80 research outputs found
Inapproximability of Truthful Mechanisms via Generalizations of the VC Dimension
Algorithmic mechanism design (AMD) studies the delicate interplay between
computational efficiency, truthfulness, and optimality. We focus on AMD's
paradigmatic problem: combinatorial auctions. We present a new generalization
of the VC dimension to multivalued collections of functions, which encompasses
the classical VC dimension, Natarajan dimension, and Steele dimension. We
present a corresponding generalization of the Sauer-Shelah Lemma and harness
this VC machinery to establish inapproximability results for deterministic
truthful mechanisms. Our results essentially unify all inapproximability
results for deterministic truthful mechanisms for combinatorial auctions to
date and establish new separation gaps between truthful and non-truthful
algorithms
Single Parameter Combinatorial Auctions with Partially Public Valuations
We consider the problem of designing truthful auctions, when the bidders'
valuations have a public and a private component. In particular, we consider
combinatorial auctions where the valuation of an agent for a set of
items can be expressed as , where is a private single parameter
of the agent, and the function is publicly known. Our motivation behind
studying this problem is two-fold: (a) Such valuation functions arise naturally
in the case of ad-slots in broadcast media such as Television and Radio. For an
ad shown in a set of ad-slots, is, say, the number of {\em unique}
viewers reached by the ad, and is the valuation per-unique-viewer. (b)
From a theoretical point of view, this factorization of the valuation function
simplifies the bidding language, and renders the combinatorial auction more
amenable to better approximation factors. We present a general technique, based
on maximal-in-range mechanisms, that converts any -approximation
non-truthful algorithm () for this problem into
and -approximate truthful
mechanisms which run in polynomial time and quasi-polynomial time,
respectively
Implementation in Advised Strategies: Welfare Guarantees from Posted-Price Mechanisms When Demand Queries Are NP-Hard
State-of-the-art posted-price mechanisms for submodular bidders with
items achieve approximation guarantees of [Assadi and
Singla, 2019]. Their truthfulness, however, requires bidders to compute an
NP-hard demand-query. Some computational complexity of this form is
unavoidable, as it is NP-hard for truthful mechanisms to guarantee even an
-approximation for any [Dobzinski and
Vondr\'ak, 2016]. Together, these establish a stark distinction between
computationally-efficient and communication-efficient truthful mechanisms.
We show that this distinction disappears with a mild relaxation of
truthfulness, which we term implementation in advised strategies, and that has
been previously studied in relation to "Implementation in Undominated
Strategies" [Babaioff et al, 2009]. Specifically, advice maps a tentative
strategy either to that same strategy itself, or one that dominates it. We say
that a player follows advice as long as they never play actions which are
dominated by advice. A poly-time mechanism guarantees an -approximation
in implementation in advised strategies if there exists poly-time advice for
each player such that an -approximation is achieved whenever all
players follow advice. Using an appropriate bicriterion notion of approximate
demand queries (which can be computed in poly-time), we establish that (a
slight modification of) the [Assadi and Singla, 2019] mechanism achieves the
same -approximation in implementation in advised
strategies
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