70 research outputs found
Tight Lower Bounds for Multiplicative Weights Algorithmic Families
We study the fundamental problem of prediction with expert advice and develop
regret lower bounds for a large family of algorithms for this problem. We
develop simple adversarial primitives, that lend themselves to various
combinations leading to sharp lower bounds for many algorithmic families. We
use these primitives to show that the classic Multiplicative Weights Algorithm
(MWA) has a regret of , there by completely closing
the gap between upper and lower bounds. We further show a regret lower bound of
for a much more general family of
algorithms than MWA, where the learning rate can be arbitrarily varied over
time, or even picked from arbitrary distributions over time. We also use our
primitives to construct adversaries in the geometric horizon setting for MWA to
precisely characterize the regret at for the case
of experts and a lower bound of
for the case of arbitrary number of experts
Lower Bounds on Revenue of Approximately Optimal Auctions
We obtain revenue guarantees for the simple pricing mechanism of a single
posted price, in terms of a natural parameter of the distribution of buyers'
valuations. Our revenue guarantee applies to the single item n buyers setting,
with values drawn from an arbitrary joint distribution. Specifically, we show
that a single price drawn from the distribution of the maximum valuation Vmax =
max {V_1, V_2, ...,V_n} achieves a revenue of at least a 1/e fraction of the
geometric expecation of Vmax. This generic bound is a measure of how revenue
improves/degrades as a function of the concentration/spread of Vmax.
We further show that in absence of buyers' valuation distributions,
recruiting an additional set of identical bidders will yield a similar
guarantee on revenue. Finally, our bound also gives a measure of the extent to
which one can simultaneously approximate welfare and revenue in terms of the
concentration/spread of Vmax.Comment: The 8th Workshop on Internet and Network Economics (WINE
Optimal Crowdsourcing Contests
We study the design and approximation of optimal crowdsourcing contests.
Crowdsourcing contests can be modeled as all-pay auctions because entrants must
exert effort up-front to enter. Unlike all-pay auctions where a usual design
objective would be to maximize revenue, in crowdsourcing contests, the
principal only benefits from the submission with the highest quality. We give a
theory for optimal crowdsourcing contests that mirrors the theory of optimal
auction design: the optimal crowdsourcing contest is a virtual valuation
optimizer (the virtual valuation function depends on the distribution of
contestant skills and the number of contestants). We also compare crowdsourcing
contests with more conventional means of procurement. In this comparison,
crowdsourcing contests are relatively disadvantaged because the effort of
losing contestants is wasted. Nonetheless, we show that crowdsourcing contests
are 2-approximations to conventional methods for a large family of "regular"
distributions, and 4-approximations, otherwise.Comment: The paper has 17 pages and 1 figure. It is to appear in the
proceedings of ACM-SIAM Symposium on Discrete Algorithms 201
Improved Revenue Bounds for Posted-Price and Second-Price Mechanisms
We study revenue maximization through sequential posted-price (SPP)
mechanisms in single-dimensional settings with buyers and independent but
not necessarily identical value distributions. We construct the SPP mechanisms
by considering the best of two simple pricing rules: one that imitates the
revenue optimal mchanism, namely the Myersonian mechanism, via the taxation
principle and the other that posts a uniform price. Our pricing rules are
rather generalizable and yield the first improvement over long-established
approximation factors in several settings. We design factor-revealing
mathematical programs that crisply capture the approximation factor of our SPP
mechanism. In the single-unit setting, our SPP mechanism yields a better
approximation factor than the state of the art prior to our work (Azar,
Chiplunkar & Kaplan, 2018). In the multi-unit setting, our SPP mechanism yields
the first improved approximation factor over the state of the art after over
nine years (Yan, 2011 and Chakraborty et al., 2010). Our results on SPP
mechanisms immediately imply improved performance guarantees for the equivalent
free-order prophet inequality problem. In the position auction setting, our SPP
mechanism yields the first higher-than approximation factor. In eager
second-price (ESP) auctions, our two simple pricing rules lead to the first
improved approximation factor that is strictly greater than what is obtained by
the SPP mechanism in the single-unit setting.Comment: Accepted to Operations Researc
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