arXiv subjects

Ashish Goel

Publications and source records attributed to Ashish Goel.

At least 55 records · Page 3Linked to original sources

arXiv2019

Markets Beyond Nash Welfare for Leontief Utilities

We study the allocation of divisible goods to competing agents via a market mechanism, focusing on agents with Leontief utilities. The majority of the economics and mechanism design literature has focused on \emph{linear} prices, meaning that the cost of a good is proportional to the quantity purchased. Equilibria for linear prices are known to be exactly the maximum Nash welfare allocations. \emph{Price curves} allow the cost of a good to be any (increasing) function of the quantity purchased. We show that price curve equilibria are not limited to maximum Nash welfare allocations with two main results. First, we show that an allocation can be supported by strictly increasing price curves if and only if it is \emph{group-domination-free}. A similarly characterization holds for weakly increasing price curves. We use this to show that given any allocation, we can compute strictly (or weakly) increasing price curves that support it (or show that none exist) in polynomial time. These results involve a connection to the \emph{agent-order matrix} of an allocation, which may have other applications. Second, we use duality to show that in the bandwidth allocation setting, any allocation maximizing a CES welfare function can be supported by price curves.

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arXiv2019

Who is in Your Top Three? Optimizing Learning in Elections with Many Candidates

Elections and opinion polls often have many candidates, with the aim to either rank the candidates or identify a small set of winners according to voters' preferences. In practice, voters do not provide a full ranking; instead, each voter provides their favorite K candidates, potentially in ranked order. The election organizer must choose K and an aggregation rule. We provide a theoretical framework to make these choices. Each K-Approval or K-partial ranking mechanism (with a corresponding positional scoring rule) induces a learning rate for the speed at which the election correctly recovers the asymptotic outcome. Given the voter choice distribution, the election planner can thus identify the rate optimal mechanism. Earlier work in this area provides coarse order-of-magnitude guaranties which are not sufficient to make such choices. Our framework further resolves questions of when randomizing between multiple mechanisms may improve learning, for arbitrary voter noise models. Finally, we use data from 5 large participatory budgeting elections that we organized across several US cities, along with other ranking data, to demonstrate the utility of our methods. In particular, we find that historically such elections have set K too low and that picking the right mechanism can be the difference between identifying the ultimate winner with only a 80% probability or a 99.9% probability after 400 voters.

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arXiv2019

Markets for Public Decision-making

A public decision-making problem consists of a set of issues, each with multiple possible alternatives, and a set of competing agents, each with a preferred alternative for each issue. We study adaptations of market economies to this setting, focusing on binary issues. Issues have prices, and each agent is endowed with artificial currency that she can use to purchase probability for her preferred alternatives (we allow randomized outcomes). We first show that when each issue has a single price that is common to all agents, market equilibria can be arbitrarily bad. This negative result motivates a different approach. We present a novel technique called "pairwise issue expansion", which transforms any public decision-making instance into an equivalent Fisher market, the simplest type of private goods market. This is done by expanding each issue into many goods: one for each pair of agents who disagree on that issue. We show that the equilibrium prices in the constructed Fisher market yield a "pairwise pricing equilibrium" in the original public decision-making problem which maximizes Nash welfare. More broadly, pairwise issue expansion uncovers a powerful connection between the public decision-making and private goods settings; this immediately yields several interesting results about public decisions markets, and furthers the hope that we will be able to find a simple iterative voting protocol that leads to near-optimum decisions.

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arXiv2019

Pruning based Distance Sketches with Provable Guarantees on Random Graphs

Measuring the distances between vertices on graphs is one of the most fundamental components in network analysis. Since finding shortest paths requires traversing the graph, it is challenging to obtain distance information on large graphs very quickly. In this work, we present a preprocessing algorithm that is able to create landmark based distance sketches efficiently, with strong theoretical guarantees. When evaluated on a diverse set of social and information networks, our algorithm significantly improves over existing approaches by reducing the number of landmarks stored, preprocessing time, or stretch of the estimated distances. On Erdös-Rényi graphs and random power law graphs with degree distribution exponent $2 < β< 3$, our algorithm outputs an exact distance data structure with space between $Θ(n^{5/4})$ and $Θ(n^{3/2})$ depending on the value of $β$, where $n$ is the number of vertices. We complement the algorithm with tight lower bounds for Erdos-Renyi graphs and the case when $β$ is close to two.

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arXiv2018

Iterative Local Voting for Collective Decision-making in Continuous Spaces

Many societal decision problems lie in high-dimensional continuous spaces not amenable to the voting techniques common for their discrete or single-dimensional counterparts. These problems are typically discretized before running an election or decided upon through negotiation by representatives. We propose a algorithm called {\sc Iterative Local Voting} for collective decision-making in this setting. In this algorithm, voters are sequentially sampled and asked to modify a candidate solution within some local neighborhood of its current value, as defined by a ball in some chosen norm, with the size of the ball shrinking at a specified rate. We first prove the convergence of this algorithm under appropriate choices of neighborhoods to Pareto optimal solutions with desirable fairness properties in certain natural settings: when the voters' utilities can be expressed in terms of some form of distance from their ideal solution, and when these utilities are additively decomposable across dimensions. In many of these cases, we obtain convergence to the societal welfare maximizing solution. We then describe an experiment in which we test our algorithm for the decision of the U.S. Federal Budget on Mechanical Turk with over 2,000 workers, employing neighborhoods defined by $\mathcal{L}^1, \mathcal{L}^2$ and $\mathcal{L}^\infty$ balls. We make several observations that inform future implementations of such a procedure.

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arXiv2018

Implementing the Lexicographic Maxmin Bargaining Solution

There has been much work on exhibiting mechanisms that implement various bargaining solutions, in particular, the Kalai-Smorodinsky solution \cite{moulin1984implementing} and the Nash Bargaining solution. Another well-known and axiomatically well-studied solution is the lexicographic maxmin solution. However, there is no mechanism known for its implementation. To fill this gap, we construct a mechanism that implements the lexicographic maxmin solution as the unique subgame perfect equilibrium outcome in the n-player setting. As is standard in the literature on implementation of bargaining solutions, we use the assumption that any player can grab the entire surplus. Our mechanism consists of a binary game tree, with each node corresponding to a subgame where the players are allowed to choose between two outcomes. We characterize novel combinatorial properties of the lexicographic maxmin solution which are crucial to the design of our mechanism.

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arXiv2018

Relating Metric Distortion and Fairness of Social Choice Rules

One way of evaluating social choice (voting) rules is through a utilitarian distortion framework. In this model, we assume that agents submit full rankings over the alternatives, and these rankings are generated from underlying, but unknown, quantitative costs. The \emph{distortion} of a social choice rule is then the ratio of the total social cost of the chosen alternative to the optimal social cost of any alternative; since the true costs are unknown, we consider the worst-case distortion over all possible underlying costs. Analogously, we can consider the worst-case \emph{fairness ratio} of a social choice rule by comparing a useful notion of fairness (based on approximate majorization) for the chosen alternative to that of the optimal alternative. With an additional metric assumption -- that the costs equal the agent-alternative distances in some metric space -- it is known that the Copeland rule achieves both a distortion and fairness ratio of at most 5. For other rules, only bounds on the distortion are known, e.g., the popular Single Transferable Vote (STV) rule has distortion $O(\log m)$, where $m$ is the number of alternatives. We prove that the distinct notions of distortion and fairness ratio are in fact closely linked -- within an additive factor of 2 for any voting rule -- and thus STV also achieves an $O(\log m)$ fairness ratio. We further extend the notions of distortion and fairness ratio to social choice rules choosing a \emph{set} of alternatives. By relating the distortion of single-winner rules to multiple-winner rules, we establish that Recursive Copeland achieves a distortion of 5 and a fairness ratio of at most 7 for choosing a set of alternatives.

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arXiv2017

Approximate Personalized PageRank on Dynamic Graphs

We propose and analyze two algorithms for maintaining approximate Personalized PageRank (PPR) vectors on a dynamic graph, where edges are added or deleted. Our algorithms are natural dynamic versions of two known local variations of power iteration. One, Forward Push, propagates probability mass forwards along edges from a source node, while the other, Reverse Push, propagates local changes backwards along edges from a target. In both variations, we maintain an invariant between two vectors, and when an edge is updated, our algorithm first modifies the vectors to restore the invariant, then performs any needed local push operations to restore accuracy. For Reverse Push, we prove that for an arbitrary directed graph in a random edge model, or for an arbitrary undirected graph, given a uniformly random target node $t$, the cost to maintain a PPR vector to $t$ of additive error $\varepsilon$ as $k$ edges are updated is $O(k + \bar{d} / \varepsilon)$, where $\bar{d}$ is the average degree of the graph. This is $O(1)$ work per update, plus the cost of computing a reverse vector once on a static graph. For Forward Push, we show that on an arbitrary undirected graph, given a uniformly random start node $s$, the cost to maintain a PPR vector from $s$ of degree-normalized error $\varepsilon$ as $k$ edges are updated is $O(k + 1 / \varepsilon)$, which is again $O(1)$ per update plus the cost of computing a PPR vector once on a static graph.

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arXiv2017

Sequential Deliberation for Social Choice

In large scale collective decision making, social choice is a normative study of how one ought to design a protocol for reaching consensus. However, in instances where the underlying decision space is too large or complex for ordinal voting, standard voting methods of social choice may be impractical. How then can we design a mechanism - preferably decentralized, simple, scalable, and not requiring any special knowledge of the decision space - to reach consensus? We propose sequential deliberation as a natural solution to this problem. In this iterative method, successive pairs of agents bargain over the decision space using the previous decision as a disagreement alternative. We describe the general method and analyze the quality of its outcome when the space of preferences define a median graph. We show that sequential deliberation finds a 1.208- approximation to the optimal social cost on such graphs, coming very close to this value with only a small constant number of agents sampled from the population. We also show lower bounds on simpler classes of mechanisms to justify our design choices. We further show that sequential deliberation is ex-post Pareto efficient and has truthful reporting as an equilibrium of the induced extensive form game. We finally show that for general metric spaces, the second moment of of the distribution of social cost of the outcomes produced by sequential deliberation is also bounded.

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arXiv2017

Metric Distortion of Social Choice Rules: Lower Bounds and Fairness Properties

We study social choice rules under the utilitarian distortion framework, with an additional metric assumption on the agents' costs over the alternatives. In this approach, these costs are given by an underlying metric on the set of all agents plus alternatives. Social choice rules have access to only the ordinal preferences of agents but not the latent cardinal costs that induce them. Distortion is then defined as the ratio between the social cost (typically the sum of agent costs) of the alternative chosen by the mechanism at hand, and that of the optimal alternative chosen by an omniscient algorithm. The worst-case distortion of a social choice rule is, therefore, a measure of how close it always gets to the optimal alternative without any knowledge of the underlying costs. Under this model, it has been conjectured that Ranked Pairs, the well-known weighted-tournament rule, achieves a distortion of at most 3 [Anshelevich et al. 2015]. We disprove this conjecture by constructing a sequence of instances which shows that the worst-case distortion of Ranked Pairs is at least 5. Our lower bound on the worst case distortion of Ranked Pairs matches a previously known upper bound for the Copeland rule, proving that in the worst case, the simpler Copeland rule is at least as good as Ranked Pairs. And as long as we are limited to (weighted or unweighted) tournament rules, we demonstrate that randomization cannot help achieve an expected worst-case distortion of less than 3. Using the concept of approximate majorization within the distortion framework, we prove that Copeland and Randomized Dictatorship achieve low constant factor fairness-ratios (5 and 3 respectively), which is a considerable generalization of similar results for the sum of costs and single largest cost objectives. In addition to all of the above, we outline several interesting directions for further research in this space.

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arXiv2017

When Hashes Met Wedges: A Distributed Algorithm for Finding High Similarity Vectors

Finding similar user pairs is a fundamental task in social networks, with numerous applications in ranking and personalization tasks such as link prediction and tie strength detection. A common manifestation of user similarity is based upon network structure: each user is represented by a vector that represents the user's network connections, where pairwise cosine similarity among these vectors defines user similarity. The predominant task for user similarity applications is to discover all similar pairs that have a pairwise cosine similarity value larger than a given threshold $τ$. In contrast to previous work where $τ$ is assumed to be quite close to 1, we focus on recommendation applications where $τ$ is small, but still meaningful. The all pairs cosine similarity problem is computationally challenging on networks with billions of edges, and especially so for settings with small $τ$. To the best of our knowledge, there is no practical solution for computing all user pairs with, say $τ= 0.2$ on large social networks, even using the power of distributed algorithms. Our work directly addresses this challenge by introducing a new algorithm --- WHIMP --- that solves this problem efficiently in the MapReduce model. The key insight in WHIMP is to combine the "wedge-sampling" approach of Cohen-Lewis for approximate matrix multiplication with the SimHash random projection techniques of Charikar. We provide a theoretical analysis of WHIMP, proving that it has near optimal communication costs while maintaining computation cost comparable with the state of the art. We also empirically demonstrate WHIMP's scalability by computing all highly similar pairs on four massive data sets, and show that it accurately finds high similarity pairs. In particular, we note that WHIMP successfully processes the entire Twitter network, which has tens of billions of edges.

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arXiv2016

Re-incentivizing Discovery: Mechanisms for Partial-Progress Sharing in Research

An essential primitive for an efficient research ecosystem is \emph{partial-progress sharing} (PPS) -- whereby a researcher shares information immediately upon making a breakthrough. This helps prevent duplication of work; however there is evidence that existing reward structures in research discourage partial-progress sharing. Ensuring PPS is especially important for new online collaborative-research platforms, which involve many researchers working on large, multi-stage problems. We study the problem of incentivizing information-sharing in research, under a stylized model: non-identical agents work independently on subtasks of a large project, with dependencies between subtasks captured via an acyclic subtask-network. Each subtask carries a reward, given to the first agent who publicly shares its solution. Agents can choose which subtasks to work on, and more importantly, when to reveal solutions to completed subtasks. Under this model, we uncover the strategic rationale behind certain anecdotal phenomena. Moreover, for any acyclic subtask-network, and under a general model of agent-subtask completion times, we give sufficient conditions that ensure PPS is incentive-compatible for all agents. One surprising finding is that rewards which are approximately proportional to perceived task-difficulties are sufficient to ensure PPS in all acyclic subtask-networks. The fact that there is no tension between local fairness and global information-sharing in multi-stage projects is encouraging, as it suggests practical mechanisms for real-world settings. Finally, we show that PPS is necessary, and in many cases, sufficient, to ensure a high rate of progress in research.

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arXiv2016

The Core of the Participatory Budgeting Problem

In participatory budgeting, communities collectively decide on the allocation of public tax dollars for local public projects. In this work, we consider the question of fairly aggregating the preferences of community members to determine an allocation of funds to projects. This problem is different from standard fair resource allocation because of public goods: The allocated goods benefit all users simultaneously. Fairness is crucial in participatory decision making, since generating equitable outcomes is an important goal of these processes. We argue that the classic game theoretic notion of core captures fairness in the setting. To compute the core, we first develop a novel characterization of a public goods market equilibrium called the Lindahl equilibrium, which is always a core solution. We then provide the first (to our knowledge) polynomial time algorithm for computing such an equilibrium for a broad set of utility functions; our algorithm also generalizes (in a non-trivial way) the well-known concept of proportional fairness. We use our theoretical insights to perform experiments on real participatory budgeting voting data. We empirically show that the core can be efficiently computed for utility functions that naturally model our practical setting, and examine the relation of the core with the familiar welfare objective. Finally, we address concerns of incentives and mechanism design by developing a randomized approximately dominant-strategy truthful mechanism building on the exponential mechanism from differential privacy.

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arXiv2016

Towards large-scale deliberative decision-making: small groups and the importance of triads

Though deliberation is a critical component of democratic decision-making, existing deliberative processes do not scale to large groups of people. Motivated by this, we propose a model in which large-scale decision-making takes place through a sequence of small group interactions. Our model considers a group of participants, each having an opinion which together form a graph. We show that for median graphs, a class of graphs including grids and trees, it is possible to use a small number of three-person interactions to tightly approximate the wisdom of the crowd, defined here to be the generalized median of participant opinions, even when agents are strategic. Interestingly, we also show that this sharply contrasts with small groups of size two, for which we prove an impossibility result. Specifically, we show that it is impossible to use sequences of two-person interactions satisfying natural axioms to find a tight approximation of the generalized median, even when agents are non-strategic. Our results demonstrate the potential of small group interactions for reaching global decision-making properties.

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arXiv2016

The Ratio Index for Budgeted Learning, with Applications

In the budgeted learning problem, we are allowed to experiment on a set of alternatives (given a fixed experimentation budget) with the goal of picking a single alternative with the largest possible expected payoff. Approximation algorithms for this problem were developed by Guha and Munagala by rounding a linear program that couples the various alternatives together. In this paper we present an index for this problem, which we call the ratio index, which also guarantees a constant factor approximation. Index-based policies have the advantage that a single number (i.e. the index) can be computed for each alternative irrespective of all other alternatives, and the alternative with the highest index is experimented upon. This is analogous to the famous Gittins index for the discounted multi-armed bandit problem. The ratio index has several interesting structural properties. First, we show that it can be computed in strongly polynomial time. Second, we show that with the appropriate discount factor, the Gittins index and our ratio index are constant factor approximations of each other, and hence the Gittins index also gives a constant factor approximation to the budgeted learning problem. Finally, we show that the ratio index can be used to create an index-based policy that achieves an O(1)-approximation for the finite horizon version of the multi-armed bandit problem. Moreover, the policy does not require any knowledge of the horizon (whereas we compare its performance against an optimal strategy that is aware of the horizon). This yields the following surprising result: there is an index-based policy that achieves an O(1)-approximation for the multi-armed bandit problem, oblivious to the underlying discount factor.

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arXiv2015

Bidirectional PageRank Estimation: From Average-Case to Worst-Case

We present a new algorithm for estimating the Personalized PageRank (PPR) between a source and target node on undirected graphs, with sublinear running-time guarantees over the worst-case choice of source and target nodes. Our work builds on a recent line of work on bidirectional estimators for PPR, which obtained sublinear running-time guarantees but in an average-case sense, for a uniformly random choice of target node. Crucially, we show how the reversibility of random walks on undirected networks can be exploited to convert average-case to worst-case guarantees. While past bidirectional methods combine forward random walks with reverse local pushes, our algorithm combines forward local pushes with reverse random walks. We also discuss how to modify our methods to estimate random-walk probabilities for any length distribution, thereby obtaining fast algorithms for estimating general graph diffusions, including the heat kernel, on undirected networks.

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arXiv2015

Personalized PageRank Estimation and Search: A Bidirectional Approach

We present new algorithms for Personalized PageRank estimation and Personalized PageRank search. First, for the problem of estimating Personalized PageRank (PPR) from a source distribution to a target node, we present a new bidirectional estimator with simple yet strong guarantees on correctness and performance, and 3x to 8x speedup over existing estimators in experiments on a diverse set of networks. Moreover, it has a clean algebraic structure which enables it to be used as a primitive for the Personalized PageRank Search problem: Given a network like Facebook, a query like "people named John", and a searching user, return the top nodes in the network ranked by PPR from the perspective of the searching user. Previous solutions either score all nodes or score candidate nodes one at a time, which is prohibitively slow for large candidate sets. We develop a new algorithm based on our bidirectional PPR estimator which identifies the most relevant results by sampling candidates based on their PPR; this is the first solution to PPR search that can find the best results without iterating through the set of all candidate results. Finally, by combining PPR sampling with sequential PPR estimation and Monte Carlo, we develop practical algorithms for PPR search, and we show via experiments that our algorithms are efficient on networks with billions of edges.

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arXiv2014

FAST-PPR: Scaling Personalized PageRank Estimation for Large Graphs

We propose a new algorithm, FAST-PPR, for estimating personalized PageRank: given start node $s$ and target node $t$ in a directed graph, and given a threshold $δ$, FAST-PPR estimates the Personalized PageRank $π_s(t)$ from $s$ to $t$, guaranteeing a small relative error as long $π_s(t)>δ$. Existing algorithms for this problem have a running-time of $Ω(1/δ)$; in comparison, FAST-PPR has a provable average running-time guarantee of ${O}(\sqrt{d/δ})$ (where $d$ is the average in-degree of the graph). This is a significant improvement, since $δ$ is often $O(1/n)$ (where $n$ is the number of nodes) for applications. We also complement the algorithm with an $Ω(1/\sqrtδ)$ lower bound for PageRank estimation, showing that the dependence on $δ$ cannot be improved. We perform a detailed empirical study on numerous massive graphs, showing that FAST-PPR dramatically outperforms existing algorithms. For example, on the 2010 Twitter graph with 1.5 billion edges, for target nodes sampled by popularity, FAST-PPR has a $20$ factor speedup over the state of the art. Furthermore, an enhanced version of FAST-PPR has a $160$ factor speedup on the Twitter graph, and is at least $20$ times faster on all our candidate graphs.

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