of approximation algorithms yields this way mechanisms with low
Price of Anarchy.
The seminal result of Lucier and Borodin [SODA 2010] shows that combining a
greedy algorithm that is an -approximation algorithm with a
pay-your-bid payment rule yields a mechanism whose Price of Anarchy is
. In this paper we significantly extend the class of algorithms for
which such a result is available by showing that this close connection between
approximation ratio on the one hand and Price of Anarchy on the other also
holds for the design principle of relaxation and rounding provided that the
relaxation is smooth and the rounding is oblivious.
We demonstrate the far-reaching consequences of our result by showing its
implications for sparse packing integer programs, such as multi-unit auctions
and generalized matching, for the maximum traveling salesman problem, for
combinatorial auctions, and for single source unsplittable flow problems. In
all these problems our approach leads to novel simple, near-optimal mechanisms
whose Price of Anarchy either matches or beats the performance guarantees of
known mechanisms.Comment: Extended abstract appeared in Proc. of 16th ACM Conference on
Economics and Computation (EC'15
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
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