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in Social Networks (with Evan
The Oxford Handbook of the Economics of Networks (2016)
of two kinds of network learning models: (i) sequential ones
in the tradition of information cascades and herding, and (ii) iterated
updating models (DeGroot), along with their variations,
foundations, and critiques. Ideal for a graduate course. [
This survey covers models of how agents update behaviors and beliefs
using information conveyed through social connections. We begin with
sequential social learning models, in which each agent makes a decision
once and for all after observing a subset of prior decisions; the
discussion is organized around the concepts of diffusion and
aggregation of information. Next, we present the DeGroot framework of
average-based repeated updating, whose long- and medium-run dynamics
can be completely characterized in terms of measures of network
centrality and segregation. Finally, we turn to various models of
repeated updating that feature richer optimizing behavior, and conclude
by urging the development of network learning theories that can deal
adequately with the observed phenomenon of persistent disagreement. The
two parts (sequential and DeGroot) may be read independently, though we
take care to relate the
different literatures conceptually.
lecture notes on DeGroot part]
- Learning from Neighbors about a Changing State (with Krishna
Dasaratha and Nir Hak)
from one another in a network about the value of a changing state. Can
aggregate information fast enough to keep up with the changes? We offer
a tractable model of learning suited to addressing that question.
In this model, a simple linear (DeGroot) rule of aggregating neighbors'
estimates is part of a Bayesian equilibrium. [
Abstract. Agents learn about a changing state using private signals and past actions of neighbors in a network. When can they learn efficiently about recent changes? We find two conditions are sufficient: (i) each individual's neighbors have sufficiently diverse types of private information; (ii) agents are sophisticated enough to use this diversity to filter out outdated information and identify recent developments. If either condition fails, learning can be bounded far from efficient levels—even in networks where, with a fixed state, learning is guaranteed to be efficient without (i) or (ii). We thus identify learning externalities that are distinctive to a dynamic environment, and argue that they can be quite severe. The model we develop to make our argument provides a Bayesian foundation for DeGroot learning in networks, permitting new counterfactual and welfare analyses for that commonly-used behavioral model.
January 6, 2018. Current
June 2019. [Non-SSRN
Interventions in Networks (with Andrea
Galeotti and Sanjeev
If a planner
has limited resources to shape incentives, whom
target, e.g., to maximize welfare? A principal component analysis
new to network games, identifies the planner's priorities across
various network intervention problems. [
We study the design of optimal interventions in network games, where
individuals' incentives to act are affected by their network neighbors'
actions. A planner shapes individuals' incentives, seeking to maximize
the group's welfare. We characterize how the planner's intervention
depends on the network structure. A key tool is the decomposition of
any possible intervention into principal components
which are determined by diagonalizing the adjacency matrix of
interactions. There is a close connection between the strategic
structure of the game and the emphasis of the optimal intervention on
various principal components: In games of strategic complements
(substitutes), interventions place more weight on the top (bottom)
principal components. For large budgets, optimal interventions are simple
a single principal component.
March 9, 2019. First
October 17, 2017.
Less is More: Experimental Evidence on Information Delivery During
India's Demonetization (with Abhijit
Banerjee, Emily Breza,
resubmit, Review of Economic Studies
are worried about how asking questions makes them look. Then giving
information to fewer people can make for greater diffusion and better
learning, in theory and in practice. [
How should information be disseminated to large populations? The options include broadcasts (e.g., via mass media) and informing a small number of "seeds" who then spread the message. While it may seem natural to try to reach the maximum number of people from the beginning, we show, theoretically and experimentally, that information frictions can reverse this result when incentives to seek are endogenous to the information policy. In a field experiment during the chaotic 2016 Indian demonetization, we varied how information about the policy was delivered to villages along two dimensions: how many people were initially informed (i.e. broadcasting versus seeding) and whether the identities of the initially informed were publicly disclosed (common knowledge). The quality of information aggregation is measured in three ways: the volume of conversations about demonetization, the level of knowledge about demonetization rules, and choice quality in a strongly incentivized decision dependent on understanding the rules. Under common knowledge, broadcasting performs worse and seeding performs better (relative to no common knowledge). Moreover, with common knowledge, seeding is the more effective strategy of the two. These comparisons hold on all three outcomes.
May 2019. First
October 20, 2017. [Non-SSRN
Shame, and Silence in Social Learning (with Arun
Chandrasekhar and He Yang)
Does the fear
appearing ignorant deter people from asking
questions, and is that an important friction in information-gathering?
In an experiment, we show that people seek information less when
needing it is related to ability. [
We examine how a social stigma of seeking information can inhibit learning. Consider a Seeker of uncertain ability who can learn about a task from an Advisor. If higher-ability Seekers need information less, then a Seeker concerned about reputation may refrain from asking to avoid signaling
low ability. Separately, low-ability individuals may feel inhibited even if their ability is known and there is nothing to signal, an effect we term shame
. Signaling and shame constitute an overall stigma
of seeking information. We distinguish the constituent parts of stigma in a simple model and then perform an experiment with treatments designed to detect both effects. Seekers have three days to retrieve information from paired Advisors in a field setting. The first arm varies whether needing information is correlated with a measure of cognitive ability; the second varies whether a Seeker's ability is revealed to the paired Advisor, irrespective of the seeking decision. We find that low-ability individuals do face large stigma inhibitions: there is a 55% decline in the probability of seeking when the need for information is correlated with ability. The second arm allows us to assess the contributions of signaling and shame, and, under structural assumptions, to estimate their relative magnitudes. We find signaling to be the dominant force overall. The shame effect is particularly pronounced among socially close pairs (in terms of network distance and caste co-membership) whereas signaling concerns dominate for more distant pairs.
May 2019. First
December 11, 2016. [Non-SSRN
Networks, and Conventions (with Stephen
in which there is both incomplete information and a network structure.
The two turn out to be, in a sense, the same thing:
A unified analysis nests
classical incomplete-information results (e.g., on common
priors) and network results (e.g. relating equilibria to network
coordination games and speculative over-the-counter financial markets,
solutions depend on higher-order average expectations
expectations about what counterparties, on average, expect their
to think, etc. We offer a unified analysis of these objects and their
for general information structures, priors, and networks of
relationships. Our key device is an interaction structure
network and agents' beliefs, which we analyze using Markov methods.
allows us to nest classical beauty contests and network games within
and unify their results. Two applications illustrate the techniques:
characterizes when slight optimism about counterparties' average
leads to contagion of optimism
and extreme asset
prices. The second
describes the tyranny of the least-informed
coordinating on the prior
expectations of the one with the worst private information, despite all
nearly common certainty, based on precise private signals, of the ex
September 10, 2017. First
April 24, 2017. [Non-SSRN
Expectations (with Stephen
their role in games, we study limits of iterated expectations with
heterogeneous priors: how priors matter, how the order in which
expectations are taken matters, and when the two enter "separably". [
We study higher-order expectations paralleling the Harsanyi (1968)
approach to higher-order beliefs—taking a basic set of random
variables as given,
and building up higher-order expectations from them. We report three
main results. First, we generalize Samet's (1998a)
characterization of the common prior assumption in terms of
higher-order expectations, resolving an apparent paradox raised by his
Second, we characterize when the limits of higher-order expectations
can be expressed in terms of agents' heterogeneous priors, generalizing
expression of limit higher-order expectations via the common prior.
Third, we study higher-order average expectations—objects that arise in
network games. We characterize when and how the network structure and
agents' beliefs enter in a separable way.
August 31, 2017. First
June 1, 2017.
Illiquidity Spirals in Coupled Over-the-Counter Markets
Aymanns and Co-Pierre
involved in two different networks simultaneously; each wants to be
active only if it has enough active neighbors in both
networks. The equilibrium outcomes are much more fragile to shocks in
such a coupled-network game than a one-network game. The leading
application is to the collapse of liquidity provision in secured
We model intermediaries trading economically coupled assets, each asset
in its own over-the-counter market—e.g., secured debt and the
underlying collateral. Incentives to provide liquidity in one market
are increasing in counterparties' activity in both markets. The
intermediaries' activity is thus the outcome of a game of strategic
complements on two coupled trading networks. We model a crisis as an
exogenous change to network structure, as well as the exogenous exit of
some intermediaries. This causes an illiquidity spiral across the two
networks. We find that in coupled networks, in contrast to uncoupled
ones, illiquidity spirals can be so severe that liquidity vanishes
discontinuously as we vary the shock. Liquidity can be improved if one
of the two OTC markets is replaced by an exchange, or if the two OTC
markets have more links in common.
September 6, 2018. First
April 8, 2017. [Non-SSRN
Network Approach to Public Goods (with Matthew
Political Economy 127(2), April 2019
eigenvalues are a natural way to measure whether an
economic system is at an efficient point, and eigenvector centrality
relates naturally to efficient negotiated outcomes.
We demonstrate these connections in a simple model of investment with
externalities, without parametric assumptions. [
agents can exert costly effort that creates nonrival,
heterogeneous benefits for each other. At each possible outcome, a
weighted, directed network describing marginal externalities is
defined. We show that Pareto efficient outcomes are those at which the
largest eigenvalue of the network is 1. An important set of efficient
solutions—Lindahl outcomes—are characterized by contributions being
proportional to agents' eigenvector centralities in the network. The
outcomes we focus on are motivated by negotiations. We apply the
results to identify who is essential for Pareto improvements, how to
efficiently subdivide negotiations, and whom to optimally add to a
January 17, 2017. First version:
November 2012. [non-SSRN
Networks and Contagion (with Matthew
Elliott and Matthew
104(10), October 2014
(more counterparties) and integration (deeper relationships with each
counterparty) have different, non-monotonic effects on financial
We model contagions and cascades of failures among organizations linked
through a network of financial interdependencies.
We identify how the network propagates discontinuous changes in asset
values triggered by failures
(e.g., bankruptcies, defaults, and other insolvencies) and use that to
study the consequences of integration(each organization becoming more
dependent on its counterparties)
and diversification (each organization interacting with a larger number
Integration and diversification have different, nonmonotonic effects on
the extent of
cascades. Initial increases in diversification connect the network
cascades to propagate further, but eventually, more diversification
makes contagion between any
pair of organizations less likely as they become less dependent on each
Integration also faces tradeoffs: increased dependence on other
versus less sensitivity to own investments. Finally, we illustrate some
aspects of the model with data on European debt cross-holdings.
September 2012. [Non-SSRN
Homophily Affects the Speed of Learning and Best-Response Dynamics
Journal of Economics, 127(3), August 2012.
segregation patterns in
networks seriously slow convergence to consensus behavior when agents'
choices are based on an average of neighbors' choices. When the process
is a simple contagion, homophily doesn't matter.
[ More] [Download] [Online Appenidx]
examine how the speed of learning and best-response processes depends
on homophily: the tendency of agents to associate disproportionately
with those having similar traits. When agents' beliefs or behaviors are
developed by averaging what they see among their neighbors, then
convergence to a consensus is slowed by the presence of homophily, but
is not influenced by network density (in contrast to other network
processes that depend on shortest paths). In deriving these results, we
propose a new, general measure of homophily based on the relative
frequencies of interactions among different groups. An application to
communication in a society before a vote shows how the time it takes
for the vote to correctly aggregate information depends on the
homophily and the initial information distribution.
November 24, 2008.
- How Sharing Information Can
Garble Experts' Advice
Elliott and Andrei Kirilenko)
Papers & Proceedings, 104(5): 463–468, 2014
Do we get
advice as our experts get more information? Two experts, who like to be
make predictions about whether an event will occur based on private
signals about its likelihood. It is possible for both
experts' information to improve unambiguously while the
usefulness of their advice to any third party unambiguously decreases. [
We model two experts who must make predictions about whether an event
will occur or not. The experts receive private signals about the
likelihood of the event occurring, and simultaneously make one of a
finite set of possible predictions, corresponding to varying degrees of
alarm. The information structure is commonly known among the experts
and the recipients of the advice. Each expert's payoff depends on
whether the event occurs and her prediction. Our main result shows that
when either or both
experts receive uniformly more informative signals, for example by
sharing their information, their predictions
can become unambiguously less informative. We call such information
improvements perverse. Suppose a third party wishes to use the experts'
recommendations to decide whether to take some costly preemptive action
to mitigate a possible bad event. Regardless of how this third party
trades off the
costs of various errors, he will be worse off after a perverse
November 21, 2010.
Random Networks and Tipping Points in Network Formation
networks in an environment of uncertainty, then arbitrarily small
changes in economic parameters (such as costs and benefits of linking)
can discontinuously change the properties
of the equilibrium networks, especially efficiency. [
Agents invest costly effort to socialize. Their effort
levels determine the probabilities of relationships, which are valuable
for their direct benefits and also because they lead to other
relationships in a later stage of ``meeting friends of friends''. In
many network formation models, there is fundamental uncertainty at the
time of investment regarding
which friendships will form. The
equilibrium outcomes are random graphs, and we characterize how their
density, connectedness, and other properties depend on the economic
fundamentals. When the value of friends of friends is low, there are
both sparse and thick equilibrium networks. But as soon as this value
crosses a key threshold, the sparse equilibria disappear completely and
only densely connected networks are possible. This transition mitigates
an extreme inefficiency.
version: November 2, 2010.
April, 2010. Working
Learning in Social Networks and the Wisdom of
Crowds (with Matthew
In what networks do agents who learn very
naively get the right answer?
[ More] [3-page
learning and influence in a setting where agents receive independent
noisy signals about the true value of a variable of interest and then
communicate according to an arbitrary social network. The agents
naively update their beliefs over time in a decentralized way by
repeatedly taking weighted averages of their neighbors'
We identify conditions determining whether the beliefs of all agents in
large societies converge to the true value of the variable, despite
their naive updating. We show that such convergence to truth
obtains if and only if the influence of the most influential agent in
the society is vanishing as the society grows. We identify
obstructions which can prevent this, including the existence of
prominent groups which receive a disproportionate share of attention.
By ruling out such obstructions, we provide structural conditions on
the social network that are sufficient for convergence to the truth.
Finally, we discuss the speed of convergence and note that whether or
not the society converges to truth is unrelated to how quickly a
society's agents reach a consensus.
January 14, 2007.
Selection Bias to Explain the Observed Structure of
Internet Diffusions (with
of the National
Academy of Sciences, 107(24):10833-10836, June 15, 2010.
and Jon Kleinberg
that the reconstructed family trees of chain letter petitions
are strangely tall and narrow. We show that this can be explained with
selection and observation biases
within a simple
model. [ More] [PNAS
Recently, large data sets stored on the Internet have enabled the
analysis of processes, such as large-scale diffusions of information,
at new levels of detail. In a recent study, Liben-Nowell and Kleinberg
((2008) Proc Natl Acad Sci USA 105:4633-4638) observed that the flow of
information on the Internet exhibits surprising patterns whereby a
chain letter reaches its typical recipient through long paths of
hundreds of intermediaries. We show that a basic
Galton-Watson epidemic model combined with the selection bias of
observing only large diffusions suffices to explain the global patterns
in the data. This demonstrates that accounting for selection
biases of which data we observe can radically change the estimation of
classical diffusion processes.
January 2010. [Download]
Homophily Predict Consensus Times? Testing a Model of Network Structure
via a Dynamic Process
network models forget most of the details of a network, focusing on
just a few dimensions of its structure. Can such models nevertheless
make good predictions about how a process would run on real networks,
in all their complexity? [
We test theoretical results from Golub
, which are based on a random network model, regarding
convergence of a learning/behavior-updating process. In particular, we
see how well those theoretical results match the process when it is
simulated on empirically observed high school friendship networks. This
tests whether a parsimonious random network model mimics real-world
networks with regard to predicting properties of a class of behavioral
processes. It also tests whether our theoretical predictions on
asymptotically large societies are accurate when applied to populations
ranging from thirty to three thousand individuals. We find that the
theoretical results account for more than half of the variation in
convergence times on the real networks. We conclude that a simple
multi-type random network model with types defined by simple observable
attributes (age, sex, race) captures aspects of real networks that are
relevant for a class of iterated updating processes.
of Learning: Measuring Homophily Based on its Consequences
Statistics, 107/108, 2012.
measure of segregation in a network (in which less popular people
matter more) predicts quite precisely how long convergence of beliefs
will take under a naive process in which agents form their own beliefs
by averaging those of their neighbors.
Homophily is the tendency of people to associate relatively more with
are similar to them than with those who are not. In Golub and Jackson
introduced degree-weighted homophily (DWH), a new measure of this
showed that it gives a lower bound on the time it takes for a certain
or learning process operating in a social network to converge. Here we
show that, in important
settings, the DWH convergence bound does substantially better than
bounds based on the Cheeger inequality. We also develop a new
bound on convergence time, tightening the relationship between DWH and
processes on networks. In doing so, we suggest that DWH is a natural
measure because it tightly tracks a key consequence of homophily
in updating processes.
of Weak Ties: How
Linking Groups Affects Inequality
weak bridges linking social groups can have arbitrarily large
consequences for inequality.
Centrality measures based on eigenvectors are important in models of
how networks affect investment decisions, the transmission of
information, and the provision of local public goods. We fully
characterize how the centrality of each member of a society changes
when initially disconnected groups begin interacting with each other
via a new bridging link. Arbitrarily weak intergroup connections can
have arbitrarily large effects on the distribution of centrality. For
instance, if a high-centrality member of one group begins interacting
symmetrically with a low-centrality member of another, the latter group
has the larger centrality in the combined network — in inverse
proportion to the centrality of its emissary! We also find that agents
who form the intergroup link, the ``bridge agents'', become relatively
more central within their own groups, while other intragroup centrality
ratios remain unchanged.
version: April 12, 2010. Working paper.
and Coffee Breaks: A Flow Model of Corporate Activity with Delays
(with R. Preston McAfee)
of Economic Design
, 15(1), March 2011.
How and when to decentralize networked
model that takes into account 'human' features of processing. [ More]
The multidivisional firm is modeled as
a system of interconnected nodes that exchange continuous flows of
projects of varying urgency and queue waiting tasks. The main
innovation over existing models is that the rate at which waiting
projects are taken into processing depends positively on both the
availability of resources and the size of the queue, capturing a
salient quality of human organizations. A transfer pricing scheme for
decentralizing the system is presented, and conditions are given to
determine which nodes can be operated autonomously. It is shown that a
node can be managed separately from the rest of the system when all of
the projects flowing through it are equally urgent.
- Ranking Agendas for Negotiations
hashing out the agenda for a summit in which each will make costly
help the others.
Should the summit focus on pollution, trade tariffs, or disarmament?
This is a theory to help them decide based on marginal costs and
benefits, without transferable
Consider a negotiation in which agents will make costly concessions to
benefit others—e.g., by implementing tariff reductions, environmental
regulations or disarmament policies. An agenda specifies which issue or
dimension each agent will make concessions on; after an agenda is
the negotiation comes down to the magnitude of each agent's
contribution. We seek a ranking of agendas based on the marginal costs
and benefits generated at the status quo, which are captured in a
matrix for each agenda. In a transferable utility (TU) setting, there
is a simple ranking based on the best available social return per unit
of cost (measured in the numeraire). Without transfers,
the problem of ranking agendas is more difficult, and we take an
axiomatic approach. First, we require the ranking not to depend on
economically irrelevant changes of units. Second, we require that the
ranking be consistent with the TU ranking on problems that are
equivalent to TU problems in a suitable sense. The unique ranking
satisfying these axioms is represented by the spectral radius
(Frobenius root) of a matrix closely related to the Jacobian, whose
entries measure the marginal benefits per unit marginal cost agents can
confer on one another.
1, 2014. Current
version: February 22, 2015. Working paper.
Brokerage (with Katherine
of the National
Academy of Sciences, 108(Suppl. 4):21326-21332, December
facilitate transactions across gaps in social structure, and there are
many reasons for their position to be unstable.
Here, we take a look, from a sociological and an economic perspective,
at what institutions stabilize brokerage. [ More]
A variety of social and economic arrangements exist to facilitate the
exchange of goods, services, and information over gaps in social
structure. Each of these arrangements bears some relationship to the
idea of brokerage, but this brokerage is rarely like the pure and
formal economic intermediation seen in some modern markets. Indeed, for
reasons illuminated by existing sociological and economic models,
brokerage is a fragile relationship. In this paper, we review the
causes of instability in brokerage and identify three social mechanisms
that can stabilize fragile brokerage relationships: social isolation,
broker capture, and organizational grafting. Each of these mechanisms
rests on the emergence or existence of supporting institutions. We
suggest that organizational grafting may be the most stable and
effective resolution to the tensions inherent in brokerage, but it
also the most institutionally demanding.
[A brief research statement]