Coordination with Local Information

CORRECTED VERSION OF RECORD; SEE LAST PAGE OF ARTICLE OPERATIONS RESEARCH Vol. 64, No. 3, May–June 2016, pp. 622–637 ISSN 0030-364X (print) ISSN 1526-5463 (online) ó http://dx.doi.org/10.1287/opre.2015.1378 © 2016 INFORMS Coordination with Local Information

Munther A. Dahleh Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, [email protected] Alireza Tahbaz-Salehi Columbia Business School, Columbia University, New York, New York 10027, [email protected] John N. Tsitsiklis Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, [email protected] Spyros I. Zoumpoulis Decision Sciences Area, INSEAD, 77300 Fontainebleau, France, [email protected]

We study the role of local information channels in enabling coordination among strategic agents. Building on the standard ﬁnite-player global games framework, we show that the set of equilibria of a coordination game is highly sensitive to how information is locally shared among different agents. In particular, we show that the coordination game has multiple equilibria if there exists a collection of agents such that (i) they do not share a common signal with any agent outside of that collection and (ii) their information sets form an increasing sequence of nested sets. Our results thus extend the results on the uniqueness and multiplicity of equilibria beyond the well-known cases in which agents have access to purely private or public signals. We then provide a characterization of the set of equilibria as a function of the penetration of local information channels. We show that the set of equilibria shrinks as information becomes more decentralized. Keywords: coordination; local information; social networks; global games. Subject classiﬁcations: games/group decisions: economics; network/graphs. Area of review: Special Issue on Information and Decisions in Social and Economic Networks. History: Received July 2013; revisions received October 2014, January 2015; accepted February 2015. Published online in Articles in Advance November 12, 2015.

1. Introduction inferior product simply because they expect other agents to 1 Coordination problems lie at the heart of many economic do the same (Argenziano 2008). and social phenomena such as bank runs, social uprisings, Given the central role of self-fulfilling beliefs in coor- and the adoption of new standards or technologies. The dination games, it is natural to expect the emergence of common feature of these phenomena is that the benefit of coordination failures to be highly sensitive to the availabil- taking a specific action to any given individual is highly ity and distribution of information across different agents. sensitive to the extent to which other agents take the same In fact, since the seminal work of Carlsson and van Damme action. The presence of such strong strategic complemen- (1993), which initiated the global games literature, it has tarities, coupled with the self-fulfilling nature of the agents’ been well known that the set of equilibria of a coordina- expectations, may then lead to coordination failures: indi- tion game depends on whether the information available to viduals may fail to take the action that is in their best the agents is public or private: the same game that exhibits collective interest. multiple equilibria in the presence of public signals may Bank runs present a concrete (and classic) example of have a unique equilibrium if the information were instead the types of coordination failures that may arise due to only privately available to the agents. the presence of strategic complementarities. In this context, The global games literature, however, has for the most Downloaded from informs.org by [128.59.83.27] on 13 July 2016, at 09:20 . For personal use only, all rights reserved. any given depositor has strong incentives to withdraw her part only focused on the role of public and private informa- money from the bank if (and only if) she expects that a tion while ignoring the effects of local information chan- large fraction of other depositors would do the same. Thus, nels in facilitating coordination. This is despite the fact that a bank run may emerge, not because of any financial dis- in many real world scenarios, local information channels tress at the bank, but rather as a result of the self-fulfilling play a key role in enabling agents to coordinate on differ- nature of the depositors’ expectations about other deposi- ent actions. For instance, it is by now conventional wis- tors’ behavior (Diamond and Dybvig 1983). Similarly, in dom that protesters in many recent antigovernment upris- the context of adoption of new technologies with strong ings throughout the Middle East used decentralized modes network effects, consumers may collectively settle for an of communication (such as word-of-mouth communications

and Internet-based social media platforms) to coordinate that the set of equilibrium strategies enlarges if information on the time and location of street protests (Ali 2011). is more centralized. In other words, as the signals become Similarly, adopters of new technologies that exhibit strong more publicly available, the set of states under which agents network effects may rely on a wide array of informa- can coordinate on either action grows. This result is due tion sources (such as blogs, professional magazines, expert to the fact that more information concentration reduces the opinions), not all of which are used by all other potential extent of strategic uncertainty among the agents, even if it adopters in the market. does not impact the level of fundamental uncertainty. Motivated by these observations, we study the role of We then use our characterization results to study the set local information channels in enabling coordination among of equilibria in large coordination games. We show that as strategic agents. Building on the standard finite-player the number of agents grows, the game exhibits multiple global games framework, we show that the set of equilibria equilibria if and only if a nontrivial fraction of the agents of a coordination game is highly sensitive to how infor- have access to the same signal. Our result thus shows that mation is locally shared among agents. More specifically, if the size of the subsets of agents with common knowledge rather than restricting our attention to cases in which infor- of a signal does not grow at the same rate as the number of mation is either purely public or private, we also allow for agents, the information structure is asymptotically isomor- the presence of local signals that are observable to subsets phic to a setting in which all signals are private. Conse- of the agents. The presence of such local sources of infor- quently, all agents face some strategic uncertainty regard- mation guarantees that some (but not necessarily all) infor- ing the behavior of almost every other agent, resulting in a mation is common knowledge among a group of agents, unique equilibrium. with important implications for the determinacy of equi- Finally, even though our benchmark model assumes that libria. Our main contribution is to provide conditions for the agents’ information sets are common knowledge, we uniqueness and multiplicity of equilibria based solely on also show that our results are robust to the introduction the pattern of information sharing among agents. Our find- of small amounts of noise in the information structure of ings thus provide a characterization of the extent to which the game. In particular, by introducing uncertainty about coordination failures may arise as a function of which piece the information sets of different agents, we show that the of information is available to each agent. equilibria of this new, perturbed game are close (in a formal As our main result, we show that the coordination game sense) to the equilibria of the original game in which all has multiple equilibria if there exists a collection of agents information sets were common knowledge. such that (i) they do not share a common signal with any In sum, our results establish that the distribution and agent outside of that collection and (ii) their information availability of information play a fundamental role in deter- sets form an increasing sequence of nested sets, which we mining coordination outcomes, thus highlighting the risk refer to as a filtration. This result is a consequence of the of abstracting from the intricate details of information dis- fact that agents in such a collection face limited strate- semination within coordination contexts. For example, on gic uncertainty—that is, uncertainty concerning the equilib- the positive side, ignoring the distribution and reach of dif- rium actions—about one another,2 which transforms their ferent information sources can lead to wrong predictions common signals into a de facto coordination device. To about potential outcomes. Similarly, on the normative side see this in the most transparent way, consider two agents (say for example, in the context of a central bank facing a whose information sets form a filtration. It is clear that the banking crisis), the interventions of policymakers may turn agent with the larger information set (say, agent 2) faces out to be counterproductive if such actions are not informed no uncertainty in predicting the equilibrium action of the by the dispersion and penetration of different information agent with the smaller information set (agent 1). More- channels within the society. over, even though the latter is not informed about other Related Literature. Our paper is part of the by now large potential signals observed by the former, the realizations literature on global games. Initiated by the seminal work of such signals may not be extreme enough to push agent of Carlsson and van Damme (1993) and later expanded by 2 toward either action, thus making it optimal for her to Morris and Shin (1998, 2003), this literature focuses on simply follow the behavior of agent 1. In other words, for how the presence of strategic uncertainty may lead to the certain realizations of signals, the agent with the smaller selection of a unique equilibrium in coordination games. Downloaded from informs.org by [128.59.83.27] on 13 July 2016, at 09:20 . For personal use only, all rights reserved. information set becomes pivotal in determining the pay- The machinery of global games has since been used exten- off maximizing action of agent 2. Consequently, the set of sively to analyze various applications that exhibit an ele- signals that are common between the two can effectively ment of coordination, such as currency attacks (Morris and function as a coordination device, leading to the emergence Shin 1998), bank runs (Goldstein and Pauzner 2005), polit- of multiple equilibria. ical protests (Edmond 2013), partnership investments (Das- We then focus on a special case of our model by assum- gupta 2007), emergence of financial crises (Vives 2014), ing that each agent observes a single signal. We provide and adoption of technologies and products that exhibit an explicit characterization of the set of equilibria in terms strong network effects (Argenziano 2008). For example, of the commonality in agents’ information sets and show Argenziano (2008) uses the global games framework to Dahleh et al.: Coordination with Local Information 624 Operations Research 64(3), pp. 622–637, © 2016 INFORMS

study a model of price competition in a duopoly with prod- Chwe (2000). Galeotti et al. (2010) focus on strategic inter- uct differentiation and network effects. She shows that the actions over networks and characterize how agents’ inter- presence of a sufficient amount of noise in the public infor- actions with their neighbors and the nature of the game mation available about the quality of the goods guarantees shape individual behavior and payoffs. They show that the that the coordination game played by the consumers for presence of incomplete information about the structure of any given set of prices has a unique equilibrium, and hence the network may lead to the emergence of a unique equi- the demand function for each product is well defined. librium. The key distinction between their model and ours All the above papers, however, restrict their attention to is in the information structure of the game and the nature the case in which all information is either public or pri- of payoffs. Whereas they study an environment in which vate. Our paper, on the other hand, focuses on more general individuals care about the actions of their neighbors (whose information structures by allowing for signals that are nei- identities they are uncertain about), we study a setting in which network effects are only reflected in the game’s ther completely public nor private and provides conditions information structure: individuals need to make deductions under which the presence of such signals may lead to equi- about an unknown parameter and the behavior of the rest librium multiplicity. of the agents while relying on local sources of information. Our paper is also related to the works of Morris et al. Chwe (2000), on the other hand, studies a coordination (2015) and Mathevet (2014), who characterize the set of game in which individuals can inform their neighbors of equilibria of coordination games in terms of the agents’ their willingness to participate in a collective risky behav- types while abstracting away from the information struc- ior. Thus, our work shares two important aspects with that ture of the game. Relatedly, Weinstein and Yildiz (2007) of Chwe (2000): not only do both papers study games with provide a critique of the global games approach by argu- strategic complementarities, but also consider information ing that any rationalizable action in a game can always structures in which information about the payoff-relevant be uniquely selected by properly perturbing the agents’ parameters are locally shared among different individuals. hierarchy of beliefs. Our work, on the other hand, pro- Nevertheless, the two papers focus on different questions. vides a characterization of the set of equilibrium strategies Whereas Chwe’s main focus is on characterizing the set of when the perturbation in the beliefs are generated by pub- networks for which, regardless of their prior beliefs, agents lic, private, or local signals that are informative about the can coordinate on a specific action, our results provide a underlying state. Despite being more restrictive in scope, characterization of how the penetration of different chan- our results shed light on the role of local information in nels of information can impact coordination outcomes. enabling coordination. Outline of the Paper. The rest of the paper is organized A different set of papers studies how the endogeneity of as follows: §2 introduces our model. In §3, we present a agents’ information structure in coordination games may series of simple examples that capture the intuition behind lead to equilibrium multiplicity, thus qualifying the refine- our results. Section 4 contains our results on the role of ment of equilibria proposed by the standard global games local information channels in the determinacy of equilibria. argument. For example, Angeletos and Werning (2006), We then provide a characterization of the set of equilibria in §5 and show that the equilibrium set shrinks as informa- and Angeletos et al. (2006, 2007) show how prices, the tion becomes more decentralized. Section 6 concludes. All action of a policymaker, or past outcomes can function proofs are presented in the appendix. as endogenous public signals that may restore equilibrium multiplicity in settings that would have otherwise exhib- 2. Model ited a unique equilibrium. We study another natural setting in which agents may rely on overlapping (but not neces- Our model is a finite-agent variant of the canonical model sarily identical) sources of information and show how the of global games studied by Morris and Shin (2003). information structure affects the outcomes of coordination 2.1. Agents and Payoffs games. Our paper also belongs to the strand of literature that Consider a coordination game played by n agents whose set focuses on the role of local information channels and social we denote by N 811 210001n9. Each agent can take one of two possible actions,= a 801 19, which we refer to as the networks in shaping economic outcomes. Some recent i 2 Downloaded from informs.org by [128.59.83.27] on 13 July 2016, at 09:20 . For personal use only, all rights reserved. examples include Acemoglu et al. (2011), Golub and Jack- safe and risky actions, respectively. The payoff of taking the son (2010), Jadbabaie et al. (2012, 2013), and Galeotti et al. safe action is normalized to zero, regardless of the actions (2013). For example, Acemoglu et al. (2011) study how of other agents. The payoff of taking the risky action, on the other hand, depends on (i) the number of other agents patterns of local information exchange among few agents who take the risky action and (ii) an underlying state of can have first-order (and long-lasting) implications for the the world à , which we refer to as the fundamental. In actions of others. ✓ particular, the2 payoff function of agent i is Within this context, however, our paper is more closely related to the subset of papers that studies coordination è4k1à5 if ai 11 ui4ai1a i1à5 = games over networks, such as Galeotti et al. (2010) and É = 0 if a 01 ( i = Dahleh et al.: Coordination with Local Information Operations Research 64(3), pp. 622–637, © 2016 INFORMS 625

n where k j 1 aj is the number of agents who take the her deposit a dominant action if the bank is sufficiently = = risky action and è2 801 110001n9 ✓ ✓ is a function that distressed (healthy). is LipschitzP continuous in its second⇥ argument.! Throughout, In summary, agents face a coordination game with strong we impose the following assumptions on the agents’ payoff strategic complementarities in which the value of coordi- function, which are standard in the global games literature. nating on the risky action depends on the underlying state of the world. Furthermore, particular values of the state Assumption 1. Function è4k1à5 is strictly increasing in k make either action strictly dominant for all agents. for all à. Furthermore, there exists a constant ê>0 such that è4k1à5 è4k 11à5>ê for all à and all k 1. É É æ 2.2. Information and Signals The above assumption captures the presence of an ele- As in the canonical global games model, agents are not ment of coordination between agents. In particular, taking aware of the realization of the fundamental. Rather, they either action becomes more attractive the more other agents hold a common prior belief on à ✓, which for simplic- take that action. For example, in the context of a bank ity we assume to be the (improper)2 uniform distribution run, each depositor’s incentive to withdraw her deposits over the real line. Furthermore, conditional on the realiza- m from a bank not only depends on the solvency of the bank tion of à, a collection 4x110001xm5 ✓ of noisy signals (i.e., its fundamental) but also increases with the number is generated, where x à é . We2 assume that the noise r = + r of other depositors who decide to withdraw. Similarly, in terms 4é110001ém5 are independent from à and are drawn the context of adoption of technologies that exhibit net- from a continuous joint probability density function with work effects, each user finds adoption more attractive the full support over ✓m. more widely the product is adopted by other users in the Not all agents, however, can observe all realized sig- market. Thus, the above assumption simply guarantees that nals. Rather, agent i has access to a nonempty subset I i ✓ the game exhibits strategic complementarities. The second 8x110001xm9 of the signals, which we refer to as her infor- part of Assumption 1 is made for technical reasons and mation set.3 This assumption essentially captures that each states that the payoff of switching to the risky action when agent may have access to multiple sources of information one more agent takes action 1 is uniformly bounded from about the underlying state. Going back to the bank run below, regardless of the value of à. example, this means that each depositor may obtain some information about the health of the bank via a variety of Assumption 2. Function è4k1à5 is strictly decreasing information sources (such as the news media, the results of in à for all k. stress tests released by the regulatory agencies, inside infor- That is, any given individual has less incentive to take mation, expert opinions, etc.). The depositor would then the risky action if the fundamental takes a higher value. use the various pieces of information available to her to Thus, taking other agents’ actions as given, each agent’s decide whether to run on the bank or not. The fact that agent i’s information set can be any optimal action is decreasing in the state. For instance, in the (nonempty) subset of 8x 10001x 9 means that the extent bank run context, if the fundamental value à corresponds to 1 m to which any given signal x is observed may vary across the financial health of the bank, the depositors’ withdrawal r signals. For example, if x I for all i, then x is essen- incentives are stronger the closer the bank is to insolvency. r i r tially a public signal observed2 by all agents. On the other Similarly, in the context of the adoption of network goods, hand, if x I for some i but x I for all j i, then x à can represent the quality of the status quo technology r i r j r is a private2 signal of agent i. Any62 signal that6= is neither relative to the new alternative in the market: regardless of private nor public can be interpreted as a local source of the market shares, a higher à makes adoption of the new information observed only by a proper subset of the agents. technology less attractive. Finally, we impose the following The potential presence of such local signals is our point of assumption: departure from the canonical global games literature, which only focuses on games with purely private or public sig- Assumption 3. There exist constants à1 à¯ ✓ satisfying 2 nals. Note that in either case, following the realization of à < à¯ such that (i) è4k1à5>0 for all k and all à<à. signals, the mean of agent i’s posterior belief about the fundamental is simply equal to her Bayesian estimate ⇧6à Ii7. Downloaded from informs.org by [128.59.83.27] on 13 July 2016, at 09:20 . For personal use only, all rights reserved. (ii) è4k1à5<0 for all k and all à>à. ¯ We remark that even though agents are uncertain aboutó Thus, each agent strictly prefers to take the safe (risky) the realizations of the signals they do not observe, the action for sufficiently high (low) states of the world, irre- information structure of the game—that is, the collection

spective of the actions of other agents. If, on the other hand, of information sets 8Ii9i N —is common knowledge among the underlying state belongs to the so-called critical region them.4 Therefore, a pure2 strategy of agent i is simply a 6à1 à7, then the optimal behavior of each agent depends on Ii ¯ mapping si2 ✓ó ó 801 19, where Ii denotes the cardinal- her beliefs about the actions of other agents. Thus, once ity of agent i’s information! set. ó ó again, in the bank run context, the above assumption sim- Finally, we impose the following mild technical assump- ply means that each depositor ﬁnds withdrawing (leaving) tion, ensuring that the agents’ payoff function is integrable. Dahleh et al.: Coordination with Local Information 626 Operations Research 64(3), pp. 622–637, © 2016 INFORMS

Assumption 4. For any collection of signals 8xr 9r H , H has an essentially unique Bayesian Nash equilibrium. To 8110001m9 and all k 8010001n9, 2 ✓ verify that the equilibrium of the game is indeed unique, 2 it is sufﬁcient to focus on the set of equilibria in threshold

⇧6 è4k1à5 8xr 9r H 7< 0 strategies, according to which each agent takes the risky ó óó 2 à action if and only if her private signal is smaller than a 5 We end this section by remarking that given that the pay- given threshold. In particular, let íi denote the threshold off functions have nondecreasing differences in the actions corresponding to the strategy of agent i. Taking the strate- and the state, the underlying game is a Bayesian game with gies of agents j and k as given, agent i’s expected pay-

strategic complementarities. Therefore, by Van Zandt and off of taking the risky action is equal to ⇧6è4k1à5 xi7 Vives (2007, Theorem 14), the game’s Bayesian Nash equi- 1 ó = 2 6⇣4xj <íj xi5 ⇣4xk <ík xi57 xi. For íi to corre- libria form a lattice whose extremal elements are monotone spond to an equilibriumó + strategyó ofÉ agent i, she has to be in types. This lattice structure also implies that the extremal indifferent between taking the safe and the risky actions equilibria are symmetric, in the sense that each agent’s strat- when xi íi. Hence, the collection of thresholds 4í11í21í35 egy only depends on the signals she observes but not on her corresponds= to a Bayesian Nash equilibrium of the game if identity (Vives 2008). Therefore, to characterize the range and only if for all permutations of i, j, and k, we have of equilibria, without loss of generality, we can restrict our

attention to symmetric equilibria in threshold strategies. 1 íj íi 1 ík íi íi Í É Í É 1 = 2 ëp2 + 2 ëp2 3. A Simple Example ✓ ◆ ✓ ◆ Before proceeding to our general results, we present a sim- where Í4 5 denotes the cumulative distribution function of · 2 ple example and show how the presence of local signals the standard normal. Note that à xi N 4xi1ë 5, and as 2 ó ⇠ determines the set of equilibria. Consider a game consisting a result, xj xi N 4xi1 2ë 5. It is then immediate to verify that í ó í ⇠ í 1/2 is the unique solution of the of n 3 agents and, for expositional simplicity, suppose 1 = 2 = 3 = that = above system of equations. Thus, in the (essentially) unique equilibrium of the game, agent i takes the risky action if k 1 she observes x < 1/2, whereas she takes the safe action è4k1à5 É à0 (1) i = n 1 É if x > 1/2. Following standard arguments from Carlsson É i and van Damme (1993) or Morris and Shin (2003), one It is easy to verify that this payoff function, which is sim- can show that this strategy profile is also the (essentially) ilar to the one in the canonical finite-player global games unique strategy profile that survives the iterated elimination literature such as Morris and Shin (2000, 2003), satisfies of strictly dominated strategies. Consequently, in contrast Assumptions 1–3 with à 0 and à 1. We consider three ¯ to the game with public information, as ë 0, all agents different information structures,= contrasting= the cases in choose the risky action if and only if à<1!/2. This obser- which agents have access to only public or private informa- vation shows that in the limit as signals become arbitrarily tion to a case with a local signal. Throughout this section, precise and all the fundamental uncertainty is removed, the we assume that the noise terms ér are mutually independent and normally distributed with mean zero and variance presence of strategic uncertainty among the agents leads to ë 2 > 0. the selection of a unique equilibrium. Public Information. First, consider a case in which all Local Information. Finally, consider the case where only agents observe the same public signal x, that is, I 8x9 two signals 4x11x25 are realized and the agents’ informa- i = tion sets are I 8x 9 and I I 8x 9; that is, agent 1 for all i 811 21 39. Thus, no agent has any private informa- 1 = 1 2 = 3 = 2 tion about2 the state. It is easy to verify that under such an observes a private signal whereas agents 2 and 3 have information structure, the coordination game has multiple access to the same local source of information. All the Bayesian Nash equilibria. In particular, for any í 601 17, information available to agents 2 and 3 is common knowl- the strategy profile in which all agents take the risky2 action edge between them, which distinguishes this case from the if and only if x<í is an equilibrium, regardless of the canonical global game model with private signals. value of ë. Consequently, as the public signal becomes To determine the extent of equilibrium multiplicity, once Downloaded from informs.org by [128.59.83.27] on 13 July 2016, at 09:20 . For personal use only, all rights reserved. infinitely accurate (i.e., as ë 0), the underlying game again it is sufficient to focus on the set of equilibria in ! threshold strategies. Let í and í í denote the thresh- has multiple equilibria as long as the underlying state à 1 2 = 3 belongs to the critical region 601 17. olds corresponding to the strategies of agents 1, 2, and 3, Private Information. Next, consider the case in which all respectively. If agent 1 takes the risky action, she obtains an agents have access to a different private signal. In particu- expected payoff of ⇧6è4k1à5 x17 ⇣4x2 <í2 x15 x1. ó = ó É 3 On the other hand, the expected payoff of taking the risky lar, suppose that three signals 4x11x21x35 ✓ are realized 2 action to agent 2 (and, by symmetry, agent 3) is given by and that xi is privately observed by agent i; that is, Ii 8xi9 for all i. As is well known from the global games= literature, the coordination game with privately observed signals ⇧6è4k1à5 x 7 1 6⇣4x <í x 5 ⌧8x <í 97 x 1 ó 2 = 2 1 1 ó 2 + 2 2 É 2 Dahleh et al.: Coordination with Local Information Operations Research 64(3), pp. 622–637, © 2016 INFORMS 627

where ⌧8 9 denotes the indicator function. Thus, for thresh- Theorem 1. Suppose that Assumptions 1–4 are satisfied. · olds í1 and í2 to correspond to a Bayesian Nash equilib- Also, suppose that there exists a subset of agents C N rium, it must be the case that such that ✓ (a) The information sets of agents in C form a filtration. í í í Í 2 É 1 0 (2) (b) Ii Ij for any i C and j C. 1 = p Then the\ coordination=ô game2 has multiple62 Bayesian Nash ✓ ë 2 ◆ equilibria. Furthermore, agents 2 and 3 should not have an incentive to deviate, which requires that their expected payoffs have The above result thus shows that, in general, the presence of a cascade of increasingly rich observations guaran- to be positive for any x2 <í2 and negative for any x2 >í2, leading to the following condition: tees equilibrium multiplicity. Such filtration provides some degree of common knowledge among the subset of agents

í1 í2 in C, reduces the extent of strategic uncertainty they face 2í2 1 ∂ Í É ∂ 2í20 (3) É ëp2 regarding each other’s behavior, and as a result leads to the ✓ ◆ emergence of multiple equilibria. Therefore, in this sense,

It is easy to verify that the pair of thresholds 4í11í25 simul- Theorem 1 generalizes the standard, well-known results in taneously satisfying (2) and (3) is not unique. In particular, the global games literature to the case when agents have as ë 0, for every í 61/31 2/37, there exists some í access to local information channels that are neither public ! 1 2 2 close enough to í1 such that the profile of threshold strate- nor private. gies 4í11í21í25 is a Bayesian Nash equilibrium. Conse- We remark that even though the presence of local signals quently, as the signals become very precise, the underlying may lead to equilibrium multiplicity, the set of Bayesian 1 2 Nash equilibria does not necessarily coincide with that of a game has multiple equilibria as long as à 6 3 1 3 7. Thus, even though there are no public signals,2 the pres- game with purely public signals. Rather, as we show in the ence of common knowledge between a proper subset of the next section, the set of equilibria crucially depends on the agents restores the multiplicity of equilibria. In other words, number of agents in C as well as the information structure the local information available to agents 2 and 3 serves as a of other agents. coordination device, enabling them to predict one another’s To see the intuition underlying Theorem 1, suppose that actions. The presence of strong strategic complementarities the information sets of agents in some set C form a fil- in turn implies that agent 1 will use his private signal as tration. It is immediate that there exists an agent i C 2 a predictor of how agents 2 and 3 coordinate their actions. whose signals are observable to all other agents in C. Con- Nevertheless, because of the presence of some strategic sequently, agents in C face no uncertainty (strategic or oth- uncertainty between agents 2 and 3 on the one hand and erwise) in predicting the equilibrium actions of agent i. agent 1 on the other, the set of rationalizable strategies is Furthermore, the signals in I I provide agent j C 8i9 j \ i 2 \ strictly smaller compared to the case where all three agents with information about the underlying state à beyond what observe a public signal. Therefore, the local signal (par- agent i has access to. Nevertheless, there exist realizations tially) refines the set of equilibria of the coordination game, of such signals such that the expected payoff of taking the though not to the extent that would lead to uniqueness. risky action to any agent j C 8i9 would be positive if and only if agent i takes the2 risky\ action. In other words, 4. Local Information and Equilibrium for such realizations of signals, agent i’s action becomes pivotal in determining the payoff maximizing actions of all Multiplicity other agents in C. Consequently, signals in Ii can essen- The examples in the previous section show that the set of tially serve as a coordination device among C’s members, equilibria in the presence of local signals may not coincide leading to multiple equilibria. with the set of equilibria under purely private or public Finally, note that condition (b) of Theorem 1 plays a cru- signals. In this section, we provide a characterization of cial role in the above argument. In particular, it guarantees the role of local information channels in determining the that agents in C effectively face an induced coordination equilibria of the coordination game presented in §2. Before game among themselves, in which they can use the signals Downloaded from informs.org by [128.59.83.27] on 13 July 2016, at 09:20 . For personal use only, all rights reserved. presenting our main result, we define the following concept: in Ii as a coordination device. Definition 1. The information sets of agents in C 8i 10001i 9 form a filtration if I I I . = 4.1. The Role of Information Filtrations 1 c i1 i2 ic ✓ ✓···✓ In this subsection, we use a series of examples to exhibit Thus, the information sets of agents in set C constitute a the main intuition underlying Theorem 1 and highlight the filtration if they form a nested sequence of increasing sets. role of its different assumptions. This immediately implies that the signals of the agent with the smallest information set is common knowledge among Example 1. Consider a two-player game with linear pay- all agents in C. We have the following result: offs è4k1à5 k 1 à as in (1). Furthermore, suppose = É É Dahleh et al.: Coordination with Local Information 628 Operations Research 64(3), pp. 622–637, © 2016 INFORMS

that the realized signals 4x11x25 are normally distributed, point, consider a three-agent variant of the above example, with agents’ information sets given by where the agents’ information sets are given by

I 8x 91 I 8x 1x 90 I 8x 1x 91 I 8x 1x 91 I 8x 1x 90 (4) 1 = 1 2 = 1 2 1 = 2 3 2 = 3 1 3 = 1 2 It is immediate that the above information structure satisfies Thus, even though there is one signal that is common the assumptions of Theorem 1. knowledge among any given pair of agents, the information As before, focusing on threshold strategies is sufficient for sets of no subset of agents form a filtration. We have the

determining the set of equilibria. Let í1 denote the thresh- following result: old corresponding to agent 1’s strategy. The fact that x is a 1 Proposition 2. Suppose that agents’ information sets are signal that is observable to both agents implies that agent 2 given by (4). There exists ë such that if ë>ë, then the can predict the equilibrium action of agent 1 with certainty game has an essentially unique¯ equilibrium. ¯ and change her behavior accordingly. On the other hand, the expected payoff of agent 2 when she takes the risky action Thus, the coordination game has a unique equilibrium

is given by a1 ⇧6à x11x27 a1 4x1 x25/2. These two despite the fact that any pair of agents shares a common observations togetherÉ ó imply that= theÉ best+ response of agent 2 signal. This is because, unlike Theorem 1 and Example 1

would be a threshold strategy 4í21í20 5 constructed as follows: above, every agent faces some strategic uncertainty about if x1 <í1, then agent 2 takes the risky action if and only if all other agents in the game, hence guaranteeing that no 4x x 5/2 <í , whereas if x í , agent 2 takes the risky collection of signals can serve as a coordination device 1 + 2 2 1 æ 1 action if and only if 4x1 x25/2 <í20 . among a subset of agents. Therefore, the expected+ payoff of taking the risky action In addition to the presence of a subset of agents C whose to agent 2 is equal to information sets form a filtration, Theorem 1 also requires that the information set of no agent outside C contain any 1 of the signals observable to agents in C. To clarify the role ⇧6è4k1à5 x11x27 ⌧8x1 <í19 2 4x1 x250 ó = É + of this assumption in generating multiple equilibria, once The fact that agent 2 has to be indifferent between taking again consider the above three-player game but instead sup- the safe and the risky actions at her two thresholds implies pose that agents’ information sets are given by that í 1 and í0 0. On the other hand, the expected 2 = 2 = payoff of taking the risky action to agent 1 is given by I1 8x191 I2 8x11x291 I3 8x290 (5) = = = It is immediate to verify that the conditions of Theorem 1 ⇧6è4k1à5 x17 ⇣4x2 < 2í2 x1 x15⌧8x1 <í19 ó = É ó are not satisfied, even though there exists a collection of ⇣4x < 2í0 x x 5⌧8x í 9 x 0 + 2 2 É 1 ó 2 1 æ 1 É 1 agents whose information sets form a filtration. We have the following result: Given that í1 captures the threshold strategy of agent 1, the Proposition 3. Suppose that agents’ information sets are above expression has to be positive if and only if x1 <í1, or equivalently: given by (5). Then the game has an essentially unique equilibrium. Furthermore, as ë 0, all agents choose the risky ! í1 2 í1 action if and only if à<1/2. Í ∂ í1 ∂ Í É 0 É p p ✓ ë 2◆ ✓ ë 2 ◆ Thus, as fundamental uncertainty is removed, information structure (5) induces the same (essentially) unique It is immediate to verify that the value of í that satisfies 1 equilibrium as the case in which all signals are private. the above inequalities is not unique, thus guaranteeing the This is despite the fact that the information sets of agents multiplicity of equilibria. in C 811 29 form a filtration. To interpret this result, note that whenever ⇧6à x 1x 7 1 2 To= understand the intuition behind this result, it is 401 15, agent 2 finds it optimal to simply follow theó behav-2 instructive to compare the above game with the game ior of agent 1, regardless of the realization of signal x . 2 described in Example 1. Notice that in both games, agents Downloaded from informs.org by [128.59.83.27] on 13 July 2016, at 09:20 . For personal use only, all rights reserved. This is because (i) within this range, agent 2’s belief about with the larger information sets do not face any uncer- the underlying state à lies inside the critical region; and tainty (strategic or otherwise) about predicting the actions (ii) agent 2 can perfectly predict the action of agent 1, mak- of agents with smaller information sets. Nevertheless, the ing agent 1 pivotal and hence leading to multiple equilibria. above game exhibits a unique equilibrium, whereas the It is important to note that the arguments following The- game in Example 1 has multiple equilibria. The key dis- orem 1 and the above example break down if agent 1 has tinction lies in the different roles played by the extra pieces access to signals that are not in the information set of of information available to the agents with larger informa- agent 2, even though there are some signals that are com- tion sets. Recall that in Example 1, there exists a set of mon knowledge between the two agents. To illustrate this realizations of signals for which agent 2 finds it optimal to Dahleh et al.: Coordination with Local Information Operations Research 64(3), pp. 622–637, © 2016 INFORMS 629

imitate the action of agent 1. Therefore, under such condi- In this section, we provide a characterization of the set of

tions, agent 2 uses x2 solely as a means of obtaining a more all Bayesian Nash equilibria as a function of the informa- precise estimate of the underlying state à. This, however, tion sets of different agents. Our characterization quantiﬁes is in sharp contrast with what happens under information the dependence of the set of rationalizable strategies on the

structure (5). In this case, signal x2 plays a second role in extent to which agents observe common signals. the information structure of agent 2: it not only provides To explicitly characterize the set of equilibria, we restrict an extra piece of information about à but also serves as our attention to a game with linear payoff functions given a perfect predictor of agent 3’s equilibrium action. Thus, by (1). We also assume that m ∂ n signals, denoted by even if observing x does not change the mean of agent 2’s 4x 10001x 5 ✓m, are realized, where the noise terms 2 1 m 2 posterior belief about à by much, it still provides her with 4é110001ém5 are mutually independent and normally dis- information about the action that agent 3 is expected to take tributed with mean zero and variance ë 2 > 0. Furthermore, in equilibrium. This extra piece of information, however, we assume that each agent observes only one of the real- is not available to agent 1. Thus, even as ë 0, agents 1 ized signals; that is, for any given agent i, her information and 3 face some strategic uncertainty regarding! the equilib- set is I 8x 9 for some 1 r m. Finally, we denote i = r ∂ ∂ rium action of one another and, as a consequence, regarding the fraction of agents that observe signal xr by cr , and let that of agent 2. The presence of such strategic uncertainties c 4c 10001c 5. Since each agent observes a single signal, = 1 m implies that the game with information structure (5) would we have c c 1. Note that as in our benchmark 1 +···+ m = exhibit a unique equilibrium. model in §2, we assume that the allocation of signals to We conclude our argument by showing that even though agents is deterministic, prespeciﬁed, and common knowl- the conditions of Theorem 1 are sufﬁcient for equilibrium edge. Our next result provides a simple characterization of multiplicity, they are not necessary. To see this, once again the set of all rationalizable strategies as ë 0. ! consider a variant of the game in Example 1 but instead Theorem 5. Let si denote a threshold strategy of agent i. assume that there are four agents whose information sets As ë 0, strategy s is rationalizable if and only if are given by ! i 1 if x<í I 8x 91 I 8x 1x 91 I 8x 91 I 8x 91 (6) si4x5 1 = 1 2 = 1 2 3 = 2 4 = 2 = 0 if x>í1 ( ¯ Note that the above information structure does not satisfy where the conditions of Theorem 1. Nevertheless, one can show n 1 that the game has multiple equilibria: í 1 í 41 c 25 = É¯= 24n 15 Éò ò2 ∂ 2 Proposition 4. Suppose that agents’ information sets are É

given by (6). Then the game has multiple Bayesian Nash and c 2 denotes the Euclidean norm of vector c. equilibria. ò ò The above theorem shows that the distance between the The importance of the above result is highlighted when thresholds of the “largest” and “smallest” rationalizable contrasted with the information structure (5) and Proposi- strategies depends on how information is locally shared tion 3. Note that in both cases, agent 2 has access to the between different agents. More speciﬁcally, a smaller c ò ò2 signals available to all other agents. However, in informa- implies that the set of rationalizable strategies would

tion structure (6), signal x2 is also available to an additional shrink. agent, namely, agent 4. The presence of such an agent Note that c is essentially a proxy for the extent to ò ò2 with an information set identical to that of agent 3 means which agents observe common signals: it takes a smaller that agents 3 and 4 face no strategic uncertainty regarding value whenever any given signal is observed by fewer one another’s actions. Therefore, even though they may be agents. Hence, a smaller value of c 2 implies that agents ò ò uncertain about the equilibrium actions of agents 1 and 2, would face higher strategic uncertainty about one another’s

there are certain realizations of signal x2 under which they actions, even when all the fundamental uncertainty is ﬁnd it optimal to imitate one another’s action, thus leading removed as ë 0. As a consequence, the set of ratio- ! to the emergence of multiple equilibria. nalizable strategies shrinks as the Euclidean norm of c Downloaded from informs.org by [128.59.83.27] on 13 July 2016, at 09:20 . For personal use only, all rights reserved. decreases. In the extreme case that agents’ information is only in the form of private signals (that is, when m n and 5. Local Information and the Extent of = cr 1/n for all r), the upper and lower thresholds coin- Multiplicity cide=4í í 1/25, implying that the equilibrium strategies Our analysis thus far was focused on the dichotomy are essentially=¯= unique. This is indeed the case that corre- between multiplicity and uniqueness of equilibria. How- sponds to maximal level of strategic uncertainty. On the ever, as the example in §3 shows, even when the game other hand, when all agents observe the same public sig-

exhibits multiple equilibria, the set of equilibria depends on nal (i.e., when m 1 and c 2 1), they face no strate- how information is locally shared between different agents. gic uncertainty about= eachò other’sò = actions, and hence all Dahleh et al.: Coordination with Local Information 630 Operations Research 64(3), pp. 622–637, © 2016 INFORMS

undominated strategies are rationalizable. Thus, to summa- Proposition 6. The sequence of games 8G4n59n exhibits rize, the above two extreme cases coincide with the stan- an (essentially) unique equilibrium asymptotically2 as dard results in the global games literature. Theorem 5 above n and in the limit as ë 0 if and only if the size of !à ! generalizes those results by characterizing the equilibria of the largest set of agents with a common observation grows the game in the intermediate cases in which signals are sublinearly in n. neither fully public nor private. Thus, as the number of agents grows, the game exhibits Recall that since the Bayesian game under consideration multiple equilibria if and only if a nontrivial fraction of the is monotone supermodular in the sense of Van Zandt and agents have access to the same signal. Even though such a Vives (2007), there exist a greatest and a smallest Bayesian signal is not public—in the sense that it is not observed by Nash equilibrium, both of which are in threshold strate- all agents—the fact that it is common knowledge among gies. Moreover, by Milgrom and Roberts (1990), all proﬁles a nonzero fraction of the agents implies that it can func- of rationalizable strategies are “sandwiched” between these tion as a powerful enough coordination device and hence two equilibria. Therefore, Theorem 5 also provides a char- induce multiple equilibria. On the other hand, if the size acterization of the set of equilibria of the game, showing of the largest subset of agents with common knowledge of that a higher level of common knowledge, captured via a a signal does not grow at the same rate as the number of

larger value for c 2, implies a larger set of equilibria. Note agents, information is diverse and effectively private: any that if m 0. This assumption guarantees even under information structures that lead to multiplicity, that any strategy of a given agent is a mapping from the the set of equilibria still depends on the extent to which same subspace of ✓n to 801 19 regardless of the realization Downloaded from informs.org by [128.59.83.27] on 13 July 2016, at 09:20 . For personal use only, all rights reserved. agents observe common signals. To further clarify the role of the information structure, thus allowing us to compare of local information in determining the size of the equilib- agents’ strategies for different probability measures å.We 7 rium set, we next study large coordination games. have the following result:

Formally, consider a sequence of games 8G4n59n pa- Theorem 7. Consider a sequence of probability distribu- 2 rametrized by the number of agents, in which each agent i tions 8ås9sà 1 such that lims ås4Q⇤5 1, and let Gs can observe a single signal, and assume that the noise terms denote the= game whose information!à structure= is drawn

in the signals are mutually independent and normally dis- according to ås. Then for almost all equilibria of G⇤, there tributed with mean zero and variance ë 2 > 0. We have the exists a large enough s such that for all s>s, that equilib- ¯ ¯ following corollary to Theorem 5. rium is also a Bayesian Nash equilibrium of Gs. Dahleh et al.: Coordination with Local Information Operations Research 64(3), pp. 622–637, © 2016 INFORMS 631

The above result thus highlights that even if the informa- Acknowledgments tion structure of the game is not common knowledge among The authors are grateful to the editors and two anonymous refer- the agents, the equilibria of the perturbed game would be ees for helpful feedback and suggestions. They also thank Daron close to the equilibria of the original game. Consequently, Acemoglu, Marios Angeletos, Antonio Penta, Ali Shourideh, and it establishes that our earlier results on the multiplicity of participants at the Workshop on the Economics of Networks, Sys- equilibria are also robust to the introduction of a small tems, and Computation and the joint Workshop on Pricing and amount of uncertainty about the information structure of Incentives in Networks and Systems, where preliminary versions the game. of this work were presented (Dahleh et al. 2011, 2013); the Inter- As a last remark, we note that in the above result, we disciplinary Workshop on Information and Decision in Social did not allow agents to obtain extra, side signals about the Networks; and the Cambridge Area Economics and Computation Day. This research was partially supported by the Air Force Office realization of the information structure of the game. Never- of Scientific Research, Grant FA9550-09-1-0420. theless, a similar argument shows that the set of equilibria is robust to small amounts of noise in the information struc- Appendix. Proofs ture of the game, even if agents obtain such informative signals. Notation and Preliminary Lemmas We first introduce some notation and prove some preliminary lem- I 6. Conclusions mas. Recall that a pure strategy of agent i is a mapping s 2 ✓ó ió i ! 801 19, where Ii denotes i’s information set. Thus, a pure strategy Many social and economic phenomena (such as bank runs, I s can equivalently be represented by the set A ✓ó ió over which adoption of new technologies, social uprisings, etc.) exhibit i i ✓ agent i takes the risky action; i.e., an element of coordination. In this paper, we focused on

how the presence of local information channels can affect I A 8y ✓ó ió2s4y 5 191 the likelihood and possibility of coordination failures in i = i 2 i i = such contexts. In particular, by introducing local signals— where yi 4xr 5r I denotes the collection of the realized signals in = 2 i signals that are neither purely public nor private—to the the information set of agent i. Hence, a strategy proﬁle can equiv- canonical global game models studied in the literature, we alently be represented by a collection of sets A 4A 10001A 5 = 1 n showed that the set of equilibria depends on how informa- over which agents take the risky action. We denote the set of all

tion is locally shared among agents. Our results establish strategies of agent i by Ai and the set of all strategy profiles by A. that the coordination game exhibits multiple equilibria if Given the strategies of other agents, A i, we denote the expected É the information sets of a group of agents form an increas- payoff to agent i of the risky action when she observes yi by ing sequence of nested sets: the presence of such a filtration Vi4A i yi5. Thus, a best response mapping BRi2 A i Ai is É ó É ! removes agents’ strategic uncertainty about one another’s naturally defined as actions and leads to multiple equilibria. Ii BRi4A i5 8yi ✓ó ó2Vi4A i yi5>090 (7) We also provided a characterization of how the extent É = 2 É ó of equilibrium multiplicity is determined by the extent to Finally, we define the mapping BR2 A A as the product of the which subsets of agents have access to common informa- ! tion. In particular, we showed that the size of the equilib- best response mappings of all agents; that is, rium set is increasing in the standard deviation of the frac- BR4A5 BR14A 15 BRn4A n50 (8) tions of agents with access to the same signal. Our result = É ⇥···⇥ É thus shows that the set of equilibria shrinks as information The BR4 5 mapping is monotone and continuous. More formally, becomes more decentralized in the society. · we have the following lemmas: On the theoretical side, our results highlight the importance of local information channels in global games by Lemma 1 (Monotonicity). Consider two strategy profiles A and A0 such that A A0. (We write A A0 whenever A A0 for underscoring how the introduction of a signal commonly ✓ ✓ i ✓ i all i.) Then BR4A5 BR4A05. observed by even a small number of agents may lead to ✓ equilibrium multiplicity in an environment that would have Proof. Fix an agent i and consider yi BRi4A i5, which by defi- 2 É otherwise exhibited a unique equilibrium. On the more nition satisfies Vi4A i yi5>0. Because of the presence of strate-

Downloaded from informs.org by [128.59.83.27] on 13 July 2016, at 09:20 . For personal use only, all rights reserved. É ó applied side, our results show that incorporating local com- gic complementarities (Assumption 1), we have Vi4A0 i yi5 æ É ó munication channels between agents may be of ﬁrst-order Vi4A i yi5 and, as a result, yi BRi4A0 i5. É É ó 2 É importance in understanding many phenomena that exhibit Lemma 2 (Continuity). Consider a sequence of strategy pro- an element of coordination. In particular, they highlight the k k k 1 ﬁles 8A 9k such that A A + for all k. Then risk of abstracting from the intricate details of information 2 ✓ dissemination within such contexts. For instance, policy à k interventions that are not informed by the dispersion and BR4A 5 BR4Aà51 k 1 = penetration of different information channels among agents [= may turn out to be counterproductive. k where Aà kà 1 A . = = S Dahleh et al.: Coordination with Local Information 632 Operations Research 64(3), pp. 622–637, © 2016 INFORMS

k k Proof. Clearly, A Aà and by Lemma 1, BR4A 5 BR4Aà5 We have the following lemma: ✓ ✓ for all k. Thus, Lemma 3. The induced coordination game between the ` agents

in C has an equilibrium strategy profile B 4B110001B`5 such à k = BR4A 5 BR4Aà50 that ã I 4B R0 5>0 for all i C, where ã r is the Lebesgue ✓ó ió i i ✓ k 1 ✓ \r 2 = measure in ✓ and the equilibrium strategy profile R0 is given by [ (12) and (13). To prove the reverse inclusion, suppose that yi BRi4Aài5, which 2 É Proof. We denote y 8x 2x I 9. Recall that any strategy implies that Vi4Aài yi5>0. On the other hand, for any à and j r r j É ó profile of agents in C =can be represented2 as a collection of sets any observation profile 4y110001yn5, we have A 4A 10001A 5, where A is the set of realizations of y over = 1 ` j j k which agent j takes the risky action. On the other hand, recall lim ui4ai1s i4y i51 à5 ui4ai1sài4y i51 à51 k É É = É É !à that since I1 I2 I`, agent j observes the realizations of signals y for all✓ i ···j✓. Therefore, any strategy A of agent j can where sk and s are strategy profiles corresponding to sets Ak and i ∂ j à be recursively captured by indexing it to the realizations of sig- Aà, respectively. Thus, by the dominated convergence theorem, nals 4y110001yi 15. More formally, given the realization of signals É k 4y110001y`5 and the strategies 4A110001Aj 15, agent j takes the lim Vi4A i yi5 Vi4Aài yi51 (9) É k É É risky action if and only if !à ó = ó 4s110001sj 15 where we have used Assumption 4. Therefore, there exists r yj Aj É 1 r 2 2 large enough such that Vi4A i yi5>0, implying that yi where k É ó 2 à BR4A 5. This completes the proof. k 1 i É 4s110001si 15 = si ⌧8yi Ai É 9 k = 2 S Throughout the rest of the proofs, we let 8R 9k denote the 2 for all i j 1. In other words, j first considers the set of sig- sequence of strategy profiles defined recursively as ∂ É nals 4y110001yj 15 and, based on their realizations, takes the risky É4s 10001s 5 1 1 j 1 R 1 action if yj Aj É . Note that in the above notation, rather 2 I =ô than being captured by a single set A ó j ó, the strategy of agent k 1 k j ✓ R + BR4R 50 (10) j 1 ✓ 4s110001sj 15 = j is captured by 2 É different sets of the form Aj É . Thus, the collection of sets Thus, any strategy profile A that survives k rounds of iterated 4s 10001s 5 k 1 1 j 1 elimination of strictly dominated strategies must satisfy R + A. A 8Aj É 91 j `1 s 801 19` 1 (14) ✓ = ∂ ∂ 2 É Consequently, A survives the iterated elimination of strictly dom- capture the strategy profile of the agents 811 0 0 0 1 `9 in the induced inated strategies only if R A, where coordination game. Finally, to simplify notation, we let sj ✓ j 1 = 4s110001sj 5 and Sj i É1 si. à = = R Rk0 (11) With the above notation at hand, we now proceed with the P = k 1 proof of the lemma. Note that since the induced coordination = [ game is a game of strategic complementarities, it has at least one Proof of Theorem 1 equilibrium. In fact, the iterated elimination of strictly dominated strategies in (12) and (13) leads to one such equilibrium A R0. Without loss of generality, let C 811 0 0 0 1 `9 and I1 I2 I`, = ✓ ···✓ Let A, represented in (14), denote one such strategy profile, where= where ` æ 2. Before proving the theorem, we first define an “induced coordination game” between agents in C 811 0 0 0 1 `9 as we assume that agents take the safe action whenever they are follows: suppose that agents in C play the game described= in §2, indifferent between the two actions. except the actions of agents outside of C are prescribed by the For the strategy profile A to be an equilibrium, the expected payoff of paying the risky action to agent j has to be positive strategy profile R defined in (10) and (11). More specifically, the s110001sj 1 s110001sj 1 for all yj Aj É , and nonpositive for yj Aj É . In other payoff of taking the risky action to agent i C is given by 2 s110001sj 1 62 2 words, if y A É , then j 2 j k 1 s 4s É 5 è4k1à5 è k ⌧8yj Rj 91 à 1 ⇧6è41 S`1à5 yj 7⇣4y` A`1yk Ak sk 11 ˜ = + j C 2 411s 10001s 5 ˜ + ó 2 2 8 = ✓ 62 ◆ j 1 r 1 + É k 1 X X 4s É 5 ` yk Ak sk 02j03 k k k k j 8R0 9 as 62 8 = ó k s110001sj 1 2 whereas, on the other hand, if y A É , then j 62 j 1 k 1 R0 1 (12) s 4s É 5 =ô ⇧6è41 S`1à5 yj 7⇣4y` A`1yk Ak sk 11 k 1 k ˜ + ó 2 2 8 = R0 + BR4R0 51 (13) 401sj 110001sr 15 + É k 1 = X 4s É 5 yk Ak sk 02j

To understand the above expressions, note that in equilibrium, and agent j faces no strategic uncertainty regarding the behavior of s agents i j 1. However, to compute her expected payoff, agent j f 04y 5 ⇧6è41 S 1à5 y 7⇣4y D 1 ∂ 1 = ˜ + ` ó 1 ` 2 ` É s 80119` 12s 0 needs to condition on different realizations of signals 4sj 110001s`5. 2 É 1= + Xk 1 k 1 Furthermore, she needs to take into account that the realizations y D4s É 5 s 11y A4s É 5 s 02k>1 y 5 of such signals not only affect the belief of agents k>j about the k 2 k 8 k = k 62 k 8 k = ó 1 4k 15 s underlying state à but also determine which set As É they would ⇧6è4S 1à5 y 7⇣4y D 1 k + ˜ ` ó 1 ` 62 ` s 80119` 12s 0 use for taking the risky action. 2 É 1= Xk 1 k 1 Given the equilibrium strategy profile A whose properties we y D4s É 5 s 11y A4s É 5 s 02k>1 y 51 just studied, we now construct a new strategy profile D as fol- k 2 k 8 k = k 62 k 8 k = ó 1 lows. Start with agent ` at the top of the chain and define a set are simply determined by evaluating inequalities (15) and (16) for 4s110001s` 15 D É to be such that it satisfies ` agent 1. Notice that unlike the definition of f4y15, in the definition 4s 10001s 5 1 ` 1 of f 04y15 we have to start from s1 0 because agent 1 is not ⇧6è41 S`1à5 y`7>0 if y` D` É 1 = ˜ + ó 2 supposed to take the risky action whenever y1 D1. 4s110001s` 15 62 ⇧6è41 S 1à5 y 7 0 if y D É 0 We have the following lemma, the proof of which is provided ˜ + ` ó ` ∂ ` y ` s later on: Note that for any given s, the set D` is essentially an extension of s I` A to the whole space ✓ó ó in the sense that the two sets coincide Lemma 4. Let X 8y 2f4y5>09 and Y 8y 2f04y 5>09. Then ` = 1 1 = 1 1 with one another over the subset of signals realizations there exists a set B such that Y B X and ã I 4B „D 5>0, 1 ✓ 1 ✓ ✓ó 1ó 1 1 si 1 si 1 where ã I refers to the Lebesgue measure and „ denotes the 8y 2y A É if s 1 and y A É if s 090 ✓ó 1ó ` i 2 i i = i y i i = symmetric difference. Similarly, (and recursively) define the sets Dj as extensions of In view of the above lemma, we now use the set B1 to construct the sets Aj such that they satisfy the following properties: if yj 4s110001sj 15 2 a new strategy profile B represented as the following collections Dj É , then of sets: k 1 s 4s É 5 6è41 S 1à5 y 7 4y D 1y D s 11 4s110001sj 15 ⇧ ` j ⇣ ` ` k k k É ˜ + ó 2 2 8 = B 4B118Dj 92∂j∂`1s 80119` 1 50 411sj 110001sr 15 = 2 É + É k 1 X 4s É 5 yk Dk sk 03k>j yj 5 In other words, agent 1 takes the risky action if and only if y B . 62 8 = ó 1 2 1 4sk 15 The rest of the agents follow the same strategies as prescribed ⇧6è4S 1à5 y 7⇣4y Ds 1y D É s 11 + ˜ ` ó j ` 62 ` k 2 k 8 k = by strategy profile D except for the fact that rather than choos- 411sj 110001sr 15 + É k 1 ing their D sets based on whether y belongs to D or not, they X 4s É 5 1 1 yk Dk sk 03k>j yj 5>03 (15) choose the sets based on whether y B or not. By construc- 62 8 = ó 1 2 1 s110001sj 1 tion, the strategy profile above is an equilibrium. Note that since and, on the other hand, if yj Dj É , then 62 Y B1 X, agent 1 is best responding to the strategies of all other k 1 j 1 s 4s É 5 ✓ ✓ 4s É 5 ⇧6è41 S`1à5 yj 7⇣4y` D`1yk Dk sk 11 agents. Furthermore, given that for j 1 we have that Bj ˜ + ó 2 2 8 = j 1 6= = 401sj 110001sr 15 4s É 5 + É k 1 Dj , inequalities (15) and (16) are also satisfied, implying that X 4s É 5 yk Dk sk 03k>j yj 5 agent j also best responds to the behavior of all other agents, 62 8 = ó 4sk 15 hence, completing the proof. É ⇧6è4S 1à5 y 7⇣4y Ds 1y D É s 11 + ˜ ` ó j ` 62 ` k 2 k 8 k = 401sj 110001sr 15 We now present the proof of Theorem 1. +X É k 1 4s É 5 Proof of Theorem 1. Lemma 3 establishes that the induced yk Dk sk 03k>j yj 5∂00 (16) 62 8 = ó coordination game between agents in C has an equilibrium Note that given that the sets D are simply extensions of sets A, B 4B 10001B 5 that is distinct from the equilibrium R0 derived = 1 ` by construction, the strategy profile 4D110001D`5 also forms a via iterated elimination of strictly dominated strategies in (12) Bayesian Nash equilibrium. In fact, it essentially is an alternative and (13). We now use this strategy profile to construct an equi- representation of the equilibrium strategy profile A. librium for the original n-agent game. Now, consider the expected payoff to agent 1, at the bottom of Define the strategy profile B 4B110001B`1R` 1 0001Rn5 for the the chain. In equilibrium, agent 1 should prefer to take the risky ˜ = + n-agent game, where Rj is defined in (10) and (11). Lemma 3 action in set D1. In other words, immediately implies that B BR4B5. Define the sequence of strat- k ˜ ✓ ˜ f4y 5>0 y D (17) egy profiles 8H 9k as 1 8 1 2 1 2 f 04y 5 0 y D 1 (18) 1 Downloaded from informs.org by [128.59.83.27] on 13 July 2016, at 09:20 . For personal use only, all rights reserved. 1 ∂ 1 1 H B 8 62 = ˜ k 1 k where H + BR4H 50 = f4y 5 6è41 S 1à5 y 7 4y Ds1 1 ⇧ ` 1 ⇣ ` ` Given that H 1 H 2, Lemma 1 implies that H k H k 1 for all k. = ` 1 ˜ + ó 2 + s 80119 É 2s1 1 ✓ k ✓ 2 = à X k 1 k 1 Thus, H k 1 H is well defined and by continuity of the BR 4s É 5 4s É 5 = = yk Dk sk 11yk Ak sk 02 1

Proof of Lemma 4. The definitions of sets X and Y imme- Proof of Proposition 2. By definition, diately imply that Y D1 X. On the other hand, the second ✓ ✓ 1 1 part of Assumption 1 implies that ã I1 4X Y5>0. Consequently, ✓ó ó Vi4R i yi5 2 6⇣4yj Rj yi5 ⇣4yl Rl yi57 2 4xj xl5 \ É ó = 2 ó + 2 ó É + there exists a set B1 distinct from D1 such that Y B X and ✓ ✓ 1 ã I 4B „D 5>0. This completes the proof. 2 6⇣4Vj 4R j yj 5>0 yi5 ✓ó 1ó 1 1 É = É ó ó 1 ⇣4Vl4R l yl5>0 yi57 2 4xj xl51 Proof of Proposition 2 + É ó ó É + k Recall the sequence of strategy profiles R and its limit R defined where R BR4R5. By Lemma 5, in (10) and (11), respectively. By Lemma 2, R BR4R5, which = = implies that yi Ri if and only if Vi4R i yi5>0. We have the 2 É ó 1 h4xl5 4xj xl5/2 following lemma. Vi4R i yi5 Í É + É ó = 2 ëp3/2 Lemma 5. There exists a strictly decreasing function h2 ✓ ✓ ✓ ◆ ! 1 h4xj 5 4xj xl5/2 1 such that Vi4R i yi5 0 if and only if xj h4xl5, where yi Í É + 4x x 50 É ó = = = + 2 ëp3/2 É 2 j + l 4xj 1xl5 and i, j, and l are different. ✓ ◆ Proof. Using an inductive argument, we first prove that for all k k Setting Vi4R i yi5 0 and any solution yi 4xj 1xl) of (i) Vi4R i yi5 is continuously differentiable in yi, É ó = = Ék ó Vi4R i yi5 0 satisfies h4xl5 xj and h4xj 5 xl imply that (ii) Vi4R i yi5 is strictly decreasing in both arguments, É ó = = = É ó xj xl 1. Therefore, 4xj 1xl5 yi, and + = = k (iii) °Vi4R i yi5/°xj 61/21Q7, ó É ó ó2 2 1 1 where j i and Ri 4xj 1xl5 ✓ 2 2 4xj xl5< 2 0 6= = 2 + ëp3è 1 Q + 0 Hence, in any strategy profile that survives iterated elimination of = 2ëp3è 2 É strictly dominated strategies, an agent takes the risky action when- The above clearly hold for k 1 because V 4 y 5 4x x 5/2. = i ôó i =É j + l ever the average of the two signals she observes is less than 1/2. Now suppose that (i)–(iii) are satisfied for some kæ1. By the 8 A symmetrical argument implies that in any strategy profile that implicit function theorem, there exists a continuously differen- survives the iterated elimination of strictly dominated strategies, tiable function h 2 ✓ ✓ such that k ! the agent takes the safe action whenever the average of her signals k Vi4R i xj 1hk4xj 55 01 is greater than 1/2. Thus, the game has an essentially unique ratio- É ó = nalizable strategy profile and hence an essentially unique Bayesian and 2Q∂hk0 4xj 5∂ 1/42Q5. (Given the symmetry between the É É Nash equilibrium. three agents, we drop the agent’s index for function hk.) The É k monotonicity of hk implies that Vi4R i yi5>0 if and only if xj < É ó hk4xl5. Therefore, Proof of Proposition 3 k 1 Vi4R +i yi5 It is sufficient to show that there exists an essentially unique equi- É ó 1 k 1 k 1 1 librium in monotone strategies. Note that because of symmetry, 2 6⇣4yj Rj + yi5 ⇣4yl Rl + yi57 2 4xj xl5 = 2 ó + 2 ó É + the strategies of agents 1 and 3 in the extremal equilibria of the 1 6⇣4x

° k 1 First suppose that í>1/2. Then the best response of agent 2 is Vi4R +i yi5 °x É ó to take the risky action if either (i) x x 1 or (ii) x 1x í j 1 + 2 ∂ 1 2 ∂ hold. On the other hand, for í to correspond to the threshold of an 1 h 4x 5 4x x 5/2 î k l É j + l equilibrium strategy of agent 1, her expected payoff of taking the =É2ëp6 ëp3/2 ✓ ◆ risky action has to be positive whenever x1 <í. In other words, 2h0 4x 5 1 h 4x 5 4x x 5/2 1 k j É î k j É j + l 1 Downloaded from informs.org by [128.59.83.27] on 13 July 2016, at 09:20 . For personal use only, all rights reserved. + p ëp3/2 É 2 1 ⇣4x <í x 5 2ë 6 ✓ ◆ 2 2 ó 1 which guarantees 1 6⇣4x x 1 x 5 ⇣41 x x í x 57 x + 2 1 + 2 ∂ ó 1 + É 1 ∂ 2 ∂ ó 1 æ 1 1 1 2Q ° k 1 1 + ∂ Vi4R +i yi5∂ 1 É 2 É 2ëp3è °xj É ó É 2 for all x1 <í. As a result, for any x1 <í, we have completing the inductive argument because 1 1 2Q 1 í x 1 1 2x + Q0 Í É 1 Í É 1 2 + 2ëp3è = 2 p + 2 p ✓ ë 2 ◆ ✓ ë 2 ◆ Now using (9) and the implicit function theorem once again com- 1 í x1 1 2x1 pletes the proof. Í É Í É æx11 É + 2 p É p  ✓ ë 2 ◆ ✓ ë 2 ◆ Dahleh et al.: Coordination with Local Information Operations Research 64(3), pp. 622–637, © 2016 INFORMS 635

which simpliﬁes to Denote the threshold corresponding to the strategy of an agent

who observes signal xr with ír . The profile of threshold strategies í x 1 corresponding to thresholds 8ír 9 is a Bayesian Nash equilibrium Í É æx10 p of the game if and only if for all r, ✓ ë 2 ◆ Taking the limit of the both sides of the above inequality as n íj xr cj Í É xr ∂0 for xr >ír x í implies that í 1/2. This, however, contradicts the original n 1 ëp2 É 1 ∂ É j r ✓ ◆ assumption" that í>1/2. A similar argument would also rule out X6= nc 1 n í x the case that í<1/2. Hence, í corresponds to the threshold of r j r É cj Í É xr æ0 for xr <ír 1 an equilibrium strategy of agents 1 and 3 only if í 1/2. As a n 1 + n 1 j r ëp2 É É É 6= ✓ ◆ consequence, in the essentially unique equilibrium of= the game, X where the first (second) inequality guarantees that the agent has agent 2 takes the risky action if and only if x1 x2 <1. This proves the first part of the proposition. The proof of+ the second part is no incentive to deviate to the risky (safe) action when the signal she observes is above (below) threshold í . Taking the limit immediate. É r as xr converges to ír from above in the first inequality implies that in any symmetric, Bayesian Nash equilibrium of the game in Proof of Proposition 4 threshold strategies, One again, we can simply focus on the set of equilibria in thresh- 4n 15í nHc1 old strategies. Let í1, í3, and í4 denote the thresholds of agents 1, É æ 3, and 4, respectively. Note that by symmetry, í3 í4. Also let m m function f denote the strategy of agent 2, in the sense= that agent 2 where í 6í110001ím7 is the vector of thresholds and H ✓ ⇥ is 2 a matrix= with zero diagonal entries, and off-diagonal entries2 given takes the risky action if and only if x

1 1 1 n 2 ⇧6è4k1à5 x27 3 ⇣4x1 <í1 x25 3 ⇣4x1 í3, which We ﬁrst prove that if the size of the largest set of agents with a implies common observation grows sublinearly in n, then asymptotically as n , the game has a unique equilibrium. We denote the 1 !à 1 1 í1 í3 1 f2É 4í35 í3 vector of the fractions of agents observing each signal by c4n5 to í2 ∂ Í É Í É ∂í20 (20) É 3 3 p + 3 p ✓ ë 2 ◆ ✓ ë 2 ◆ make explicit the dependence on the number of agents n. Note that if the size of the largest set of agents with a common observation It is easy to verify that the triplet 4í11f21í35 that satisﬁes condi- grows sublinearly in n, then tions (19) and (20) simultaneously is not unique. É lim c4n5 01 n ò òà = Proof of Theorem 5 !à

Downloaded from informs.org by [128.59.83.27] on 13 July 2016, at 09:20 . For personal use only, all rights reserved. where z is the maximum element of vector z. On the other As already mentioned, the Bayesian game under consideration hand, byò ò Hölder’sà inequality, is monotone supermodular in the sense of Van Zandt and Vives (2007), which ensures that the set of equilibria has well-deﬁned 2 c4n5 2 ∂ c4n5 1 c4n5 0 maximal and minimal elements, each of which is in threshold ò ò ò ò ·ò òà strategies. Moreover, by Milgrom and Roberts (1990), all proﬁles Given that the elements of vector c4n5 add up to one,

of rationalizable strategies are “sandwiched” between these two limn c4n5 2 0. Consequently, Theorem 5 implies that thresh- equilibria. Hence, to characterize the set of rationalizable strate- olds!àí4nò5 andòí4=n5 characterizing the set of rationalizable strategies, it sufﬁces to focus on threshold strategies and determine the gies of the game¯ of size n satisfy smallest and largest thresholds that correspond to Bayesian Nash equilibria of the game. lim í4n5 lim í4n5 1/20 n n !à = !à ¯ = Dahleh et al.: Coordination with Local Information 636 Operations Research 64(3), pp. 622–637, © 2016 INFORMS

Thus, asymptotically, the game has an essentially unique Bayesian 4. We relax this assumption in §5.2. Nash equilibrium. 5. As already mentioned, this is a consequence of the underlying To prove the converse, suppose that the size of the largest game being a Bayesian game with strategic complementarities, set of agents with a common observation grows linearly in n, and hence the extremal equilibria are monotone in types. For a which means that c4n5 remains bounded away from zero as detailed study of Bayesian games with strategic complementari- ò òà n . Furthermore, the inequality c4n5 ∂ c4n5 2 immedi- ties, see Van Zandt and Vives (2007). !à ò òà ò ò ately implies that c4n5 also remains bounded away from zero 6. Note that since the information set of any given agent is a sub- ò ò2 as n . Hence, by Theorem 5, set of the finitely many realized signals, å is a discrete probability !à measure with finite support. limsupí4n5<1/2 7. We thank an anonymous referee for suggesting this result. n !à 8. See, for example, Hadamard’s global implicit function theorem liminf í4n5>1/21 n in Krantz and Parks (2002). !à ¯ guaranteeing asymptotic multiplicity of equilibria as n . !à É Proof of Theorem 7 References Pick a nonextremal equilibrium in the game with information Acemoglu D, Dahleh M, Lobel I, Ozdaglar A (2011) Bayesian learning structure Q 8I 10001I 9 and denote it by 4A 10001A 5, according in social networks. Rev. Econom. Stud. 78(4):1201–1236. ⇤ 1 n 1 n Ali AH (2011) The power of social media in developing nations: New to which agent= i takes the risky action if and only if y A .We i 2 i tools for closing the global digital divide and beyond. Harvard denote the expected payoff of taking the risky action to agent i Human Rights J. 24(1):185–219. under information structure Q⇤ by Vi⇤4A i yi5. Given that each Angeletos G-M, Werning I (2006) Crises and prices: Information signal is observed by at least two agents andÉ ó that the strategy pro- aggregation, multiplicity, and volatility. Amer. Econom. Rev. 96(5): 1720–1736. file 4A110001An5 is a nonextremal equilibrium, there exists Ö>0 such that Angeletos G-M, Hellwig C, Pavan A (2006) Signaling in a global game: Coordination and policy traps. J. Political Econom. 114(3):452–484. Angeletos G-M, Hellwig C, Pavan A (2007) Dynamic global games of Vi⇤4A i yi5>Ö for all yi Ai É ó 2 regime change: Learning, multiplicity, and the timing of attacks. Econometrica 75(3):711–756. Vi⇤4A i yi5< Ö for all yi Ai0 É ó É 62 Argenziano R (2008) Differentiated networks: Equilibrium and efficiency. RAND J. Econom. 39(3):747–769. Now consider the sequence of games 8Gs9 in which the information structure is drawn according to the sequence of probability Carlsson H, van Damme E (1993) Global games and equilibrium selection. Econometrica 61(5):989–1018. distributions 8å 9. It is easy to see that in the game indexed by s Chwe MS-Y (2000) Communication and coordination in social networks. s, agent i’s expected payoff of taking the risky action, when all Rev. Econom. Stud. 67(1):1–16. other agents follow strategies A i, is given by Dahleh MA, Tahbaz-Salehi A, Tsitsiklis JN, Zoumpoulis SI (2011) On É global games in social networks of information exchange. Proc. Sixth 4s5 Q Vi 4A i yi5 ås4Q5Vi 4A i yi51 Workshop Econom. Networks, Systems, and Comput. (NetEcon), ACM É ó = Q É ó Federated Comput. Res. Conf., San Jose, CA. X Dahleh MA, Tahbaz-Salehi A, Tsitsiklis JN, Zoumpoulis SI (2013) Coor- where we denote the expected payoff of taking the risky action Q dination with local information. ACM Performance Eval. Rev., Proc. to agent i under information structure Q by Vi 4A i yi5. Since joint Workshop on Pricing and Incentives in Networks and Systems É ó lims ås4Q⇤5 1, it is immediate that the above expression con- (W-PIN NetEcon), ACM Sigmetrics, Pittsburgh, PA. !à = + verges to Vi⇤4A i yi5. Dasgupta A (2007) Coordination and delay in global games. J. Econom. É ó Theory 134(1):195–225. Consequently, given that Vi⇤4A i yi5 is uniformly bounded away from zero for all y , there existsÉ ó a large enough s such that Diamond DW, Dybvig PH (1983) Bank runs, deposit insurance, and li- i quidity. J. Political Econom. 91(3):401–419. for s>s, the expected payoff of taking the risky action¯ in game ¯ Edmond C (2013) Information manipulation, coordination and regime Gs when other agents follow strategies A i is strictly positive if Rev. Econom. Stud. É change. 80(4):1422–1458. y A , whereas it would be strictly negative if y A . Thus, for Galeotti A, Ghiglino C, Squintani F (2013) Strategic information trans- i 2 i i 62 i any s>s, the strategy profile 4A110001An5 is also a Bayesian Nash mission networks. J. Econom. Theory 148(5):1751–1769. ¯ Galeotti A, Goyal S, Jackson MO, Vega-Redondo F, Yariv L (2010) Net- equilibrium of game Gs. É work games. Rev. Econom. Stud. 77(1):218–244. Goldstein I, Pauzner A (2005) Demand deposit contracts and the proba- Endnotes bility of bank runs. J. Finance 60(3):1293–1328. 1. See Morris and Shin (2003) for more examples. Golub B, Jackson MO (2010) Naïve learning in social networks and the 2. Note that strategic uncertainty is distinct from fundamental wisdom of crowds. Amer. Econom. J. Microeconomics 2(1):112–149.

Downloaded from informs.org by [128.59.83.27] on 13 July 2016, at 09:20 . For personal use only, all rights reserved. Jadbabaie A, Molavi P, Tahbaz-Salehi A (2013) Information heterogeneity uncertainty. Whereas fundamental uncertainty simply refers to the and the speed of learning in social networks. Columbia Business agents’ uncertainty about the underlying, payoff-relevant state of School Working Paper 13-28, Columbia University, New York. the world, strategic uncertainty refers to their uncertainty concern- Jadbabaie A, Molavi P, Sandroni A, Tahbaz-Salehi A (2012) Non- ing the equilibrium actions of one another. Bayesian social learning. Games Econom. Behav. 76(1):210–225. Krantz SG, Parks HR (2002) The Implicit Function Theorem: History, 3. With some abuse of notation, we use Ii to denote both the set of actual signals observed by agent i as well as the set of Theory, and Applications, 1st ed. (Birkhäuser, Boston). Mathevet L (2014) Beliefs and rationalizability in games with comple- indices of the signals observed by that agent. Zoumpoulis (2014) mentarities. Games Econom. Behav. 85:252–271. discusses the special case where the agents are organized in a Milgrom P, Roberts J (1990) Rationalizability, learning, and equilib- network, such that each agent observes a signal that pertains to rium in games with strategic complementarities. Econometrica 58(6): her and the signals of her neighbors. 1255–1277. Dahleh et al.: Coordination with Local Information Operations Research 64(3), pp. 622–637, © 2016 INFORMS 637

Morris S, Shin HS (1998) Unique equilibrium in a model of self-fulfilling and Director-designate of a new organization addressing major currency attacks. Amer. Econom. Rev. 88(3):587–597. societal problems through unification of the intellectual pillars Morris S, Shin HS (2000) Rethinking multiple equilibria in macroeco- in Statistics, Information and Decision Systems, and Human and nomics. NBER Macroeconomics Ann. 15:139–161. Morris S, Shin HS (2003) Global games: Theory and applications. Dewa- Institution Behavior. tripont M, Hansen LP, Turnovsky SJ, eds. Advances Econom. Econo- Alireza Tahbaz-Salehi is Daniel W. Stanton Associate Pro- metrics. Proc. Eighth World Congress of the Econometric Soc., Vol. 1 fessor of Business at Columbia Business School. His research (Cambridge University Press, Cambridge, UK), 56–114. focuses on the implications of network economies for business Morris S, Shin HS, Yildiz M (2015) Common belief foundations of global cycle fluctuations and financial stability. games. William S. Dietrich II Economic Theory Center Research Paper No. 69, Princeton University, Princeton, NJ. John N. Tsitsiklis is the Clarence J. Lebel Professor of Elec- Van Zandt T, Vives X (2007) Monotone equilibria in Bayesian games of trical Engineering and Computer Science at MIT. He chairs the strategic complementarities. J. Econom. Theory 134(1):339–360. Council of the Harokopio University in Athens, Greece. His Vives X (2008) Supermodularity and supermodular games. Durlauf SN, research interests are in systems, optimization, communications, Blume LE eds. The New Palgrave: A Dictionary of Economics (Pal- control, and operations research. He has been a recipient of an grave Macmillan, New York). Vives X (2014) Strategic complementarity, fragility, and regulation. Rev. Outstanding Paper Award from the IEEE Control Systems Society Financial Stud. 27(12):3547–3592. (1986), the M.I.T. Edgerton Faculty Achievement Award (1989), Weinstein J, Yildiz M (2007) A structure theorem for rationalizability the Bodossakis Foundation Prize (1995), an ACM Sigmetrics with application to robust predictions of refinements. Econometrica Best Paper Award (2013), and a co-recipient of two INFORMS 75(2):365–400. Computing Society prizes (1997, 2012). He is a member of the Zoumpoulis SI (2014) Networks, decisions, and outcomes: Coordination with local information and the value of temporal data for learning National Academy of Engineering. In 2008, he was conferred the influence networks. Unpublished doctoral dissertation, Massachusetts title of Doctor honoris causa, from the Université Catholique de Institute of Technology, Cambridge, MA. Louvain. Spyros I. Zoumpoulis is an assistant professor of Decision Munther A. Dahleh is the William A. Coolidge Professor of Sciences at INSEAD. He is broadly interested in problems at the Electrical Engineering and Computer Science at MIT. He is cur- interface of networks, learning with data, information, and deci- rently the acting director of the Engineering Systems Division sion making.

CORRECTION In the print version of this article, “Coordination with Local Information” by Munther A. Dahleh, Alireza Tahbaz- Salehi, John N. Tsitsiklis, and Spyros I. Zoumpoulis (Operations Research, Vol. 64, No. 3, pp. 622–637), the research article was mislabeled as a technical note. “Technical Note” has been removed from the title in the online version and an erratum will be printed in the July–August issue. Downloaded from informs.org by [128.59.83.27] on 13 July 2016, at 09:20 . For personal use only, all rights reserved.