1、Bayesian Learning in Social Networks1Douglas Gale (Corresponding Author)Department of Economics, New York University269 Mercer St., 7th Floor, New York, NY, 10003-6687.E-mail: douglas.galenyu.eduUrl: http:/www.econ.nyu.edu/user/galed/Phone: (212) 998-8944Fax: (212) 995-3932andShachar KarivDepartment
2、 of Economics, New York University269 Mercer St., 7th Floor, New York, NY, 10003-6687.E-mail: sk510nyu.eduUrl: http:/home.nyu.edu/sk510Version: March 13, 2003.We extend the standard model of social learning in two ways. First, weintroduce a social network and assume that agents can only observe the
3、actionsof agents to whom they are connected by this network. Secondly, we allow agentsto choose a dierent action at each date. If the network satisfies a connectednessassumption, the initial diversity resulting from diverse private information iseventually replaced by uniformity of actions, though n
4、ot necessarily of beliefs,in finite time with probability one. We look at particular networks to illustratethe impact of network architecture on speed of convergence and the optimalityof absorbing states. Convergence is remarkably rapid, so that asymptotic resultsare a good approximation even in the
5、 medium run.Journal of Economic Literature Classification Numbers: D82, D83Key Words: Networks, Social learning, Herd behavior, Informationalcascades.Running Title: Bayesian Learning in Social Networks.1One of us discussed this problem with Bob Rosenthal several years ago, when wewere both at Boston
6、 University. At that time, we found the problem of learning innetworks fascinating but made no progress and were eventually diverted into working onboundedly rational learning, which led to our paper on imitation and experimentation.We thank seminar participants at NYU, DELTA, INSEAD, Cergy, Cornell
7、 and Iowafor their comments. The financial support of the National Science Foundation throughGrant No. SES-0095109 is gratefully acknowledged.11. INTRODUCTIONThe canonical model of social learning comprises a set of agents I,afinite set of actions A,asetofstatesofnature,andacommonpayofunction U(a,),
8、wherea is the action chosen and is the state of nature.Each agent i receives a private signal i(), a function of the state of nature, and uses this private information to identify a payo-maximizing action.Thissetupprovidesanexampleofapure information externality.Eachagents payo depends on his own ac
9、tion and on the state of nature. Itdoes not depend directly on the actions of other agents. However, eachagents action reveals something about his private signal, so an agent cangenerally improve his decision by observingwhatothersdobeforechoosinghis own action. In social settings, where agents can
10、observe one anothersactions, it is rational for them to learn from one another.This kind of social learning was first studied by Banerjee (1992) andBikhchandani, Hirshleifer and Welch (1992). Their work was extended bySmith and Srensen (2000). These models of social learning assume a sim-ple sequent
11、ial structure, in which the order of play is fixed and exogenous.They also assume that the actions of all agents are public information.Thus, at date 1, agent 1 chooses an action a1, based on his private in-formation; at date 2, agent 2 observes the action chosen by agent 1 andchooses an action a2ba
12、sed on his private information and the informationrevealed by agent 1s action; at date 3, agent 3 observes the actions chosenby agents 1 and 2 and chooses an action a3.; and so on. In what followswe refer to this structure as the sequential social-learning model (SSLM).One goal of the social learnin
13、g literature is to explain the striking uni-formity of social behavior that occurs in fashion, fads, “mob psychology”,and so forth. In the context of the SSLM, this uniformity takes the formof herd behavior.2Smith and Srensen (2000) have shown that, in theSSLM, herd behavior arises in finite time wi
14、th probability one. Once theproportion of agents choosing a particular action is large enough, the pub-lic information in favor of this action outweighs the private information ofany single agent. So each subsequent agent “ignores” his own signal and“follows the herd”.This is an important result and
15、 it helps us understand the basis foruniformity of social behavior.3Atthesametime,theSSLMhasseveral2A herd occurs if, after some finite date t, every agent chooses the same action. Aninformational cascade occurs if, after some finite date t,everyagentfinds it optimal tochoose the same action regardl
16、ess of the value of his private signal. An informationalcascade implies herd behavior, but a herd can arise without a cascade.3The most interesting property of the models of Bikhchandani, Hirshleifer and Welch(1992) and Banerjee (1992) is that informational cascades arise very rapidly, before muchin
17、formation has been revealed. For example, in these models if the first two agents makethe same choice, all subsequent agents will ignore their information and imitate the firsttwo. The behavior of a potential infinity of agents is determined by the behavior of thefirst two. This is both informationa
18、lly inecient and Pareto inecient.2special features that deserve further examination: (i) each agent makesa single, irreversible decision; (ii) the timing of the agents decision (hisposition in the decision-making queue) is fixed and exogenous; (iii) agentsobserve the actions of all their predecessor
19、s; and (iv) the number of signals,like the number of agents, is infinite, so once a cascade begins the amountof information lost is large. These features simplify the analysis of theSSLM, but they are quite restrictive.In this paper, we study the uniformity of behavior in a framework thatallows for
20、a richer pattern of social learning. We depart from the SSLMin two ways. First, we drop the assumption that actions are public infor-mation and assume that agents can observe the actions of some, but notnecessarily all, of their neighbors. Second, we allow agents to make deci-sions simultaneously, r
21、ather than sequentially, and to revise their decisionsrather than making a single, irreversible decision. We refer to this structureas the social network model (SNM). For empirical examples that illustratethe important role of networks in social learning, see Bikhchandani, Hirsh-leifer and Welch (19
22、98).Onthefaceofit,uniformbehaviorseemslesslikelyintheSNM,whereagents have very dierent information sets, than in the SSLM. However,uniformity turns out to be a robust feature of connected social networks.4The following results are established for any connected network:Uniformity of behavior: Initial
23、ly, diversity of private information leads todiversity of actions. Over time, as agents learn by observing the actionsof their neighbors, some convergence of beliefs is inevitable. A centralquestion is whether agents can rationally choose dierent actions forever.Disconnected agents can clearly disag
24、ree forever. Also, there may becases where agents are indierent between two actions and disagreementof actions is immaterial. However, apart from cases of disconnectednessand indierence, all agents must eventually choose the same action. Thus,learning occurs through diversity but is eventually replaced by uniformity.Optimality: We are interested in whether thecommon action chosen asymp-totically is optimal, in the sense that the same action would be chosen ifall the signals wer
