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Register a new userSeparating the Communication Complexity of Truthful and Non-Truthful Combinatorial Auctions
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Sepehr Assadi
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We provide the first separation in the approximation guarantee achievable by truthful and non-truthful combinatorial auctions with polynomial communication. Specifically, we prove that any truthful mechanism guaranteeing a $(frac{3}{4}-frac{1}{240}+varepsilon)$-approximation for two buyers with XOS valuations over $m$ items requires $exp(Omega(varepsilon^2 cdot m))$ communication, whereas a non-truthful algorithm by Dobzinski and Schapira [SODA 2006] and Feige [2009] is already known to achieve a $frac{3}{4}$-approximation in $poly(m)$ communication. We obtain our separation by proving that any {simultaneous} protocol ({not} necessarily truthful) which guarantees a $(frac{3}{4}-frac{1}{240}+varepsilon)$-approximation requires communication $exp(Omega(varepsilon^2 cdot m))$. The taxation complexity framework of Dobzinski [FOCS 2016] extends this lower bound to all truthful mechanisms (including interactive truthful mechanisms).
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We study a central problem in Algorithmic Mechanism Design: constructing truthful mechanisms for welfare maximization in combinatorial auctions with submodular bidders. Dobzinski, Nisan, and Schapira provided the first mechanism that guarantees a non-trivial approximation ratio of $O(log^2 m)$ [STOC06], where $m$ is the number of items. This was subsequently improved to $O(log mlog log m)$ [Dobzinski, APPROX07] and then to $O(log m)$ [Krysta and Vocking, ICALP12]. In this paper we develop the first mechanism that breaks the logarithmic barrier. Specifically, the mechanism provides an approximation ratio of $O(sqrt {log m})$. Similarly to previous constructions, our mechanism uses polynomially many value and demand queries, and in fact provides the same approximation ratio for the larger class of XOS (a.k.a. fractionally subadditive) valuations. We also develop a computationally efficient implementation of the mechanism for combinatorial auctions with budget additive bidders. Although in general computing a demand query is NP-hard for budget additive valuations, we observe that the specific form of demand queries that our mechanism uses can be efficiently computed when bidders are budget additive.
We consider the problem of fitting a linear model to data held by individuals who are concerned about their privacy. Incentivizing most players to truthfully report their data to the analyst constrains our design to mechanisms that provide a privacy guarantee to the participants; we use differential privacy to model individuals privacy losses. This immediately poses a problem, as differentially private computation of a linear model necessarily produces a biased estimation, and existing approaches to design mechanisms to elicit data from privacy-sensitive individuals do not generalize well to biased estimators. We overcome this challenge through an appropriate design of the computation and payment scheme.
We consider a fundamental dynamic allocation problem motivated by the problem of $textit{securities lending}$ in financial markets, the mechanism underlying the short selling of stocks. A lender would like to distribute a finite number of identical copies of some scarce resource to $n$ clients, each of whom has a private demand that is unknown to the lender. The lender would like to maximize the usage of the resource $mbox{---}$ avoiding allocating more to a client than her true demand $mbox{---}$ but is constrained to sell the resource at a pre-specified price per unit, and thus cannot use prices to incentivize truthful reporting. We first show that the Bayesian optimal algorithm for the one-shot problem $mbox{---}$ which maximizes the resources expected usage according to the posterior expectation of demand, given reports $mbox{---}$ actually incentivizes truthful reporting as a dominant strategy. Because true demands in the securities lending problem are often sensitive information that the client would like to hide from competitors, we then consider the problem under the additional desideratum of (joint) differential privacy. We give an algorithm, based on simple dynamics for computing market equilibria, that is simultaneously private, approximately optimal, and approximately dominant-strategy truthful. Finally, we leverage this private algorithm to construct an approximately truthful, optimal mechanism for the extensive form multi-round auction where the lender does not have access to the true joint distributions between clients requests and demands.
We study the following communication variant of local search. There is some fixed, commonly known graph $G$. Alice holds $f_A$ and Bob holds $f_B$, both are functions that specify a value for each vertex. The goal is to find a local maximum of $f_A+f_B$ with respect to $G$, i.e., a vertex $v$ for which $(f_A+f_B)(v)geq (f_A+f_B)(u)$ for every neighbor $u$ of $v$. Our main result is that finding a local maximum requires polynomial (in the number of vertices) bits of communication. The result holds for the following families of graphs: three dimensional grids, hypercubes, odd graphs, and degree 4 graphs. Moreover, we provide an emph{optimal} communication bound of $Omega(sqrt{N})$ for the hypercube, and for a constant dimensional greed, where $N$ is the number of vertices in the graph. We provide applications of our main result in two domains, exact potential games and combinatorial auctions. First, we show that finding a pure Nash equilibrium in $2$-player $N$-action exact potential games requires polynomial (in $N$) communication. We also show that finding a pure Nash equilibrium in $n$-player $2$-action exact potential games requires exponential (in $n$) communication. The second domain that we consider is combinatorial auctions, in which we prove that finding a local maximum in combinatorial auctions requires exponential (in the number of items) communication even when the valuations are submodular. Each one of the results demonstrates an exponential separation between the non-deterministic communication complexity and the randomized communication complexity of a total search problem.
We consider the problem of allocating a set on indivisible items to players with private preferences in an efficient and fair way. We focus on valuations that have dichotomous marginals, in which the added value of any item to a set is either 0 or 1, and aim to design truthful allocation mechanisms (without money) that maximize welfare and are fair. For the case that players have submodular valuations with dichotomous marginals, we design such a deterministic truthful allocation mechanism. The allocation output by our mechanism is Lorenz dominating, and consequently satisfies many desired fairness properties, such as being envy-free up to any item (EFX), and maximizing the Nash Social Welfare (NSW). We then show that our mechanism with random priorities is envy-free ex-ante, while having all the above properties ex-post. Furthermore, we present several impossibility results precluding similar results for the larger class of XOS valuations. To gauge the robustness of our positive results, we also study $epsilon$-dichotomous valuations, in which the added value of any item to a set is either non-positive, or in the range $[1, 1 + epsilon]$. We show several impossibility results in this setting, and also a positive result: for players that have additive $epsilon$-dichotomous valuations with sufficiently small $epsilon$, we design a randomized truthful mechanism with strong ex-post guarantees. For $rho = frac{1}{1 + epsilon}$, the allocations that it produces generate at least a $rho$-fraction of the maximum welfare, and enjoy $rho$-approximations for various fairness properties, such as being envy-free up to one item (EF1), and giving each player at least her maximin share.
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