113 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
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
Mechanism Design for Perturbation Stable Combinatorial Auctions
Motivated by recent research on combinatorial markets with endowed valuations
by (Babaioff et al., EC 2018) and (Ezra et al., EC 2020), we introduce a notion
of perturbation stability in Combinatorial Auctions (CAs) and study the extend
to which stability helps in social welfare maximization and mechanism design. A
CA is if the optimal solution is resilient to
inflation, by a factor of , of any bidder's valuation for any
single item. On the positive side, we show how to compute efficiently an
optimal allocation for 2-stable subadditive valuations and that a Walrasian
equilibrium exists for 2-stable submodular valuations. Moreover, we show that a
Parallel 2nd Price Auction (P2A) followed by a demand query for each bidder is
truthful for general subadditive valuations and results in the optimal
allocation for 2-stable submodular valuations. To highlight the challenges
behind optimization and mechanism design for stable CAs, we show that a
Walrasian equilibrium may not exist for -stable XOS valuations for any
, that a polynomial-time approximation scheme does not exist for
-stable submodular valuations, and that any DSIC mechanism that
computes the optimal allocation for stable CAs and does not use demand queries
must use exponentially many value queries. We conclude with analyzing the Price
of Anarchy of P2A and Parallel 1st Price Auctions (P1A) for CAs with stable
submodular and XOS valuations. Our results indicate that the quality of
equilibria of simple non-truthful auctions improves only for -stable
instances with
- …