9,209 research outputs found

    Learning Strong Substitutes Demand via Queries

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    This paper addresses the computational challenges of learning strong substitutes demand when given access to a demand (or valuation) oracle. Strong substitutes demand generalises the well-studied gross substitutes demand to a multi-unit setting. Recent work by Baldwin and Klemperer shows that any such demand can be expressed in a natural way as a finite list of weighted bid vectors. A simplified version of this bidding language has been used by the Bank of England. Assuming access to a demand oracle, we provide an algorithm that computes the unique list of weighted bid vectors corresponding to a bidder's demand preferences. In the special case where their demand can be expressed using positive bids only, we have an efficient algorithm that learns this list in linear time. We also show super-polynomial lower bounds on the query complexity of computing the list of bids in the general case where bids may be positive and negative. Our algorithms constitute the first systematic approach for bidders to construct a bid list corresponding to non-trivial demand, allowing them to participate in `product-mix' auctions

    Combinatorial Assortment Optimization

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    Assortment optimization refers to the problem of designing a slate of products to offer potential customers, such as stocking the shelves in a convenience store. The price of each product is fixed in advance, and a probabilistic choice function describes which product a customer will choose from any given subset. We introduce the combinatorial assortment problem, where each customer may select a bundle of products. We consider a model of consumer choice where the relative value of different bundles is described by a valuation function, while individual customers may differ in their absolute willingness to pay, and study the complexity of the resulting optimization problem. We show that any sub-polynomial approximation to the problem requires exponentially many demand queries when the valuation function is XOS, and that no FPTAS exists even for succinctly-representable submodular valuations. On the positive side, we show how to obtain constant approximations under a "well-priced" condition, where each product's price is sufficiently high. We also provide an exact algorithm for kk-additive valuations, and show how to extend our results to a learning setting where the seller must infer the customers' preferences from their purchasing behavior

    Learning Economic Parameters from Revealed Preferences

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    A recent line of work, starting with Beigman and Vohra (2006) and Zadimoghaddam and Roth (2012), has addressed the problem of {\em learning} a utility function from revealed preference data. The goal here is to make use of past data describing the purchases of a utility maximizing agent when faced with certain prices and budget constraints in order to produce a hypothesis function that can accurately forecast the {\em future} behavior of the agent. In this work we advance this line of work by providing sample complexity guarantees and efficient algorithms for a number of important classes. By drawing a connection to recent advances in multi-class learning, we provide a computationally efficient algorithm with tight sample complexity guarantees (Θ(d/ϵ)\Theta(d/\epsilon) for the case of dd goods) for learning linear utility functions under a linear price model. This solves an open question in Zadimoghaddam and Roth (2012). Our technique yields numerous generalizations including the ability to learn other well-studied classes of utility functions, to deal with a misspecified model, and with non-linear prices

    Computational Efficiency Requires Simple Taxation

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    We characterize the communication complexity of truthful mechanisms. Our departure point is the well known taxation principle. The taxation principle asserts that every truthful mechanism can be interpreted as follows: every player is presented with a menu that consists of a price for each bundle (the prices depend only on the valuations of the other players). Each player is allocated a bundle that maximizes his profit according to this menu. We define the taxation complexity of a truthful mechanism to be the logarithm of the maximum number of menus that may be presented to a player. Our main finding is that in general the taxation complexity essentially equals the communication complexity. The proof consists of two main steps. First, we prove that for rich enough domains the taxation complexity is at most the communication complexity. We then show that the taxation complexity is much smaller than the communication complexity only in "pathological" cases and provide a formal description of these extreme cases. Next, we study mechanisms that access the valuations via value queries only. In this setting we establish that the menu complexity -- a notion that was already studied in several different contexts -- characterizes the number of value queries that the mechanism makes in exactly the same way that the taxation complexity characterizes the communication complexity. Our approach yields several applications, including strengthening the solution concept with low communication overhead, fast computation of prices, and hardness of approximation by computationally efficient truthful mechanisms

    Learning Convex Partitions and Computing Game-theoretic Equilibria from Best Response Queries

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    Suppose that an mm-simplex is partitioned into nn convex regions having disjoint interiors and distinct labels, and we may learn the label of any point by querying it. The learning objective is to know, for any point in the simplex, a label that occurs within some distance ϵ\epsilon from that point. We present two algorithms for this task: Constant-Dimension Generalised Binary Search (CD-GBS), which for constant mm uses poly(n,log(1ϵ))poly(n, \log \left( \frac{1}{\epsilon} \right)) queries, and Constant-Region Generalised Binary Search (CR-GBS), which uses CD-GBS as a subroutine and for constant nn uses poly(m,log(1ϵ))poly(m, \log \left( \frac{1}{\epsilon} \right)) queries. We show via Kakutani's fixed-point theorem that these algorithms provide bounds on the best-response query complexity of computing approximate well-supported equilibria of bimatrix games in which one of the players has a constant number of pure strategies. We also partially extend our results to games with multiple players, establishing further query complexity bounds for computing approximate well-supported equilibria in this setting.Comment: 38 pages, 7 figures, second version strengthens lower bound in Theorem 6, adds footnotes with additional comments and fixes typo

    When Are Welfare Guarantees Robust?

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    Computational and economic results suggest that social welfare maximization and combinatorial auction design are much easier when bidders\u27 valuations satisfy the "gross substitutes" condition. The goal of this paper is to evaluate rigorously the folklore belief that the main take-aways from these results remain valid in settings where the gross substitutes condition holds only approximately. We show that for valuations that pointwise approximate a gross substitutes valuation (in fact even a linear valuation), optimal social welfare cannot be approximated to within a subpolynomial factor and demand oracles cannot be simulated using a subexponential number of value queries. We then provide several positive results by imposing additional structure on the valuations (beyond gross substitutes), using a more stringent notion of approximation, and/or using more powerful oracle access to the valuations. For example, we prove that the performance of the greedy algorithm degrades gracefully for near-linear valuations with approximately decreasing marginal values; that with demand queries, approximate welfare guarantees for XOS valuations degrade gracefully for valuations that are pointwise close to XOS; and that the performance of the Kelso-Crawford auction degrades gracefully for valuations that are close to various subclasses of gross substitutes valuations

    Is Google the next Microsoft? Competition, Welfare and Regulation in Internet Search

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    Internet search (or perhaps more accurately `web-search') has grown exponentially over the last decade at an even more rapid rate than the Internet itself. Starting from nothing in the 1990s, today search is a multi-billion dollar business. Search engine providers such as Google and Yahoo! have become household names, and the use of a search engine, like use of the Web, is now a part of everyday life. The rapid growth of online search and its growing centrality to the ecology of the Internet raise a variety of questions for economists to answer. Why is the search engine market so concentrated and will it evolve towards monopoly? What are the implications of this concentration for different `participants' (consumers, search engines, advertisers)? Does the fact that search engines act as `information gatekeepers', determining, in effect, what can be found on the web, mean that search deserves particularly close attention from policy-makers? This paper supplies empirical and theoretical material with which to examine many of these questions. In particular, we (a) show that the already large levels of concentration are likely to continue (b) identify the consequences, negative and positive, of this outcome (c) discuss the possible regulatory interventions that policy-makers could utilize to address these

    A Unifying Hierarchy of Valuations with Complements and Substitutes

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    We introduce a new hierarchy over monotone set functions, that we refer to as MPH\mathcal{MPH} (Maximum over Positive Hypergraphs). Levels of the hierarchy correspond to the degree of complementarity in a given function. The highest level of the hierarchy, MPH\mathcal{MPH}-mm (where mm is the total number of items) captures all monotone functions. The lowest level, MPH\mathcal{MPH}-11, captures all monotone submodular functions, and more generally, the class of functions known as XOS\mathcal{XOS}. Every monotone function that has a positive hypergraph representation of rank kk (in the sense defined by Abraham, Babaioff, Dughmi and Roughgarden [EC 2012]) is in MPH\mathcal{MPH}-kk. Every monotone function that has supermodular degree kk (in the sense defined by Feige and Izsak [ITCS 2013]) is in MPH\mathcal{MPH}-(k+1)(k+1). In both cases, the converse direction does not hold, even in an approximate sense. We present additional results that demonstrate the expressiveness power of MPH\mathcal{MPH}-kk. One can obtain good approximation ratios for some natural optimization problems, provided that functions are required to lie in low levels of the MPH\mathcal{MPH} hierarchy. We present two such applications. One shows that the maximum welfare problem can be approximated within a ratio of k+1k+1 if all players hold valuation functions in MPH\mathcal{MPH}-kk. The other is an upper bound of 2k2k on the price of anarchy of simultaneous first price auctions. Being in MPH\mathcal{MPH}-kk can be shown to involve two requirements -- one is monotonicity and the other is a certain requirement that we refer to as PLE\mathcal{PLE} (Positive Lower Envelope). Removing the monotonicity requirement, one obtains the PLE\mathcal{PLE} hierarchy over all non-negative set functions (whether monotone or not), which can be fertile ground for further research

    Oracles and query lower bounds in generalised probabilistic theories

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    We investigate the connection between interference and computational power within the operationally defined framework of generalised probabilistic theories. To compare the computational abilities of different theories within this framework we show that any theory satisfying three natural physical principles possess a well-defined oracle model. Indeed, we prove a subroutine theorem for oracles in such theories which is a necessary condition for the oracle to be well-defined. The three principles are: causality (roughly, no signalling from the future), purification (each mixed state arises as the marginal of a pure state of a larger system), and strong symmetry existence of non-trivial reversible transformations). Sorkin has defined a hierarchy of conceivable interference behaviours, where the order in the hierarchy corresponds to the number of paths that have an irreducible interaction in a multi-slit experiment. Given our oracle model, we show that if a classical computer requires at least n queries to solve a learning problem, then the corresponding lower bound in theories lying at the kth level of Sorkin's hierarchy is n/k. Hence, lower bounds on the number of queries to a quantum oracle needed to solve certain problems are not optimal in the space of all generalised probabilistic theories, although it is not yet known whether the optimal bounds are achievable in general. Hence searches for higher-order interference are not only foundationally motivated, but constitute a search for a computational resource beyond that offered by quantum computation.Comment: 17+7 pages. Comments Welcome. Published in special issue "Foundational Aspects of Quantum Information" in Foundations of Physic
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