

Survey
Learning
in Social Networks
(with Evan
Sadler)
The Oxford
Handbook of the
Economics of Networks, (Yann Bramoullé, Andrea Galeotti,
and Brian Rogers, eds.), 2016
A broad overview
of two kinds of network learning models: (i) sequential ones
in the tradition of information cascades and herding, and (ii) iterated
linear
updating models (DeGroot), along with their variations,
foundations, and critiques. Ideal for a graduate course. [
More]
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
averagebased repeated updating, whose long and mediumrun 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.
[Companion handwritten
lecture notes on DeGroot part]
Papers
 Bayesian
Social Learning in a Dynamic Environment
(with Krishna
Dasaratha and Nir Hak)
When agents
learn repeatedly from one
another in a network, Bayesian updating is complicated, so nonoptimal
behavioral rules are often assumed instead for tractability. In a
stationary, dynamic model we present, the tension between tractability
and rationality goes away: updating in equilibrium turns out to be as
tractable as in the simplest heuristic (the DeGroot rule). Using this,
we can take a standard economic approach to the welfare analysis of
repeated learning processes. [
More]
Bayesian
agents learn about a moving target, such as a commodity price, using
private signals and their network neighbors' estimates. The weights
agents place on these sources of information are endogenously
determined by the precisions and correlations of the sources; the
weights, in turn, determine future correlations. We study stationary
equilibria—ones in which all of these quantities are constant over
time. Equilibria in linear updating rules always exist, yielding a
Bayesian learning model as tractable as the commonlyused DeGroot
heuristic. Equilibria and the comparative statics of learning outcomes
can be readily computed even in large networks. Substantively, we
identify pervasive inefficiencies in Bayesian learning. In any
stationary equilibrium where agents put positive weights on neighbors'
actions, learning is Pareto inefficient in a generic network: agents
rationally overuse social information and underuse their private
signals.
Current
version:
January 6, 2018. [NonSSRN
download]

 Targeting
Interventions in Networks
(with Andrea
Galeotti and Sanjeev
Goyal)
If a planner
has limited resources to shape incentives, whom
should she
target, e.g., to maximize welfare? A principal component analysis, new to network games, identifies the planner's priorities across various network intervention problems. [
More]
Individuals
interact strategically with their network neighbors. A planner can
shape incentives in pursuit of an aggregate goal, such as maximizing
welfare or minimizing volatility. We analyze a variety of targeting
problems by identifying how a given profile of incentive changes is
amplified or attenuated by the strategic spillovers in the network. The
optimal policies are simplest when the budget for intervention is
large. If actions are strategic complements, the optimal intervention
changes all agents' incentives in the same direction and does so in
proportion to their eigenvector centralities. In games of strategic
substitutes, the optimal intervention is very different: it moves
neighbors' incentives in opposite directions, dividing local
communities into positively and negatively targeted agents, with few
links across these two categories. To derive these results and
characterize optimal interventions more generally, we introduce a
method of decomposing any potential intervention into principal
components determined by the network. A particular ordering
of
principal components describes the planner's priorities across a range
of network intervention problems.
Submitted. Current
version:
March 12, 2018. First
version:
October 17, 2017.
[NonSSRN
download]


Illiquidity Spirals in Coupled OvertheCounter Markets
(with Christoph
Aymanns and CoPierre
Georg)
(Updated
Dec. 25, 2017.) Consider two
related overthecounter markets: e.g., one for secured debt and
another for
the collateral. If liquidityprovision decisions are strategic
complements across
the markets, liquidity can evaporate discontinuously
in
the size of a shock, in contrast to standard (singlenetwork) models.
Making networks more similar across the two OTC markets increases
their resilience.
[
More]
Banks provide intermediation of two economically coupled assets, each
traded on an OTC market—e.g., secured debt and the underlying
collateral. We model banks' decisions to provide liquidity as a game of
strategic complements on two coupled
trading networks: incentives to be active in one network are increasing
in its neighbors' activity in both networks. When an exogenous shock
renders some banks inactive, other banks follow in an illiquidity
spiral across the two networks. Liquidity can be improved if one of the
two OTC markets is replaced by an exchange. For a class of market
structures associated with random graphs, liquidity changes discontinuously
in the size of an exogenous shock, in contrast to contagion on one
network.
Submitted. Current
version:
December 25, 2017. First
version:
April 8, 2017. [NonSSRN
download]
 Expectations,
Networks, and Conventions
(with Stephen
Morris)
We study
certain games
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 incompleteinformation results (e.g., on common
priors) and network results (e.g. relating equilibria to network
centralities). [
More] [Slides]
In
coordination games and speculative overthecounter financial markets,
solutions depend on higherorder average expectations:
agents'
expectations about what counterparties, on average, expect their
counterparties
to think, etc. We offer a unified analysis of these objects and their
limits,
for general information structures, priors, and networks of
counterparty
relationships. Our key device is an interaction structure
combining the
network and agents' beliefs, which we analyze using Markov methods.
This device
allows us to nest classical beauty contests and network games within
one model
and unify their results. Two applications illustrate the techniques:
The first
characterizes when slight optimism about counterparties' average
expectations
leads to contagion of optimism and extreme asset
prices. The second
describes the tyranny of the leastinformed: agents
coordinating on the prior
expectations of the one with the worst private information, despite all
having
nearly common certainty, based on precise private signals, of the ex
post
optimal action.
Current
version:
September 10, 2017. First
version: April 24, 2017. [NonSSRN
download]

 HigherOrder
Expectations
(with Stephen
Morris)
Motivated by
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". [
More]
We study higherorder expectations paralleling the Harsanyi (1968)
approach to higherorder beliefs—taking a basic set of random
variables as given,
and building up higherorder expectations from them. We report three
main results. First, we generalize Samet's (1998a)
characterization of the common prior assumption in terms of
higherorder expectations, resolving an apparent paradox raised by his
result.
Second, we characterize when the limits of higherorder expectations
can be expressed in terms of agents' heterogeneous priors, generalizing
Samet's
expression of limit higherorder expectations via the common prior.
Third, we study higherorder average expectations—objects that arise in
network games. We characterize when and how the network structure and
agents' beliefs enter in a separable way.
Current
version:
August 31, 2017. First
version:
June 1, 2017.
[NonSSRN
download]
 Signaling,
Stigma, and Silence in Social Learning
(with Arun
Chandrasekhar and He Yang)
Does the fear
of
appearing ignorant deter people from asking
questions, and is that an important friction in informationgathering?
In an experiment, we show that people seek information less when
needing it is related to ability. [
More]
We model the stigma associated with seeking information as a potential
friction in social learning. A Seeker has information relevant to a
task, and may choose to rely on it or seek better information from an
Advisor. Higherskill Seekers need help less. Thus, to signal skill,
Seekers may refrain from asking Advisors questions. To test predictions
of the model, we conduct a 2by2 experiment across 70 villages in
India, where Seekers had three days to choose to retrieve information
from Advisors. The first treatment arm varies whether needing help is
correlated with skill, and the second varies whether a Seeker’s skill
is revealed to the paired Advisor. When information depends on skill,
lowskill Seekers seek information substantially less often in the
cases when they most need it. We also identify two additional
behavioral effects that arise when a low Seeker score is revealed to
the Advisor: a direct "shame" effect—the Seeker dislikes meeting the
Advisor in this circumstance; and an “effort signaling” effect—the
Seeker is more likely to seek information in the skillbased treatment,
where the (known) low score implies a need for help. These effects are
particularly pronounced among friends. We argue that these forces,
taken together, can stabilize and magnify homophily in communication
networks and explain the observed phenomenon of persistent differences
in information among physically proximate peers.
Current
version:
April 5, 2017. First
version: December 11, 2016.

 A
Network Approach to Public Goods
(with Matthew
Elliott)
Accepted, Journal of
Political Economy
Perron
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. [
More] [Slides]
[4page
version]
Suppose
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
team.
Submitted.
Current
version: January 17, 2017. First version:
November 2012. [nonSSRN
download]

 Ranking Agendas for Negotiations
(with Matthew
Elliott)
Countries are
hashing out the agenda for a summit in which each will make costly
concessions to
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
utility. [
More]
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
chosen,
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
Jacobian
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.
First version:
May
1, 2014. Current
version: February 22, 2015. Working paper.
 Financial
Networks and Contagion
(with Matthew
Elliott and Matthew
O.
Jackson
)
American
Economic Review,
104(10), October 2014
Diversification
(more counterparties) and integration (deeper relationships with each
counterparty) have different, nonmonotonic effects on financial
contagions. [
More] [Slides]
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
of counterparties).
Integration and diversification have different, nonmonotonic effects on
the extent of
cascades. Initial increases in diversification connect the network
which permits
cascades to propagate further, but eventually, more diversification
makes contagion between any
pair of organizations less likely as they become less dependent on each
other.
Integration also faces tradeoffs: increased dependence on other
organizations
versus less sensitivity to own investments. Finally, we illustrate some
aspects of the model with data on European debt crossholdings.
First
version:
September 2012. [NonSSRN
Version] [Online
Appenix]

 How
Homophily Affects the Speed of Learning and BestResponse Dynamics
(with Matthew
O.
Jackson)
Quarterly
Journal of Economics, 127(3), August 2012.
Grouplevel
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]
[Slides]
We
examine how the speed of learning and bestresponse 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.
First
version:
November 24, 2008.

 How Sharing Information Can
Garble Experts' Advice
(with Matthew
Elliott and Andrei Kirilenko)
American
Economic Review:
Papers & Proceedings, 104(5): 463–468, 2014
Do we get
better
advice as our experts get more information? Two experts, who like to be
right,
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. [
More]
[Long Version]
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
information
improvement.
First version:
November 21, 2010.
 Strategic
Random Networks and Tipping Points in Network Formation
(with
Yair Livne)
If agents
form
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. [
More]
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
contrast to
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.
Current
version: November 2, 2010.
First
version:
April, 2010. Working
paper.
 Naive
Learning in Social Networks and the Wisdom of
Crowds (with Matthew
O.
Jackson)

American
Economic
Journal: Microeconomics,
2(1):112149,
February 2010.
In what networks do agents who learn very
naively get the right answer?

[ More] [3page
version] [Slides]
We study
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'
opinions.
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.
First
version:
January 14, 2007.
 Using
Selection Bias to Explain the Observed Structure of
Internet Diffusions (with
Matthew
O.
Jackson)
Proceedings
of the National
Academy of Sciences, 107(24):1083310836, June 15, 2010.
David
LibenNowell and Jon Kleinberg
have
observed
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
blurb]
Recently, large data sets stored on the Internet have enabled the
analysis of processes, such as largescale diffusions of information,
at new levels of detail. In a recent study, LibenNowell and Kleinberg
((2008) Proc Natl Acad Sci USA 105:46334638) 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
GaltonWatson 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.
First
version:
January 2010. [Download]
 Does
Homophily Predict Consensus Times? Testing a Model of Network Structure
via a Dynamic Process
(with
Matthew
O.
Jackson)
Review of
Network Economics,
11(3), 2012.
Many random
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? [
More]
We test theoretical results from Golub
and Jackson
(2012a), which are based on a random network model, regarding
time to
convergence of a learning/behaviorupdating 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 realworld
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
multitype 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.
First
version:
February 2012.
 Network
Structure and
the Speed
of Learning: Measuring Homophily Based on its Consequences (with
Matthew
O.
Jackson)
Annals of
Economics and
Statistics, 107/108, 2012.
A
simple
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.
[
More]
Homophily is the tendency of people to associate relatively more with
those who
are similar to them than with those who are not. In Golub and Jackson
(2010a), we
introduced degreeweighted homophily (DWH), a new measure of this
phenomenon, and
showed that it gives a lower bound on the time it takes for a certain
natural bestreply
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
previous
bounds based on the Cheeger inequality. We also develop a new
complementary upper
bound on convergence time, tightening the relationship between DWH and
updating
processes on networks. In doing so, we suggest that DWH is a natural
homophily
measure because it tightly tracks a key consequence of homophily
—
namely, slowdowns
in updating processes.
First
version:
April 2010.
 The
Leverage
of Weak Ties: How
Linking Groups Affects Inequality
(with
Carlos
Lever)
Arbitrarily
weak bridges linking social groups can have arbitrarily large
consequences for inequality.
[
More]
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 highcentrality member of one group begins interacting
symmetrically with a lowcentrality 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.
Current
version: April 12, 2010. Working paper.
 Firms,
Queues,
and Coffee Breaks: A Flow Model of Corporate Activity with Delays
(with R. Preston McAfee)

Review
of Economic Design, 15(1), March 2011.
How and when to decentralize networked
production —
in a
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.
First
version: May
2006.
 Stabilizing
Brokerage (with Katherine
Stovel and
Eva
Meyersson Milgrom)
Proceedings
of the National
Academy of Sciences, 108(Suppl. 4):2132621332, December
27, 2011.
Brokers
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]

