155 research outputs found
Randomization beats Second Price as a Prior-Independent Auction
Designing revenue optimal auctions for selling an item to symmetric
bidders is a fundamental problem in mechanism design. Myerson (1981) shows that
the second price auction with an appropriate reserve price is optimal when
bidders' values are drawn i.i.d. from a known regular distribution. A
cornerstone in the prior-independent revenue maximization literature is a
result by Bulow and Klemperer (1996) showing that the second price auction
without a reserve achieves of the optimal revenue in the worst case.
We construct a randomized mechanism that strictly outperforms the second
price auction in this setting. Our mechanism inflates the second highest bid
with a probability that varies with . For two bidders we improve the
performance guarantee from to of the optimal revenue. We also
resolve a question in the design of revenue optimal mechanisms that have access
to a single sample from an unknown distribution. We show that a randomized
mechanism strictly outperforms all deterministic mechanisms in terms of worst
case guarantee
Envy Freedom and Prior-free Mechanism Design
We consider the provision of an abstract service to single-dimensional
agents. Our model includes position auctions, single-minded combinatorial
auctions, and constrained matching markets. When the agents' values are drawn
from a distribution, the Bayesian optimal mechanism is given by Myerson (1981)
as a virtual-surplus optimizer. We develop a framework for prior-free mechanism
design and analysis. A good mechanism in our framework approximates the optimal
mechanism for the distribution if there is a distribution; moreover, when there
is no distribution this mechanism still performs well.
We define and characterize optimal envy-free outcomes in symmetric
single-dimensional environments. Our characterization mirrors Myerson's theory.
Furthermore, unlike in mechanism design where there is no point-wise optimal
mechanism, there is always a point-wise optimal envy-free outcome.
Envy-free outcomes and incentive-compatible mechanisms are similar in
structure and performance. We therefore use the optimal envy-free revenue as a
benchmark for measuring the performance of a prior-free mechanism. A good
mechanism is one that approximates the envy free benchmark on any profile of
agent values. We show that good mechanisms exist, and in particular, a natural
generalization of the random sampling auction of Goldberg et al. (2001) is a
constant approximation
Prior-Independent Auctions for Heterogeneous Bidders
We study the design of prior-independent auctions in a setting with
heterogeneous bidders. In particular, we consider the setting of selling to
bidders whose values are drawn from independent but not necessarily
identical distributions. We work in the robust auction design regime, where we
assume the seller has no knowledge of the bidders' value distributions and must
design a mechanism that is prior-independent. While there have been many strong
results on prior-independent auction design in the i.i.d. setting, not much is
known for the heterogeneous setting, even though the latter is of significant
practical importance. Unfortunately, no prior-independent mechanism can hope to
always guarantee any approximation to Myerson's revenue in the heterogeneous
setting; similarly, no prior-independent mechanism can consistently do better
than the second-price auction. In light of this, we design a family of
(parametrized) randomized auctions which approximates at least one of these
benchmarks: For heterogeneous bidders with regular value distributions, our
mechanisms either achieve a good approximation of the expected revenue of an
optimal mechanism (which knows the bidders' distributions) or exceeds that of
the second-price auction by a certain multiplicative factor. The factor in the
latter case naturally trades off with the approximation ratio of the former
case. We show that our mechanism is optimal for such a trade-off between the
two cases by establishing a matching lower bound. Our result extends to selling
identical items to heterogeneous bidders with an additional -factor in our trade-off between the two cases
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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