Herd Behavior and the Quality of Opinions

Journal of Socio-Economics 32 (2003) 661–673

Herd behavior and the quality of opinions

Shinji Teraji

Department of Economics, Yamaguchi University, 1677-1 Yoshida, Yamaguchi 753-8514, Japan Accepted 14 October 2003

Abstract

This paper analyzes a decentralized decision model by adding some inertia in the social leaning process. Before making a decision, an agent can observe the group opinion in a society. Social learning can result in a variety of equilibrium behavioral patterns. For insufficient ranges of quality (precision) of opinions, the chosen stationary state is unique and globally accessible, in which all agents adopt the superior action. Sufficient quality of opinions gives rise to multiple stationary states. One of them will be characterized by inefficient herding. The confidence in the majority opinion then has serious welfare consequences. © 2003 Elsevier Inc. All rights reserved.

JEL classiﬁcation: D83

Keywords: Herd behavior; Social learning; Opinions; Equilibrium selection

1. Introduction

Missing information is ubiquitous in our society. Product alternatives at the store, in catalogs, and on the Internet are seldom fully described, and detailed speciﬁcations are often hidden in manuals that are not easily accessible. In fact, which product a person decides to buy will depend on the experience of other purchasers. Learning from others is a central feature of most cognitive and choice activities, through which a group of interacting agents deals with environmental uncertainty. The effect of observing the consumption of others is described as the socialization effect. The pieces of information are processed by agents to update their assessments. Here people may change their preferences as a result of

E-mail address: [email protected] (S. Teraji).

1053-5357/$ – see front matter © 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.socec.2003.10.004 662 S. Teraji / Journal of Socio-Economics 32 (2003) 661Ð673 interpersonal contact. Then information externalities arise that drive towards the emergence of some patterns of influences among individuals.1 Recently there has been increasing interest in economic models in the presence of information externalities, which put a great emphasis on the notion of social learning. In social and economic situations, we are often influenced in our decision making by what others around us are doing.2 Mimetic contagion was thought of as an irrational behavior that is responsible for pathological dynamics such as financial bubbles. Some studies have been seen as a way to formulate rigorously a number of challenges to standard economic doctrine.3 Individual behavior may be governed by herd externality. What everyone else is doing is rational because their decisions may reflect information that they have and we do not. Everyone may do what everyone else is doing, even when his or her private information suggests doing something different. The sort of herding behavior corresponds to human behavior reported by Becker (1991). When faced with two apparently very similar restaurants on either side of a street, a large majority chose one rather than the other, even though this involved waiting in line. The consequences will indeed be different due to the behavior of other agents in a society.4 An individual’s choice will then depend on his or her degree of confidence in the majority view concerning the state of the world. It is important to observe the self-reinforcing nature of confidence in the group opinion in a society. In this paper, the quality (precision) of opinions, which reflects the confidence in the majority opinion, is a central parameter. Several questions arise regarding the relationship between the quality of opinions and herd externality. Specifically, by comparing situations where each person’s decision is more or less responsive to the majority view, this paper considers how the quality of opinions has an impact on the efficiency of the long-run outcome in the social learning. The social learning model of Banerjee (1992) and Bikhchandani et al. (1992) describes the decision problem faced by a sequence of exogenously ordered individuals each acting under the state of the world.5 An agent conditions the decision in a Bayes-rational fashion on both one’s privately observed information and the ordered history of all predecessors’ decisions. The rest of the population is then allowed to choose sequentially, with each agent observing the choices made by all the predecessors. An information cascade occurs when agents ignore their own information completely and simply take the same action as predecessors have taken. The aggregate information that is available in the population is not correctly revealed by the sequence of decisions. This may eventually lead the whole

1 This terminology may be a close parallel to what Leibenstein (1950) called the “bandwagon effect” and “snob effect” in his classic study on the static market demand curve. By the bandwagon (snob) effect he referred to the extent to which the demand for a commodity is increased (decreased) because others are consuming the same commodity. 2 Note that the decision may not be optimal from the social point of view since the individual does not take account of the effect of his or her decision on the information of others. 3 See, for example, the analysis of ‘rumours’ in Banerjee (1993) and ‘fashions’ in Karni and Schmeidler (1990). 4 See also Kirman (1993) for an explanation of asymmetric aggregate behavior arising from the interaction between identical individuals. 5 Anderson and Holt (1997) induce the emergence of information cascades in a laboratory setting. Their results seem to support the hypothesis that agents tend to decide by combining their private information with the information conveyed by the previous choices made by other agents, in conformity with Bayesian updating of beliefs. S. Teraji / Journal of Socio-Economics 32 (2003) 661Ð673 663 population to take the wrong decision, and therefore to a socially inefficient outcome. However, the formation of a cascade is strongly influenced by the initial decisions. Orléan (1995) considers the dynamics of imitative decision processes in non-sequential contexts where agents are interacting simultaneously and modifying their decisions at each period in time. Such a framework is better suited for the modeling of herd behavior. In many circumstances, agents are always present and they revise their opinions in a continuous mode, not one for all. In this setting, opinions are modified endogenously as a result of interaction between agents. He describes the herd behavior as corresponding to the stationary distribution of a stochastic process rather than to switching between multiple equilibria. In this paper, to offer a clear justification for possible equilibrium patterns, I provide an explanation for switches from one to the other in the social learning dynamics.6 It is worth to offer an equilibrium selection criterion of how a particular equilibrium will emerge collectively in a non-sequential context. In this paper, I consider a choice between two competing alternatives with unequal payoffs, and show how individuals are likely to herd onto a single choice. I specify a simple model for boundedly rational choice given the information conveyed by the majority opinion. I consider the society consisting of a continuum of agents each of whom are Bayesian optimizers. They must make a short-run commitment to the action they chose. Opportunities to switch actions arrive at random, which are identical and independent across agents.7 This friction reflects uncertainty that leads to inertia. Following this interpretation, no agent can change his or her choice at every point in time because of the inability to make an assessment of the current configuration of opinions continuously. It seems to capture a certain aspect of boundedly rational behavior. The social dynamics generate equilibrium paths of behavioral patterns in the presence of herd externality. In some situation, the stability properties of stationary states depend not on the initial decisions, but rather on the degree of quality of opinions defined below. What may be surprising is that the two stationary states, corresponding to the two long-run configu- rations of opinions, possess different stability properties when we change the quality of opinions. I show that sufficient quality of opinions tends to yield inefficient herding in the long-run. A key element is the multiplicity of equilibrium paths of behavioral patterns.8 Then one has to determine which equilibrium actually gets established. The emphasis on the quality of opinions also distinguishes my work from other explanations of equilibrium selection in economic problems (e.g. Keynesian macroeconomics in Cooper and John, 1988 and

6 Akerlof (1980) says that there are multiple equilibria in the sense that different customs, once established, could be followed in equilibrium. In one of these equilibria a custom is obeyed, and the values underlying the custom are widely subscribed to by members of the community. In the other equilibrium, the custom has disappeared, no one believes in the values underlying it, and it is not obeyed. 7 The equilibrium dynamics of this kind is used in Matsuyama (1992). However, the interpretation attached to this dynamics here is quite different. 8 The fundamental argument in the path dependency literature (David, 1985; Arthur, 1989) is that the free market typically generates sub-optimal equilibrium solutions to a variety of economic problems and the probability of sub-optimal equilibrium outcomes increases where increasing returns prevail. It is even possible for efficient and inefficient (sub-optimal) solutions to prevail simultaneously in the world of path dependency. For this reason, one cannot expect the free market to force the economy to converge to unique equilibrium. 664 S. Teraji / Journal of Socio-Economics 32 (2003) 661Ð673 economic development in Murphy et al., 1989; Matsuyama, 1992).9 The literature offers very few formal approaches to the process through which interacting agents’ beliefs are formed. Most approaches in the literature on equilibrium selection have nothing to say about the self-reinforcing nature of confidence concerning the state of the world. My point is that many aspects of herd behavior can be explained by equilibrium selection problems based on the quality (precision) of information conveyed by the group opinion. The selection is based on the stability properties of the stationary states. The stability property, which I call globally accessible below, assures us that, for any initial behavioral patterns, there exists an equilibrium path along which the opinions converge to it. As the quality of opinions is smaller than a certain threshold, the society will reach the state that is globally accessible. The consequence is desirable from the social point of view. On the contrary, as the quality is larger than a certain threshold, the society will tend to reach the state where inefficient herding occurs. This situation corresponds to a self-reinforcing process of confidence in the majority opinion. Then the reduction of confidence in the majority view may be socially beneficial in ex ante welfare sense. Thus, a population exhibits inefficient herding if in the long-run everyone uses the inferior product, and it exhibits efficient social learning if in the long-run everyone uses the superior choice. I find it useful to present the parameter space, which depends on agents’ degree of confidence, to obtain the social dynamics of their beliefs. Accordingly, the emergence of inefficient herding results from a slight modification in the individual’s level of confidence in the group opinion. The rest of this paper is organized as follows: the basic framework is presented in Section 2. I deal with individual Bayesian behavior in Section 3. The decision rule itself will change if we change the quality of opinions. Section 4 examines how the quality of opinions affects the aggregate properties in the resulting social learning dynamics. I explore the long-run properties of the evolution of collective opinions: one which exhibits efficient herding and another which gives rise to inefficient herding. Section 5 offers an example, web herd behavior. Section 6 concludes by discussing some implications of the main results.

2. The framework

There is a continuum of identical agents who are faced with a choice between two alternative products, technologies, or practices, labeled by A and B. The decision making is non-sequential; at every point in time t, each agent is drawn randomly from the overall population and has to make a new choice of either adopting A or adopting B. All agents choose one of the two alternatives to maximize expected payoffs. There are many situations in which such a binary restriction seems quite reasonable. For example, when we talk about economic thought, we often think in terms of two alternative schools or approaches, such as Monetarist versus Keynesian, Historical versus Analytical, Rational versus Evolutionary. Even within the ﬁeld of economic theory, we often debate

9 Many studies on evolutionary games also address the question of how a particular equilibrium will emerge in a dynamic context; see, for example, Friedman (1991) and Gilboa and Matsui (1991). These studies do not, however, offer an equilibrium selection criterion, since all strict Nash equilibria share the same dynamic properties in their models. S. Teraji / Journal of Socio-Economics 32 (2003) 661Ð673 665 the pros and cons of two alternative styles of writing, such as algebraic versus geometrical approach. At certain times, the generality of a model tends to be values, at other times, the simplicity tends to be regarded as a virtue. In many of theses situations, pursuing a middle ground may not be a practical option. The agents in this model are assumed to follow boundedly rational behavioral rules that incorporate the notions that there is inertia in consumer choices, and that they do not take into account the information they could have observed before. Inertia, which introduces the sluggishness in the social learning process, is modeled with the assumption that agents cannot switch actions at every point in time. Each agent must make a commitment to a particular choice in the short-run. Opportunities to switch actions arrive randomly; at each point in time, some fraction α,0<α<1, of the agents decide to reevaluate their choice. Thus, some agents are simultaneously present, and can make decisions. Here, α is the expected frequency of the conscious decision made by an agent per unit of time, and could be interpreted as the planning horizon. Alternatively, α reflects ease of coordination, since a large α implies that a large fraction of agents can switch to an alternative action over a given time interval. In making decisions, agents do not incorporate the entire history of their observations. It is justified if we suppose that each agent does not want to keep track of historical information because of the infrequent opportunity to switch. In each period, all agents using the same brand receive the same payoff. Let uA or uB be the payoff to each agent’s choice A or B. I suppose that (uA − uB) has two possible values, where θ+ > 0 >θ−. Agents do not know the true value of (uA − uB). Each agent assigns common prior probability q>0.5toevent(uA − uB) = θ+. I just assume that the value q is given. Furthermore, I suppose that qθ+ + (1 − q)θ− > 0, i.e. ex ante brand A is better than brand B for each agent. Thus, the prior odds ratio, q/(1 − q), satisfies that q/(1 − q) > −θ+/θ−≡k. The aggregate behavior of the population at each point in time t can be summarized by a state variable xt, giving the fraction of the population who are using brand A. This variable will also be called the group opinion. It is a macroscopic datum that aggregates all agents having opinion (A) in the population. Here I take the initial state x0 to be given exogenously, or by “history.” Since agents do not know the true payoff realization, they will value the other source of information about it. Before making a choice, an agent is allowed to observe the group opinion as of t in a society. Then he or she can benefit from the information contained in it and will find out which alternative other agents have chosen in the population. I will consider some specifications of the decision rules for each agent, who is drawn randomly from the population at t and observes xt. An agent receives a signal about the state of the world by observing the group opinion in a society. The information needs not, of course, be true, and it may be false. However, each agent believes that the majority side in the population, {xt > 0.5} or {xt < 0.5}, may convey some implicit information about the realization of (uA −uB). Then the agents incorporate observations of the relative popularity of the two choices in the decision making. Agents weigh observations of others’ experience because others’ decisions might reflect the information that they have and he or she does not. 666 S. Teraji / Journal of Socio-Economics 32 (2003) 661Ð673

Though the relative popularity of the two choices in the population conveys some information, it is “noisy” and does not give true information about the payoff realization. I assume that the relative popularity of the two choices, {xt > 0.5} or {xt < 0.5},atevery point in time, is linked to the realization of (uA − uB) through the following conditional probabilities: + − Prob(xt > 0.5|θ ) = µ, Prob(xt > 0.5|θ ) = 1 − µ, + − Prob(xt < 0.5|θ ) = 1 − µ, Prob(xt < 0.5|θ ) = µ.

Within the present framework, {xt > 0.5} (respectively, {xt < 0.5}) is better correlated with the state θ+ (respectively, θ−) than with θ− (respectively, θ+), i.e. µ>1−µ. The closer µ is to 1, the more precise information the majority opinion conveys. Here, µ>1/2, which I call the quality of opinions, reflects the degree of confidence in the majority view concerning the realization. It is assumed to be public information for the decision makers. Indeed, by making the quality of opinions large, we can make the precision of the information as higher + − as we like. Furthermore, I assume that Prob(xt = 0.5|θ ) = Prob(xt = 0.5|θ ) = 0.5, i.e. the group opinion such that {xt = 0.5} is uncorrelated with the states. The task is to determine exactly how these factors influence each agent’s decision making. In the next section, I examine how the quality of opinions affects the decision rule itself. What I try to understand in Section 4 is how the collective interactions will affect the formation of the group opinion in the social learning process.

3. Behavior

An important characteristic of this social learning process is that revision of beliefs can be expressed as a simple updating of the agent’s prior belief by Bayes’ theorem. Once each agent observes the popularity of each choice, he or she updates the prior on the basis of it and then chooses whichever product has the highest current score given the posterior. This process of revision of probabilities is called Bayesian, after Bayes’ theorem. Thus, we have the posterior probability that an individual should attach to the state of the world after receiving the signal. The agent can estimate the precision of the group opinion and decide whether to follow the majority side of the group. In the present framework, if agents observe the current information such that {xt > 0.5} or {xt < 0.5}, they update their prior beliefs according to Bayes’ rule. Then they must decide whether to follow the majority side of the population. + The posterior probability of event θ given the group opinion in a society, {xt > 0.5},is (x > . |θ+)q (θ+|x > . ) = Prob t 0 5 Prob t 0 5 + − Prob(xt > 0.5|θ )q + Prob(xt> 0.5|θ )(1 − q) (µ/( − µ))q = 1 . (µ/(1 − µ))q + (1 − q) Here, a central rule of probabilistic calculus is that the relative weight assigned to collective opinion is controlled by the degree of conﬁdence in the majority opinion. This is a consequence of Bayes’ rule. When µ increases, the relative weight that agents assign to the opinion of others increases. This paper analyzes the way µ affects the decentralized S. Teraji / Journal of Socio-Economics 32 (2003) 661Ð673 667 collective learning process. Similarly, it follows that:

− 1 − q Prob(θ |xt > 0.5) = , (µ/(1 − µ))q + (1 − q) + q Prob(θ |xt < 0.5) = , q + (µ/(1 − µ))(1 − q) − (µ/(1 − µ))(1 − q) Prob(θ |xt < 0.5) = . q + (µ/(1 − µ))(1 − q)

The agents, who observe the collective configuration of opinions such that {xt > 0.5} at t, + + − − choose brand A when Prob(θ |xt > 0.5)θ +Prob(θ |xt > 0.5)θ > 0. This is equivalent + − to the posterior odds ratio, Prob(θ |xt > 0.5)/Prob(θ |xt > 0.5), being strictly greater than −θ−/θ+, which is denoted by k defined above. That is, (µ/(1 − µ))(q/(1 − q))>k. Note that the above assumption that brand A is optimal under the prior beliefs (the prior odds ratio q/(1 − q) exceeds k) and that µ/(1 − µ) > 1. Thus the agents, who observe the group opinion such that {xt > 0.5} at t, will choose A with probability 1. Next, the agents, who observe the group opinion such that {xt < 0.5} at t, choose A + + − − when Prob(θ |xt < 0.5)θ + Prob(θ |xt < 0.5)θ > 0, which implies the posterior odds ratio (µ/(1 − µ))(q/(1 − q)) exceeds k defined above. It should be noted that this condition is equivalent to q/(1 − q)k > µ/(1 − µ) or: q/( − q)k 1 ≡ Λ>µ. 1 + q/(1 − q)k Then the degree of quality of opinions is smaller than a certain threshold Λ. Thus, when the information conveyed by the majority opinion is imprecise, each agent should always follow his or her own prior belief. The agents are not imitative at all. On the other hand, the agents will choose B when Λ<µ, i.e. the degree of quality of opinions is larger than a certain threshold Λ. Thus, if the signal is precise, each agent should follow the majority side of the group, not his own prior belief. The population is then more sensitive to the information conveyed by the group opinion. When µ is exactly Λ, agents are indifferent and hence randomize between the two choices. (If this happens, there is a chance that the agent will flip a coin to decide either to choose A or to choose B.) Furthermore, the agents, who observe the collective opinion {xt = 0.5} at t, will choose A with probability 1 because it follows that qθ+ + (1 − q)θ− > 0 in this setting. Let ψt ∈ [0, 1] be the probability that each agent chooses brand A at t. Then there are two different forms corresponding to {xt ≥ 0.5} and {xt < 0.5}. The form for {xt < 0.5} becomes complicated in this framework. Namely, we have:

ψt ={1}, where xt ≥ 0.5, (1) and   {1}, if Λ>µ ψt = [0, 1], if Λ = µ, where xt < 0.5. (2)  {0}, if Λ<µ The following proposition gives a summary of the results discussed above. 668 S. Teraji / Journal of Socio-Economics 32 (2003) 661Ð673

Proposition 1. Consider the situation in which ex ante brand A is better than brand B for each agent. (i) Suppose that the agents, given the opportunity to switch the action, observe the group opinion such that {xt > 0.5} at t. Then, they choose A with probability 1. (ii) Suppose that the agents, given the opportunity to switch the action, observe the group opinion such that {xt < 0.5} at t. Then, for a certain threshold Λ, if Λ>µ, they choose A with probability 1; if Λ<µ, they choose B with probability 1.

An implication of the above result is the following. For the group opinion such that {xt ≥ 0.5}, each agent adopts the action that is socially efficient. On the other hand, for the group opinion such that {xt < 0.5}, each agent will adopt the superior choice with a sufficiently small degree of quality of opinions; and with a sufficiently large degree of it, he or she will adopt the choice that is inefficient in the ex ante welfare sense. Thus, the decision making will depend on the relative estimation of the parameters. When µ increases, the importance of imitation increases in the population.

4. The collective conﬁguration of opinions

I consider the social learning system, where the behavioral patterns in the population evolve continuously over time. I analyze a deterministic dynamical system, in which the system variables are the population fractions that use A in each state of the world. A complete characterization is provided to determine the long-run behavior of the system. In particular, the analysis focuses on an equilibrium path that will converge to one of its endpoints, where the population exhibits “conformity.” In Banerjee (1992) and Bikhchandani et al. (1992), the agents are essentially in a line, the order of which is exogenously ﬁxed and known to all, and they are able to observe the binary actions of all the agents ahead of them. An information cascade is a sequence of decisions where it is optimal for agents to ignore their own preferences and imitate the decisions of all those who have entered ahead of them. This paper analyzes non-sequential situations, where agents are interacting simultaneously and modifying their decisions at each period in time. All the agents are always present and they revise their decisions in a continuous mode, and not once for all. Such a decision structure is better suited for the modeling of market situations. I consider how herding occurs in the process of collective decision making. Let us recall that, at every point in time, a fraction α of the agents decide to reevaluate their choice. Since a fraction (1−xt) of the agents are currently using B, the probability that they will choose A is given by ψtα(1 − xt). Similarly, a fraction xt of these are currently using A, the probability that they will choose B is given by (1 − ψt)αxt. The dynamics of the fraction of the population, in the continuous time, is then characterized by: dxt = ψtα(1 − xt) − (1 − ψt)αxt, dt for all t ∈ [0, ∞). This is the basic equation for the dynamics of social learning. Hence, for all t ∈ [0, ∞), it satisﬁes an equilibrium path from x0. S. Teraji / Journal of Socio-Economics 32 (2003) 661Ð673 669

From Eq. (1), the dynamic process, which corresponds to {xt ≥ 0.5}, is then determined by: dxt = α(1 − xt), (3) dt for all t ∈ [0, ∞). It is straightforward to show that this process has a degenerate stationary state x = 1, where all agents eventually adopt the same choice A. Similarly, from Eq. (2), the dynamic process, which corresponds to {xt < 0.5},is characterized by:   α(1 − xt), if Λ>µ dxt = [ − αxt,α(1 − xt)], if Λ = µ, (4) dt  −αxt, if Λ<µ for all t ∈ [0, ∞). It is straightforward to show that this process has a degenerate stationary state x = 1 (respectively, x = 0), where all agents eventually adopt the same choice A (respectively, B), when Λ>µ(respectively, Λ<µ) for a certain threshold Λ. The goal is to study the stability of the stationary states in the dynamical system deﬁned by Eqs. (3) and (4) and to demonstrate that two stationary states have different stability properties. Since there are generally multiple equilibrium paths from a given condition, I must be speciﬁc about what stability means. It is thus necessary to introduce some terminology.

Deﬁnitions. (i) x ∈ [0, 1] is accessible from x ∈ [0, 1], if there exists an equilibrium path from x that reaches or converges to x. (ii) x ∈ [0, 1] is globally accessible if it is accessible from any x ∈ [0, 1].

The second stability property, which I call globally accessible, states that, for any initial behavioral patterns, there exists an equilibrium path along which the behavioral patterns converge to a degenerate stationary state. For the dynamics corresponding to {xt ≥ 0.5}, it follows that ∂( x / t) d t d < , ∂x 0 t x=1 from Eq. (3). The path implies that xt → 1 for any xt ∈ [0.5, 1]. Thus, x = 1 is accessible from any x0 ∈ [0.5, 1). For the dynamics corresponding to {xt < 0.5}, it follows that, from Eq. (4), ∂( x / t) d t d < , ∂x 0 t x=1 when Λ>µ, and ∂( x / t) d t d < , ∂x 0 t x=0 670 S. Teraji / Journal of Socio-Economics 32 (2003) 661Ð673 when Λ<µ. The induced path is xt → 1 (respectively, xt → 0) for any x0 ∈ [0, 0.5) when Λ>µ(respectively, Λ<µ). Thus, x = 0 is accessible from any x0 ∈ [0, 0.5) when Λ<µ. Then there are multiple equilibrium paths (which converge to x = 1 and x = 0) with different equilibrium selection mechanisms. Furthermore, x = 1 is globally accessible when Λ>µbecause it is then accessible from any x0 ∈ [0, 1]. Then there is a unique equilibrium path of the behavioral patterns, which converges to x = 1. All agents will adopt the superior alternative in the limit. As a consequence, I can identify classes of environment for which the dynamic in the population is such that an homogeneous population arises in the limit. The following key proposition gives a formal summary of the results discussed above.

Proposition 2. Consider the situation in which ex ante brand A is better than brand B for each agent.

(i) x = 1 is accessible from any x0 ∈ [0.5, 1]. And for a certain threshold Λ, (ii) x = 0 is accessible from any x0 ∈ [0, 0.5) if Λ<µ. (iii) x = 1 is globally accessible if Λ>µ.

Thus, the system will exhibit two possible patterns of behavior in the long-run. First, there is efficient social learning, allowing convergence to the superior extreme (x = 1). Second, there is inefficient herding (x = 0) if everyone eventually adopts the same choice but the common choice is not optimal in the ex ante welfare sense. The process leads to inefficient information aggregation. This suggests why herd behavior may be undesirable from the social point of view. Proposition 2 shows how these regions arise. For {xt ≥ 0.5} at t, the force toward the superior extreme is overwhelming, and efficient social learning occurs. Then, even if the quality of opinions is at the lower level, the efficient state is globally accessible. A low µ implies that the agent might not have confidence in the majority view. The formation is strongly influenced by the initial conditions. On the other hand, for {xt < 0.5} at t, the two stationary states have different stability properties. The long-run behavior of the system is then determined by how a force pushing all agents toward using the same choice combines with the degree of quality of opinions µ. As a result, the system exhibits conformity toward the superior choice when agents have little confidence in the majority view, and exhibits inefficient herding when they have much confidence. These conditions show the possibilities that a society evolves toward efficient herding or inefficient herding. The equilibrium selection mechanism is then based on the presence of the herding externality. The equilibrium selection problem that arises is solved by agents who follow a sluggish social learning rule. Then the coordination problem, which may prevent the society from attaining the efficient equilibrium, may be alleviated by the reduction of confidence of the majority view. Thus, mass behavior is often fragile in the sense that the mere possibility of a value change can shatter herding. There are multiple equilibria for some parameter values, and the social learning system leads to fragility only when agents happen to balance at a knife-edge. S. Teraji / Journal of Socio-Economics 32 (2003) 661Ð673 671

5. Web herd behavior: an example

Digital auctions on the Internet present not only a new market place for transactions, but also a new domain for consumer decision making. Perhaps the most important factor is that a digital auction is typically a multi-stage process that involves multiple periods. A consumer decides whether to choose or enter a particular auction, which is often followed by a sequence of bidders. The online auction environment provides particular value cues that bidders can rely on. Consumers can update their value assessments based on others’ bids. It is important to note that reliance on others’ bids in online auctions may often lead buyers to overestimate the value of the auctioned item. Dholakia and Soltysinski (2001) provide evidence of the herding bias—many buyers tend to bid for items with existing bids, ignoring more attractive items within the same category. Susceptibility to the herding bias implies that the buyer may end up paying a higher price or winning a less favorable item than necessary. Furthermore, there is a possibility that an item, competitive in all respects, may remain unnoticed and fail to find buyers. Because of the herding bias, the buyers, who participate in these so-called efficient digital market places, routinely violate principles of consistency and make sub-optimal bidding decisions. Behavioral decision research has shown that in many instances, consumers are influenced by contextual informational cues when making choices and exhibit inconsistent preferences in different choice contexts (Simonson and Tversky, 1992). In fact, online bidders tend to have a concern for other persons and the outcomes derived from them. When bidding in a digital auction, others’ preceding behavior may provide valuable information and may be perceived as having greater credibility than seller-originating content such as descriptions or pictures. The process of social identification with other bidders may further increase the influence of this cue. Such a spiraling escalation may magnify the bias, elevating the inferior item’s market value furiously. This situation corresponds to a self-reinforcing process of confidence in the majority opinion. In this situation, each person’s decision is more responsive to the majority opinion. The reduction of confidence in the majority view may be socially beneficial in an ex ante welfare sense. The society may be better off by encouraging the consumers to use their own information, not joining the herd.

6. Concluding remarks

Many studies clearly explain how increasing returns to adoption can lead potential adopters to a situation of lock-in. The sources of lock-in are well established, and are principally network externalities, informational increasing returns, technological interrelat- edness and evaluation norms. However, the interaction between the agents and the resulting aggregate phenomena is often not clear. One feature of economic activity is the tendency for agents to form coalitions. The aggregate behavior is the result of the interaction between individuals and coalitions. A good example of this sort of approach is the analysis proposed by De Vany (1996) who examines the emergence of self-organized coalitions among decentralized agents playing a network coordination game. The network evolves to optimal or sub-optimal coalition structures for 672 S. Teraji / Journal of Socio-Economics 32 (2003) 661Ð673 some parameter settings. With some settings, the system freezes on sub-optimal states. But noisy evolution of coalitions, which is implemented by a Boltzman network, can overcome lock-in and reach global optima by leaping to new paths. The “temperature” parameter effectively adds “noise” to the signals reaching the agent, and it exponentially increases the paths over which the system may evolve. We can see it as analogous to the model presented in this paper. The present paper has identified the long-run properties of herd behavior in the economic environment with multiple equilibria. In the model, there are two equilibrium patterns of choices; one is efficient and the other is inefficient in the ex ante welfare sense. This paper has found the relation between the degree of quality of opinions and the properties of equilibrium. With the small degree of quality of opinions, the efficient equilibrium is unique and globally accessible in the model. That is, as long as the signal is imprecise, the system can escape sub-optimal states. However, for sufficient ranges of quality of opinions, there are multiple steady states. In one of them, there is inefficient herding in the long-run. High signal accuracy gives too much credibility to the majority opinion in a society. Each person’s decision is then more responsive to the majority view. This is the self-reinforcing nature of confidence in the majority opinion. The reduction of confidence in the majority view has the serious consequences in terms of ex ante welfare. This is essential in leading the whole society away from inefficient herding. Because of the simplicity of this model, I have assumed that the confidence an agent attaches to the other agent’s opinions is uniform in a society. It may be more realistic to consider the quality of opinions as private information. Furthermore, I have assumed that the opportunities to switch actions arrive at random: a more natural assumption is to assume that agents can decide when to decide. Waiting incurs some costs but it may allow an agent to make a better decision in a later period of time.10 Finally, choice decisions made by an individual depend crucially on the perceptions of choice objects. To the extent that such perceptions are affected by social elements, they cannot be independent of a particular social environment, in which decisions are made. An individual participates in the economy not simply as an economic abstract with idiosyncratic tastes but as a whole person with a variety of social concerns and motivations.

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10 See, for strategic delays caused by information externalities, Chamley and Gale (1994). S. Teraji / Journal of Socio-Economics 32 (2003) 661Ð673 673

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