Foundations

Abstract

In this chapter explains some fundamental concepts used in this book. The concepts are as follows: social norms; economic behavior; rationality; cognition; institutions; path dependence; and institutional change. An understanding of social norms is critical to predict and explain human behavior. People incorporate in themselves a set of social norms from their surroundings. Norms govern behavior, and are self-sustaining in an interdependent system. Norms specify a limited range of behavior that is acceptable in a situation, and facilitate confidence in the course of action. Norms enable individuals to deal with the complexity and incompleteness of information, and make them stick to prescribed behavior. Norms, thus, describe the uniformities of behavior that characterize groups.

Appendices

Appendix 2 is based on Teraji (2003).

Appendix 1

1.1.1 Hyperbolic Discounting

Most people complain that they do not have enough of self-control. Self-control problems exist when a choice is between a currently available good and an option that requires bearing a current cost associated with a deferred benefit. Agents have a tendency to succumb to short-run impulses at the expense of their long-run interests. They tend to over-consume goods that provide an immediate reward but affect negatively their future welfare, through a deterioration of health (drinking, smoking, overeating, etc.) or a decrease in wealth (compulsive credit card purchases, consumption of luxury goods, gambling, etc.). That is, the individual’s current ‘self’ overweighs the present relative to the future.

Inter-temporal choices are decisions with consequences that play out over time. The standard economic theory has analyzed inter-temporal decision-making by using the discounted utility model, which assumes that people evaluate the pleasures and pains resulting from a decision in much the same way that financial markets evaluate the gains and losses. The discounted utility model involves an agent who chooses a feasible consumption plan to maximize his or her present exponentially discounted utility. The critical feature of exponential discounting is that it preserves dynamic consistency. Inter-temporal choices are not different from any other type of choice except that some consequences are delayed. Psychologists have questioned the validity of the assumption of exponential discounting on the basis of experimental evidence (Ainslie 1975; Kirby and Herrnstein 1995). Events in the near future tend to be discounted at a higher rate than events that occur in the long run. Self-control and procrastination are now topics of growing interests to behavioral economic theorists (Laibson 1997; O’Donoghue and Rabin 1999a, 1999b). The behavioral economic approach has criticized the traditional economic approach, suggesting an alternative specification of discounting, that is, hyperbolic discounting.

The reversal of preferences is called time inconsistency. A person may reverse the decisions formulated in the earlier stage. An individual who is offered a choice today of receiving $50 in 100 days or $60 in 101 days will take the $60, but when the options are moved forward, so that he or she can either have the $50 immediately or $60 tomorrow, he or she may choose the $50. Subjects exhibit a reversal of preferences when choosing between a smaller-sooner reward and a larger-later one in experiments. The earlier reward is preferred when it offers an immediate payoff whereas the latter reward is preferred when both rewards are received with delay. Time-inconsistent preferences are proposed by Stroz (1956), and further developed by Phelps and Pollak (1968).¹⁴ Time inconsistency implies that an ex ante optimal decision is not carried out because a latter reevaluation suggests that it is not optimal anymore.

Since Stroz’s (1956) analysis, economists have known that inter-temporal choices are time-consistent only if agents discount exponentially using a discount rate that is constant over time. People are more impatient when they make short-run tradeoffs than when they make long-run tradeoffs. Hence, the rates of time preferences would be very high in the short run but much lower in the long run, as viewed from today’s perspective. The contrast between short-run impatience and long-run patience has been modeled using hyperbolic discounting. Hyperbolic discounting rationalizes the preferences for present consumption as a form of time inconsistency. This creates a conflict between an individual agent’s preferences at different points in time. Ainslie (2001) explains violations of delay-independence in terms of the steepness of a function which discounts the utility of a reward.¹⁵ In contrast to exponential discount curves, hyperbolic discount curves depict a strong but temporary tendency to prefer a smaller-sooner reward to a larger-later one in the period just before the smaller-sooner reward is due. The time-inconsistency approach postulates that people may have a variety of selves that become dominant at different points because of their timing. For example, before the party, one self favors a strict limit of two beers during the evening, but at the party, having already consumed the two beers, the current self feels certain of the desirability of having just one more beer. An agent who discounts rewards hyperbolically is not a straightforward value estimator, but a succession of estimators whose conclusions differ. The hyperbolic discounter is then involved in a decision that has ‘intra-personal’ strategic dimensions. Individual choices are taken to be the outcome of an intra-personal game in which the same individual agent is represented by a different player at every date. Intra-personal games are played between the agent’s temporally situated selves. A variety of conflicting reward-getting processes will grow and survive within the individual, sometimes leading to harmful choices in the long run.

In formal discounting models, a consumer’s welfare can be represented as a discounted sum of current and future utilities. The model assumes that, at each point in time t, the agent consumes goods c _t . The subjective value to the consumer is given by a (instantaneous) utility function u(c _t ), which translates the consumption measure, c _t , into a single measure of utility at period t. The discount rate measures the rate of decline of the discount function. The discount function measures the value of utility, as perceived from the present, at each future time period. Exponential discount functions have a constant discount rate. Specifically, a util delayed τ periods is worth δτ as much as a util enjoyed immediately (τ = 0). Here, δ is assumed to be less than one (future utils are worth less than current utils). Exponential discount functions also have the property of not generating preference reversals. For a time-consistent agent, discounted utility at period t takes the familiar form:

$$ {\displaystyle \sum}_{\tau =0}^{\infty }{\delta}^{\tau }u\left({c}_{i+\tau}\right). $$

(1.1)

An agent who discounts exponentially maximizes his or her inter-temporal utility in expression (1.1). The standard economic procedure to compare outcomes at different times, introduced by Samuelson (1937), is to discount them at a constant rate and sum them up. The standard economic analysis requires that devaluation occurs as an exponential function of decay. The weight at which future costs and benefits influence today’s decisions declines exponentially over time. If delayed outcomes are devaluated exponentially, the relative preference of expected future outcomes does not change as the individual moves closer in time to those outcomes. Exponential discounting preserves time consistency. Time consistency means that the future actions required to maximize the current value of utility remain optimal in the periods when the actions are to be taken.

Most experimental evidence suggests that the discount rate declines with the length of the delay horizon. When two rewards are both far away in time, people act relatively patiently. But when both rewards are brought forward in time, preferences exhibit a reversal, reflecting more impatience. The actual discount function declines at a greater rate in the short run than in the long run. The agent behaves given the predicted behavior of his or her subsequent selves. Such discounting patterns play a role in generating self-control problems.¹⁶

An individual is modeled as a sequence of autonomous temporal selves. Temporal selves are indexed by periods t = 0, 1, 2, … Self t’s discounted utility from present and future consumption is:

$$ u\left({c}_i\right)+\beta {\displaystyle \sum}_{\tau =1}^{\infty }{\delta}^{\tau }u\left({c}_{i+\tau}\right), $$

(1.2)

where β is assumed to be between zero and one. The parameter β < 1 represents the ‘bias for the present.’ For β = 1, these preferences are time-consistent. Laibson (1997) calls the discount structure in self t’s lifetime utility ‘quasi-hyperbolic.’ An agent who discounts quasi-hyperbolically maximizes his or her inter-temporal utility in expression (1.2). The quasi-hyperbolic discount function is a discrete time function with values {1, βδ, βδ ², βδ ³, ...} (Phelps and Pollak 1968). The discount factor between consecutive future periods (δ) is larger than between the current period and the next one (βδ). Since β is taken to be less than one, a short-term discount factor is less than the long-term discount factor. In this functional form, the rate of substitution between today and tomorrow is smaller than that between any other pair of successive periods. Self t uses the lifetime utility function to evaluate the stream of payments from t onward. This implies that time inconsistency since δ (the marginal rate of substitution between t and t + 1 from the point of view of any previous period) is replaced by βδ at t. The quasi-hyperbolic discount function is only ‘hyperbolic’ in the sense that it captures the key qualitative property of the hyperbolic functions: a faster rate of decline in the short run than in the long run. Inter-temporal tradeoffs shift with the mere passage of time, and plans or projects that seemed optimal yesterday need no longer be optimal today. In the original psychological research, hyperbolic discount functions are like 1/τ and 1/(1+ατ) with α > 0, which generalizes simpler hyperbolas.

The dynamic inconsistency forces the hyperbolic consumer to grapple with intra-personal strategic conflict. Let us simply consider a three-period hyperbolic model, the shortest possible that actually generates time inconsistency effect. The periods are labeled 0, 1, 2, and subscripts on c refer to the period in question. In period 0, self 0’s discounted utility is:

$$ u\left({c}_0\right)+\beta \delta u\left({c}_1\right)+\beta {\delta}^2u\left({c}_2\right), $$

(1.3)

while in period 1, self 1’s discounted utility is:

$$ u\left({c}_1\right)+\beta \delta u\left({c}_2\right). $$

(1.4)

For self 0, the discount factor between period 0 and period 1 is βδ, and between period 1 and period 2 it is δ. Self 1’s discount factor between period 1 and period 2 is βδ. Therefore, there is a conflict between different selves about how much to consume in a given period. The consumer is dynamically inconsistent when β differs from one. When β = 1, we are back to the case of a dynamically consistent consumer, with exponential discounting. This hyperbolic model captures a specific type of self-control problem. Since the discount factor self 0 applies between periods 1 and 2 (or δ) is greater than the discount factor self 1 applies between the same two periods (or βδ), self 0 would like to behave more patiently in period 1 than self 1 actually does. Self 0 benefits from self-control that brings period-1 behavior in line with his or her wishes. Since each self consumes too much from earlier selves’ point of view, each of them would agree to increase savings a little bit in exchange for later selves doing the same. All inter-temporal selves could be made better off if all of them saved a little bit more. By making the implicit parameter restriction β = 1, standard economic models assume that no such self-control problem exists.

A single individual is modeled as many separate selves, one for each period. The future selves will control his or her future behavior. There is an important problem whether a person has beliefs about how his or her future selves will behave.

A ‘sophisticated’ person knows exactly what his or her future selves’ preferences will be. The sophisticated consumer chooses c ₀ to maximize (1.3), knowing that in period 1 the consumption levels c ₁and c ₂ will be chosen by self 1 with the discounted utility given by (1.4). Thus, self 0 takes actions that seek to constrain self 1. Sophisticates realize that they have hyperbolic preferences, and are fully aware of their future self-control problems. However, self 0 might expect self 1 to carry out the wishes of self 0. Consumers might make current choices under the false beliefs that future selves will act in the interests of the current self.

A ‘naive’ person believes that future selves’ preferences will be identical to those of the current self. Naives are fully unaware of their future self-control problems. The hyperbolic consumer may or may not be aware that his or her preferences will change over time. O’Donoghue and Rabin (1999a) make a comparison between these two extreme types.¹⁷ Actual behavior is likely best described by something between naives and sophisticates, not just the two extremes. Self 0 may believe with certainty that the discount factor between periods 1 and 2 is β*δ. The parameter β ^* reflects self 0’s beliefs about his or her future β, so that β ^* = β corresponds to sophisticates, and β ^* = 1 corresponds to naives.

Procrastination occurs when present costs are unduly salient in comparison with future costs, leading individuals to postpone tasks until tomorrow without foreseeing that when tomorrow comes, the required action will be delayed yet again (Akerlof 1991). Akerlof (1991) implicitly corresponds to a model of time-inconsistent procrastination, highlighting the role of naive beliefs. Hyperbolic agents procrastinate because they think that whatever they will be doing later will not be as important as what they are doing now. O’Donoghue and Rabin (1999b) explore procrastination in terms of time-inconsistent preferences. Time-consistent agents do not procrastinate. Time-inconsistent agents, on the other hand, procrastinate.¹⁸

1.1.2 Addiction

It is often difficult to wait for a delayed reward when an immediately gratifying alternative is available. This is related to theories of addiction. Although people initially decide to take a far-sighted course of action (quitting smoking), they subsequently succumb to temptation. One element common to addictions is the addict’s inability to escape when he or she wants to. The immediacy of reward associated with smoking clearly has some importance in understanding why quitting can be difficult. The rewards from smoking are immediate and the adverse consequences tend to be delayed. People who become addicted may get greater (or longer) pleasure from their substances, and may also discount future rewards more sharply. Self-control can be defined as efforts made by the individual to avoid or resist behaving inconsistently.

Becker and Murphy (1988) study consumption of an addictive good in a standard exponential discounting model. Exponential discount functions predict consistency of preferences over time. A single utility function describes both the addict’s and the nonaddict’s behavior. Rational addiction means that the addict’s behavior, as well as the nonaddict’s behavior, maximizes utility in the long run according to some particular utility function. In Becker and Murphy’s (1988) model, people are aware of the addictive nature of cigarette smoking, and choose to smoke simply because the lifetime benefits are greater than the costs. Greater past consumption of addictive goods increases the desire for present consumption (‘reinforcement’). The consumption of addictive goods at different times are complements. Consumption today increases the likelihood of consumption tomorrow, and the fear of addiction may generate conducts that resemble compulsive behavior. An increase in either past or expected future prices decreases current consumption. The price-theoretic framework yields the prediction that addicts will respond much more to permanent than to temporary price changes. Becker and Murphy (1988) show that the existence of multiple stable steady-state equilibria where agents either abstain or consume substantially.

Gruber and Köszegi (2001) incorporate time-inconsistent preferences into Becker and Murphy’s (1988) rational addiction model. Forward-looking behavior does not imply time consistency. The purpose of Gruber and Köszegi (2001) is to show that a time-inconsistent model, where agents are forward-looking but have time-inconsistent preferences, can generate the Becker and Murphy’s (1988) prediction that current consumption depends positively upon future consumption. An implication of time-inconsistent preferences is that individuals, who realize their self-control problems, have a demand for commitment devices.

Let a _t and c _t denote the consumption of the addictive and the ‘ordinary’ (nonaddictive) goods in period t, respectively. Furthermore, let S _t denote the period t stock of past consumption (the stock of ‘addictive capital’). S _t depends upon both past consumption of a and life cycle events, and it evolves as follows:

$$ {S}_{i+1}=\left(1-\eta \right)\left({S}_i+{a}_i\right), $$

where η is the depreciation rate of the stock. Instantaneous utility is additively separable in these two goods:

$$ {U}_i=U\left({a}_i,{c}_i,{S}_i\right)=v\left({a}_i,{S}_i\right)+u\left({c}_i\right). $$

There are two extreme kinds of hyperbolic discounters: naives and sophisticates. This can be modeled as a sequential game played by the successive inter-temporal selves. For a naive agent, self t does not realize that self t + 1 will in turn overvalue period t + 1. A naive agent maximizes (1.2), unconscious of the fact that his or her future selves will change the plans. On the other hand, for a sophisticated agent, self t knows that self t + 1 will want to do something other than what self t would have him or her do. Sophisticated agents have a demand for self-control devices if they want to quit.¹⁹

In the context of hyperbolic discounting, players are not separate individuals, but they are different inter-temporal incarnations of the same individual. This form of discounting sets up a conflict between the preferences of different inter-temporal selves. The agent’s long-run preferences are taken as those relevant for social welfare maximization. In the standard rational addiction model, the optimal tax on addictive goods should depend only on the externalities that their use imposes on society. Sophisticates try to influence the future selves through their current consumption of addictive goods. They feel a need to exert control on the future selves by consuming less. This effect helps the government. Smokers who have an intention to quit smoking will support anti-smoking public policies. In the time-inconsistent alternative model, optimal government policy should also depend on the ‘internality’ that consumers impose on themselves. Due to the existence of internality, there is room for government intervention, for example, a tax hike could be welfare improving.

We can think about price interventions of the following form. The hyperbolic consumer lives for three periods, t = 0, 1, 2. Consumption only occurs in period 1. An adjustment is added to the per-unit price that the consumer faces in period 1. In that period, the consumer has to make a choice between goods a and c. Good a may be pleasant to consume in period 1, but it causes harm in period 2. Consumption of one unit of good a leads to harm h > 0 in period 2. Good c is an ordinary good that is pleasant to consume at the moment and that has no implications for future utility. Normalize the price of good c in period 1 to 1.

We consider the per-period utility functions as follows:

$$ {u}_t=\Big\{\begin{array}{cc}0& \mathrm{in}\ \mathrm{period}\ t=0\\ {}v(a)+c& \mathrm{in}\ \mathrm{period}\ t=1\\ {}- ha& \mathrm{in}\ \mathrm{period}\ t=2\end{array} $$

Self 0’s discounted utility function is:

$$ {u}_0+\beta \delta {u}_1+\beta {\delta}^2{u}_2, $$

while self 1’s discounted utility function is:

$$ {u}_1+\beta \delta {u}_2. $$

The parameters β and δ are between 0 and 1. Then, self 0 would like self 1 to discount period 2 relative to period 1 by a factor of δ. Hence, self 0 prefers that self 1 counts the future implications of consuming each unit of good a at a value of δh. However, self 1’s discount factor between period 1 and period 2 is βδ. Self 1 only counts the future implications of consuming a at a value of βδh. To align self 1’s interests with those of self 0, we can impose a price adjustment of δh – βδh = (1– β) δh per unit of good a. Due to this change in price, self 1 is forced to fully internalize the future harm of consumption.

In hyperbolic models, a person discounts heavily between the present and all future periods, but does not discount so heavily between future periods. Following Kan (2007), let us consider a simple model of decision-making by a smoker living for a long period. In the model, a smoker faces two choices: to continue smoking or to quit. The two choices yield different paths of lifetime utilities:

$$ \left\{\mathrm{S,S,S,} \dots \right\},\mathrm{if}\ \mathrm{a}\ \mathrm{smoker}\;\mathrm{continues}\;\mathrm{smoking}, $$

and

$$ \left\{\mathrm{Q,N,N,} \dots \right\},\mathrm{if}\ \mathrm{a}\ \mathrm{smoker}\;\mathrm{quits}\;\mathrm{smoking}, $$

where S, Q, and N are per-period utilities, with Q < S < N. The utility in the period when a smoker quits, Q, is assumed to be less than that if he or she smokes in that period, S (it is the disutility from quitting). S < N is due to the fact that smoking is harmful.

A smoker quits in the current period if:

$$ \begin{array}{l}\mathrm{Q}+\beta {\displaystyle \sum}_{t=1}^{\infty }{\delta}^t\mathrm{N}>\mathrm{S}+\beta {\displaystyle \sum}_{t=1}^{\infty }{\delta}^t\mathrm{S},\hfill \\ {}\mathrm{S}-\mathrm{Q}<\frac{\beta \delta }{1-\delta}\left(\mathrm{N}-\mathrm{S}\right).\hfill \end{array} $$

Thus, the cost of quitting is smaller than the lifetime gain from not smoking.

Instead of quitting in the current period, a smoker may consider quitting in the next period. From the perspective of the current period, he or she plans to quit in the next period if

$$ \begin{array}{l}\beta \mathrm{Q}+\beta {\displaystyle \sum}_{t=1}^{\infty }{\delta}^t\mathrm{N}>\beta \mathrm{S}+\beta {\displaystyle \sum}_{t=1}^{\infty }{\delta}^t\mathrm{S},\hfill \\ {}\mathrm{S}-\mathrm{Q}<\frac{\delta }{1-\delta}\left(\mathrm{N}-\mathrm{S}\right).\hfill \end{array} $$

Since β < 1, a smoker is more likely to plan to quit in the next period than to actually quit in the current period.

If the following inequality holds, the individual is a perpetual procrastinator.

$$ \frac{\beta \delta }{1-\delta}\left(\mathrm{N}-\mathrm{S}\right)<\mathrm{S}-\mathrm{Q}<\frac{\delta }{1-\delta}\left(\mathrm{N}-\mathrm{S}\right). $$

When the above holds, a smoker will not quit in the current period, but will plan to quit in the next period. However, when the next period arrives, the smoker will again postpone the plan to quit for one more period. This postponing will continue forever.

Appendix 2: Herd Behavior and the Quality of Opinions

Appendix 2 is based on Teraji (2003).

1.1.1 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 interpersonal contact. Then, information externalities arise that drive toward the emergence of some patterns of influences among individuals.²⁰

Recently, there has been increasing interest in economic models in the presence of information externalities, which has put 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.²¹ 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.²²

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.²³ 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 appendix, 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 appendix 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 exogenously ordered individuals, each acting sequentially under the state of the world.²⁴ 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 that the 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 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 ‘nonsequential’ contexts in which 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 manner, not holding one opinion for all decisions. In his 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 appendix, 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.²⁵ It is worth offering an equilibrium selection criterion of how a particular equilibrium will emerge collectively in a nonsequential context.

In this appendix, 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.²⁶ 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 situations, 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 configurations of opinions, possess different stability properties when people 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.²⁷ 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, for example, Keynesian macroeconomics in Cooper and John (1988) and economic development in Murphy et al. (1989).²⁸ 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 the 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. We 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 basic framework is presented in section “The Framework” in Appendix 2. I deal with individual Bayesian behavior in section “Behavior” in Appendix 2. The decision rule itself will change if we change the quality of opinions. Section “The Collective Configuration of Opinions” in Appendix 2 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 “Web Herd Behavior: An Example” in Appendix 2 offers an example, web herd behavior. Section “Concluding Remarks” in Appendix 2 concludes by discussing some implications of the main results.

1.1.2 The Framework

There is a continuum of identical agents who are faced with a choice between two alternative products, technologies and practices, labeled by A and B. The decision-making is ‘nonsequential’; 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, or Rational versus Behavioral. Even within the field of economic theory, we often debate 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 these 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 u ^A or u ^B be the payoff to each agent’s choice A or B. I suppose that (u ^A − u ^B) has two possible values, where θ ⁺ > 0 > θ ⁻. Agents do not know the true value of (u ^A − u ^B). Each agent assigns common prior probability q > 0.5 to the event (u ^A − u ^B) = θ ⁺. The value q is assumed to be given. Furthermore, I suppose that qθ ⁺ + (1 − q)θ ⁻ > 0, that is, 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 x _t , 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 x ₀ 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 x _t . An agent receives a signal about the state of the world by observing the group opinion in a society. The information need not, of course, be true, and it may be false. However, each agent believes that the majority side in the population, {x _t > 0.5} or {x _t < 0.5}, may convey some implicit information about the realization of (u ^A − u ^B). Then the agents incorporate observations of the relative popularity of the two choices in the decision-making. The agents weigh observations of others’ experience because others’ decisions might reflect the information that they have and he or she does not.

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, {x _t > 0.5} or {x _t < 0.5}, at every point in time, is linked to the realization of (u ^A − u ^B) through the following conditional probabilities:

$$ \mathrm{Prob}\left({x}_t>0.5\Big|{\theta}^{+}\right)=\mu, \kern2em \mathrm{Prob}\left({x}_t>0.5\Big|{\theta}^{-}\right)=1-\mu, $$

$$ \mathrm{Prob}\left({x}_t<0.5\Big|{\theta}^{+}\right)=1-\mu, \kern2em \mathrm{Prob}\left({x}_t<0.5\Big|{\theta}^{-}\right)=\mu . $$

Within the present framework, {x _t > 0.5} (respectively, {x _t < 0.5}) is better correlated with the state θ ⁺ (respectively, θ ⁻) than with θ ⁻ (respectively, θ ⁺), that is, μ > 1 − μ. The closer μ is to 1, the more precise the 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 high as we like. Furthermore, I assume that Prob(x _t = 0.5|θ ⁺) = Prob(x _t = 0.5|θ ⁻) = 0.5, that is, the group opinion such that {x _t = 0.5} is uncorrelated with the states. The task is to determine exactly how these factors influence each agent’s decision-making. In section “Behavior” in Appendix 2, I examine how the quality of opinions affects the decision rule itself. What I try to understand in section “The Collective Configuration of Opinions” in Appendix 2 is how the collective interactions will affect the formation of the group opinion in the social learning process.

1.1.3 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 {x _t > 0.5} or {x _t < 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 the event θ ⁺ given the group opinion in a society, {x _t > 0.5}, is:

$$ \begin{array}{l}\mathrm{Prob}\left({\theta}^{+}\Big|{x}_t>0.5\right)\hfill \\ {}=\frac{\mathrm{Prob}\left({x}_t>0.5\Big|{\theta}^{+}\right)q}{\mathrm{Prob}\left({x}_t>0.5\Big|{\theta}^{+}\right)q+\mathrm{Prob}\left({x}_t>0.5\Big|{\theta}^{-}\right)\left(1-q\right)}\hfill \\ {}=\frac{\left(\mu /\left(1-\mu \right)\right)q}{\left(\mu /\left(1-\mu \right)\left)q+\right(1-q\right)}.\hfill \end{array} $$

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 collective learning process. Similarly, it follows that:

$$ \begin{array}{l}\mathrm{Prob}\left({\theta}^{-}\Big|{x}_t>0.5\right)=\frac{1-q}{\left(\mu /\left(1-\mu \right)\left)q+\right(1-q\right)},\hfill \\ {}\mathrm{Prob}\left({\theta}^{+}\Big|{x}_t<0.5\right)=\frac{q}{q+\left(\mu /\left(1-\mu \right)\right)\left(1-q\right)},\hfill \\ {}\mathrm{Prob}\left({\theta}^{-}\Big|{x}_t<0.5\right)=\frac{\left(\mu /\left(1-\mu \right)\left)\right(1-q\right)}{q+\left(\mu /\left(1-\mu \right)\right)\left(1-q\right)}.\hfill \end{array} $$

The agents, who observe the collective configuration of opinions such that {x _t > 0.5} at t, choose brand A when Prob(θ ⁺|x _t > 0.5)θ ⁺ + Prob(θ ⁻|x _t >0.5)θ ⁻ > 0. This is equivalent to the posterior odds ratio, Prob(θ ⁺|x _t > 0.5)/Prob(θ ⁻|x _t > 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 {x _t > 0.5} at t will choose A with probability 1.

Next, the agents, who observe the group opinion such that {x _t < 0.5} at t, choose A when Prob(θ ⁺|x _t < 0.5)θ ⁺+Prob(θ ⁻|x _t < 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

$$ \frac{q/\left(1-q\right)k}{1+q/\left(1-q\right)k}\equiv \varLambda >\mu . $$

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, that is, 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 {x _t = 0.5} at t, will choose A with probability 1 because it follows that qθ ⁺ + (1−q)θ ⁻ > 0 in this setting.

Let ψ _t ∈[0,1] be the probability that each agent chooses brand A at t. Then there are two different forms corresponding to {x _t ≥ 0.5} and {x _t < 0.5}. The form for {x _t < 0.5} becomes complicated in this framework. Namely, we have:

$$ {\psi}_t=\left\{1\right\},\mathrm{where}{x}_t\ge 0.5, $$

(1.5)

and

$$ {\psi}_t=\Big\{\begin{array}{cc}\left\{1\right\},& \mathrm{if}\ \varLambda >\mu \\ {}\left[0,1\right],& \mathrm{if}\ \varLambda =\mu \\ {}\left\{0\right\},& \mathrm{if}\ \varLambda <\mu \end{array},\kern2em \mathrm{where}{x}_t<0.5. $$

(1.6)

The following proposition gives a summary of the results discussed above.

Proposition 1.1

[Teraji (2003)]. Consider the situation in which ex ante brand A is better than brand B for each agent.

• (1) Suppose that the agents, given the opportunity to switch the action, observe the group opinion such that {x _t > 0.5} at t. Then, they choose A with probability 1.

• (2) Suppose that the agents, given the opportunity to switch the action, observe the group opinion such that {x _t < 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 {x _t ≥ 0.5}, each agent adopts the action that is socially efficient. On the other hand, for the group opinion such that {x _t < 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.

1.1.4 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 appendix analyzes nonsequential 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−x _t ) of the agents are currently using B, the probability that they will choose A is given by ψ _t α(1−x _t ). Similarly, a fraction x _t of these are currently using A, the probability that they will choose B is given by (1−ψ _t )αx _t .

The dynamics of the fraction of the population, in the continuous time, is then characterized by:

$\frac{\mathrm{d}{x}_t}{\mathrm{d}t}={\psi}_t\alpha \left(1-{x}_t\right)-\left(1-{\psi}_t\right)\alpha {x}_t,$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 x ₀.

From (1.5), the dynamic process, which corresponds to {x _t ≥ 0.5}, is determined by:

$$ \frac{\mathrm{d}{x}_t}{\mathrm{d}t}=\alpha \left(1-{x}_t\right), $$

(1.7)

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 (1.6), the dynamic process, which corresponds to {x _t < 0.5}, is characterized by:

$$ \frac{\mathrm{d}{x}_t}{\mathrm{d}t}=\Big\{\begin{array}{cc}\alpha \left(1-{x}_t\right),& \mathrm{if}\ \varLambda >\mu \\ {}\left[-\alpha {x}_t,\alpha \left(1-{x}_t\right)\right],& \mathrm{if}\ \varLambda =\mu \\ {}-\alpha {x}_t,& \mathrm{if}\ \varLambda <\mu \end{array}, $$

(1.8)

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 (1.7) and (1.8), 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 necessary to introduce some terminology.

Definitions.

• (1) x ∈[0, 1] is accessible from x′ ∈[0, 1], if there exists an equilibrium path from x′ that reaches or converges to x.

• (2) 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 {x _t ≥ 0.5}, it follows that:

$$ \frac{\partial \left(\mathrm{d}{x}_t/\mathrm{d}t\right)}{\partial {x}_t}\Big|{}_{x=1}<0, $$

from (1.7). The path implies that x _t → 1 for any x _t ∈[0.5, 1]. Thus, x = 1 is accessible from any x ₀ ∈[0.5, 1).

For the dynamics corresponding to {x _t < 0.5}, it follows that, from (1.8),

$$ \frac{\partial \left(\mathrm{d}{x}_t/\mathrm{d}t\right)}{\partial {x}_t}\Big|{}_{x=1}<0, $$

when Λ > μ, and

$$ \frac{\partial \left(\mathrm{d}{x}_t/\mathrm{d}t\right)}{\partial {x}_t}\Big|{}_{x=0}<0, $$

when Λ < μ. The induced path is x _t → 1 (respectively, x _t → 0) for any x ₀ ∈[0, 0.5) when Λ > μ (respectively, Λ < μ). Thus, x = 0 is accessible from any x ₀ ∈[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 > μ because it is then accessible from any x ₀ ∈[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 1.2

[Teraji (2003)]. Consider the situation in which ex ante brand A is better than brand B for each agent.

• (1) x = 1 is accessible from any x ₀ ∈[0.5, 1]. And for a certain threshold Λ,

• (2) x = 0 is accessible from any x ₀ ∈[0, 0.5) if Λ < μ.

• (3) 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 1.2 shows how these regions arise. For {x _t ≥ 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 {x _t < 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.

1.1.5 Web Herd Behavior: An Example

Digital auctions on the Internet present not only a new marketplace for transactions, but also a new domain for consumer decision-making. Perhaps the most important factor is that a digital auction is typically a multistage 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.

1.1.6 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 interrelatedness, 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 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 appendix.

The model has identified the long-run properties of herd behavior in the economic environment with multiple equilibria. In this appendix, there are two equilibrium patterns of choices; one is efficient and the other is inefficient, in the ex ante welfare sense. The model 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 situation. 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 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 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.²⁹

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.