Nima Anari

Mechanism Design for Crowdsourcing: An Optimal 1-1/e Competitive Budget-Feasible Mechanism for Large Markets

In this paper we consider a mechanism design problem in the context of large-scale crowdsourcing markets such as Amazons Mechanical Turk mturk, ClickWorker clickworker, CrowdFlower crowdflower. In these markets, there is a requester who wants to hire workers to accomplish some tasks. Each worker is assumed to give some utility to the requester on getting hired. Moreover each worker has a minimum cost that he wants to get paid for getting hired. This minimum cost is assumed to be private information of the workers. The question then is -- if the requester has a limited budget, how to design a direct revelation mechanism that picks the right set of workers to hire in order to maximize the requesters utility? We note that although the previous work (Singer (2010) chen et al. (2011)) has studied this problem, a crucial difference in which we deviate from earlier work is the notion of large-scale markets that we introduce in our model. Without the large market assumption, it is known that no mechanism can achieve a competitive ratio better than 0.414 and 0.5 for deterministic and randomized mechanisms respectively (while the best known deterministic and randomized mechanisms achieve an approximation ratio of 0.292 and 0.33 respectively). In this paper, we design a budget-feasible mechanism for large markets that achieves a competitive ratio of 1 - 1/e ≃ 0.63. Our mechanism can be seen as a generalization of an alternate way to look at the proportional share mechanism, which is used in all the previous works so far on this problem. Interestingly, we can also show that our mechanism is optimal by showing that no truthful mechanism can achieve a factor better than 1 - 1/e, thus, fully resolving this setting. Finally we consider the more general case of submodular utility functions and give new and improved mechanisms for the case when the market is large.

Equilibrium Pricing with Positive Externalities (Extended Abstract)

We study the problem of selling an item to strategic buyers in the presence of positive historical externalities, where the value of a product increases as more people buy and use it. This increase in the value of the product is the result of resolving bugs or security holes after more usage. We consider a continuum of buyers that are partitioned into types where each type has a valuation function based on the actions of other buyers. Given a fixed sequence of prices, or price trajectory, buyers choose a day on which to purchase the product, i.e., they have to decide whether to purchase the product early in the game or later after more people already own it. We model this strategic setting as a game, study existence and uniqueness of the equilibria, and design an FPTAS to compute an approximately revenue-maximizing pricing trajectory for the seller in two special cases: the symmetric settings in which there is just a single buyer type, and the linear settings that are characterized by an initial type-independent bias and a linear type-dependent influenceability coefficient.

A generalization of permanent inequalities and applications in counting and optimization

A polynomial pΕℝ[z1,…,zn] is real stable if it has no roots in the upper-half complex plane. Gurvitss permanent inequality gives a lower bound on the coefficient of the z1z2…zn monomial of a real stable polynomial p with nonnegative coefficients. This fundamental inequality has been used to attack several counting and optimization problems. Here, we study a more general question: Given a stable multilinear polynomial p with nonnegative coefficients and a set of monomials S, we show that if the polynomial obtained by summing up all monomials in S is real stable, then we can lower bound the sum of coefficients of monomials of p that are in S. We also prove generalizations of this theorem to (real stable) polynomials that are not multilinear. We use our theorem to give a new proof of Schrijvers inequality on the number of perfect matchings of a regular bipartite graph, generalize a recent result of Nikolov and Singh, and give deterministic polynomial time approximation algorithms for several counting problems.

Effective-Resistance-Reducing Flows, Spectrally Thin Trees, and Asymmetric TSP

We show that the integrality gap of the natural LP relaxation of the Asymmetric Traveling Salesman Problem is polyloglog(n). In other words, there is a polynomial time algorithm that approximates the value of the optimum tour within a factor of polyloglog(n), where polyloglog(n) is a bounded degree polynomial of loglog(n). We prove this by showing that any k-edge-connected unweighted graph has a polyloglog(n)/k-thin spanning tree. Our main new ingredient is a procedure, albeit an exponentially sized convex program, that “transforms” graphs that do not admit any spectrally thin trees into those that provably have spectrally thin trees. More precisely, given a k-edge-connected graph G = (V, E) where k ≥ 7 log(n), we show that there is a matrix D that “preserves” the structure of all cuts of G such that for a set F ⊆ E that induces an Ω(k)-edge-connected graph, the effective resistance of every edge in F w.r.t. D is at most polylog(k)/k. Then, we use our extension of the seminal work of Marcus, Spielman, and Srivastava [1], fully explained in [2], to prove the existence of a polylog(k)/k-spectrally thin tree with respect to D. Such a tree is polylog(k)/k-combinatorially thin with respect to G as D preserves the structure of cuts of G.

Nash Social Welfare, Matrix Permanent, and Stable Polynomials

We study the problem of allocating

Simply Exponential Approximation of the Permanent of Positive Semidefinite Matrices

m

Euclidean movement minimization

items to

Graph Clustering using Effective Resistance.

n

Budget Feasible Procurement Auctions

agents subject to maximizing the Nash social welfare (NSW) objective. We write a novel convex programming relaxation for this problem, and we show that a simple randomized rounding algorithm gives a

Equilibrium pricing with positive externalities

1/e