arXiv subjects

Noam Nisan

Publications and source records attributed to Noam Nisan.

At least 37 records · Page 2Linked to original sources

arXiv2020

On the Effectiveness of Tracking and Testing in SEIR Models

We study the effectiveness of tracking and testing in mitigating or suppressing epidemic outbreaks, in combination with or as an alternative to quarantines and global lockdowns. We study these intervention methods on a network-based SEIR model, augmented with an additional probability to model symptomatic, asymptomatic and pre-symptomatic cases. Our focus is on the basic trade-offs between economic costs and human lives lost, and how these trade-offs change under different lockdown, quarantine, tracking and testing policies. Our main findings are as follows: (i) Tests combined with patient quarantines reduce both economic costs and mortality, but require a large-scale testing capacity to achieve a significant improvement; (ii) Tracking significantly reduces both economic costs and mortality; (iii) Tracking combined with a limited number of tests can achieve containment without lockdowns; (iv) If there is a small flow of new incoming infections, dynamic "On-Off" lockdowns are more efficient than fixed lockdowns. Our simulation results underline the extreme effectiveness of tracking and testing policies in reducing both economic costs and mortality and their potential to contain epidemic outbreaks without imposing social distancing restrictions. This highlights the difficult social question of trading-off these gains with the privacy loss that tracking necessarily entails.

q-bio.PE↗

arXiv2020

Bipartite Perfect Matching as a Real Polynomial

We obtain a description of the Bipartite Perfect Matching decision problem as a multilinear polynomial over the Reals. We show that it has full degree and $(1-o_n(1))\cdot 2^{n^2}$ monomials with non-zero coefficients. In contrast, we show that in the dual representation (switching the roles of 0 and 1) the number of monomials is only exponential in $Θ(n \log n)$. Our proof relies heavily on the fact that the lattice of graphs which are "matching-covered" is Eulerian.

cs.DM↗

arXiv2019

Competitive Equilibrium with Generic Budgets: Beyond Additive

We study competitive equilibrium in the canonical Fisher market model, but with indivisible goods. In this model, every agent has a budget of artificial currency with which to purchase bundles of goods. Equilibrium prices match between demand and supply---at such prices, all agents simultaneously get their favorite within-budget bundle, and the market clears. Unfortunately, a competitive equilibrium may not exist when the goods are indivisible, even in extremely simple markets such as two agents with exactly the same budget and a single item. Yet in this example, once the budgets are slightly perturbed---i.e., made generic---a competitive equilibrium is guaranteed to exist. In this paper we explore the extent to which generic budgets can guarantee equilibrium existence (and thus related fairness guarantees) in markets with multiple items. We complement our results in [Babaioff et al., 2019] for additive preferences by exploring the case of general monotone preferences, establishing positive results for small numbers of items and mapping the limits of our approach. We then consider cardinal preferences, define a hierarchy of such preference classes and establish relations among them, and for some classes prove equilibrium existence under generic budgets.

cs.GT↗

arXiv2019

Universal Growth in Production Economies

We consider a simple variant of the von Neumann model of an expanding economy, in which multiple producers make goods according to their production function. The players trade their goods at the market and then use the bundles acquired for the production in the next round. We study a simple decentralized dynamic---known as proportional response---in which players update their bids proportionally to how useful the investments were in the past round. We show this dynamic leads to growth of the economy in the long term (whenever growth is possible) but also creates unbounded inequality, i.e. very rich and very poor players emerge. We analyze several other phenomena, such as how the relation of a player with others influences its development and the Gini index of the system. One of the key technical findings is that the players learn a global feature of the network (the optimal cycle) in a decentralized way, while interacting locally with their direct neighbors. We obtain this by studying the volume in the resulting dynamical system and showing that the volume of each cycle expands or contracts by a constant factor in each round.

cs.GT↗

arXiv2019

The Demand Query Model for Bipartite Matching

We introduce a `concrete complexity' model for studying algorithms for matching in bipartite graphs. The model is based on the "demand query" model used for combinatorial auctions. Most (but not all) known algorithms for bipartite matching seem to be translatable into this model including exact, approximate, sequential, parallel, and online ones. A perfect matching in a bipartite graph can be found in this model with O(n^{3/2}) demand queries (in a bipartite graph with n vertices on each side) and our main open problem is to either improve the upper bound or prove a lower bound. An improved upper bound could yield "normal" algorithms whose running time is better than the fastest ones known, while a lower bound would rule out a faster algorithm for bipartite matching from within a large class of algorithms. Our main result is a lower bound for finding an approximately maximum size matching in parallel: A deterministic algorithm that runs in n^{o(1)} rounds, where each round can make at most n^{1.99} demand queries cannot find a matching whose size is within n^{o(1)} factor of the maximum. This is in contrast to randomized algorithms that can find a matching whose size is $99\%$ of the maximum in O(\log n) rounds, each making n demand queries.

cs.CC↗

arXiv2018

Selling Multiple Correlated Goods: Revenue Maximization and Menu-Size Complexity (old title: "The Menu-Size Complexity of Auctions")

We consider the well known, and notoriously difficult, problem of a single revenue-maximizing seller selling two or more heterogeneous goods to a single buyer whose private values for the goods are drawn from a (possibly correlated) known distribution, and whose valuation is additive over the goods. We show that when there are two (or more) goods, _simple mechanisms_ -- such as selling the goods separately or as a bundle -- _may yield only a negligible fraction of the optimal revenue_. This resolves the open problem of Briest, Chawla, Kleinberg, and Weinberg (JET 2015) who prove the result for at least three goods in the related setup of a unit-demand buyer. We also introduce the menu size as a simple measure of the complexity of mechanisms, and show that the revenue may increase polynomially with _menu size_ and that no bounded menu size can ensure any positive fraction of the optimal revenue. The menu size also turns out to "pin down" the revenue properties of deterministic mechanisms.

cs.GT↗

arXiv2018

The Communication Complexity of Local Search

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 $Ω(\sqrt{N})$ 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.

cs.GT↗

arXiv2018

Competitive Equilibrium with Indivisible Goods and Generic Budgets

Competitive equilibrium from equal incomes (CEEI) is a classic solution to the problem of fair and efficient allocation of goods [Foley'67, Varian'74]. Every agent receives an equal budget of artificial currency with which to purchase goods, and prices match demand and supply. However, a CEEI is not guaranteed to exist when the goods are indivisible, even in the simple two-agent, single-item market. Yet, it is easy to see that once the two budgets are slightly perturbed (made generic), a competitive equilibrium does exist. In this paper we aim to extend this approach beyond the single-item case, and study the existence of equilibria in markets with two agents and additive preferences over multiple items. We show that for agents with equal budgets, making the budgets generic -- by adding vanishingly small random perturbations -- ensures the existence of an equilibrium. We further consider agents with arbitrary non-equal budgets, representing non-equal entitlements for goods. We show that competitive equilibrium guarantees a new notion of fairness among non-equal agents, and that it exists in cases of interest (like when the agents have identical preferences) if budgets are perturbed. Our results open opportunities for future research on generic equilibrium existence and fair treatment of non-equals.

cs.GT↗

arXiv2018

Communication Complexity of Cake Cutting

We study classic cake-cutting problems, but in discrete models rather than using infinite-precision real values, specifically, focusing on their communication complexity. Using general discrete simulations of classical infinite-precision protocols (Robertson-Webb and moving-knife), we roughly partition the various fair-allocation problems into 3 classes: "easy" (constant number of rounds of logarithmic many bits), "medium" (poly-logarithmic total communication), and "hard". Our main technical result concerns two of the "medium" problems (perfect allocation for 2 players and equitable allocation for any number of players) which we prove are not in the "easy" class. Our main open problem is to separate the "hard" from the "medium" classes.

cs.GT↗

arXiv2018

A Stable Marriage Requires Communication

The Gale-Shapley algorithm for the Stable Marriage Problem is known to take $Θ(n^2)$ steps to find a stable marriage in the worst case, but only $Θ(n \log n)$ steps in the average case (with $n$ women and $n$ men). In 1976, Knuth asked whether the worst-case running time can be improved in a model of computation that does not require sequential access to the whole input. A partial negative answer was given by Ng and Hirschberg, who showed that $Θ(n^2)$ queries are required in a model that allows certain natural random-access queries to the participants' preferences. A significantly more general - albeit slightly weaker - lower bound follows from Segal's general analysis of communication complexity, namely that $Ω(n^2)$ Boolean queries are required in order to find a stable marriage, regardless of the set of allowed Boolean queries. Using a reduction to the communication complexity of the disjointness problem, we give a far simpler, yet significantly more powerful argument showing that $Ω(n^2)$ Boolean queries of any type are indeed required for finding a stable - or even an approximately stable - marriage. Notably, unlike Segal's lower bound, our lower bound generalizes also to (A) randomized algorithms, (B) allowing arbitrary separate preprocessing of the women's preferences profile and of the men's preferences profile, (C) several variants of the basic problem, such as whether a given pair is married in every/some stable marriage, and (D) determining whether a proposed marriage is stable or far from stable. In order to analyze "approximately stable" marriages, we introduce the notion of "distance to stability" and provide an efficient algorithm for its computation.

cs.GT↗

arXiv2018

The Query Complexity of Cake Cutting

We study the query complexity of cake cutting and give lower and upper bounds for computing approximately envy-free, perfect, and equitable allocations with the minimum number of cuts. The lower bounds are tight for computing connected envy-free allocations among n=3 players and for computing perfect and equitable allocations with minimum number of cuts between n=2 players. We also formalize moving knife procedures and show that a large subclass of this family, which captures all the known moving knife procedures, can be simulated efficiently with arbitrarily small error in the Robertson-Webb query model.

cs.GT↗

arXiv2018

Optimal Deterministic Mechanisms for an Additive Buyer

We study revenue maximization by deterministic mechanisms for the simplest case for which Myerson's characterization does not hold: a single seller selling two items, with independently distributed values, to a single additive buyer. We prove that optimal mechanisms are submodular and hence monotone. Furthermore, we show that in the IID case, optimal mechanisms are symmetric. Our characterizations are surprisingly non-trivial, and we show that they fail to extend in several natural ways, e.g. for correlated distributions or more than two items. In particular, this shows that the optimality of symmetric mechanisms does not follow from the symmetry of the IID distribution.

cs.GT↗

arXiv2017

The Query Complexity of Correlated Equilibria

We consider the complexity of finding a correlated equilibrium of an $n$-player game in a model that allows the algorithm to make queries on players' payoffs at pure strategy profiles. Randomized regret-based dynamics are known to yield an approximate correlated equilibrium efficiently, namely, in time that is polynomial in the number of players $n$. Here we show that both randomization and approximation are necessary: no efficient deterministic algorithm can reach even an approximate correlated equilibrium, and no efficient randomized algorithm can reach an exact correlated equilibrium. The results are obtained by bounding from below the number of payoff queries that are needed.

cs.GT↗

arXiv2017

Approximate Revenue Maximization with Multiple Items

Maximizing the revenue from selling _more than one_ good (or item) to a single buyer is a notoriously difficult problem, in stark contrast to the one-good case. For two goods, we show that simple "one-dimensional" mechanisms, such as selling the goods separately, _guarantee_ at least 73% of the optimal revenue when the valuations of the two goods are independent and identically distributed, and at least $50\%$ when they are independent. For the case of $k>2$ independent goods, we show that selling them separately guarantees at least a $c/\log^2 k$ fraction of the optimal revenue; and, for independent and identically distributed goods, we show that selling them as one bundle guarantees at least a $c/\log k$ fraction of the optimal revenue. Additional results compare the revenues from the two simple mechanisms of selling the goods separately and bundled, identify situations where bundling is optimal, and extend the analysis to multiple buyers.

cs.GT↗

arXiv2017

Selling Complementary Goods: Dynamics, Efficiency and Revenue

We consider a price competition between two sellers of perfect-complement goods. Each seller posts a price for the good it sells, but the demand is determined according to the sum of prices. This is a classic model by Cournot (1838), who showed that in this setting a monopoly that sells both goods is better for the society than two competing sellers. We show that non-trivial pure Nash equilibria always exist in this game. We also quantify Cournot's observation with respect to both the optimal welfare and the monopoly revenue. We then prove a series of mostly negative results regarding the convergence of best response dynamics to equilibria in such games.

cs.GT↗

arXiv2017

Efficient Empirical Revenue Maximization in Single-Parameter Auction Environments

We present a polynomial-time algorithm that, given samples from the unknown valuation distribution of each bidder, learns an auction that approximately maximizes the auctioneer's revenue in a variety of single-parameter auction environments including matroid environments, position environments, and the public project environment. The valuation distributions may be arbitrary bounded distributions (in particular, they may be irregular, and may differ for the various bidders), thus resolving a problem left open by previous papers. The analysis uses basic tools, is performed in its entirety in value-space, and simplifies the analysis of previously known results for special cases. Furthermore, the analysis extends to certain single-parameter auction environments where precise revenue maximization is known to be intractable, such as knapsack environments.

cs.GT↗

arXiv2017

An Experimental Evaluation of Regret-Based Econometrics

Using data obtained in a controlled ad-auction experiment that we ran, we evaluate the regret-based approach to econometrics that was recently suggested by Nekipelov, Syrgkanis, and Tardos (EC 2015). We found that despite the weak regret-based assumptions, the results were (at least) as accurate as those obtained using classic equilibrium-based assumptions. En route we studied to what extent humans actually minimize regret in our ad auction, and found a significant difference between the "high types" (players with a high valuation) who indeed rationally minimized regret and the "low types" who significantly overbid. We suggest that correcting for these biases and adjusting the regret-based econometric method may improve the accuracy of estimated values.

cs.GT↗

arXiv2017

ERA: A Framework for Economic Resource Allocation for the Cloud

Cloud computing has reached significant maturity from a systems perspective, but currently deployed solutions rely on rather basic economics mechanisms that yield suboptimal allocation of the costly hardware resources. In this paper we present Economic Resource Allocation (ERA), a complete framework for scheduling and pricing cloud resources, aimed at increasing the efficiency of cloud resources usage by allocating resources according to economic principles. The ERA architecture carefully abstracts the underlying cloud infrastructure, enabling the development of scheduling and pricing algorithms independently of the concrete lower-level cloud infrastructure and independently of its concerns. Specifically, ERA is designed as a flexible layer that can sit on top of any cloud system and interfaces with both the cloud resource manager and with the users who reserve resources to run their jobs. The jobs are scheduled based on prices that are dynamically calculated according to the predicted demand. Additionally, ERA provides a key internal API to pluggable algorithmic modules that include scheduling, pricing and demand prediction. We provide a proof-of-concept software and demonstrate the effectiveness of the architecture by testing ERA over both public and private cloud systems -- Azure Batch of Microsoft and Hadoop/YARN. A broader intent of our work is to foster collaborations between economics and system communities. To that end, we have developed a simulation platform via which economics and system experts can test their algorithmic implementations.

cs.GT↗