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Bayesian Optimal Auctions via Multi to Singleagent 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 multidimensional 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 10 (3 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 multidimensional 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 singleagent 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 multiagent mechanism design problem to collection of singleagent problems. We focus on maximizing revenue, but our results can be applied to other objectives (e.g., welfare). An optimal multiagent mechanism can be computed by a linear/convex program on interim allocation rules by simultaneously optimizing several singleagent mechanisms subject to joint feasibility of the allocation rules. For singleunit auctions, Border (1991) showed that the space of all jointly feasible interim allocation rules for n agents is a Ddimensional convex polytope which can be specified by 2D linear constraints, where D is the total number of all agents’
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 quasilinearutility 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 2 (1 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 quasilinearutility 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. Thispaperconsidersmechanismdesigninenvironmentswheretheagentshavemultidimensional and nonlinear 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.,
Optimal Pricing Is Hard
, 2012
"... We show that computing the revenueoptimal deterministic auction in unitdemand singlebuyer Bayesian settings, i.e. the optimal itempricing, is computationally hard even in singleitem settings where the buyer’s value distribution is a sum of independently distributed attributes, or multiitem se ..."
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Cited by 1 (1 self)
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We show that computing the revenueoptimal deterministic auction in unitdemand singlebuyer Bayesian settings, i.e. the optimal itempricing, is computationally hard even in singleitem settings where the buyer’s value distribution is a sum of independently distributed attributes, or multiitem settings where the buyer’s values for the items are independent. We also show that it is intractable to optimally price the grand bundle of multiple items for an additive bidder whose values for the items are independent. These difficulties stem from implicit definitions of a value distribution. We provide three instances of how different properties of implicit distributions can lead to intractability: the first is a #Phardness proof, while the remaining two are reductions from the SQRTSUM problem of Garey, Graham, and Johnson [14]. While simple pricing schemes can oftentimes approximate the best scheme in revenue, they can have drastically different underlying structure. We argue therefore that either the specification of the input distribution must be highly restricted in format, or it is necessary for the goal to be mere approximation to the optimal scheme’s revenue instead of computing properties of the scheme itself.
Mechanisms and Allocations with Positive Network Externalities
, 2012
"... With the advent of social networks such as Facebook and LinkedIn, and online offers/deals web sites, network externalties raise the possibility of marketing and advertising to users based on influence they derive from their neighbors in such networks. Indeed, a user’s knowledge of which of his neigh ..."
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Cited by 1 (0 self)
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With the advent of social networks such as Facebook and LinkedIn, and online offers/deals web sites, network externalties raise the possibility of marketing and advertising to users based on influence they derive from their neighbors in such networks. Indeed, a user’s knowledge of which of his neighbors “liked ” the product, changes his valuation for the product. Much of the work on the mechanism design under network externalities has addressed the setting when there is only one product. We consider a more natural setting when there are multiple competing products, and each node in the network is a unitdemand agent. We first consider the problem of welfare maximization under various different types of externality functions. Specifically we get a O(lognlog(nm)) approximation for concave externality functions, 2 O(d)approximation for convex externality functions that are bounded above by a polynomial of degree d, and we give a O(log 3 n)approximation when the externality function is submodular. Our techniques involve formulating nontrivial linear relaxations in each case, and developing novel rounding schemes that yield bounds vastly superior to those obtainable by directly applying results from combinatorial welfare maximization. We then consider the problem of Nash equilibrium where each node in the network is a player whose strategy space corresponds to selecting an item. We develop tight characterization of the conditions under which a Nash equilibrium exists in this game. Lastly, we consider the question of pricing and revenue optimization
Additional Key Words and Phrases: Optimal Multidimensional Mechanism Design
"... We solve the optimal multidimensional mechanism design problem when either the number of bidders is a constant or the number of items is a constant. In the first setting, we need that the values of each bidder for the items are i.i.d., but allow different distributions for each bidder. In the secon ..."
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We solve the optimal multidimensional mechanism design problem when either the number of bidders is a constant or the number of items is a constant. In the first setting, we need that the values of each bidder for the items are i.i.d., but allow different distributions for each bidder. In the second setting, we allow the values of each bidder for the items to be arbitrarily correlated, but assume that the bidders are i.i.d. For all ɛ> 0, we obtain an efficient additive ɛapproximation, when the value distributions are bounded, or a multiplicative (1−ɛ)approximation when the value distributions are unbounded, but satisfy the Monotone Hazard Rate condition. When there is a single bidder, we generalize these results to independent but not necessarily identically distributed value distributions, and to independent regular distributions.
Abstract
"... Valiant’s theory of holographic algorithms is a novel methodology to achieve exponential speedups in computation. A fundamental parameter in holographic algorithms is the dimension of the linear basis vectors. We completely resolve the problem of the power of higher dimensional bases. We prove that ..."
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Valiant’s theory of holographic algorithms is a novel methodology to achieve exponential speedups in computation. A fundamental parameter in holographic algorithms is the dimension of the linear basis vectors. We completely resolve the problem of the power of higher dimensional bases. We prove that 2dimensional bases are universal for holographic algorithms. 1