## Surveys

### Learning in Social Networks (with Evan Sadler)

**The Oxford Handbook of the Economics of Networks**, 2016 SSRN

**The Oxford Handbook of the Economics of Networks**

##### 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 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

### Networks in Economic Development (with Emily Breza, Arun Chandrasekhar, and Aneesha Parvathaneni)

**Oxford Review of Exonomic Policy** 35(4), Winter 2019 Publisher's Site

**Oxford Review of Exonomic Policy**

##### A survey of network models, especially on information diffusion and risk sharing, in the context of economic development.

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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.

## Working Papers

### Supply Network Formation and Fragility (with Matt Elliot and Matt V. Leduc)

##### 2021, Revise and resubmit, **American Economic Review** arXiv SSRN Appendix

**American Economic Review**

Firms source inputs through failure-prone relationships with other firms. Equilibrium supply networks are naturally drawn to a “robust yet fragile” configuration: insured against idiosyncratic risk but arbitrarily sensitive to aggregate shocks.

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We model the production of complex goods in a large supply network. Firms source several essential inputs through relationships with other firms. Due to the risk of such supply relationships being idiosyncratically disrupted, firms multisource inputs and invest to make relationships with suppliers stronger. In equilibrium, aggregate production is robust to idiosyncratic disruptions. However, depending on parameters, the supply network may be robust or arbitrarily sensitive to small aggregate shocks that affect the functioning of relationships. We give conditions under which the equilibrium network is driven to a fragile configuration, where arbitrarily small aggregate shocks cause discontinuous losses. We use the model to provide a unified account of a number of stylized facts about complex economies.

**Current version**: February 2021. **First version**: September 2019.

### Learning from Neighbors about a Changing State (with Krishna Dasaratha and Nir Hak)

##### 2021, Revise and resubmit, **Review of Economic Studies** arXiv SSRN EC 18 conference version

**Review of Economic Studies**

A rational model of learning in a network. Simple time-invariant learning rules emerge in equilibrium. Whether learning is good depends on properties very different from the ones that matter with a fixed state.

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Agents learn about a changing state using private signals and past actions of neighbors in a network. We characterize equilibrium learning and social influence in this setting. We then examine when agents can aggregate information well, responding quickly to recent changes. A key sufficient condition for good aggregation is that each individual’s neighbors have sufficiently different types of private information. In contrast, when signals are homogeneous, aggregation is suboptimal on any network. We also examine behavioral versions of the model, and show that achieving good aggregation requires a sophisticated understanding of correlations in neighbors’ actions. The model provides a Bayesian foundation for a tractable learning dynamic in networks, closely related to the DeGroot model, and offers new tools for counterfactual and welfare analyses.

**Current version**: November 30, 2021. **First version**: January 6, 2018.

### Games on Endogenous Networks (with Evan Sadler)

##### 2021 arXiv

### More

We study network games in which players both create spillovers for one another and choose with whom to associate. The endogenous outcomes include both the strategic actions (e.g., effort levels) and the network in which spillovers occur. We introduce a framework and two solution concepts that extend standard approaches — Nash equilibrium in actions and pairwise (Nash) stability in links. Our main results show that under suitable monotonicity assumptions on incentives, stable networks take simple forms. Our first condition concerns whether links create positive or negative payoff spillovers. Our second condition concerns whether actions and links are strategic complements or substitutes. Together, these conditions allow for a taxonomy of how network structure depends on economic primitives. 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**: May 14, 2021. **First Version:** February 2, 2021

### Discord and Harmony in Networks (with Andrea Galeotti, Sanjeev Goyal and Rithvik Rao )

##### 2021 arXiv

### More

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

### More

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.

**Current version**: May 2019. **First version**: October 20, 2017.

### Signaling, Shame, and Silence in Social Learning (with Arun Chandrasekhar and He Yang)

##### 2019 SSRN NBER

### More

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.

### Illiquidity Spirals in Coupled Over-the-Counter Markets (with Christoph Aymanns and Co-Pierre Georg)

##### 2020, Revise and resubmit, **Operations Research** SSRN

**Operations Research**

Traders are 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 lending.

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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.

**Current version**: August 2020. **First version**: April 8, 2017.

### Expectations, Networks, and Conventions (with Stephen Morris)

##### 2017 SSRN Slides

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).

### More

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.

### Higher-Order Expectations (with Stephen Morris)

##### 2017 SSRN

### More

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.

## Published research papers

### Targeting Interventions in Networks (with Andrea Galeotti and Sanjeev Goyal)

**Econometrica** 88(6), November 2020 Online Appendix Slides

**Econometrica**

##### 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

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*—targeting a single principal component.

**Current version: **November 12, 2019.** First version: **October 17, 2017.

### A Network Approach to Public Goods (with Matt Elliott)

**Journal of Political Economy** 127(2), April 2019 Online Appendix Slides 4-page version SSRN

**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.

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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.

### Financial Networks and Contagion (with Matt Elliott and Matthew O. Jackson)

**American Economic Review** 104(10), October 2014 Slides Online Appendix SSRN

**American Economic Review**

Diversification (more counterparties) and integration (deeper relationships with each counterparty) have different, non-monotonic effects on financial contagions.

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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.

### How Homophily Affects the Speed of Learning and Best-Response Dynamics (with Matthew O. Jackson)

**Quarterly Journal of Economics** 127(3), August 2012 Online Appendix Slides Publisher's Site

**Quarterly Journal of Economics**

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.

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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

**American Economic Review: Papers & Proceedings**

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.

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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.

### Naive Learning in Social Networks and the Wisdom of Crowds (with Matthew O. Jackson)

**American Economic Journal: Microeconomics** 2(1), February 2010 3-page version Slides

**American Economic Journal: Microeconomics**

In what networks do agents who learn very naively get the right answer?

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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

**Proceedings of the National Academy of Sciences**

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.

### More

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

**Review of Network Economics**

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/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

**Annals of Economics and Statistics**

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 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

**Review of Economic Design**

How and when to decentralize networked production — in a model that takes into account ‘human’ features of processing.

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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), December 27, 2011 PNAS

**Proceedings of the National Academy of Sciences**

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

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.

## Older Working Papers

### The Leverage of Weak Ties: How Linking Groups Affects Inequality (with Carlos Lever)

##### 2010

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 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)

##### 2011

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.

### Ranking Agendas for Negotiations (with Matt Elliott)

##### 2015

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.