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39
Optimal multi-dimensional mechanism design: Reducing revenue to welfare maximization
, 2012
"... Bayesian auctions with arbitrary (possibly combinatorial) feasibility constraints and independent bidders with arbitrary (possibly combinatorial) demand constraints, appropriately extending Myerson’s single-dimensional result [24] to this setting. We also show that every feasible Bayesian auction ca ..."
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Cited by 38 (13 self)
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Bayesian auctions with arbitrary (possibly combinatorial) feasibility constraints and independent bidders with arbitrary (possibly combinatorial) demand constraints, appropriately extending Myerson’s single-dimensional result [24] to this setting. We also show that every feasible Bayesian auction can be implemented as a distribution over virtual VCG allocation rules. A virtual VCG allocation rule has the following simple form: Every bidder’s type ti is transformed into a virtual type fi(ti), via a bidder-specific function. Then, the allocation maximizing virtual welfare is chosen. Using this characterization, we show how to find and run the revenue-optimal auction given only black box access to an implementation of the VCG allocation rule. We generalize this result to arbitrarily correlated bidders, introducing the notion of a second-order VCG allocation rule. We obtain our reduction from revenue to welfare optimization via two algorithmic results on reduced form auctions in settings with arbitrary feasibility and demand constraints. First, we provide a separation oracle for determining feasibility of a reduced form auction. Second, we provide a geometric algorithm to decompose any feasible reduced form into a distribution over virtual VCG allocation rules. In addition, we show how to execute both algorithms given only
Bayesian Optimal Auctions via Multi- to Single-agent Reduction
, 1203
"... We study an abstract optimal auction problem for a single good or service. This problem includes environments where agents have budgets, risk preferences, or multi-dimensional preferences over several possible configurations of the good (furthermore, it allows an agent’s budget and risk preference t ..."
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Cited by 23 (6 self)
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We study an abstract optimal auction problem for a single good or service. This problem includes environments where agents have budgets, risk preferences, or multi-dimensional preferences over several possible configurations of the good (furthermore, it allows an agent’s budget and risk preference to be known only privately to the agent). These are the main challenge areas for auction theory. A single-agent problem is to optimize a given objective subject to a constraint on the maximum probability with which each type is allocated, a.k.a., an allocation rule. Our approach is a reduction from multi-agent mechanism design problem to collection of single-agent problems. We focus on maximizing revenue, but our results can be applied to other objectives (e.g., welfare). An optimal multi-agent mechanism can be computed by a linear/convex program on interim allocation rules by simultaneously optimizing several single-agent mechanisms subject to joint feasibility of the allocation rules. For single-unit auctions, Border (1991) showed that the space of all jointly feasible interim allocation rules for n agents is a D-dimensional convex polytope which can be specified by 2D linear constraints, where D is the total number of all agents’
Matroid Prophet Inequalities
, 2012
"... Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent observation. The famous prophet inequality of Krengel, Sucheston, and Garling asserts that a gambler who knows the distri ..."
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Cited by 19 (4 self)
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Consider a gambler who observes a sequence of independent, non-negative random numbers and is allowed to stop the sequence at any time, claiming a reward equal to the most recent observation. The famous prophet inequality of Krengel, Sucheston, and Garling asserts that a gambler who knows the distribution of each random variable can achieve at least half as much reward, in expectation, as a “prophet ” who knows the sampled values of each random variable and can choose the largest one. We generalize this result to the setting in which the gambler and the prophet are allowed to make more than one selection, subject to a matroid constraint. We show that the gambler can still achieve at least half as much reward as the prophet; this result is the best possible, since it is known that the ratio cannot be improved even in the original prophet inequality, which corresponds to the special case of rank-one matroids. Generalizing the result still further, we show that under an intersection of p matroid constraints, the prophet’s reward exceeds the gambler’s by a factor of at most O(p), and this factor is also tight. Beyond their interest as theorems about pure online algoritms or optimal stopping rules, these results also have applications to mechanism design. Our results imply improved bounds on the ability of sequen-tial posted-price mechanisms to approximate Bayesian optimal mechanisms in both single-parameter and multi-parameter settings. In particular, our results imply the first efficiently computable constant-factor approximations to the Bayesian optimal revenue in certain multi-parameter settings.
Reducing revenue to welfare maximization: Approximation algorithms and other generalizations
- IN SODA
, 2013
"... It was recently shown in [12] 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 comb ..."
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Cited by 13 (6 self)
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It was recently shown in [12] 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 blackbox 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 [12], our results here are obtained through novel uses of the Ellipsoid algorithm and other optimization techniques over non-convex regions.
Simple and Nearly Optimal Multi-Item Auctions
"... We provide a Polynomial Time Approximation Scheme (PTAS) for the Bayesian optimal multi-item multibidder auction problem under two conditions. First, bidders are independent, have additive valuations and are from the same population. Second, every bidder’s value distributions of items are independen ..."
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Cited by 11 (5 self)
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We provide a Polynomial Time Approximation Scheme (PTAS) for the Bayesian optimal multi-item multibidder auction problem under two conditions. First, bidders are independent, have additive valuations and are from the same population. Second, every bidder’s value distributions of items are independent but not necessarily identical monotone hazard rate (MHR) distributions. For non-i.i.d. bidders, we also provide a PTAS when the number of bidders is small. Prior to our work, even for a single bidder, only constant factor approximations are known. Another appealing feature of our mechanism is the simple allocation rule. Indeed, the mechanism we use is either the second-price auction with reserve price on every item individually, or VCG allocation with a few outlying items that requires additional treatments. It is surprising that such simple allocation rules suffice to obtain nearly optimal revenue. 1
Extreme-value theorems for optimal multidimensional pricing
, 2011
"... We provide a Polynomial Time Approximation Scheme for the multi-dimensional unit-demand pricing problem, when the buyer’s values are independent (but not necessarily identically distributed.) For all ɛ> 0, we obtain a (1 + ɛ)-factor approximation to the optimal revenue in time polynomial, when th ..."
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Cited by 11 (3 self)
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We provide a Polynomial Time Approximation Scheme for the multi-dimensional unit-demand pricing problem, when the buyer’s values are independent (but not necessarily identically distributed.) For all ɛ> 0, we obtain a (1 + ɛ)-factor approximation to the optimal revenue in time polynomial, when the values are sampled from Monotone Hazard Rate (MHR) distributions, quasi-polynomial, when sampled from regular distributions, and polynomial in n poly(log r) , when sampled from general distributions supported on a set [umin, rumin]. We also provide an additive PTAS for all bounded distributions. Our algorithms are based on novel extreme value theorems for MHR and regular distributions, and apply probabilistic techniques to understand the statistical properties of revenue distributions, as well as to reduce the size of the search space of the algorithm. As a byproduct of our techniques, we establish structural properties of optimal solutions. We show that, for all ɛ> 0, g(1/ɛ) distinct prices suffice to obtain a (1+ɛ)-factor approximation to the optimal revenue for MHR distributions, where g(1/ɛ) is a quasi-linear function of 1/ɛ that does not depend on the number of items. Similarly, for all ɛ> 0 and n> 0, g(1/ɛ · log n) distinct prices suffice for regular distributions,
Prior-independent multi-parameter mechanism design
- In Workshop on Internet and Network Economics (WINE
, 2011
"... Abstract. In a unit-demand multi-unit multi-item auction, an auctioneer is selling a collection of different items to a set of agents each interested in buying at most unit. Each agent has a different private value for each of the items. We consider the problem of designing a truthful auction that m ..."
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Cited by 11 (3 self)
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Abstract. In a unit-demand multi-unit multi-item auction, an auctioneer is selling a collection of different items to a set of agents each interested in buying at most unit. Each agent has a different private value for each of the items. We consider the problem of designing a truthful auction that maximizes the auctioneer’s profit in this setting. Previously, there has been progress on this problem in the setting in which each value is drawn from a known prior distribution. Specifically, it has been shown how to design auctions tailored to these priors that achieve a constant factor approximation ratio [2, 5]. In this paper, we present a prior-independent auction for this setting. This auction is guaranteed to achieve a constant fraction of the optimal expected profit for a large class of, so called, “regular ” distributions, without specific knowledge of the distributions. 1
Understanding incentives: Mechanism design becomes algorithm design
, 2013
"... We provide a computationally efficient black-box reduction from mechanism design to algorithm design in very general settings. Specifically, we give an approximation-preserving reduction from truthfully maximizing any objective under arbitrary feasibility constraints with arbitrary bidder types to ( ..."
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Cited by 10 (6 self)
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We provide a computationally efficient black-box reduction from mechanism design to algorithm design in very general settings. Specifically, we give an approximation-preserving reduction from truthfully maximizing any objective under arbitrary feasibility constraints with arbitrary bidder types to (not necessarily truthfully) maximizing the same objective plus virtual welfare (under the same feasibility constraints). Our reduction is based on a fundamentally new approach: we describe a mechanism’s behavior indirectly only in terms of the expected value it awards bidders for certain behavior, and never directly access the allocation rule at all. Applying our new approach to revenue, we exhibit settings where our reduction holds both ways. That is, we also provide an approximation-sensitive reduction from (non-truthfully) maximizing virtual welfare to (truthfully) maximizing revenue, and therefore the two problems are computationally equivalent. With this equivalence in hand, we show that both problems are NP-hard to approximate within any polynomial factor, even for a single monotone submodular bidder. We further demonstrate the applicability of our reduction by providing a truthful mechanism maximizing fractional max-min fairness. This is the first instance of a truthful mechanism that optimizes a non-linear objective.
The Complexity of Optimal Mechanism Design
, 1211
"... Myerson’s seminal work provides a computationally efficient revenue-optimal auction for selling one item to multiple bidders [17]. Generalizing this work to selling multiple items at once has been a central question in economics and algorithmic game theory, but its complexity has remained poorly und ..."
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Cited by 10 (4 self)
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Myerson’s seminal work provides a computationally efficient revenue-optimal auction for selling one item to multiple bidders [17]. Generalizing this work to selling multiple items at once has been a central question in economics and algorithmic game theory, but its complexity has remained poorly understood. We answer this question by showing that a revenue-optimal auction in multi-item settings cannot be found and implemented computationally efficiently, unless ZPP ⊇ P #P. This is true even for a single additive bidder whose values for the items are independently distributedon tworationalnumberswith rationalprobabilities. Ourresult isvery general: we show that it is hard to compute any encoding of an optimal auction of any format (direct or indirect, truthful or non-truthful) that can be implemented in expected polynomial time. In particular, under well-believed complexity-theoreticassumptions, revenue-optimization in very simple multi-item settings can only be tractably approximated. We note that our hardness result applies to randomized mechanisms in a very simple setting, and is not an artifact of introducing combinatorial structure to the problem by allowing correlation among item values, introducing combinatorial valuations, or requiring the mechanism to be deterministic (whose structure is readily combinatorial). Our proof is enabled by a flowinterpretation of the solutions of an exponential-size linear program for revenue maximization with an additional supermodularity constraint.
The Simple Economics of Approximately Optimal Auctions Arvix
, 2012
"... The intuition that profit is optimized by maximizing marginal revenue is a guiding principle in microeconomics. In the classical auction theory for agents with quasi-linearutility and singledimensional preferences, Bulow and Roberts (1989) show that the optimal auction of Myerson (1981) is in fact o ..."
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Cited by 7 (3 self)
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The intuition that profit is optimized by maximizing marginal revenue is a guiding principle in microeconomics. In the classical auction theory for agents with quasi-linearutility and singledimensional preferences, Bulow and Roberts (1989) show that the optimal auction of Myerson (1981) is in fact optimizing marginal revenue. In particular Myerson’s virtual values are exactly the derivative of an appropriate revenue curve. Thispaperconsidersmechanismdesigninenvironmentswheretheagentshavemulti-dimensional and non-linear preferences. Understanding good auctions for these environments is considered to be the main challenge in Bayesian optimal mechanism design. In these environments maximizing marginal revenue may not be optimal, and furthermore, there is sometimes no direct way to implementing the marginal revenue maximization mechanism. Our contributions are three fold: we characterize the settings for which marginal revenue maximization is optimal, we give simple procedures for implementing marginal revenue maximization in general, and we show that marginal revenue maximization is approximately optimal. Our approximation factor smoothly degrades in a term that quantifies how far the environment is from an ideal one (i.e.,
