26,245 research outputs found
On Simultaneous Two-player Combinatorial Auctions
We consider the following communication problem: Alice and Bob each have some
valuation functions and over subsets of items,
and their goal is to partition the items into in a way that
maximizes the welfare, . We study both the allocation
problem, which asks for a welfare-maximizing partition and the decision
problem, which asks whether or not there exists a partition guaranteeing
certain welfare, for binary XOS valuations. For interactive protocols with
communication, a tight 3/4-approximation is known for both
[Fei06,DS06].
For interactive protocols, the allocation problem is provably harder than the
decision problem: any solution to the allocation problem implies a solution to
the decision problem with one additional round and additional bits of
communication via a trivial reduction. Surprisingly, the allocation problem is
provably easier for simultaneous protocols. Specifically, we show:
1) There exists a simultaneous, randomized protocol with polynomial
communication that selects a partition whose expected welfare is at least
of the optimum. This matches the guarantee of the best interactive, randomized
protocol with polynomial communication.
2) For all , any simultaneous, randomized protocol that
decides whether the welfare of the optimal partition is or correctly with probability requires
exponential communication. This provides a separation between the attainable
approximation guarantees via interactive () versus simultaneous () protocols with polynomial communication.
In other words, this trivial reduction from decision to allocation problems
provably requires the extra round of communication
Computer-aided verification in mechanism design
In mechanism design, the gold standard solution concepts are dominant
strategy incentive compatibility and Bayesian incentive compatibility. These
solution concepts relieve the (possibly unsophisticated) bidders from the need
to engage in complicated strategizing. While incentive properties are simple to
state, their proofs are specific to the mechanism and can be quite complex.
This raises two concerns. From a practical perspective, checking a complex
proof can be a tedious process, often requiring experts knowledgeable in
mechanism design. Furthermore, from a modeling perspective, if unsophisticated
agents are unconvinced of incentive properties, they may strategize in
unpredictable ways.
To address both concerns, we explore techniques from computer-aided
verification to construct formal proofs of incentive properties. Because formal
proofs can be automatically checked, agents do not need to manually check the
properties, or even understand the proof. To demonstrate, we present the
verification of a sophisticated mechanism: the generic reduction from Bayesian
incentive compatible mechanism design to algorithm design given by Hartline,
Kleinberg, and Malekian. This mechanism presents new challenges for formal
verification, including essential use of randomness from both the execution of
the mechanism and from the prior type distributions. As an immediate
consequence, our work also formalizes Bayesian incentive compatibility for the
entire family of mechanisms derived via this reduction. Finally, as an
intermediate step in our formalization, we provide the first formal
verification of incentive compatibility for the celebrated
Vickrey-Clarke-Groves mechanism
Mechanisms for Risk Averse Agents, Without Loss
Auctions in which agents' payoffs are random variables have received
increased attention in recent years. In particular, recent work in algorithmic
mechanism design has produced mechanisms employing internal randomization,
partly in response to limitations on deterministic mechanisms imposed by
computational complexity. For many of these mechanisms, which are often
referred to as truthful-in-expectation, incentive compatibility is contingent
on the assumption that agents are risk-neutral. These mechanisms have been
criticized on the grounds that this assumption is too strong, because "real"
agents are typically risk averse, and moreover their precise attitude towards
risk is typically unknown a-priori. In response, researchers in algorithmic
mechanism design have sought the design of universally-truthful mechanisms ---
mechanisms for which incentive-compatibility makes no assumptions regarding
agents' attitudes towards risk.
We show that any truthful-in-expectation mechanism can be generically
transformed into a mechanism that is incentive compatible even when agents are
risk averse, without modifying the mechanism's allocation rule. The transformed
mechanism does not require reporting of agents' risk profiles. Equivalently,
our result can be stated as follows: Every (randomized) allocation rule that is
implementable in dominant strategies when players are risk neutral is also
implementable when players are endowed with an arbitrary and unknown concave
utility function for money.Comment: Presented at the workshop on risk aversion in algorithmic game theory
and mechanism design, held in conjunction with EC 201
Reducing Revenue to Welfare Maximization: Approximation Algorithms and other Generalizations
It was recently shown in [http://arxiv.org/abs/1207.5518] that revenue
optimization can be computationally efficiently reduced to welfare optimization
in all multi-dimensional Bayesian auction problems with arbitrary (possibly
combinatorial) feasibility constraints and independent additive bidders with
arbitrary (possibly combinatorial) demand constraints. This reduction provides
a poly-time solution to the optimal mechanism design problem in all auction
settings where welfare optimization can be solved efficiently, but it is
fragile to approximation and cannot provide solutions to settings where welfare
maximization can only be tractably approximated. In this paper, we extend the
reduction to accommodate approximation algorithms, providing an approximation
preserving reduction from (truthful) revenue maximization to (not necessarily
truthful) welfare maximization. The mechanisms output by our reduction choose
allocations via black-box calls to welfare approximation on randomly selected
inputs, thereby generalizing also our earlier structural results on optimal
multi-dimensional mechanisms to approximately optimal mechanisms. Unlike
[http://arxiv.org/abs/1207.5518], our results here are obtained through novel
uses of the Ellipsoid algorithm and other optimization techniques over {\em
non-convex regions}
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