433 research outputs found

    Inapproximability of Truthful Mechanisms via Generalizations of the VC Dimension

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    Algorithmic mechanism design (AMD) studies the delicate interplay between computational efficiency, truthfulness, and optimality. We focus on AMD's paradigmatic problem: combinatorial auctions. We present a new generalization of the VC dimension to multivalued collections of functions, which encompasses the classical VC dimension, Natarajan dimension, and Steele dimension. We present a corresponding generalization of the Sauer-Shelah Lemma and harness this VC machinery to establish inapproximability results for deterministic truthful mechanisms. Our results essentially unify all inapproximability results for deterministic truthful mechanisms for combinatorial auctions to date and establish new separation gaps between truthful and non-truthful algorithms

    FlexAuc: Serving Dynamic Demands in a Spectrum Trading Market with Flexible Auction

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    In secondary spectrum trading markets, auctions are widely used by spectrum holders (SHs) to redistribute their unused channels to secondary wireless service providers (WSPs). As sellers, the SHs design proper auction schemes to stimulate more participants and maximize the revenue from the auction. As buyers, the WSPs determine the bidding strategies in the auction to better serve their end users. In this paper, we consider a three-layered spectrum trading market consisting of the SH, the WSPs and the end users. We jointly study the strategies of the three parties. The SH determines the auction scheme and spectrum supplies to optimize its revenue. The WSPs have flexible bidding strategies in terms of both demands and valuations considering the strategies of the end users. We design FlexAuc, a novel auction mechanism for this market to enable dynamic supplies and demands in the auction. We prove theoretically that FlexAuc not only maximizes the social welfare but also preserves other nice properties such as truthfulness and computational tractability.Comment: 11 pages, 7 figures, Preliminary version accepted in INFOCOM 201

    Mechanisms for Risk Averse Agents, Without Loss

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    Auctions in which agents' payoffs are random variables have received increased attention in recent years. In particular, recent work in algorithmic mechanism design has produced mechanisms employing internal randomization, partly in response to limitations on deterministic mechanisms imposed by computational complexity. For many of these mechanisms, which are often referred to as truthful-in-expectation, incentive compatibility is contingent on the assumption that agents are risk-neutral. These mechanisms have been criticized on the grounds that this assumption is too strong, because "real" agents are typically risk averse, and moreover their precise attitude towards risk is typically unknown a-priori. In response, researchers in algorithmic mechanism design have sought the design of universally-truthful mechanisms --- mechanisms for which incentive-compatibility makes no assumptions regarding agents' attitudes towards risk. We show that any truthful-in-expectation mechanism can be generically transformed into a mechanism that is incentive compatible even when agents are risk averse, without modifying the mechanism's allocation rule. The transformed mechanism does not require reporting of agents' risk profiles. Equivalently, our result can be stated as follows: Every (randomized) allocation rule that is implementable in dominant strategies when players are risk neutral is also implementable when players are endowed with an arbitrary and unknown concave utility function for money.Comment: Presented at the workshop on risk aversion in algorithmic game theory and mechanism design, held in conjunction with EC 201

    Single Parameter Combinatorial Auctions with Partially Public Valuations

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    We consider the problem of designing truthful auctions, when the bidders' valuations have a public and a private component. In particular, we consider combinatorial auctions where the valuation of an agent ii for a set SS of items can be expressed as vif(S)v_if(S), where viv_i is a private single parameter of the agent, and the function ff is publicly known. Our motivation behind studying this problem is two-fold: (a) Such valuation functions arise naturally in the case of ad-slots in broadcast media such as Television and Radio. For an ad shown in a set SS of ad-slots, f(S)f(S) is, say, the number of {\em unique} viewers reached by the ad, and viv_i is the valuation per-unique-viewer. (b) From a theoretical point of view, this factorization of the valuation function simplifies the bidding language, and renders the combinatorial auction more amenable to better approximation factors. We present a general technique, based on maximal-in-range mechanisms, that converts any α\alpha-approximation non-truthful algorithm (α1\alpha \leq 1) for this problem into Ω(αlogn)\Omega(\frac{\alpha}{\log{n}}) and Ω(α)\Omega(\alpha)-approximate truthful mechanisms which run in polynomial time and quasi-polynomial time, respectively

    On Simultaneous Two-player Combinatorial Auctions

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    We consider the following communication problem: Alice and Bob each have some valuation functions v1()v_1(\cdot) and v2()v_2(\cdot) over subsets of mm items, and their goal is to partition the items into S,SˉS, \bar{S} in a way that maximizes the welfare, v1(S)+v2(Sˉ)v_1(S) + v_2(\bar{S}). We study both the allocation problem, which asks for a welfare-maximizing partition and the decision problem, which asks whether or not there exists a partition guaranteeing certain welfare, for binary XOS valuations. For interactive protocols with poly(m)poly(m) communication, a tight 3/4-approximation is known for both [Fei06,DS06]. For interactive protocols, the allocation problem is provably harder than the decision problem: any solution to the allocation problem implies a solution to the decision problem with one additional round and logm\log m additional bits of communication via a trivial reduction. Surprisingly, the allocation problem is provably easier for simultaneous protocols. Specifically, we show: 1) There exists a simultaneous, randomized protocol with polynomial communication that selects a partition whose expected welfare is at least 3/43/4 of the optimum. This matches the guarantee of the best interactive, randomized protocol with polynomial communication. 2) For all ε>0\varepsilon > 0, any simultaneous, randomized protocol that decides whether the welfare of the optimal partition is 1\geq 1 or 3/41/108+ε\leq 3/4 - 1/108+\varepsilon correctly with probability >1/2+1/poly(m)> 1/2 + 1/ poly(m) requires exponential communication. This provides a separation between the attainable approximation guarantees via interactive (3/43/4) versus simultaneous (3/41/108\leq 3/4-1/108) protocols with polynomial communication. In other words, this trivial reduction from decision to allocation problems provably requires the extra round of communication

    Implementation in Advised Strategies: Welfare Guarantees from Posted-Price Mechanisms When Demand Queries Are NP-Hard

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    State-of-the-art posted-price mechanisms for submodular bidders with mm items achieve approximation guarantees of O((loglogm)3)O((\log \log m)^3) [Assadi and Singla, 2019]. Their truthfulness, however, requires bidders to compute an NP-hard demand-query. Some computational complexity of this form is unavoidable, as it is NP-hard for truthful mechanisms to guarantee even an m1/2εm^{1/2-\varepsilon}-approximation for any ε>0\varepsilon > 0 [Dobzinski and Vondr\'ak, 2016]. Together, these establish a stark distinction between computationally-efficient and communication-efficient truthful mechanisms. We show that this distinction disappears with a mild relaxation of truthfulness, which we term implementation in advised strategies, and that has been previously studied in relation to "Implementation in Undominated Strategies" [Babaioff et al, 2009]. Specifically, advice maps a tentative strategy either to that same strategy itself, or one that dominates it. We say that a player follows advice as long as they never play actions which are dominated by advice. A poly-time mechanism guarantees an α\alpha-approximation in implementation in advised strategies if there exists poly-time advice for each player such that an α\alpha-approximation is achieved whenever all players follow advice. Using an appropriate bicriterion notion of approximate demand queries (which can be computed in poly-time), we establish that (a slight modification of) the [Assadi and Singla, 2019] mechanism achieves the same O((loglogm)3)O((\log \log m)^3)-approximation in implementation in advised strategies
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