13,989 research outputs found
Revenue Maximization and Ex-Post Budget Constraints
We consider the problem of a revenue-maximizing seller with m items for sale
to n additive bidders with hard budget constraints, assuming that the seller
has some prior distribution over bidder values and budgets. The prior may be
correlated across items and budgets of the same bidder, but is assumed
independent across bidders. We target mechanisms that are Bayesian Incentive
Compatible, but that are ex-post Individually Rational and ex-post budget
respecting. Virtually no such mechanisms are known that satisfy all these
conditions and guarantee any revenue approximation, even with just a single
item. We provide a computationally efficient mechanism that is a
-approximation with respect to all BIC, ex-post IR, and ex-post budget
respecting mechanisms. Note that the problem is NP-hard to approximate better
than a factor of 16/15, even in the case where the prior is a point mass
\cite{ChakrabartyGoel}. We further characterize the optimal mechanism in this
setting, showing that it can be interpreted as a distribution over virtual
welfare maximizers.
We prove our results by making use of a black-box reduction from mechanism to
algorithm design developed by \cite{CaiDW13b}. Our main technical contribution
is a computationally efficient -approximation algorithm for the algorithmic
problem that results by an application of their framework to this problem. The
algorithmic problem has a mixed-sign objective and is NP-hard to optimize
exactly, so it is surprising that a computationally efficient approximation is
possible at all. In the case of a single item (), the algorithmic problem
can be solved exactly via exhaustive search, leading to a computationally
efficient exact algorithm and a stronger characterization of the optimal
mechanism as a distribution over virtual value maximizers
On the legitimacy of coercion for the nancing of public goods
The literature on public goods has shown that e?cient outcomes are impossible if participation constraints have to be respected. This paper addresses the question whether they should be imposed. It asks under what conditions e?ciency considerations justify that individuals are forced to pay for public goods that they do not value. It is shown that participation constraints are desirable if public goods are provided by a malevolent Leviathan. By contrast, with a Pigouvian planner, e?ciency can be achieved. Finally, the paper studies the delegation of public goods provision to a pro?t-maximizing ?rm. This also makes participation constraints desirable
Near-Optimal and Robust Mechanism Design for Covering Problems with Correlated Players
We consider the problem of designing incentive-compatible, ex-post
individually rational (IR) mechanisms for covering problems in the Bayesian
setting, where players' types are drawn from an underlying distribution and may
be correlated, and the goal is to minimize the expected total payment made by
the mechanism. We formulate a notion of incentive compatibility (IC) that we
call {\em support-based IC} that is substantially more robust than Bayesian IC,
and develop black-box reductions from support-based-IC mechanism design to
algorithm design. For single-dimensional settings, this black-box reduction
applies even when we only have an LP-relative {\em approximation algorithm} for
the algorithmic problem. Thus, we obtain near-optimal mechanisms for various
covering settings including single-dimensional covering problems, multi-item
procurement auctions, and multidimensional facility location.Comment: Major changes compared to the previous version. Please consult this
versio
Optimal Auctions vs. Anonymous Pricing: Beyond Linear Utility
The revenue optimal mechanism for selling a single item to agents with
independent but non-identically distributed values is complex for agents with
linear utility (Myerson,1981) and has no closed-form characterization for
agents with non-linear utility (cf. Alaei et al., 2012). Nonetheless, for
linear utility agents satisfying a natural regularity property, Alaei et al.
(2018) showed that simply posting an anonymous price is an e-approximation. We
give a parameterization of the regularity property that extends to agents with
non-linear utility and show that the approximation bound of anonymous pricing
for regular agents approximately extends to agents that satisfy this
approximate regularity property. We apply this approximation framework to prove
that anonymous pricing is a constant approximation to the revenue optimal
single-item auction for agents with public-budget utility, private-budget
utility, and (a special case of) risk-averse utility.Comment: Appeared at EC 201
A geometric approach to mechanism design
We develop a novel geometric approach to mechanism design using an important result in convex analysis: the duality between a closed convex set and its support function. By deriving the support function for the set of feasible interim values we extend the wellknown Maskin-Riley-Matthews-Border conditions for reduced-form auctions to social choice environments. We next refine the support function to include incentive constraints using a geometric characterization of incentive compatibility. Borrowing results from majorization theory that date back to the work of Hardy, Littlewood, and Polya (1929) we elucidate the "ironing" procedure introduced by Myerson (1981) and Mussa and Rosen (1978). The inclusion of Bayesian and dominant strategy incentive constraints result in the same support function, which establishes equivalence between these implementation concepts. Using Hotelling's lemma we next derive the optimal mechanism for any social choice problem and any linear objective, including revenue and surplus maximization. We extend the approach to include general concave objectives by providing a fixed-point condition characterizing the optimal mechanism. We generalize reduced-form implementation to environments with multi-dimensional, correlated types, non-linear utilities, and interdependent values. When value interdependencies are linear we are able to include incentive constraints into the support function and provide a condition when the second-best allocation is ex post incentive compatible.Mechanism design, convex set, support function, duality, majorization, ironing, Hotelling's lemma, reduced-from implementation, BIC-DIC equivalence, concave objectives, interdependent values, second-best mechanisms
Budget Constrained Auctions with Heterogeneous Items
In this paper, we present the first approximation algorithms for the problem
of designing revenue optimal Bayesian incentive compatible auctions when there
are multiple (heterogeneous) items and when bidders can have arbitrary demand
and budget constraints. Our mechanisms are surprisingly simple: We show that a
sequential all-pay mechanism is a 4 approximation to the revenue of the optimal
ex-interim truthful mechanism with discrete correlated type space for each
bidder. We also show that a sequential posted price mechanism is a O(1)
approximation to the revenue of the optimal ex-post truthful mechanism when the
type space of each bidder is a product distribution that satisfies the standard
hazard rate condition. We further show a logarithmic approximation when the
hazard rate condition is removed, and complete the picture by showing that
achieving a sub-logarithmic approximation, even for regular distributions and
one bidder, requires pricing bundles of items. Our results are based on
formulating novel LP relaxations for these problems, and developing generic
rounding schemes from first principles. We believe this approach will be useful
in other Bayesian mechanism design contexts.Comment: Final version accepted to STOC '10. Incorporates significant reviewer
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