2,178 research outputs found
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
On Revenue Maximization with Sharp Multi-Unit Demands
We consider markets consisting of a set of indivisible items, and buyers that
have {\em sharp} multi-unit demand. This means that each buyer wants a
specific number of items; a bundle of size less than has no value,
while a bundle of size greater than is worth no more than the most valued
items (valuations being additive). We consider the objective of setting
prices and allocations in order to maximize the total revenue of the market
maker. The pricing problem with sharp multi-unit demand buyers has a number of
properties that the unit-demand model does not possess, and is an important
question in algorithmic pricing. We consider the problem of computing a revenue
maximizing solution for two solution concepts: competitive equilibrium and
envy-free pricing.
For unrestricted valuations, these problems are NP-complete; we focus on a
realistic special case of "correlated values" where each buyer has a
valuation v_i\qual_j for item , where and \qual_j are positive
quantities associated with buyer and item respectively. We present a
polynomial time algorithm to solve the revenue-maximizing competitive
equilibrium problem. For envy-free pricing, if the demand of each buyer is
bounded by a constant, a revenue maximizing solution can be found efficiently;
the general demand case is shown to be NP-hard.Comment: page2
Networks of Complements
We consider a network of sellers, each selling a single product, where the
graph structure represents pair-wise complementarities between products. We
study how the network structure affects revenue and social welfare of
equilibria of the pricing game between the sellers. We prove positive and
negative results, both of "Price of Anarchy" and of "Price of Stability" type,
for special families of graphs (paths, cycles) as well as more general ones
(trees, graphs). We describe best-reply dynamics that converge to non-trivial
equilibrium in several families of graphs, and we use these dynamics to prove
the existence of approximately-efficient equilibria.Comment: An extended abstract will appear in ICALP 201
Mechanism Design via Correlation Gap
For revenue and welfare maximization in single-dimensional Bayesian settings,
Chawla et al. (STOC10) recently showed that sequential posted-price mechanisms
(SPMs), though simple in form, can perform surprisingly well compared to the
optimal mechanisms. In this paper, we give a theoretical explanation of this
fact, based on a connection to the notion of correlation gap.
Loosely speaking, for auction environments with matroid constraints, we can
relate the performance of a mechanism to the expectation of a monotone
submodular function over a random set. This random set corresponds to the
winner set for the optimal mechanism, which is highly correlated, and
corresponds to certain demand set for SPMs, which is independent. The notion of
correlation gap of Agrawal et al.\ (SODA10) quantifies how much we {}"lose" in
the expectation of the function by ignoring correlation in the random set, and
hence bounds our loss in using certain SPM instead of the optimal mechanism.
Furthermore, the correlation gap of a monotone and submodular function is known
to be small, and it follows that certain SPM can approximate the optimal
mechanism by a good constant factor.
Exploiting this connection, we give tight analysis of a greedy-based SPM of
Chawla et al.\ for several environments. In particular, we show that it gives
an -approximation for matroid environments, gives asymptotically a
-approximation for the important sub-case of -unit
auctions, and gives a -approximation for environments with
-independent set system constraints
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