Annual Review of Economics, in press 2022
Bad shocks, even fairly small ones, can cause a lot of damage because of network propagation mechanisms. This is a survey of relevant facts and theories, with a focus on extensive margin channels: firms or relationships between them not functioning for a while.
Many firms, banks, or other economic agents embedded in a network of codependencies may experience a contemporaneous, sharp drop in functionality or productivity following a shock—even if that shock is localized or moderate in magnitude. We offer an extended review of motivating evidence that such fragility is a live concern in supply networks and in financial systems. Network models of fragility are then reviewed, focusing on the forces that make aggregate functionality especially sensitive to the economic environment. We consider the key structural features of networks that determine their fragility, emphasizing the importance of phase transitions. We then turn to endogenous decisions, both by agents in the models (e.g., firms investing in network formation and robustness) and by planners (e.g., an authority undertaking macroprudential regulation), and discuss some distinctive implications of fragility phenomena for such decisions.
This chapter surveys the implications of studies in network economics for economic development. We focus on information flow and risk-sharing—two topics where work in theory, empirics, and policy analysis have been especially intensive and complementary. In analysing information, we distinguish models of information diffusion and aggregation, and highlight how different models imply very different guidance regarding the right way to seed information. In discussing risk-sharing, we look at the key frictions that impede efficient informal insurance, and some potential unintended consequences when policymakers intervene to help. Throughout, we stress practical insights that can be used with limited measurement of the details of networks.
The Oxford Handbook of the Economics of Networks, 2016 SSRN
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.
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.
Published and forthcoming research papers
Firms source inputs through failure-prone relationships with other firms. Equilibrium supply networks are drawn to a precipice: they are very sensitive to small aggregate shocks (e.g., through global shipping congestion or energy prices).
We model the production of complex goods in a large supply network. Each firm sources several essential inputs through relationships with other firms. Individual supply relationships are at risk of idiosyncratic failure, which threatens to disrupt production. To protect against this, firms multisource inputs and strategically invest to make relationships stronger, trading off the cost of investment against the benefits of increased robustness. A supply network is called fragile if aggregate output is very sensitive to small aggregate shocks. We show that supply networks of intermediate productivity are fragile in equilibrium, even though this is always inefficient. The endogenous configuration of supply networks provides a new channel for the powerful amplification of shocks.
Current version: April 2022. First version: September 2019.
Review of Economic Studies, 2022, forthcoming arXiv SSRN EC 18 conference version Slides
A model of rational learning in a network where agents use neighbors’ estimates simply by taking weighted sums. The dynamics are linear (as in the famously tractable DeGroot model) but the players are fully Bayesian and their learning weights are an equilibrium outcome. The rational modeling makes a big difference for predictions both about learning and social influence.
Agents learn about a changing state using private signals and their neighbors’ past estimates of the state. We present a model in which Bayesian agents in equilibrium use neighbors’ estimates simply by taking weighted sums with time-invariant weights. The dynamics thus parallel those of the tractable DeGroot model of learning in networks, but arise as an equilibrium outcome rather than a behavioral assumption. We examine whether information aggregation is nearly optimal as neighborhoods grow large. A key condition for this is signal diversity: each individual’s neighbors have private signals that not only contain independent information, but also have sufficiently different distributions. Without signal diversity—e.g., if private signals are i.i.d.—learning is suboptimal in all networks and highly inefficient in some. Turning to social influence, we find it is much more sensitive to one’s signal quality than to one’s number of neighbors, in contrast to standard models with exogenous updating rules.
Current version: November 1, 2022. First version: January 6, 2018.
Operations Research, 2022, forthcoming SSRN
Strategic agents are involved in two different networks simultaneously. Activity is complementary across networks, and across neighbors. Equilibrium activity in a coupled-network can be very fragile to an aggregate shock, though either network on its own would be robust.
Strategic agents choose whether to be active in networked markets. The value of being active depends on the activity choices of specific counterparties. Several markets are coupled when agents’ participation decisions are complements across markets. We model the problem of an analyst assessing the robustness of coupled networked markets during a crisis—an exogenous negative payoff shock—based only on partial information about the network structure. We give conditions under which exit spirals emerge—abrupt collapses of activity following shocks. Market coupling is a pervasive cause of fragility, creating exit spirals even between networks that are individually robust. The robustness of a coupled network system can be improved if one of two markets is replaced by a centralized one, or if links become more correlated across markets.
Current version: September 2022. First version: April 8, 2017.
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.
First version: October 20, 2017.
Econometrica 88(6), November 2020 Online Appendix Slides
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.
We study games in which a network mediates strategic spillovers and externalities among the players. How does a planner optimally target interventions that change individuals’ private returns to investment? We analyze this question by decomposing any intervention into orthogonal principal components, which are determined by the network and are ordered according to their associated eigenvalues. There is a close connection between the nature of spillovers and the representation of various principal components in the optimal intervention. In games of strategic complements (substitutes), interventions place more weight on the top (bottom) principal components, which reflect more global (local) network structure. For large budgets, optimal interventions are simple–they essentially involve only a single principal component.
First version: October 17, 2017.
Journal of Political Economy 127(2), April 2019 4-page version SSRN Online Appendix Slides
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.
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.
First version: November 2012.
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 cross-holdings.
First version: September 2012.
Quarterly Journal of Economics 127(3), August 2012 Publisher's Site Online Appendix Slides
Group-level 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.
We 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.
First version: November 24, 2008.
How Sharing Information Can Garble Experts’ Advice (with Matt Elliott and Andrei Kirilenko)
American Economic Review: Papers & Proceedings 104(5), 2014 Long Version
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.
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.
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), June 15, 2010 PNAS Blurb
David Liben-Nowell 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.
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.
First version: January 2010.
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?
We test theoretical results from Golub and Jackson (2012a), which are based on a random network model, regarding time to 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.
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.
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 degree-weighted 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 best-reply 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.
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.
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.
Proceedings of the National Academy of Sciences 108(Suppl. 4), December 27, 2011 PNAS
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.
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 is also the most institutionally demanding.
Suppliers of differentiated goods make simultaneous pricing decisions, which are strategically linked due to consumer preferences and the structure of production. Because of market power, the equilibrium is inefficient. We study how a policymaker should target a budget-balanced tax-and-subsidy policy to increase welfare. A key tool is a certain basis for the goods space, determined by the network of interactions among suppliers. It consists of eigenbundles—orthogonal in the sense that a tax on any eigenbundle passes through only to its own price—with pass-through coefficients determined by associated eigenvalues. Our basis permits a simple characterization of optimal interventions. For example, a planner maximizing consumer welfare should tax eigenbundles with low pass-through and subsidize ones with high pass-through. We interpret these results in terms of the network structure of the market.
We study network games where agents form a network, and then spillovers (e.g., peer effects) occur in that network. We define stable outcomes and show that under natural conditions, stable networks are simple, having either a hierarchical structure or consisting of overlapping cliques.
We study network games in which players choose both the partners with whom they associate and an action level (e.g., effort) that creates spillovers for those partners. We introduce a framework and two solution concepts, extending standard approaches for analyzing each choice in isolation: Nash equilibrium in actions and pairwise stability in links. Our main results show that, under suitable order conditions on incentives, stable networks take simple forms. The first condition concerns whether links create positive or negative payoff spillovers. The second concerns whether actions are strategic complements to links, or strategic substitutes. Together, these conditions yield a taxonomy of the relationship between network structure and economic primitives organized around two network architectures: ordered overlapping cliques and nested split graphs. We apply our model to understand the consequences of competition for status, to microfound matching models that assume clique formation, and to interpret empirical findings that highlight unintended consequences of group design.
Current version: September 12, 2022. First Version: February 2, 2021
We consider the utilitarian planner’s problem in network coordination games. Welfare is most sensitive to interventions proportional to the last principal component of the network, which relates to local disagreement. Notably, improving welfare worsens other canonical measures of discord.
Consider a coordination game played on a network, where agents prefer taking actions closer to those of their neighbors and to their own ideal points in action space. We explore how the welfare outcomes of a coordination game depend on network structure and the distribution of ideal points throughout the network. To this end, we imagine a benevolent or adversarial planner who intervenes, at a cost, to change ideal points in order to maximize or minimize utilitarian welfare subject to a constraint. A complete characterization of optimal interventions is obtained by decomposing interventions into principal components of the network’s adjacency matrix. Welfare is most sensitive to interventions proportional to the last principal component, which focus on local disagreement. A welfare-maximizing planner optimally works to reduce local disagreement, bringing the ideal points of neighbors closer together, whereas a malevolent adversary optimally drives neighbors’ ideal points apart to decrease welfare. Such welfare-maximizing/minimizing interventions are very different from ones that would be done to change some traditional measures of discord, such as the cross-sectional variation of equilibrium actions. In fact, an adversary sowing disagreement to maximize her impact on welfare will minimize her impact on global variation in equilibrium actions, underscoring a tension between improving welfare and increasing global cohesion of equilibrium behavior.
Current version: February 26, 2021
Signaling, Shame, 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 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.
Current version: May 2019. First version: December 11, 2016.
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 incomplete-information results (e.g., on common priors) and network results (e.g. relating equilibria to network centralities).
In coordination games and speculative over-the-counter financial markets, solutions depend on higher-order 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 least-informed: 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.
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 result. Second, we characterize when the limits of higher-order expectations can be expressed in terms of agents’ heterogeneous priors, generalizing Samet’s 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.
Current version: August 31, 2017. First version: June 1, 2017.
Older Working Papers
Arbitrarily 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.
Current version: April 12, 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.
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.
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.
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.