Showing papers by "Richard Cole published in 2013"

Mechanism design for fair division: allocating divisible items without payments

Abstract: We revisit the classic problem of fair division from a mechanism design perspective and provide an elegant truthful mechanism that yields surprisingly good approximation guarantees for the widely used solution of Proportional Fairness. This solution, which is closely related to Nash bargaining and the competitive equilibrium, is known to be not implementable in a truthful fashion, which has been its main drawback. To alleviate this issue, we propose a new mechanism, which we call the Partial Allocation mechanism, that discards a carefully chosen fraction of the allocated resources in order to incentivize the agents to be truthful in reporting their valuations. This mechanism introduces a way to implement interesting truthful outcomes in settings where monetary payments are not an option.For a multi-dimensional domain with an arbitrary number of agents and items, and for the very large class of homogeneous valuation functions, we prove that our mechanism provides every agent with at least a 1/e ≈ 0.368 fraction of her Proportionally Fair valuation. To the best of our knowledge, this is the first result that gives a constant factor approximation to every agent for the Proportionally Fair solution. To complement this result, we show that no truthful mechanism can guarantee more than 0.5 approximation, even for the restricted class of additive linear valuations. In addition to this, we uncover a connection between the Partial Allocation mechanism and VCG-based mechanism design.We also ask whether better approximation ratios are possible in more restricted settings. In particular, motivated by the massive privatization auction in the Czech republic in the early 90s we provide another mechanism for additive linear valuations that works really well when all the items are highly demanded.

Tatonnement beyond gross substitutes?: gradient descent to the rescue

Abstract: Tatonnement is a simple and natural rule for updating prices in Exchange (Arrow-Debreu) markets. In this paper we define a class of markets for which tatonnement is equivalent to gradient descent. This is the class of markets for which there is a convex potential function whose gradient is always equal to the negative of the excess demand and we call it Convex Potential Function (CPF) markets. We show the following results. CPF markets contain the class of Eisenberg Gale (EG) markets, defined previously by Jain and Vazirani. The subclass of CPF markets for which the demand is a differentiable function contains exactly those markets whose demand function has a symmetric negative semi-definite Jacobian. We define a family of continuous versions of tatonnement based on gradient descent using a Bregman divergence. As we show, all processes in this family converge to an equilibrium for any CPF market. This is analogous to the classic result for markets satisfying the Weak Gross Substitutes property. A discrete version of tatonnement converges toward the equilibrium for the following markets of complementary goods; its convergence rate for these settings is analyzed using a common potential function. Fisher markets in which all buyers have Leontief utilities. The tatonnement process reduces the distance to the equilibrium, as measured by the potential function, to an e fraction of its initial value in O(1/e) rounds of price updates. Fisher markets in which all buyers have complementary CES utilities. Here, the distance to the equilibrium is reduced to an e fraction of its initial value in O(log(1/e)) rounds of price updates.This shows that tatonnement converges for the entire range of Fisher markets when buyers have complementary CES utilities, in contrast to prior work, which could analyze only the substitutes range, together with a small portion of the complementary range.

Analysis of Randomized Work Stealing with False Sharing

Abstract: This paper analyzes the overhead due to false sharing when parallel tasks are scheduled using randomized work stealing (RWS). We obtain high-probability bounds on the cache miss overhead, including the overhead due to false sharing, for several parallel cache-efficient algorithms when scheduled using RWS. These include algorithms for fundamental problems, such as matrix computations, FFT, sorting, basic dynamic programming, list ranking and graph connected components. Our main technical contribution, from which these results follow, is the derivation of nontrivial high-probability bounds on the number of steals incurred by these algorithms in the presence of false sharing, when using RWS.

Positive results for mechanism design without money

Abstract: Consider the problem of allocating multiple divisible goods to two agents in a strategy-proof fashion without the use of payments or priors. Previous work has aimed at implementing allocations that are competitive with respect to an appropriately defined measure of social welfare. These results have mostly been negative, proving that no dictatorial mechanism can achieve an approximation factor better than 0.5, and leaving open the question of whether there exists a non-dictatorial mechanism that outperforms this bound. We provide a positive answer to this question by presenting an interesting non-dictatorial mechanism that achieves an approximation factor of 2/3 for this measure of social welfare. In proving this bound we also touch on the issue of fairness: we show that the proportionally fair solution, a well known fairness concept for money-free settings, is highly competitive with respect to social welfare. We then show how to use the proportionally fair solution to design our non-dictatorial strategy-proof mechanism.

Suffix Trays and Suffix Trists: Structures for Faster Text Indexing

Abstract: Suffix trees and suffix arrays are two of the most widely used data structures for text indexing. Each uses linear space and can be constructed in linear time for polynomially sized alphabets. However, when it comes to answering queries with worst-case deterministic time bounds, the prior does so in $O(m\log|\Sigma|)$ time, where $m$ is the query size, $|\Sigma|$ is the alphabet size, and the latter does so in $O(m+\log n)$ time, where $n$ is the text size. If one wants to output all appearances of the query, an additive cost of $O(occ)$ time is sufficient, where $occ$ is the size of the output. We propose a novel way of combining the two into, what we call, a {\em suffix tray}. The space and construction time remain linear and the query time improves to $O(m+\log|\Sigma|)$ for integer alphabets from a linear range, i.e. $\Sigma \subset \{1,\cdots, cn\}$, for an arbitrary constant $c$. The construction and query are deterministic. Here also an additive $O(occ)$ time is sufficient if one desires to output all appearances of the query. We also consider the online version of indexing, where the text arrives online, one character at a time, and indexing queries are answered in tandem. In this variant we create a cross between a suffix tree and a suffix list (a dynamic variant of suffix array) to be called a {\em suffix trist}; it supports queries in $O(m+\log|\Sigma|)$ time. The suffix trist also uses linear space. Furthermore, if there exists an online construction for a linear-space suffix tree such that the cost of adding a character is worst-case deterministic $f(n,|\Sigma|)$ ($n$ is the size of the current text), then one can further update the suffix trist in $O(f(n,|\Sigma|)+\log |\Sigma|)$ time. The best currently known worst-case deterministic bound for $f(n,|\Sigma|)$ is $O(\log n)$ time.

Incentive-centered design of money-free mechanisms

Abstract: This thesis serves as a step toward a better understanding of how to design fair and efficient multiagent resource allocation systems by bringing the incentives of the participating agents to the center of the design process. As the quality of these systems critically depends on the ways in which the participants interact with each other and with the system, an ill-designed set of incentives can lead to severe inefficiencies. The special focus of this work is on the problems that arise when the use of monetary exchanges between the system and the participants is prohibited. This is a common restriction that substantially complicates the designer's task; we nevertheless provide a sequence of positive results in the form of mechanisms that maximize efficiency or fairness despite the possibly self-interested behavior of the participating agents. The first part of this work is a contribution to the literature on approximate mechanism design without money. Given a set of divisible resources, our goal is to design a mechanism that allocates them among the agents. The main complication here is due to the fact that the agents' preferences over different allocations may not be known to the system. Therefore, the mechanism needs to be designed in such a way that it is in the best interest of every agent to report the truth about her preferences; since monetary rewards and penalties cannot be used in order to elicit the truth, a much more delicate regulation of the resource allocation is necessary. The second part of this work concerns the design of money-free resource allocation mechanisms for decentralized multiagent systems. As the world has become increasingly interconnected, such systems are using more and more resources that are geographically dispersed; to provide scalability in these systems, the mechanisms need to be decentralized. That is, the allocation decisions for any given resource should not assume global information regarding the system's resources or participants. We approach this restriction by using coordination mechanisms: simple resource allocation policies, each of which controls only one of the resources and uses only local information regarding the state of the system.