11,832 research outputs found
Pricing Multi-Unit Markets
We study the power and limitations of posted prices in multi-unit markets,
where agents arrive sequentially in an arbitrary order. We prove upper and
lower bounds on the largest fraction of the optimal social welfare that can be
guaranteed with posted prices, under a range of assumptions about the
designer's information and agents' valuations. Our results provide insights
about the relative power of uniform and non-uniform prices, the relative
difficulty of different valuation classes, and the implications of different
informational assumptions. Among other results, we prove constant-factor
guarantees for agents with (symmetric) subadditive valuations, even in an
incomplete-information setting and with uniform prices
The Combinatorial World (of Auctions) According to GARP
Revealed preference techniques are used to test whether a data set is
compatible with rational behaviour. They are also incorporated as constraints
in mechanism design to encourage truthful behaviour in applications such as
combinatorial auctions. In the auction setting, we present an efficient
combinatorial algorithm to find a virtual valuation function with the optimal
(additive) rationality guarantee. Moreover, we show that there exists such a
valuation function that both is individually rational and is minimum (that is,
it is component-wise dominated by any other individually rational, virtual
valuation function that approximately fits the data). Similarly, given upper
bound constraints on the valuation function, we show how to fit the maximum
virtual valuation function with the optimal additive rationality guarantee. In
practice, revealed preference bidding constraints are very demanding. We
explain how approximate rationality can be used to create relaxed revealed
preference constraints in an auction. We then show how combinatorial methods
can be used to implement these relaxed constraints. Worst/best-case welfare
guarantees that result from the use of such mechanisms can be quantified via
the minimum/maximum virtual valuation function
Prophet Inequalities with Limited Information
In the classical prophet inequality, a gambler observes a sequence of
stochastic rewards and must decide, for each reward ,
whether to keep it and stop the game or to forfeit the reward forever and
reveal the next value . The gambler's goal is to obtain a constant
fraction of the expected reward that the optimal offline algorithm would get.
Recently, prophet inequalities have been generalized to settings where the
gambler can choose items, and, more generally, where he can choose any
independent set in a matroid. However, all the existing algorithms require the
gambler to know the distribution from which the rewards are
drawn.
The assumption that the gambler knows the distribution from which
are drawn is very strong. Instead, we work with the much simpler
assumption that the gambler only knows a few samples from this distribution. We
construct the first single-sample prophet inequalities for many settings of
interest, whose guarantees all match the best possible asymptotically,
\emph{even with full knowledge of the distribution}. Specifically, we provide a
novel single-sample algorithm when the gambler can choose any elements
whose analysis is based on random walks with limited correlation. In addition,
we provide a black-box method for converting specific types of solutions to the
related \emph{secretary problem} to single-sample prophet inequalities, and
apply it to several existing algorithms. Finally, we provide a constant-sample
prophet inequality for constant-degree bipartite matchings.
We apply these results to design the first posted-price and multi-dimensional
auction mechanisms with limited information in settings with asymmetric
bidders
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