1,580 research outputs found
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
Approximate Revenue Maximization with Multiple Items
Maximizing the revenue from selling _more than one_ good (or item) to a
single buyer is a notoriously difficult problem, in stark contrast to the
one-good case. For two goods, we show that simple "one-dimensional" mechanisms,
such as selling the goods separately, _guarantee_ at least 73% of the optimal
revenue when the valuations of the two goods are independent and identically
distributed, and at least when they are independent. For the case of
independent goods, we show that selling them separately guarantees at
least a fraction of the optimal revenue; and, for independent and
identically distributed goods, we show that selling them as one bundle
guarantees at least a fraction of the optimal revenue. Additional
results compare the revenues from the two simple mechanisms of selling the
goods separately and bundled, identify situations where bundling is optimal,
and extend the analysis to multiple buyers.Comment: Presented in ACM EC conference, 201
Knightian Analysis of the Vickrey Mechanism
We analyze the Vickrey mechanism for auctions of multiple identical goods
when the players have both Knightian uncertainty over their own valuations and
incomplete preferences. In this model, the Vickrey mechanism is no longer
dominant-strategy, and we prove that all dominant-strategy mechanisms are
inadequate. However, we also prove that, in undominated strategies, the social
welfare produced by the Vickrey mechanism in the worst case is not only very
good, but also essentially optimal.Comment: To appear in Econometric
Mixed Bundling Auctions
We study multi-object auctions where agents have private and additive valuations for heterogeneous objects. We focus on the revenue properties of a class of dominant strategy mechanisms where a weight is assigned to each partition of objects. The weights influence the probability with which partitions are chosen in the mechanism. This class contains efficient auctions, pure bundling auctions, mixed bundling auctions, auctions with reserve prices and auctions with pre-packaged bundles. For any number of objects and bidders, both the pure bundling auction and separate, efficient auctions for the single objects are revenue-inferior to an auction that involves mixed bundling
Fee-Setting Mechanisms: On Optimal Pricing by Intermediaries and Indirect Taxation
Mechanisms according to which private intermediaries or governments charge
transaction fees or indirect taxes are prevalent in practice. We consider a setup with
multiple buyers and sellers and two-sided independent private information about
valuations. We show that any weighted average of revenue and social welfare can be
maximized through appropriately chosen transaction fees and that in increasingly
thin markets such optimal fees converge to linear fees. Moreover, fees decrease with
competition (or the weight on welfare) and the elasticity of supply but decrease
with the elasticity of demand. Our theoretical predictions fit empirical observations
in several industries with intermediaries
Truthful implementation and preference aggregation in restricted domains
In a setting where agents have quasi-linear utilities over social alternatives and a transferable commodity,we consider three properties that a social choice function may possess: truthful implementation (in dominant strategies); monotonicity in differences; and lexicographic affine maximization. We introduce the notion of a flexible domain of preferences that allows elevation of pairs and study which of these conditions implies which others in such domain. We provide a generalization of the theorem of Roberts (1979) [36] in restricted valuation domains. Flexibility holds (and the theorem is not vacuous) if the domain of valuation profiles is restricted to the space of continuous functions defined on a compact metric space, or the space of piecewise linear functions defined on an affine space, or the space of smooth functions defined on a compact differentiable manifold. We provide applications of our results to public goods allocation settings, with finite and infinite alternative sets
Efficient Design with Interdependent Valuations
We study efficient, Bayes-Nash incentive compatible mechanisms in a social choice setting that allows for informational and allocative externalities. We show that such mechanisms exist only if a congruence condition relating private and social rates of information substitution is satisfied. If signals are multidimensional, the congruence condition is determined by an integrability constraint, and it can hold only in non-generic cases such as the private value case or the symmetric case. If signals are one-dimensional, the congruence condition reduces to a monotonicity constraint and it can be generically satisfied. We apply the results to the study of multi-object auctions, and we discuss why such auctions cannot be reduced to one-dimensional models without loss of generality.
Cake Cutting Algorithms for Piecewise Constant and Piecewise Uniform Valuations
Cake cutting is one of the most fundamental settings in fair division and
mechanism design without money. In this paper, we consider different levels of
three fundamental goals in cake cutting: fairness, Pareto optimality, and
strategyproofness. In particular, we present robust versions of envy-freeness
and proportionality that are not only stronger than their standard
counter-parts but also have less information requirements. We then focus on
cake cutting with piecewise constant valuations and present three desirable
algorithms: CCEA (Controlled Cake Eating Algorithm), MEA (Market Equilibrium
Algorithm) and CSD (Constrained Serial Dictatorship). CCEA is polynomial-time,
robust envy-free, and non-wasteful. It relies on parametric network flows and
recent generalizations of the probabilistic serial algorithm. For the subdomain
of piecewise uniform valuations, we show that it is also group-strategyproof.
Then, we show that there exists an algorithm (MEA) that is polynomial-time,
envy-free, proportional, and Pareto optimal. MEA is based on computing a
market-based equilibrium via a convex program and relies on the results of
Reijnierse and Potters [24] and Devanur et al. [15]. Moreover, we show that MEA
and CCEA are equivalent to mechanism 1 of Chen et. al. [12] for piecewise
uniform valuations. We then present an algorithm CSD and a way to implement it
via randomization that satisfies strategyproofness in expectation, robust
proportionality, and unanimity for piecewise constant valuations. For the case
of two agents, it is robust envy-free, robust proportional, strategyproof, and
polynomial-time. Many of our results extend to more general settings in cake
cutting that allow for variable claims and initial endowments. We also show a
few impossibility results to complement our algorithms.Comment: 39 page
Mixed Bundling Auctions
We study multi-object auctions where agents have private and additive valuations for heterogeneous objects. We focus on the revenue properties of a class of dominant strategy mechanisms where a weight is assigned to each partition of objects. The weights influence the probability with which partitions are chosen in the mechanism. This class contains efficient auctions, pure bundling auctions, mixed bundling auctions, auctions with reserve prices and auctions with pre-packaged bundles. For any number of objects and bidders, both the pure bundling auction and separate, efficient auctions for the single objects are revenue-inferior to an auction that involves mixed bundling.
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