Costly word-of-mouth learning in networks

Transcription

1 Costly word-of-mouth learning in networks By Daniel C. Opolot and Théophile T. Azomahou This paper develops a framework for an information externality game in which individuals interacting through a network choose an optimal time to act. Individuals exchange information through word-of-mouth communication and waiting to become more informed is beneficial but costly. We characterize equilibrium exit times and asymptotic learning. The main result is that asymptotic learning with rational agents obtains in networks that exhibit exponentially growing neighbourhood sizes and/or bounded diameter. Asymptotic learning by naïve agents on the other hand obtains in networks that are asymptotically balanced with the second largest eigenvalue bounded away from one. JEL: C72, D82, D83, D85 Keywords: Information externalities, word-of-mouth communication, social learning, networks, exit times. I. Introduction This paper studies the process of social learning in networks in which: (a) The process of information exchange involves word-of-mouth communication, whereby individuals communicate the summary statistic (posterior belief) of their private information. (b) Gathering information is costly and Opolot: Maastricht University School of Business and Economics, Tongersestraat 53, 6211 LM Maastricht, the Netherlands, d.opolot@maastrichtuniversity.nl. Azomahou: Maastricht University UNU-MERIT, Maastricht, the Netherlands, azomahou@merit.unu.edu. We thank Dimitri Vayanos for the constructive comments on an earlier version of this paper that has circulated under the title Belief dynamics in communication networks. This research was supported by UNU-MERIT. The usual disclaimer applies. 1

2 2 OCTOBER 2015 individuals face a choice between waiting to become more informed, which increases confidence in their opinion, and taking an action early enough to avoid costs associated with waiting. Such costs can result from either impatience by agents due to urgency of the decision at hand, or material costs associated to loss of potential gains. Many social decisions, ranging from product and investment choices to political and occupations decisions, fit into this framework as demonstrated by the following motivational examples The first example concerns public programs such as public health. Consider a society of individuals to whom a new vaccination program has been introduced. The willingness to vaccinate depends on individual perceptions or beliefs about vaccine safety and its consequences for disease control. Each individual starts with a prior belief about the benefits of the vaccination program. If the level of confidence in their belief is low, they will wait to receive more information from those in their social network, which includes media sources. Once their confidence reaches a sufficient level, they then decide on whether to participate in the vaccination program. There exists empirical evidence in support of social learning in such settings. Henrich and Holmes (2011) conduct an empirical study of online discussions regarding the vaccination program for the 2009 H1N1 pandemic in Canada and find that they reflected actual public opinion and hence decision making process about vaccination. 1 A second example of social decisions that fit into our framework is product, technology and services choices by consumers. Consider a group of individuals contemplating the adoption of one among new insurance services. Each individual will possess a prior belief about the benefits of avail- 1 Miguel et al. (2003) examine social learning using data from a program that promoted use of deworming medicine in Kenyan schools. They find positive effects of information exchange through social networks, which in turn influenced individual beliefs and decisions.

3 COSTLY WORD-OF-MOUTH LEARNING 3 able options. Only those with an infinite level of confidence in their belief will make a choice initially; otherwise each individual would gather information regarding the available options until their confidence is sufficiently high. Cai et al. (2015) study the influence of social networks on weather insurance adoption and the mechanisms through which social networks operate in rural China. They find positive evidence of social network effect, which is not driven by the diffusion of information on purchase decisions, but instead by the diffusion of knowledge about insurance. Their result thus also highlights evidence of word-of-mouth communication rather than observation of others actions. 2 The third example is political behaviour in elections. Consider an electoral process with two or more candidates. Each voter initially possess prior beliefs regarding each candidate. As the campaign period progresses, voters gather/receive more information via both traditional and internet media sources as well as direct word-of-mouth communication with friends and colleagues. Some voters who start with high levels of confidence in their opinion about particular candidates make final choice even before the end of campaign period; these would typically be voters who pledge allegiance to a particular political party or ideologies. The rest of the voters would continue gathering information until their confidence is sufficiently high. In both cases however, voters final choice depends on their belief resulting from the process of information exchange. 3 The framework we propose to study the above mentioned social deci- 2 Duflo and Saez (2002, 2003) provide evidence of social networks (that is information received from social connections) effects in retirement decisions by employees in a university. A similar study and result is established by Sorensen (2006). There is also strong evidence of word-of-mouth communication and social network effects in adoption of new technologies (Bandiera and Rasul, 2006; Conley and Udry, 2010) and investment decisions (Hong et al., 2004). 3 Tumasjan et al. (2010) investigate whether Twitter is used as a forum for political deliberation and whether online messages on Twitter validly mirror offline political sentiment in the case of German federal election. They find that the mere number of messages mentioning a party reflects the election result and that the content of Twitter messages plausibly reflects the offline political landscape.

4 4 OCTOBER 2015 sions can be summarized as follows. Agents interactions are governed by a network whose edges depict the directions of information exchange. Here, the network nodes do not have to be homogeneous; they can be a mix of individuals making decisions and news channels that are additional sources of information. Each agent is endowed with prior beliefs about the subject at hand. We distinguish between the cases in which prior beliefs of neighbours are observable from that in which they are not. In addition to possessing a prior belief, each agent also observes a private signal that is informative about the state of nature. Prior beliefs together with private signals form agents private beliefs. Agents then share information through word-of-mouth communication by sharing their posterior beliefs. When incorporating new information into their beliefs, agents follow a Bayesian updating rule. We distinguish between Bayesian learning by rational from that by naïve agents. Rational agents are capable of distinguishing between new and old information from their neighbours messages while naïve agents are incapable of such complex process and hence simply take weighted average of neighbours messages. We make this distinction since our focus is not on which among the two learning mechanisms best describes reality. Moreover, the empirical evidence in this regard is not conclusive. A field experiment by Möbius et al. (2010) points to evidence of Bayesian rational learning in social networks while a lab experiment by Chandrasekhar et al. (2012) indicates that individuals tend to be naïve in their belief updating. Waiting to collect information is costly but necessary to increase ones confidence in their belief. Each agent is then faced with two choices at each period, wait and gather more information or stop gathering information and take an irreversible action. The above framework can be viewed as an information externality game whose equilibrium behaviour consists of agents choosing optimal exit time; that is the time to stop gathering information and take an irreversible

5 COSTLY WORD-OF-MOUTH LEARNING 5 action. We first characterize the nature of exit times given the network topology and model parameters: prior belief and signal precisions, and discount rate. We then focus on asymptotic learning, that is whether a large population of agents make correct choices at the end of the learning process. Asymptotic learning both with rational and naïve agents depends on the network topology. We provide a direct relationship between exit times, which is a function of model parameters, and parameters associated with the network topology, such as network diameter and eigenvalue spectrum. The main findings of this paper can be summarized as follows. Asymptotic learning with rational agents occurs in networks with either exponentially growing neighbourhood sizes and/or bounded network diameter. An example of the former is tree network families, which generally exhibit a characteristic of hierarchies, while and example of the later is Erdös-Rényi family of random networks in which the average degree is unbounded. This result is true for both the case in which prior beliefs of neighbours are observable and when they are not. Asymptotic learning with naïve agents occurs under three conditions. (a) Prior beliefs must be informative of the true state of nature. (b) The convergence time to a consensus must be asymptotically bounded. This implies that the second largest eigenvalue of the associated normalized adjacency matrix of the network must be asymptotically bounded away from one. (c) The network must be asymptotically balanced. A balanced network is that in which the total influence each agent exerts on her first-order neighbours is equal to the total influence her neighbours exert on her. Asymptotic balancedness then implies that a network becomes balanced for a large population. Among networks topologies satisfying both conditions (b) and (c) is the Erdös-Rényi family of random networks. Compared to conditions for asymptotic learning by rational agents, fewer network topologies support asymptotic learning by naïve agents. This findings underscore

6 6 OCTOBER 2015 the superiority of rational agents in aggregating decentralized information when compared to naïve agents. In addition to the empirical literature discussed above, this paper is related to other strands of existing literature in the following ways. First, it is related to the literature on Bayesian learning with rational agents in social networks (e.g. Gale and Kariv (2003), Rosenberg et al. (2009) and Mueller-Frank (2013)). Just as we do in this paper, these papers also consider simultaneous communication among agents. 4 They however consider social learning whereby agents observe actions of their neighbours. And the primary focus of analysis is on the uniformity and local indifference in the actions chosen by agents at equilibrium. On a contrary, our interest is in costly word-of-mouth communication and we focus on asymptotic learning. Secondly, this paper is closely related to models of information percolation in which rational agents exchange private signals. The notable contributions are Duffie et al. (2009) and Acemoglu et al. (2014). In the model of Duffie et al. (2009), a continuum of agents are randomly matched according to search intensities determined by individual specific effort to gather information. At each period, the set of agents that happen to meet share their signals. They characterize equilibrium search intensities as a function of information gathered. The model of Acemoglu et al. (2014) is similar to ours in that interactions are governed by a social network and communication is costly. But as in the case of Duffie et al. (2009), agents exchange signals rather than communicate their beliefs. They obtain the interesting result that, learning in large societies occurs in the presence of information hubs, which receive and distribute a large amount of information. 4 There also exists a literature on sequential Bayesian learning in which agents make a decision once in a lifetime in an exogenously predefined order. When it is an agent s turn to act, he observes the history of actions of all agents that acted before him. The primary concern of this literature is establishing conditions under which informational cascades and herds behavior occurs. The main contributions are Banerjee (1992), Bikhchandani et al. (1992), Smith and Sorensen (2000) and Acemoglu et al. (2011).

7 COSTLY WORD-OF-MOUTH LEARNING 7 In addition to the fundamental difference that we consider word-of-mouth communication, which enables us to study the effect of prior belief uncertainty on asymptotic learning, our analysis is explicit in characterizing the relationship between equilibrium exit times and general network properties. In doing so, we provide concrete analytical tools for characterizing a range of parameter value within which asymptotic learning occurs. There are two main contributions in this regard. (a) The analytical tools provided pave a way for empirical analysis in such social decision environments. We demonstrate this possibility with three examples of empirical social networks from the Stanford Large Network Data set Collection (Leskovec and Krevl, 2014). (b) Our analytical tools make it feasible to establish conditions for asymptotic learning within any given network topology. For example, we find that in additions to information hubs being essential, asymptotic learning also occurs in other forms of networks that assume a random and/or regular structures, such as the Erdös-Rényi family of random networks and the regular-tree networks respectively. Moreover, we show that the desirable properties for asymptotic learning with rational agents are that either the neighbourhood sizes grow exponentially and/or the network diameter is bounded. Thirdly, this paper is related to the literature on word-of-mouth learning by naïve agents. Notable contributions include Ellison and Fudenberg (1995) and Banerjee and Fudenberg (2004) who consider the case in which agents observe and take weighted average of their neighbours payoffs, and DeGroot (1974), Demarzo et al. (2003) and Golub and Jackson (2010) in which agents observe and take weighted average of neighbours opinions. The main outcome in both cases is that learning converges to a consensus in choices made (for the former models) and in opinions (for the later models). This paper is more closely related to the later set of models and in particular Golub and Jackson (2010) who also study learning in large societies but

8 8 OCTOBER 2015 in which communication is costless. They find an interesting result that, asymptotic learning in such a framework occurs in networks where the total influence of the most influential agent decays with population size. In comparison to this paper, the first fundamental difference is that we consider costly word-of-mouth communication, that is when the discount rate r 0. The framework in Golub and Jackson (2010) thus corresponds to the case in which r = 0. In addition, we develop alternative analytical methods for characterizing asymptotic learning that directly relate to the parameters of first-order influence among agents. The measure we developed, the balancedness of a network, is computationally less cumbersome and analytically tractable making it suitable for empirical analysis. Secondly, and perhaps most importantly, our framework takes into account convergence rates in examining asymptotic learning as reflected in exit times. We provide examples of cases where asymptotic learning obtains when r = 0 but the convergence time of learning is asymptotically infinite, which makes it an important factor to take into account in such analysis. Finally, our framework is also related to the literature on exit games, in which players learn from their own private experiences and by observing the actions of other players. Players then choose an optimal time to exit play. This framework has been applied to model investment decisions by firms and notable contributions are Caplin and Leahy (1994), Bulow and Klemperer (1994), Chamley and Gale (1994), and Murto and Välimäki (2011). Although we have focused on word-of-mouth communication as opposed to observation of others actions, our framework has a potential of contributing to the aforementioned literature. The predictions of these models are that equilibrium behaviour exhibits two extremes in which either all agents participate in the game or massively exit. Such predictions suitably capture the phenomena observable in market crushes. Our model however, is that in which the likelihood to exit the game depends on the level of confidence

9 COSTLY WORD-OF-MOUTH LEARNING 9 in ones belief. Starting from an initial state where agents possess heterogeneous information precisions, it is feasible to obtain intermediate (interior) equilibria in which some agents exit while others continue participating in the game. Hence, providing a richer understanding of collective behaviour in such decision environments. The remainder of the paper is organized as follows. Section II outlines a framework of costly word-of-mouth learning and provides the structure of equilibrium exit times. Section III presents a characterization of asymptotic learning by rational agents when prior beliefs of opponents are observable. Section IV characterizes asymptotic learning by naïve agents. Section V characterizes asymptotic learning by rational agents when prior beliefs of opponents are not observable. A conclusion is offered in Section VI and lengthy proofs are relegated to the Appendix. II. The model This section outlines the specifications of a game of informational externalities, in which agents exchange information through word-of-mouth communication. A. Actions, payoffs, informational and communication structures We consider an information externality game played by a set N of agents, each of whom chooses an irreversible action x R. The payoff (more specifically the loss function) U(x, θ) to choice x is a function of x and a state of nature θ; that is U(x, θ) = (x θ) 2. We consider this simple structure of U(x, θ) for the sake of concreteness. As will become clear in the following sections, what matters is that the optimal value of U(x, θ) must in one way or the other reflect individual confidence in their opinion; so U(x, θ) could take any form provided this property is preserved. The true state of nature θ is unknown to all agents and is drawn from a normal

10 10 OCTOBER 2015 distribution with mean θ and variance σθ 2. Each agent observes a noisy signal s that is informative about θ of the form s i = θ + ε i, where across all i, θ and ε1,, ε n knowledge that ε i N (0, σ 2 ε). are independently distributed, and it is common Agents exchange information through word-of-mouth communication and their interactions are governed by a network G n = (N n, E n ), where for the population of size n, N n is the set of agents and E n is the set of links connecting them. The network of interactions imposes constraints on information exchange in that agents can only communicate to those in their neighbourhood. We write G n, with elements g ij, for the adjacency matrix of G n. If g ij > 0, then j communicates or sends a message to i and g ij = 0 implies otherwise. Communication is simultaneous and deterministic. The following definitions and notations regarding networks will be used through out the paper. The t-order neighbourhood of i will be denoted by N i,t such that N i,1 is the set of agents that directly communicate to i. We write b n i,t and k n i,t for the size of t-order and t-th neighbourhood respectively. That is k n i,t is the number of agents at the t radius from i and b n i,t is the number of all agents within and including those at the t radius from i. A network is said to be strongly connected if for every pair of agents i and j, there exists a directed path from i to j and vice versa. The distance or equivalently the geodesic d ij (G n ) is the shortest path from i to j. The diameter d(g n ) is the longest geodesic of G n. B. Dynamics and learning mechanisms In addition to the constraints imposed by the communication network on information exchange, there is a cost associated with delays in taking an irreversible action. At every period, each agent i chooses between taking an irreversible action and exit the game or wait to collect more information. Waiting is costly in that the payoffs are discounted over time with discount

11 COSTLY WORD-OF-MOUTH LEARNING 11 rate of r. That is at time t, i s loss function to taking action x i is e rt (x i θ) 2. Under such a set-up, agents endogenously choose the optimal time to exit the game. Our analysis can however also directly extend to situations where the time to exit the game is exogenously given. An example of such a situation occurs in electoral processes where the campaigns period is the time interval within which information exchange occurs. Let I n i,t denote the information set of i at time t; where I n i,0 = {s i }. The information set at any t > 0 depends on the network structure and the learning mechanism. If agents are rational in the sense that they follow Bayesian updating rule, and in addition the network structure is common knowledge and that prior beliefs are observable (we relax the assumption of observable prior beliefs in section V), then I n i,t will consist of all signals from agents within the radius of t. That is Ii,t n = {s i, {s j } bn i,t j=1 }. The above argument follows directly from Bayes rule. Let µ n i,t and ρ n i,t denote the mean and precision (that is the reciprocal of the variance var n i,t) of i s posterior belief at t. From Bayes rule, j s posterior belief after observing a signal s j has mean and precision of 5 µ j,1 = σ2 ε σθ 2 + µ j,0 + σ2 θ σ2 ε σθ 2 + s i and ρ n σ2 j,1 = ρ θ + ρ ε. ε If j is a first order neighbour of i and prior beliefs of neighbours are observable, then i can correctly deduce the signal of j after observing j posterior belief. Consequently, at the end of period t the posterior belief 5 The relation follows from the Bayesian rule that given θ N (µ, σ 2 θ ) and ε N (0, σ 2 ε), if s = θ + ε then σ2 ε E [θ s] = σθ 2 + σ2 ε with a conditional variance of var [θ s] = σ2 ε σ2 θ σ 2 θ +σ2 ε µ + σ2 θ σθ 2 + s. σ2 ε

12 12 OCTOBER 2015 of i has mean and precision of (1) µ n i,t = σ 2 ε σ 2 θ b n i,t σ2 θ + µ j,0 + σ2 ε b n i,t σ2 θ + σ2 ε b n i,t s j and ρ n i,t = ρ θ + b n i,tρ ε. j=1 If on the other hand agents are naïve or boundedly rational, then their information sets at t consist of individual signals and the summary statistic of other agents private information within the tth order neighbourhood. More specifically, we assume that agents are not able to disentangle between old and new information from messages of their first-order neighbours. After receiving private signals, each agent updates their prior belief in accordance to Bayes rule. The resulting private beliefs are then communicated to the first-order neighbours. From the second period onwards, agents incorporate information from their first-order neighbours by taking the weighted average of their posterior beliefs. That is, at period t i s posterior mean is given by (2) µ n i,t+1 = n g ij (t)µ n j,t j=1 i = 1,, n If G n (t) is the associated matrix of interactions at time t, that is the normalized adjacency matrix of G n, then (2) can be written as (3) µ n t+1 = G n (t)µ n t where µ t is a vector of posterior means in the t-th period. We assume that individual belief precisions evolve in a similar manner as in (1). The learning dynamics in (3) is often referred to as the DeGroot model after DeGroot (1974). The main properties of the outcomes of this model have been extensively studied in the literature (e.g. Demarzo et al. (2003), Golub and Jackson (2010) and Jadbabaie et al. (2012)). Contrary to the analysis

13 COSTLY WORD-OF-MOUTH LEARNING 13 in the literature, we are interested in learning outcomes when waiting to collect information is costly (i.e. when the discount rate r > 0), and to establish conditions for correct learning in large societies. by Given information set I n i,t, i s optimal actions if he decides to act is given x n i,t = arg min x R E [ (x θ) 2 I n i,t] = E [ θ I n i,t ] = µ n i,t and the payoff corresponding to the optimal action is U it (Ii,t) n = E [ e rt (x n it θ) 2 ] Ii,t n = e rt var n i,t = e rt 1 ρ θ + b n i,t ρ ε where var n i,t is the variance of i s posterior belief at t. If on the other hand i decides to wait for one more period before taking an irreversible action, then the respective expected loss function is U i,t+1 (I n i,t+1) = E [ e r (x θ) 2 I n i,t+1]. C. Equilibrium of information externality game The dynamic process described above entails an information externality game. Individual actions depend on those of other agents through information sharing. For each i and t, let Ii,t n denote the set of all possible information sets. Agent i s action at t is then a mapping a n i,t : Ii,t n R { wait }, from an information set to an action set. We write a n t for the action profile at time t, and a n i,t for an action profile that excludes i s strategy. Then the respective value function for i at t is var n i,t(ii,t) n i,t = max e r E [ ] U i,t+1 (Ii,t+1) I n i,t n V n when a n i,t = x when a n i,t = wait

14 14 OCTOBER 2015 Given the value function above, an equilibrium of information externality game is then defined as follows. Definition 1. An action profile a n, is a pure-strategy perfect Bayesian equilibrium of the information externality game if for every i N, t and action profile a n, i, action an, i,t yields to i an expected payoff equal to i s value function at t, V n i,t(i n i,t). We denote the set of equilibria of the game by A (G n ) An equilibrium strategy profile induces an equilibrium timing or equivalently exit time profile t n,a e. Each t n,a is the time at which i takes an irreversible action and exits the game. After taking an irreversible action i stops acquiring new information and can only transmit information acquired until t = t n,a. Since an agent s t-order neighbourhood is influenced by others actions, we then write b n,a i,t when the strategy profile is a n. for a t-order neighbourhood size of i The exit times are a function of discount rate, precision of agents beliefs, which in itself is a function of the network topology. Lemma 1 below shows the relationship between these components. Lemma 1: Suppose that the precision of agents beliefs evolve as in (1). The exit time for each i N is given by the solution to the equation (4) rb n,a i,t d dt ( ) b n,a rρ θ i,t + = 0 ρ ε where d b is the derivative of b with respect to t. dt PROOF: See Appendix VII.A The size of an agent s t-order neighbourhood b n,a i,t is non-decreasing function of time. For each i, the variation of b n,a i,t with time depends on network topology and i s position in the network. After a relation between b n,a i,t t is established, t n,a can be obtained by solving (4). and

15 COSTLY WORD-OF-MOUTH LEARNING 15 Consider the two infinite regular networks in Figures 1 and 2. For the network in Figure 1, it is easy to see that for each i, b n,a i,t = 4t 1. That is, at t = 1 each i receives three signals from the first order neighbours and from t 2 onward receives four signals periodically. Substituting into (4) and solving for t n,a yields (5) t n,a = 1 r 1 ( ) ρθ 1 4 ρ ε The precisions ρ θ, ρ ε and discount rate r are all exogenous parameters. The exits times are thus readily computable once the network topology is known. The exit time decreases with discount rate in that the larger the cost of waiting the earlier agents take irreversible action and exit the game. It is a decreasing function of agents prior belief precision in that the less confident they are in their prior belief, the longer or more signals it takes to be convinced in their opinion regarding the correct choice. Figure 1. : A 3-regular infinite grid network. Figure 2. : A 3-regular infinite tree network. For the network in Figure 2 b n,a i,t = 3 (2 t 1). In fact for a family of tree networks as that in Figure 2, if the first-order degree is k then b n,a i,t =

16 16 OCTOBER 2015 k ((k 1) t 1). Substituting into (4) and solving for t n,a yields (6) t n,a ( ) ( ln 1 ρ θ = 3ρ ε ln 1 ln(2) r ln(2) Clearly, what matters in determining the exit time is not just the firstorder connectivity or degree of an agent but rather the growth of their t-order neighbourhood. For the two networks in Figure 1 and 2, although the first-order degree for each agent is three in both networks, the exit times are not equivalent. For example, when r = 0.7 and ρ θ = ρ ε = 1, t n,a = 1.4 and 4.2 for Figure 1 and 2 respectively. The neighbourhood of network in Figure 2 grows exponentially as opposed to that of Figure 1 which is linear. Agents in Figure 2 benefit from waiting for a few more periods as the number of signals grows exponentially. Through their influence on the exit times, the discount rate, precisions and network topology all influence information aggregation. The exit times are interdependent in that as some agents exit earlier than others, it affects the speed and how much information the remaining agents receive. Consequently, these externalities determine whether or not correct learning (as defined in the next section) occurs. ) D. Learning in large societies Given the learning mechanism, equilibrium strategies and network structure, we are interested in determining whether correct learning occurs at equilibrium; equivalently, the conditions under which equilibrium behaviour leads to correct aggregation of decentralized information. We particularly focus on the network topologies that lead to correct learning. We define correct learning or more accurately asymptotic learning as the convergence in probability to the correct equilibrium action as prescribed by the true state of nature. That is whether equilibrium choices coincide with

17 COSTLY WORD-OF-MOUTH LEARNING 17 that prescribed by the true state of nature when the population size is infinitely large. Definition 2. Given a sequence of networks {G n } n 2, asymptotic learning is said to occur if lim n P ( x n,a θ > ɛ ) = 0 for all i N, where x n,a is the choice made by i after exiting play. Note that the correct action for the model considered here is equivalent to the true state of nature θ. Our next objective is then to characterize, for each learning mechanism, the conditions for asymptotic learning given a sequence of networks. III. Asymptotic learning: rational agents This section establishes conditions for asymptotic learning by rational agents. We focus on the relationship between belief confidence, discount rate and network characteristics in influencing correct learning. As highlighted in section II.C, the interdependence between exit times (induced by Bayesian perfect equilibrium) of agents through informational externalities influences information aggregation and hence asymptotic learning. The key to asymptotic learning is that no agents are forced to exit the game too early. By too early we mean before fully aggregating information from other agents. The analysis for asymptotic learning thus anchors on comparing for each agent their exit time to the time it would take to aggregate information if waiting were not costly. The time it takes an agent to fully aggregate information when waiting is not costly is related to the maximum geodesic of that agent. The following theorem establishes this relationship. Theorem 1: Given a sequence {G n } n 2 of strongly connected communication networks, let {d i (G n )} i N,n 2 be the corresponding sequence of agents maximum geodesics. Then asymptotic learning with rational agents

18 18 OCTOBER 2015 and observable priors obtains if (7) 1 n n i=1 ( t n,a d i(g n ) ) 0 as n, in probability PROOF: See Appendix VII.B Theorem 1 establishes a condition under which asymptotic learning oc- t curs. It can equivalently be stated as lim n,a n d i = 1 for all i. The (G n ) strength of Theorem 1 lies in the fact that for every sequence of networks one can derive a range of parameter values under which asymptotic learning occurs. For a given network structure, exit times t n,a can be derived from (4) and geodesics d i (G n ) directly from the network. Once expressions for these two measures are derived, the parameter values of r, ρ ε and ρ θ within which asymptotic learning obtains are those that satisfy condition (7). In the next sections, we demonstrate the application of Theorem 1 to various families of networks, starting with deterministic, then random/empirical networks. A. Deterministic networks In this section, we show how to determine whether or not asymptotic learning occurs in a given network where r > 0. Consider the networks of Figure 1, the exit time for all agents is (5) and diameter when the network is finite and of size n is d(g) = n. An agent i with the shortest geodesic 2 thus has d i (G) = n. From Theorem 1, asymptotic learning occurs if 4 (8) lim n t n,a d i (G n ) = lim n t n,a e d(g n ) = 1 for all i N where t n,a e = max i N t n,a is the exit time of agents at opposite ends of the network diameter. Condition (8) implies that asymptotic learning occurs if and only if r = 0 and fails in this family of networks whenever r > 0.

19 COSTLY WORD-OF-MOUTH LEARNING 19 For the network in Figure 2, the exit times are given by (6). The diameter when the network is finite and of size n is given by 2 ln ( ) n+2 ln 2 3. Clearly, in this network, asymptotic learning occurs 6 whenever r = ln 2. The following corollary generalizes this result to infinite k-regular tree networks. 7 Corollary 1: Asymptotic learning with rational agents and observable prior beliefs occurs in k-regular infinite tree networks whenever r = ln(k 1). Corollary 1 shows that asymptotic learning occurs in tree networks, and this result generally applies to all networks that exhibit the property of exponentially growing neighbourhood sizes just as tree networks do. The underlying reason being that when r = ln(k 1), the exponential growth of neighbourhood size balances out the exponential decay of the payoff loss associated with the cost of waiting. This makes agents indifferent between waiting to collect more information or take an irreversible action and exit the game. Examples of tree networks include hierarchical networks that are typically the organizational structures found in both public and corporate institutions. Acemoglu et al. (2014) show that under a similar set-up, asymptotic learning occurs in a special form of random hierarchical networks that are bounded and exhibit homophily. That is agents are connected to both those in the upper hierarchy and those at the same 6 This follows from Theorem 1, that asymptotic learning occurs when ( ) ( ) t n,a ln 1 ρ θ e lim n d(g n ) = lim 3ρ ε ln 1 ln(2) r n 2 ln ( ) n+2 = 1, 3 a condition that is satisfied whenever r = ln 2. 7 The proof follows from the fact that for a k-regular infinite tree network, b n,a i,t = k ((k 1) t 1) for each i N. Substituting into (4) yields for each i. t n,a = ln ( 1 ρ θ kρ ε ) ln ( ln(k 1) 1 ln(k 1) r ( ) The diameter when the network is finite and of size n is given by (k 1) ln(k 1) ln n+(k 1) k. It follows from condition (8) that asymptotic learning occurs if and only if r = ln(k 1). )

20 20 OCTOBER 2015 hierarchical level. Here, we have shown that homophily is not a necessary condition for asymptotic learning in hierarchical networks. B. Random and empirical networks This section demonstrates how to derive parameter values of r within which asymptotic learning occurs in random networks such as the Erdös- Rényi and scale-free families of networks. The Erdös-Rényi family of random networks G n = (N, p) are constructed by connecting every pair of nodes randomly and independently with probability p. Chung and Lu (2001) show that if the average degree np grows with n, that is np, then the diameter is approximately equal to ln n/ ln(np). Moreover, if np/ ln n > 8 then the diameter is concentrated around two values. This implies that the diameter for this family of networks is bounded. A parameter space thus exists for r > 0 within which asymptotic learning is feasible in Erdös-Rényi family of random networks provided np and that the network is strongly connected. Chung and Lu (2001) also show that when np 1, the diameter of Erdös-Rényi family of random networks is approximately ln n. Similarly, for scale-free networks (the topology that is representative of many real world networks), Newman et al. (2001) show the diameter to be approximately ln n. A scale-free network G n = (N, γ) has a degree distribution that follows a power law and results from dynamic process that exhibit preferential attachment. That is p(k) = k γ, where p(k) is the number of agents with degree k. The derivation of neighbourhood size growth function and hence exit times can be done empirically. For Erdös-Rényi family of random networks, we demonstrate this process with network parameter values n = 5000 and p = Through numerical simulations, we find that for these parameter values, the network diameter to be on average 7. The growth of

21 COSTLY WORD-OF-MOUTH LEARNING 21 neighbourhood sizes follows a log-normal distribution. That is ln(b n,a i,t ) = t f(τ)dτ, where 1 f(t) = 1 ( σ 2π exp 1 ) (t µ)2 2σ2 with σ and µ as the standard deviation and mean respectively. Figure 3 shows the quantile-quantile plot of the log values of b n,a i,t for a randomly chosen agent. The distribution of points around the 45-degree line implies that the neighbourhood growth indeed follows a log-normal distribution. The following lemma shows the structure of exit time for Erdös-Rényi family of random networks, and any other networks whose neighbourhood sizes growth follows a log-normal distribution. Normal Q Q Plot Sample Quantiles Theoretical Quantiles Figure 3. : A quantile-quantile plot of the log values of b n,a i,t Erdös-Rényi random networks. for a randomly chosen agent in an Lemma 2: For networks in which b n,a i,t times are of the form t n,a is log-normally distributed, the exit = A + Br + C ln r. where A, B and C are some constants that depend on the parameters values of ρ θ, ρ ε, σ and µ. PROOF: See Appendix VII.C Corollary 2: For a sequence {G} n 2 of strongly connected networks

22 22 OCTOBER 2015 whose neighbourhood sizes grow log-normally, let the network diameter be asymptotically bounded. Under rational learning with observable priors, there exists a real number r > 0 such that asymptotic learning occurs whenever r < r. PROOF: Given that for such networks t n,a Theorem 1 that asymptotic learning occurs whenever = A + Br + C ln r, it follows from from lim n t n,a e d(g n ) = lim A + Br + C ln r n d(g n ) = 1, If the network diameter is asymptotically bounded, that is lim n d(g n ) = c, where c is some constant, then there exists a value of r = r > 0 below which asymptotic learning occurs. Corollary 2 shows that what matters for asymptotic learning in networks with log-normally growing neighbourhood sizes is for the network diameter to be asymptotically bounded. Examples of such networks is the Erdös- Rényi family of random networks in which np as discussed above. It is easy to show that this result also holds for networks in which neighbourhood sizes growth follows a normal distribution. In the case of scale-free networks, the neighbourhood sizes growth follows a type II Pareto distribution. That is b n,a i,t f(t) = ( 1 + t 1 ) α t m = 1 f(t), where A parameter of the distribution, α > 0, determines its shape, and t m > 0 is the minimum possible value of t. The following corollary provides the structure of exit times for networks whose neighbourhood sizes growth follows a type II Pareto distribution, and that asymptotic learning fails in scale-free networks

23 COSTLY WORD-OF-MOUTH LEARNING 23 Corollary 3: For networks in which b n,a i,t follows a type II Pareto distribution, exit times are of the form t n,a = 1 t m ( 1 ( r 1 + r ) ) 1/α ρ ε + ρ θ ρ ε This implies that asymptotic learning fails in scale-free networks whenever r > 0. PROOF: See Appendix VII.D Unlike the case of Erdös-Rényi family of random networks where there exists a range of values of parameter p within which the network diameter is asymptotically bounded, scale-free networks have asymptotically infinite network diameter. This makes complete aggregation of decentralized information infeasible in such networks. In the supplementary appendix VII.I, we explore the properties of neighbourhood size growth for three empirical networks the Stanford Large Network Data set Collection (Leskovec and Krevl, 2014). We conclude this section by highlighting the novelty of the above analysis in relation to existing literature. As already mentioned above, a closely related study is Acemoglu et al. (2014). In addition to the fundamental differences discussed in Section I, we have been explicit in laying out the relationship between exit times and network variables. We have developed tools for characterizing conditions on the network topology and parameter values under which asymptotic learning occurs that can be applied to any family of networks. Acemoglu et al. (2014) find that asymptotic learning obtains in the presence of information hubs which receive and distribute a large amount of information. Here, we have been able to show that the key properties for asymptotic learning are that either the neighbourhood sizes grow exponentially or the diameter of the network is asymptotically

24 24 OCTOBER 2015 bounded. Hence, asymptotic learning can occur even in random networks provided either of these conditions is satisfied. IV. Asymptotic learning: Naïve agents This section characterizes conditions for asymptotic learning by naïve agents. We start by briefly reviewing properties of equilibrium outcomes of learning dynamics given by (3). A well known result for such a dynamic process is that a consensus in beliefs emerges provided the network is not disconnected; the network need not be strongly connected. For the case of dynamic networks, Azomahou and Opolot (2014) show that a consensus obtains if for every interval [t, t + τ), the sequence of networks {G n } t+τ t is jointly-connected. For a given time interval [t, t + τ), a corresponding sequence of networks Gt n, Gt+1, n, Gt+τ n is said to be jointly-connected if t+τ the network resulting from the union is connected. The following Lemma summarizes these results. Lemma 3: For some integer τ 0, if for every time interval [t, t + τ) t+τ the network is connected, then j=t (9) µ n i, = G n j [ lim T ] T Gγ(t)µ n 1 t=0 i j=t where (v n ) is the transpose of the vector v n. G n j = (v n ) µ 1 for every i N The implication of Lemma 3 is that for dynamic networks, the network G t at any given time t need not be connected provided it is jointly connected with either the preceding or succeeding sequence of networks. The vector v n is the influence vector and reflects the level of influence each agent exerts on the resulting beliefs. The overall influence of each agent, vi n depends not only on the level of influence that i exerts on the first-order

25 COSTLY WORD-OF-MOUTH LEARNING 25 c b a e f 0.4 c b a 0.4 d f e d (a) A connected network: G 1 (b) Network with prominent family : G 2 Figure 4. neighbours but also on the second-order neighbours, third-order neighbours, and so forth. 8 Consider an example of the two graphs in Figures 4a and 4b, call them G 1 and G 2 respectively. The corresponding influence vectors are respectively v 1 = (0.363, 0.204, 0.191, 0.073, 0.121, 0.048) and v 2 = (0, 0, 0, 0.359, 0.381, 0.260). From v 1 and v 1, the overall influence of each agent clearly depends on their first-order connectivity, the connectivity of their second-order neighbours, and so forth. Take for example the network G 1 in which agents d and f both observe the posterior beliefs of only one other agent and both communicate to one other agent. Although the first-order neighbour of agent f attaches more weight to f s messages than does the first-order neighbour of agent d, agent d is more influential than f in overall. This is precisely the effect of being connected to other agents who themselves are highly connected. In the case of network G 2, there are two subgroups (that is {a, b, c} and {d, e, f}) each of whom form a complete subgroup. The inter-subgroup communication on the other hand is unidirectional, that is members of subgroup {a, b, c} receive messages of those in subgroup {d, e, f} and not vice versa. As a consequence, a consensus emerges in a long-run in which members of subgroup {a, b, c} adopt private beliefs of subgroup {d, e, f}. 8 see Demarzo et al. (2003) and Golub and Jackson (2010) for a detailed study of the properties of the influence vector

26 26 OCTOBER 2015 This example highlights the effect of presence of prominent families. For the learning dynamics in this paper however, agents can exit the game before fully aggregating information such that the influence vectors can be different for each i depending on the perfect Bayesian equilibrium. To distinguish this characteristic of the learning process, we write v n,a i for the vector of influence of all agents on i s beliefs at the end of the learning process when the equilibrium choice profile is a. Where no confusion arise, we write v n,a to imply an influence vector with a consensus; that is whenever v n,a i identical for all i. The above two examples also act to illustrate how the network topology may affect asymptotic learning. Consider the case in which no agent exits the game before fully aggregating information. Under such a setting, the irreversible action taken by agent i at the end of the learning process is given by x n,a I n,a and µ n,a = lim t x n,a i,t = E[θ I n,a ] = µn,a is = (vn,a i ) µ 1, where are the information set and posterior mean of i at the end of the learning process respectively. Recall also that µ j,1 = It then follows that [ x n,a = 1 σθ 2 + σ σ2 ε 2 ε n j=1 v n,a ij µ j,0 + σθ 2 n j=1 σ2 ε µ σθ 2 j,0 + σ2 θ s +σ2 ε σθ 2 j. +σ2 ε v n,a ij s j Clearly, whether or not asymptotic learning occurs, that is x n,a ] = θ (the true state of the nature) depends on the values of influence vector. Therefore, in addition to restrictions on network topology imposed by exit times, there are also restrictions imposed by influence vector. Before stating the main result in this regard, the following definitions are used in the subsections that follow. Definition 3. A matrix G n is said to be doubly stochastic if n j=1 gn ij = n i=1 gn ij = 1 for all (i, j) N Definition 4. (a) A communication network G n is said to be perfectly balanced if the corresponding transition matrix G n is doubly stochas-

27 COSTLY WORD-OF-MOUTH LEARNING 27 tic. (b) A sequence of networks {G n } n 2 is said to be asymptotically balanced if lim n G n = S, where S is an infinite doubly stochastic matrix. The following theorem provides conditions on network topology and prior belief distribution for asymptotic learning with naïve agents to occur. Theorem 2: Let {G n } n 2 be a sequence of networks that are jointly connected, then asymptotic learning with naïve agents occurs if and only if the following conditions hold: (i) µ 0 N ( θ, σ 2 pi), where θ is an n dimensional vector of θ, µ 0 is a vector of prior beliefs and I is the identity matrix. (ii) lim n t n,a = t s = lim n ln(2ɛv n min ) ln λ 2 (G n ) for all i, where λ 2 (G n ) is the second largest eigenvalue of the transition matrix associated with the network, v n min = min i N v n,a i and ɛ < 1 2 that defines how close the vector µ n,a t influence vector v n,a. (iii) lim n G n = S PROOF: See Appendix VII.E. is an arbitrary real number of posterior beliefs at t is to the Theorem 2 highlights three main conditions for asymptotic learning with naïve agents. The first relates to the structure of prior beliefs. Unlike in the case of rational learning where the distribution of prior beliefs can be independent of the true state of nature, asymptotic learning with naïve agents occurs only if prior beliefs are informative about the true state of nature. That is the prior belief of each agent must be normally distributed with mean equal to the true state of nature. This condition generally highlights the superiority of rational over naïve agents in signal interpretation and hence in aggregating decentralized information.

28 28 OCTOBER 2015 The second condition concerns the relationship between exit times and network properties; and in particular the second largest eigenvalue of the transition matrix associated with the network. It states that the exit times for all agents must be asymptotically equal to the time t s at which a consensus obtains or equivalently, the time learning stops when the discount rate is zero. The second largest eigenvalue of graphs is a well studied concept in the literature of graph theory. Generally, the more sparsely connected a network is, the larger is λ 2. For example, a complete network (that in which each agent is connected to every other agent) of size n has λ 2 = 1 n 1, while a cyclic network (in which agents are arranged around a circle each connected to two neighbours on the left and right) of size n has λ 2 = 1 for odd value of n. Since for a complete network v n i = v n min = 1 n for all i, it follows that t s = 1. This value is fairly accurate since in a complete network, agents update their beliefs only once and in the second period a consensus obtains. The third condition relates to the general properties of network structure. That is, asymptotic learning occurs if and only if the network is asymptotically balanced. To fully characterize implications of condition (iii) of Theorem 2, we develop a measure of balancedness for arbitrary network in the next subsection. A. Asymptotically balanced networks To begin with, balanced networks are also asymptotically balanced but not vice versa. To characterize the class of networks that are asymptotically balanced, we need to construct a measure of balancedness of a network/matrix, which we denote by φ n. Definition 5. Let S n be closest doubly stochastic matrix in terms of the Frobenius norm to the matrix G n. Then G n is said to be φ n -balanced if given S n, S n G n 1 = φ n, where. 1 is the L 1 norm.