Benjamin Golub

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.

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.

Companion handwritten lecture notes on DeGroot part

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.

Bolletta (Mathematical Social Sciences, 2021) studies a model in which a network is strategically formed and then agents play a linear best-response investment game in it. The model is motivated by an application in which people choose both their study partners and their levels of educational effort. Agents have different one-dimensional types—private returns to effort. A main result claims that (pairwise Nash) stable networks have a locally complete structure consisting of possibly overlapping cliques: if two agents are linked, they are part of a clique composed of all agents with types between theirs. A counterexample shows that the claimed characterization is incorrect. We specify where the analysis errs and discuss implications for network formation models.

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.

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.

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.

A principal uses payments conditioned on stochastic outcomes of a team project to elicit costly effort from the team members. We develop a multi-agent generalization of a classic first-order approach to contract optimization by leveraging methods from network games. The main results characterize the optimal allocation of incentive pay across agents and outcomes. Incentive optimality requires equalizing, across agents, a product of (i) individual productivity (ii) organizational centrality and (iii) responsiveness to monetary incentives.

A large differentiated oligopoly yields inefficient market equilibria. An authority with imprecise information about the primitives of the market aims to design tax/subsidy interventions that increase efficiency \emph{robustly}—i.e., with high probability. We identify a condition on demand that guarantees the existence of such interventions, and we show how to construct them using noisy estimates of demand complementarities and substitutabilities across products. The analysis works by deriving a novel description of the incidence of market interventions in terms of spectral statistics of a Slutsky matrix. Our notion of recoverable structure ensures that parts of the spectrum that are useful for the design of interventions are statistically recoverable from noisy demand estimates.

Workers contribute effort toward a team output. Each worker’s effort is complementary to the efforts of specific collaborators. A principal motivates workers by paying them shares of the output. We characterize the principal’s optimal allocation of shares. It satisfies a balance condition: for any agent exerting effort, the (complementarity-weighted) sum of shares held by that agent’s collaborators is equal. Moreover, the subset of agents induced to work have tight-knit complementarities: any two members are collaborators or share a common collaborator. We apply our results to study how compensation and output depend on the exogenously given network of complementarities.

We study network games in which players choose the partners with whom they associate as well as an effort level that creates spillovers for those partners. New stability definitions extend standard solution concepts for each choice in isolation: pairwise stability in links and Nash equilibrium in actions. Focusing on environments in which all agents agree on the desirability of potential partners, we identify conditions that determine the shapes of stable networks. The first condition concerns whether higher actions create positive or negative spillovers for neighbors. The second concerns whether actions are strategic complements or substitutes to links. Depending on which combination of these conditions occurs, equilibrium networks are either ordered overlapping cliques or nested split graphs—highly structured forms that facilitate the computation and analysis of outcomes. We apply the framework to examine the consequences of competition for status, to microfound matching models that assume clique formation, and to explain empirical findings in which designing groups to leverage peer effects backfired.

Current version: February 5, 2024. First Version: February 2, 2021

Complex organizations accomplish tasks through many steps of collaboration among workers. Corporate culture supports collaborations by establishing norms and reducing misunderstandings. Because a strong corporate culture relies on costly, voluntary investments by many workers, we model it as an organizational public good, subject to standard free-riding problems, which become severe in large organizations. Our main finding is that voluntary contributions to culture can nevertheless be sustained, because an organization’s equilibrium productivity is endogenously highly sensitive to individual contributions. However, the completion of complex tasks is then necessarily fragile to small shocks that damage the organization’s culture.

Current version: February 23, 2023. First Version: January 21, 2023.

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

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.

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.

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.