8 Search Results for "Lucier, Brendan"

Document
Extended Abstract
DOI: 10.4230/LIPIcs.ITCS.2024.57
Communicating with Anecdotes (Extended Abstract)

Authors: Nika Haghtalab, Nicole Immorlica, Brendan Lucier, Markus Mobius, and Divyarthi Mohan

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract
We study a communication game between a sender and receiver. The sender chooses one of her signals about the state of the world (i.e., an anecdote) and communicates it to the receiver who takes an action affecting both players. The sender and receiver both care about the state of the world but are also influenced by personal preferences, so their ideal actions can differ. We characterize perfect Bayesian equilibria. The sender faces a temptation to persuade: she wants to select a biased anecdote to influence the receiver’s action. Anecdotes are still informative to the receiver (who will debias at equilibrium) but the attempt to persuade comes at the cost of precision. This gives rise to informational homophily where the receiver prefers to listen to like-minded senders because they provide higher-precision signals. Communication becomes polarized when the sender is an expert with access to many signals, with the sender choosing extreme outlier anecdotes at equilibrium (unless preferences are perfectly aligned). This polarization dissipates all the gains from communication with an increasingly well-informed sender when the anecdote distribution is heavy-tailed. Experts therefore face a curse of informedness: receivers will prefer to listen to less-informed senders who cannot pick biased signals as easily.
Cite as

Nika Haghtalab, Nicole Immorlica, Brendan Lucier, Markus Mobius, and Divyarthi Mohan. Communicating with Anecdotes (Extended Abstract). In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 57:1-57:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{haghtalab_et_al:LIPIcs.ITCS.2024.57,
  author =	{Haghtalab, Nika and Immorlica, Nicole and Lucier, Brendan and Mobius, Markus and Mohan, Divyarthi},
  title =	{{Communicating with Anecdotes}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{57:1--57:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.57},
  URN =		{urn:nbn:de:0030-drops-195852},
  doi =		{10.4230/LIPIcs.ITCS.2024.57},
  annote =	{Keywords: Communication game, Equilibrium, Polarization, Signalling}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.9
Making Auctions Robust to Aftermarkets

Authors: Moshe Babaioff, Nicole Immorlica, Yingkai Li, and Brendan Lucier

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
A prevalent assumption in auction theory is that the auctioneer has full control over the market and that the allocation she dictates is final. In practice, however, agents might be able to resell acquired items in an aftermarket. A prominent example is the market for carbon emission allowances. These allowances are commonly allocated by the government using uniform-price auctions, and firms can typically trade these allowances among themselves in an aftermarket that may not be fully under the auctioneer’s control. While the uniform-price auction is approximately efficient in isolation, we show that speculation and resale in aftermarkets might result in a significant welfare loss. Motivated by this issue, we consider three approaches, each ensuring high equilibrium welfare in the combined market. The first approach is to adopt smooth auctions such as discriminatory auctions. This approach is robust to correlated valuations and to participants acquiring information about others' types. However, discriminatory auctions have several downsides, notably that of charging bidders different prices for identical items, resulting in fairness concerns that make the format unpopular. Two other approaches we suggest are either using posted-pricing mechanisms, or using uniform-price auctions with anonymous reserves. We show that when using balanced prices, both these approaches ensure high equilibrium welfare in the combined market. The latter also inherits many of the benefits from uniform-price auctions such as price discovery, and can be introduced with a minor modification to auctions currently in use to sell carbon emission allowances.
Cite as

Moshe Babaioff, Nicole Immorlica, Yingkai Li, and Brendan Lucier. Making Auctions Robust to Aftermarkets. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 9:1-9:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{babaioff_et_al:LIPIcs.ITCS.2023.9,
  author =	{Babaioff, Moshe and Immorlica, Nicole and Li, Yingkai and Lucier, Brendan},
  title =	{{Making Auctions Robust to Aftermarkets}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{9:1--9:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.9},
  URN =		{urn:nbn:de:0030-drops-175122},
  doi =		{10.4230/LIPIcs.ITCS.2023.9},
  annote =	{Keywords: carbon markets, aftermarkets, price of anarchy, multi-unit auctions, posted prices}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.52
Budget Pacing in Repeated Auctions: Regret and Efficiency Without Convergence

Authors: Jason Gaitonde, Yingkai Li, Bar Light, Brendan Lucier, and Aleksandrs Slivkins

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
Online advertising via auctions increasingly dominates the marketing landscape. A typical advertiser may participate in thousands of auctions each day with bids tailored to a variety of signals about user demographics and intent. These auctions are strategically linked through a global budget constraint. To help address the difficulty of bidding, many major online platforms now provide automated budget management via a flexible approach called budget pacing: rather than bidding directly, an advertiser specifies a global budget target and a maximum willingness-to-pay for different types of advertising opportunities. The specified maximums are then scaled down (or "paced") by a multiplier so that the realized total spend matches the target budget. These automated bidders are now near-universally adopted across all mature advertising platforms, raising pressing questions about market outcomes that arise when advertisers use budget pacing simultaneously. In this paper we study the aggregate welfare and individual regret guarantees of dynamic pacing algorithms in repeated auctions with budgets. We show that when agents simultaneously use a natural form of gradient-based pacing, the liquid welfare obtained over the course of the dynamics is at least half the optimal liquid welfare obtainable by any allocation rule, matching the best possible bound for static auctions even in pure Nash equilibria [Aggarwal et al., WINE 2019; Babaioff et al., ITCS 2021]. In contrast to prior work, these results hold without requiring convergence of the dynamics, circumventing known computational obstacles of finding equilibria [Chen et al., EC 2021]. Our result is robust to the correlation structure among agents' valuations and holds for any core auction, a broad class that includes first-price, second-price, and GSP auctions. We complement the aggregate guarantees by showing that an agent using such pacing algorithms achieves an O(T^{3/4}) regret relative to the value obtained by the best fixed pacing multiplier in hindsight in stochastic bidding environments. Compared to past work, this result applies to more general auctions and extends to adversarial settings with respect to dynamic regret.
Cite as

Jason Gaitonde, Yingkai Li, Bar Light, Brendan Lucier, and Aleksandrs Slivkins. Budget Pacing in Repeated Auctions: Regret and Efficiency Without Convergence. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, p. 52:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gaitonde_et_al:LIPIcs.ITCS.2023.52,
  author =	{Gaitonde, Jason and Li, Yingkai and Light, Bar and Lucier, Brendan and Slivkins, Aleksandrs},
  title =	{{Budget Pacing in Repeated Auctions: Regret and Efficiency Without Convergence}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{52:1--52:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.52},
  URN =		{urn:nbn:de:0030-drops-175557},
  doi =		{10.4230/LIPIcs.ITCS.2023.52},
  annote =	{Keywords: repeated auctions with budgets, pacing, learning in auctions}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.46
Prophet Matching in the Probe-Commit Model

Authors: Allan Borodin, Calum MacRury, and Akash Rakheja

Published in: LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

Abstract
We consider the online bipartite stochastic matching problem with known i.d. (independently distributed) online vertex arrivals. In this problem, when an online vertex arrives, its weighted edges must be probed (queried) to determine if they exist, based on known edge probabilities. Our algorithms operate in the probe-commit model, in that if a probed edge exists, it must be used in the matching. Additionally, each online node has a downward-closed probing constraint on its adjacent edges which indicates which sequences of edge probes are allowable. Our setting generalizes the commonly studied patience (or time-out) constraint which limits the number of probes that can be made to an online node’s adjacent edges. Most notably, this includes non-uniform edge probing costs (specified by knapsack/budget constraint). We extend a recently introduced configuration LP to the known i.d. setting, and also provide the first proof that it is a relaxation of an optimal offline probing algorithm (the offline adaptive benchmark). Using this LP, we establish the following competitive ratio results against the offline adaptive benchmark: 1) A tight 1/2 ratio when the arrival ordering π is chosen adversarially. 2) A 1-1/e ratio when the arrival ordering π is chosen u.a.r. (uniformly at random). If π is generated adversarially, we generalize the prophet inequality matching problem. If π is u.a.r., we generalize the prophet secretary matching problem. Both results improve upon the previous best competitive ratio of 0.46 in the more restricted known i.i.d. (independent and identically distributed) arrival model against the standard offline adaptive benchmark due to Brubach et al. We are the first to study the prophet secretary matching problem in the context of probing, and our 1-1/e ratio matches the best known result without probing due to Ehsani et al. This result also applies to the unconstrained bipartite matching probe-commit problem, where we match the best known result due to Gamlath et al.
Cite as

Allan Borodin, Calum MacRury, and Akash Rakheja. Prophet Matching in the Probe-Commit Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 46:1-46:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{borodin_et_al:LIPIcs.APPROX/RANDOM.2022.46,
  author =	{Borodin, Allan and MacRury, Calum and Rakheja, Akash},
  title =	{{Prophet Matching in the Probe-Commit Model}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{46:1--46:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-249-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{245},
  editor =	{Chakrabarti, Amit and Swamy, Chaitanya},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.46},
  URN =		{urn:nbn:de:0030-drops-171686},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.46},
  annote =	{Keywords: Stochastic probing, Online algorithms, Bipartite matching, Optimization under uncertainty}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.77
Buying Data over Time: Approximately Optimal Strategies for Dynamic Data-Driven Decisions

Authors: Nicole Immorlica, Ian A. Kash, and Brendan Lucier

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
We consider a model where an agent has a repeated decision to make and wishes to maximize their total payoff. Payoffs are influenced by an action taken by the agent, but also an unknown state of the world that evolves over time. Before choosing an action each round, the agent can purchase noisy samples about the state of the world. The agent has a budget to spend on these samples, and has flexibility in deciding how to spread that budget across rounds. We investigate the problem of choosing a sampling algorithm that optimizes total expected payoff. For example: is it better to buy samples steadily over time, or to buy samples in batches? We solve for the optimal policy, and show that it is a natural instantiation of the latter. Under a more general model that includes per-round fixed costs, we prove that a variation on this batching policy is a 2-approximation.
Cite as

Nicole Immorlica, Ian A. Kash, and Brendan Lucier. Buying Data over Time: Approximately Optimal Strategies for Dynamic Data-Driven Decisions. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 77:1-77:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{immorlica_et_al:LIPIcs.ITCS.2021.77,
  author =	{Immorlica, Nicole and Kash, Ian A. and Lucier, Brendan},
  title =	{{Buying Data over Time: Approximately Optimal Strategies for Dynamic Data-Driven Decisions}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{77:1--77:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.77},
  URN =		{urn:nbn:de:0030-drops-136161},
  doi =		{10.4230/LIPIcs.ITCS.2021.77},
  annote =	{Keywords: Online Algorithms, Value of Data, Markov Processes}
}
Document
Extended Abstract
DOI: 10.4230/LIPIcs.ITCS.2021.84
Non-Quasi-Linear Agents in Quasi-Linear Mechanisms (Extended Abstract)

Authors: Moshe Babaioff, Richard Cole, Jason Hartline, Nicole Immorlica, and Brendan Lucier

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
Mechanisms with money are commonly designed under the assumption that agents are quasi-linear, meaning they have linear disutility for spending money. We study the implications when agents with non-linear (specifically, convex) disutility for payments participate in mechanisms designed for quasi-linear agents. We first show that any mechanism that is truthful for quasi-linear buyers has a simple best response function for buyers with non-linear disutility from payments, in which each bidder simply scales down her value for each potential outcome by a fixed factor, equal to her target return on investment (ROI). We call such a strategy ROI-optimal. We prove the existence of a Nash equilibrium in which agents use ROI-optimal strategies for a general class of allocation problems. Motivated by online marketplaces, we then focus on simultaneous second-price auctions for additive bidders and show that all ROI-optimal equilibria in this setting achieve constant-factor approximations to suitable welfare and revenue benchmarks.
Cite as

Moshe Babaioff, Richard Cole, Jason Hartline, Nicole Immorlica, and Brendan Lucier. Non-Quasi-Linear Agents in Quasi-Linear Mechanisms (Extended Abstract). In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, p. 84:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{babaioff_et_al:LIPIcs.ITCS.2021.84,
  author =	{Babaioff, Moshe and Cole, Richard and Hartline, Jason and Immorlica, Nicole and Lucier, Brendan},
  title =	{{Non-Quasi-Linear Agents in Quasi-Linear Mechanisms}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{84:1--84:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.84},
  URN =		{urn:nbn:de:0030-drops-136230},
  doi =		{10.4230/LIPIcs.ITCS.2021.84},
  annote =	{Keywords: Return on investment, Non-quasi-linear agents, Transferable Welfare, Simultaneous Second-Price Auctions}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2020.15
Reducing Inefficiency in Carbon Auctions with Imperfect Competition

Authors: Kira Goldner, Nicole Immorlica, and Brendan Lucier

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract
We study auctions for carbon licenses, a policy tool used to control the social cost of pollution. Each identical license grants the right to produce a unit of pollution. Each buyer (i.e., firm that pollutes during the manufacturing process) enjoys a decreasing marginal value for licenses, but society suffers an increasing marginal cost for each license distributed. The seller (i.e., the government) can choose a number of licenses to put up for auction, and wishes to maximize the societal welfare: the total economic value of the buyers minus the social cost. Motivated by emission license markets deployed in practice, we focus on uniform price auctions with a price floor and/or price ceiling. The seller has distributional information about the market, and their goal is to tune the auction parameters to maximize expected welfare. The target benchmark is the maximum expected welfare achievable by any such auction under truth-telling behavior. Unfortunately, the uniform price auction is not truthful, and strategic behavior can significantly reduce (even below zero) the welfare of a given auction configuration. We describe a subclass of "safe-price" auctions for which the welfare at any Bayes-Nash equilibrium will approximate the welfare under truth-telling behavior. We then show that the better of a safe-price auction, or a truthful auction that allocates licenses to only a single buyer, will approximate the target benchmark. In particular, we show how to choose a number of licenses and a price floor so that the worst-case welfare, at any equilibrium, is a constant approximation to the best achievable welfare under truth-telling after excluding the welfare contribution of a single buyer.
Cite as

Kira Goldner, Nicole Immorlica, and Brendan Lucier. Reducing Inefficiency in Carbon Auctions with Imperfect Competition. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 15:1-15:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{goldner_et_al:LIPIcs.ITCS.2020.15,
  author =	{Goldner, Kira and Immorlica, Nicole and Lucier, Brendan},
  title =	{{Reducing Inefficiency in Carbon Auctions with Imperfect Competition}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{15:1--15:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.15},
  URN =		{urn:nbn:de:0030-drops-117006},
  doi =		{10.4230/LIPIcs.ITCS.2020.15},
  annote =	{Keywords: welfare, price of anarchy, mechanism design, equilibrium, costs}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2014.192
Reaching Consensus via Non-Bayesian Asynchronous Learning in Social Networks

Authors: Michal Feldman, Nicole Immorlica, Brendan Lucier, and S. Matthew Weinberg

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Abstract
We study the outcomes of information aggregation in online social networks. Our main result is that networks with certain realistic structural properties avoid information cascades and enable a population to effectively aggregate information. In our model, each individual in a network holds a private, independent opinion about a product or idea, biased toward a ground truth. Individuals declare their opinions asynchronously, can observe the stated opinions of their neighbors, and are free to update their declarations over time. Supposing that individuals conform with the majority report of their neighbors, we ask whether the population will eventually arrive at consensus on the ground truth. We show that the answer depends on the network structure: there exist networks for which consensus is unlikely, or for which declarations converge on the incorrect opinion with positive probability. On the other hand, we prove that for networks that are sparse and expansive, the population will converge to the correct opinion with high probability.
Cite as

Michal Feldman, Nicole Immorlica, Brendan Lucier, and S. Matthew Weinberg. Reaching Consensus via Non-Bayesian Asynchronous Learning in Social Networks. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 192-208, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{feldman_et_al:LIPIcs.APPROX-RANDOM.2014.192,
  author =	{Feldman, Michal and Immorlica, Nicole and Lucier, Brendan and Weinberg, S. Matthew},
  title =	{{Reaching Consensus via Non-Bayesian Asynchronous Learning in Social Networks}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{192--208},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.192},
  URN =		{urn:nbn:de:0030-drops-46976},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.192},
  annote =	{Keywords: Information Cascades, Social Networks, non-Bayesian Asynchronous Learning, Expander Graphs, Stochastic Processes}
}
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