The Implications of Pricing on Social Learning

05/09/2019
by   Itai Arieli, et al.
0

We study the implications of endogenous pricing for learning and welfare in the classic herding model . When prices are determined exogenously, it is known that learning occurs if and only if signals are unbounded. By contrast, we show that learning can occur when signals are bounded as long as non-conformism among consumers is scarce. More formally, learning happens if and only if signals exhibit the vanishing likelihood property introduced bellow. We discuss the implications of our results for potential market failure in the context of Schumpeterian growth with uncertainty over the value of innovations.

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1 Introduction

In many markets of substitute products, the value of the various alternatives may depend on some unknown variable. This may take the form of some future change in regulation, a technological shock, environmental developments, or prices in related upstream markets. Although this information is unknown, individual consumers may receive some private information about these fundamentals. We ask whether, in such an environment, markets aggregate information correctly and the ex-post superior product eventually dominates the market.

In this work we focus on the role of social learning in such environments. We study whether the learning process guarantees an efficient outcome. We isolate the role of learning by introducing a simple duopoly model of common value in which consumers, with a unit demand, choose between two substitute products, each with zero marginal cost of production. The timing of the interaction is as follows. Nature randomly chooses one of two states of nature, and so determines the identity of the firm with the superior product. At each stage the two firms observe the entire history of the market – past prices and consumption decisions – and simultaneously set prices. Thereafter, a single consumer arrives and receives a private signal regarding the state of nature. The consumer makes his consumption decision based on his signal, the pair of prices for each product, and the entire history of prices and consumption decisions. The consumer can also choose to opt out of buying any product. Our main goal is to identify conditions under which asymptotic learning holds; that is, information is fully aggregated in the market asymptotically.

When prices are set exogenously and fixed throughout, the above model is exactly the standard herding model [6, 8]. In that model, as shown by Smith and Sørensen [34]

, the characterization of asymptotic learning crucially depends upon the quality of agents’ private signals. In particular, one must distinguish between two families of signals: bounded versus unbounded. In the unbounded case the private beliefs of the agents are, with positive probability, arbitrarily close to zero and one. Therefore, no matter how many people herd on one of the alternatives, there is always a positive probability that the next agent will receive a signal that will make him break away from the herd toward the other alternative. This property, as shown by Smith and Sørensen

[34], entails asymptotic learning. The same logic applies in our model as well: when signals are unbounded, even if the prior is extremely in favor of one product, with positive probability there will be a consumer who gets a sufficiently strong signal,that tilts the consumption decision toward the a priori inferior product; thus, under strategic pricing and unbounded signals, asymptotic learning holds.

The learning results in our model depart from those of the canonical model when signals are bounded. In the herding model there is always a positive probability that the suboptimal alternative will eventually be chosen by all agents. However, intuition suggests that when prices are endogenized they serve to prevent such a herding phenomenon. Hypothetically, once society stops learning and a herd develops on the product of one firm, the other firm will lower its product price to attract new consumers and learning will not cease. It turns out that this intuition, although not entirely correct, does have some merit. In order for the intuitive argument to hold, signals must exhibit a property referred to here as vanishing likelihood.

When signals are bounded the posterior belief of any agent, given his signal, is bounded away from zero and one for any interior prior. The proportion of agents whose posterior lies within of the posterior distribution’s boundaries obviously shrinks to zero as goes to zero. We say that signals exhibit vanishing likelihood if the density of consumers at the posterior belief distribution’s boundaries goes to zero.

Consumers who receive signals that induce such extreme posterior beliefs are those consumers who are likely to go against a herd and purchase the less popular product. We refer to such consumers as nonconformists. With this interpretation in mind the property of “vanishing likelihood” serves as a measure of the prevalence of nonconformism. More particularly, we associate vanishing likelihood with a negligible level of nonconformism while signals that do not exhibit vanishing likelihood are associated with significant nonconformism.

When society herds, each agent follows in the footsteps of his predecessors and thus, intuitively, one expects nonconformism (when signals do not exhibit vanishing likelihood) to induce learning. Our main result shows that the opposite occurs: in the presence of strategic pricing asymptotic learning holds if and only if signals have the vanishing likelihood property.

1.1 Schumpeterian Growth

It is widely agreed that innovation and the evolution of technology constantly propel the economy forward. New technologies replace older ones and may improve product quality, reduce production costs, and often completely disrupt an industry.

However, not every innovation entails improvement. Arguably, innovations that do not entail improvement will naturally be driven out of the market and only those that do will prevail. This argument forms the basis of the evolutionary economics literature that dates back to Marx, Veblen, and Schumpeter.

In his seminal work, Schumpeter [33] described the process of economic growth, which he refers to as “Capitalism,” as an evolutionary process that is shaped by “gales of creative destruction.” Some of his contemporaries had argued that large and profitable firms are the source of innovation and so regulation protecting them was essential to R&D investments. By contrast, Schumpeter argued that incumbent firms, anticipating innovation by potential entrants, invest in R&D to stay ahead of the game. Therefore such regulation is unwarranted, and may even be detrimental. However, such profitable incumbents may also use their power to drive innovation away by lowering prices. This is true in particular when it is hard to identify which innovation constitutes an improvement and which does not.

Does the evolutionary process guarantee that the economy will successfully separate the wheat from the chaff? This question becomes more acute with the accelerated pace of innovation witnessed in the past two decades [29].

Our theoretical results shed light on this issue and relate the outcome of the evolutionary process to the market structure. Our model shows that whenever the proportion of nonconformist consumers (often referred to as “early adopters” in the context of technological revolutions) is insignificant (a phenomenon captured by the technical notion of vanishing likelihood), the evolutionary process successfully sieves the better technologies. However, whenever this proportion is significant then the evolutionary process may fail and policies to support entry may be warranted in order to sustain Schumpeterian growth.

In Section 6 we discuss two case studies from the late related to technological innovations. In both cases the innovative technology turned out to be the better than the incumbent one but only in one of these cases was it adopted by society. Our model provides an explanation as it distinguishes between the two cases, based on whether the underlying information structure complied with the vanishing likelihood condition.

1.2 Related Literature

Our work primarily contributes to the work of Bikhchandani, Hirshleifer, and Welch [8] and Banerjee [6], who introduced models of social learning with agents who act sequentially. Their primary contribution was to point out the possibility of information cascades and market failure when signals are bounded. Smith and Sørensen [34] characterize the information structure that induces such a potential market failure. In these and many of the follow-up papers, prices are assumed exogenous and fixed throughout. The primary departure of our model from this line of research is that our model incorporates endogenous pricing. We associate a favorite firm (product) with each state of nature and allow for the firms to set prices dynamically, based on the information available in the market.

Avery and Zemsky [4] incorporate dynamic pricing into herding models. They consider a single firm whose product value is associated with an (unknown) state of nature. Instead of offering the product at a fixed price, as in the earlier papers, they assume that the price is set dynamically to be the expected value of the product conditional on all the publicly available information. Since their primary interest is to study financial markets, they assume that there is a market maker that uses all the publicly available data to set prices. By contrast, we assume that the firms themselves set the prices.

A model that is reminiscent of our model is that of Bose et al. [9, 10] who study a monopoly, with a good of uncertain quality, that dynamically adjusts prices to compete against an outside option. Consumers arrive sequentially and make a consumption decision based on their predecessors’ decisions, past prices, and an additional private signal. In [9] the authors restrict attention to signal structures with finitely many signals whereas in [10] they further restrict attention to symmetric binary signal structures. In both models it is shown that herding is inevitable and if the public belief is sufficiently in favor of the monopoly, then the monopoly will price low enough to attract all consumers, regardless of their realized signal. As vanishing likelihood never holds when the signal space is finite (see the discussion in Section 5.1), the results of Bose et al., albeit in a monopoly framework, hold in our duopolistic model (see the additional discussion in Section 5.3). Restricting the analysis to finite signal spaces would not allow the unraveling of the vanishing likelihood property that we introduce and that characterizes learning when prices are endogenous.

Moscarini and Ottaviani [25] study the duopoly case and their paper focuses on a single-stage interaction with two firms and a single knowledgeable consumer. In fact, it is exactly the model of the stage game () we study in Section 3.1, except that their model is restricted to a binary and symmetric signal space. Unsurprisingly, whenever the prior belief is above (or below) some threshold, all equilibria in their model form a deterrence equilibrium (see definition 6), where one firm prices out the other firm. Clearly, the emergence of a deterrence equilibrium implies that learning stops in the repeated model. The authors go on to provide comparative statics over the threshold public belief for which learning stops as a function of the informativeness of the signal (here is where they leverage the restricted signal space). As signals become more informative the thresholds move to the extremes.

Our main result on the one-shot game, Theorem 2, argues that learning stops whenever the vanishing likelihood condition does not hold. As this condition can never hold for a finite signal space the result in [25] follows as a corollary. The main take-home message from comparing our work with [25] is that learning is determined not by the level of informativeness of the signals but rather by the vanishing likelihood condition. In particular, signal distributions that satisfy vanishing likelihood need not be highly informative. The restriction to a binary model, in this case, is misleading. A similar distinction is valid in the monopoly setting of Bose et al. [9, 10].

Mueller-Frank [26, 27] introduces a pair of models with dynamic pricing of a monopoly [27] and a duopoly [26]. The model is very similar to ours with the distinction that the firms have the informational advantage and know the true state of the world.111When firms have an informational advantage the equilibrium analysis, as Mueller-Frank points out, crucially hinges on consumers’ off-equilibrium beliefs. This is not the case in our model, which consequently allows for robust observations. Mueller-Frank does not characterize the informational conditions that entail learning, as we do. Rather, he studies the connection between welfare and learning and shows that learning is not sufficient for welfare maximization (see Corollary 1). It is worth noting that in our model, by contrast, learning is necessary and sufficient for welfare maximization.

The paper is organized as follows. Section 2 presents the model and the main theorem for the case where firms are myopic. Section 3 provides the proof of the main result. Section 4 is an extension of our model to the case where firms are farsighted. Section 5 informally discusses some extensions and Section 7 concludes.

2 Social Learning and Myopic Pricing

Our model comprises a countably infinite number of consumers, indexed by , and two firms: Firm and Firm . There are two states of nature . In state , firm produces the superior product. We normalize the value of the superior product to and the value of the inferior product to . In every time period the two firms first set (non-negative) prices for their products. Then consumer receives a private signal and must decide whether to buy product , product , or neither product. Formally, the action set of every consumer is , where the action corresponds to the decision to buy from firm and the action corresponds to the decision to exit the market and not to buy either product. The payoff of every consumer

, given the price vector

as a function of the realized state , is

(1)

For simplicity we assume that both firms have no marginal cost of production. Hence, firm ’s stage payoff, given a price vector , can be described as a function of the consumer’s decision as follows:

(2)

We assume that the state is drawn at stage according to a commonly known prior distribution, such that . The state is unknown to both the firms and the consumers. Each consumer forms a belief on the state using two sources of information: the history of prices and actions, , and a private signal (where is some abstract measurable signal space).222An alternative model is to assume that consumers do not observe prices. In this alternative formulation our results go through when restricting attention to pure equilibria. This follows from the observation that the consumption history determines the corresponding equilibrium prices at each stage. The firms observe only the realized history at every time and receive no private information. Conditional on the state , signals are independently drawn according to a probability measure . We refer to the tuple as a signal structure. We assume throughout that and are mutually absolutely continuous with respect to each other.333 and are mutually absolutely continuous whenever for any measurable set

. Note that with this assumption the probability of a fully revealing signal, for which the posterior probability is either

or , is zero. The prior and the functions and are common knowledge among consumers and firms.

We let be the set of all finite histories and let be the set of all infinite histories. We let be the set of decision rules for the consumer; i.e., is the set of all measurable functions that map pairs consisting of a price vector and a signal to a consumption decision. A (pure) strategy for consumer is a measurable function that maps every history and signal to a decision rule. We denote by a pure strategy profile for the consumers. We can view as a function A (behavioral) strategy for firm is a (measurable) mapping . We note that the strategy profile together with the prior and the signal structure

induce a probability distribution

over .

Let be the probability that the state is conditional on the realized history . We call the public belief at time . The following observation regarding the sequence of public beliefs, is straightforward.

Observation 1.

is a martingale. Thus, by the martingale convergence theorem, it must converge almost surely to a limit random variable

.

A strategy profile and a history induce both an expected payoff for every firm and an expected consumer utility We can now define the notion of a Bayesian Nash equilibrium for myopic firms.

Definition 1.

A strategy profile constitutes a myopic Bayesian Nash equilibrium if for every time the following conditions hold for almost every history that is realized in accordance with :

  • For every ,

  • For every price vector , and every decision rule

In words, a strategy profile constitutes a myopic Bayesian Nash equilibrium if for every time and almost every history , maximizes the conditional expected stage payoff to every firm and maximizes the conditional expected payoff to consumer with respect to every price vector .

Note that our notion of equilibrium is weaker than the notion of a subgame perfect equilibrium; however, it still eliminates equilibria with non-credible threats by consumers. One such equilibrium with non-credible threats is the following equilibrium: both firms ask for price in every time period. Every consumer never buys a product (i.e., plays ) unless both firms ask for a price of in which case he buys product whenever and product if . Note that this equilibrium is sustained by non-credible threats made by the consumer. Such threats are eliminated by the second condition, which requires that conditional on the realized history the decision rule be optimal with respect to every price vector , and not just with respect to . We note that since our results in the sequel hold for all myopic equilibria they are in particular valid for subgame perfect equilibria.444A natural question is whether restricting attention to the stronger solution concept yields a weaker condition for learning. We conjecture that this is not the case, particularly when firms are myopic.

As is common in the literature, we define asymptotic learning as follows.

Definition 2.

Fix a signal structure Let be the prior and let be a strategy profile of the corresponding game. We say that learning holds for and if the belief martingale sequence converges almost surely to a point belief assigning probability to the realized state. Asymptotic learning holds for the signal structure if learning holds for every prior and every corresponding myopic Bayesian Nash equilibrium . By contrast, asymptotic learning never holds if for every prior and a corresponding myopic Bayesian Nash equilibrium the limit public belief lies in with positive probability.

Thus, when asymptotic learning holds, it must be the case that consumers and firms eventually learn the superior product. In our case, even for a very strong public belief in favor of one firm, it is not a priori clear that the strong firm will dominate the market as the weak firm can always lower its price. However, we show in Lemma 2 that the probability of buying from the superior firm converges to one when asymptotic learning occurs. Whenever asymptotic learning doesn’t hold, only one firm prevails (from some time on, all consumers buy from one firm but are not 100% certain that it is the superior one). As a result, there is positive probability that the prevailing firm is the inferior one.

Definition 3.

Let denote the Radon–Nikodym derivative of with respect to the probability measure . We consider the random variable , which is the posterior probability that , conditional on the signal , when the prior over is . Let , be the two cumulative distribution correspondences of the random variable induced by the two probability distributions, , over . Define the bounds of the signal distribution as follows: and .555Since and are mutually absolutely continuous it follows that and have the same support.

The main goal of our paper is to provide a characterization of asymptotic learning under strategic pricing in terms of the signal structure Such a characterization is provided by Smith and Sørensen [34] for the standard herding model where prices are set exogenously. We start by presenting the formal definition of bounded and unbounded signals due to Smith and Sørensen [34].

Definition 4.

The signal structure is called unbounded if and . The signal structure is bounded if and .

In words, a signal structure is unbounded if for every the two sets and have positive probability under . Smith and Sørensen’s characterization shows that in the standard herding model asymptotic learning holds under an unbounded signal structure and fails under a bounded signal structure.

2.1 Characterization of Asymptotic Learning

For ease of exposition we make the following assumption on . We refer the reader to Section 5 for the general case.

Assumption 1.

We assume that the functions are differentiable on with continuous derivatives .

Definition 5.

A signal structure exhibits vanishing likelihood if

We next show how information aggregation depends on the vanishing likelihood property. The following theorem provides a full characterization of asymptotic learning in our model.

Theorem 1.

If signals are unbounded or if signals are bounded and exhibit vanishing likelihood then asymptotic learning holds. If signals are bounded and do not exhibit vanishing likelihood then asymptotic learning never holds.

Although Theorem 1 formally hinges on Assumption 1, an analog of this theorem carries over completely to a model without such an assumption. This requires an alternative formulation of the vanishing likelihood condition. We elaborate on this in Section 5.

Remark 2.1.

In fact, the proof of Theorem 1 shows that whenever the limit public belief , conditional on state , lies in with probability . Analogously, whenever the limit public belief , conditional on the state , lies in with probability .

3 Proof of the Main Result

In the proof of Theorem 1 we rely on the analysis of the following three-player stage game . The game comprises two firms and a single consumer and is derived from our sequential game by restricting the game to a single period. That is, in the state is realized according to the prior (state is realized with probability and state with probability ). The two firms post a price simultaneously (possibly at random) and a single consumer receives a private signal in accordance with and based on his private signal and the realized vector of prices takes an action . The utility for the consumer is determined by equation (1) and the utility for the firms is determined by equation (2).

In the following observation, which is a direct implication of Definition 1, we connect the stage game with the sequential model.

Observation 2.

A strategy profile constitutes a myopic Bayesian Nash equilibrium if and only if for every time for and for almost every history the tuple is a subgame perfect equilibrium (SPE) of .

The strong connection of to our sequential game allows us to derive some insight into information aggregation from the subgame perfect equilibrium properties of , which we analyze next.

3.1 Analysis of

We begin by studying the consumer’s best-reply strategy in . We denote the consumer’s posterior belief after the consumer observes the signal by . It follows readily from Bayes’ rule that

(3)

The bounds and , together with equation (3), imply that with probability one, where:

(4)

Fix a price vector and note that the consumer optimizes his expected utility against if and only if he follows the following strategy:

(5)

We note that, under Assumption 1, in every perfect Bayesian Nash equilibrium of the game the strategy constitutes a unique action for the consumer almost surely.

Note further that every mixed strategy induces two possible market scenarios: a full market scenario, where, under the consumer always buys from one of the firms for almost all signal realizations and almost every realized price vector , and a non-full market scenario, where holds with positive probability.

We can infer from (5) that the consumer buys from Firm whenever and the market is full or whenever and the market is not full.

Given a prior and a pair of prices , we let be the threshold in terms of the private belief discussed above that Firm is chosen. That is, choosing Firm is uniquely optimal for the consumer if and only if . One can easily see from the above equations that has the following form:

(6)

Note that is a continuous function of

We can therefore suppress the behavior of the consumer, which, under Assumption 1, is determined uniquely for every price vector and almost every signal realization . Hence we can write the expected utility of Firm in the game for the price vector as follows:

(7)

A similar equation can be derived for , the profit of Firm .

For a strategy profile let be the probability over ; the state, the price vector, and the signal set are induced by , and .

In what follows we make a distinction between two forms of perfect Bayesian Nash equilibria of the game : a deterrence equilibrium, where only a single firm sells with positive probability, and a non-deterrence equilibrium, where both firms sell with positive probability. That is,

Definition 6.

A deterrence equilibrium in is a Bayesian Nash SPE, , in which there exists a unique firm such that

A non-deterrence equilibrium is an equilibrium that is not a deterrence equilibrium.

The following theorem summarizes the main characteristics of equilibria in the stage game . This characterization is the driving force behind the proof of Theorem 1.

Theorem 2.

Let and let be a Bayesian Nash subgame perfect equilibrium of the game :

  1. If signals are unbounded, then no firm is deterred.

  2. If signals are bounded and exhibit the vanishing likelihood property, then no firm is deterred.

  3. If signals are bounded and do not exhibit the vanishing likelihood property, then:

    1. If then for some sufficiently high prior, whenever Firm is deterred and Firm captures the whole market.

    2. If then for some sufficiently low prior, whenever Firm is deterred and Firm captures the whole market.

The proof of Theorem 2 as well as the complete analysis of this stage game is relegated to Appendices A and B.

3.2 Proof of Theorem 1

We next introduce the formal proof of Theorem 1, based on Theorem 2. Consider the case where signals exhibit vanishing likelihood. In this case, by Theorem 2, at every time no firm is deterred in . Therefore the decision of the consumer remains informative throughout the play. This implies that asymptotic learning always holds. By contrast, if signals do not exhibit vanishing likelihood and then for all equilibria of are deterrence equilibria. This implies that when crosses all subsequent consumers buy from Firm and learning stops.

In order to establish Theorem 1 we start with the following corollary of Lemma 5 (the proof can be found at Appendix B).

Corollary 1.

If signals exhibit vanishing likelihood or if signals are unbounded, then for every there exists some and such that if and is a SPE of , then

A similar condition holds for Firm

In words, by Theorem 2, if signals exhibit vanishing likelihood or if signals are unbounded, then the probability of a consumer going against the herd is positive. Corollary 1 argues that this probability cannot be arbitrarily close to zero if the prior is bounded away from the edges.

Proof of Theorem 1.

First we show that if the signal structure does not exhibit vanishing likelihood, then the martingale of the public belief must converge to an interior point. Indeed, let us assume without loss of generality that . Let be a myopic equilibrium. By Observation 2, for almost every history the profile is a SPE of . By Theorem 2, there exists such that there is a unique Bayesian Nash subgame perfect equilibrium of in which the consumer almost surely chooses Firm (Firm is deterred by Firm ). This implies that if , then with probability one. We note that since signals are never fully informative it must be the case that for all with probability one. Therefore, if the vanishing likelihood property does not hold, then asymptotic learning fails.

Next we show that if vanishing likelihood holds, then the public belief martingale converges to a limit belief in which the true state is assigned probability one. It follows from Observation 2 that is a SPE of for almost every history . Corollary 1 implies that if then for some and the realized price vector satisfies with probability at least .

Since the distribution first-order stochastically dominates (see Lemma 10 in Appendix D), under any such price vector there exists a probability at least that the consumer will not buy from Firm Note that (again by Lemma 10)

Therefore, it follows from Bayes’ rule that with probability at least the public belief satisfies

(8)

Hence, in particular, if , then there exists a positive constant such that, with probability at least , it holds that .

By Observation 1, the limit exists and by the above argument with probability . This shows that asymptotic learning holds. ∎

4 Social Learning and Farsighted Firms

In this section we show that our main result carries forward to a setting where the firms are farsighted and maximize a discounted expected revenue stream. We extend our sequential model to the non-myopic case by defining the non-myopic sequential consumption game. In this model, as in the myopic case, each firm sets a price in every time period, except that now each firm tries to maximize its discounted sum of the stream of payoffs. We still retain the perfection assumption with respect to consumers. Given a strategy profile , in the repeated game, denote by the expected payoff to Firm when the discount factor is . Namely,

Define a Bayesian Nash equilibrium as follows.

Definition 7.

A strategy profile constitutes a Bayesian Nash equilibrium if:

  • For every strategy of Firm ,

  • For every time the following condition holds for almost every history that is realized in accordance with , every price vector , and every decision rule :

Let be a strategy profile, let , and denote by the continuation payoff to Firm in the subgame starting at history . Note that is well defined for almost every history that is realized in accordance with . By Definition 7, if constitutes a Bayesian Nash equilibrium, then maximizes the continuation payoff of Firm for almost every history that is realized in accordance with .

We can now state our first result for farsighted firms.

Theorem 3.

If the signal structure exhibits the vanishing likelihood property or if signals are unbounded, then asymptotic learning holds for every discount factor .666Recall this implies that learning takes place for any prior and any corresponding Bayesian Nash equilibrium.

By Theorem 3, vanishing likelihood is a sufficient condition for asymptotic learning.777The fact that asymptotic learning holds for unbounded signals carries forward to the farsighted case under a similar proof to that of the myopic case.

For the converse direction we establish the following weaker statement.

Theorem 4.

If signals are bounded and do not exhibit vanishing likelihood, then asymptotic learning fails in the following sense: for every prior there exists a Bayesian Nash equilibrium for which learning fails.

That is, in order for asymptotic learning always to hold in a Bayesian Nash equilibrium it is necessary and sufficient for the signal structure to exhibit vanishing likelihood. We conjecture that, in fact, if vanishing likelihood fails then learning fails in any equilibrium.

5 Discussion

We turn to discuss four natural questions that arise from our model and analysis:888We thank anonymous reviewers for prompting these questions.

  • Do our conclusions hold when the differentiability assumption on the signal distribution (Assumption 1) is relaxed?

  • What are the implications of vanishing likelihood in the monopolistic setting?

  • What are the social welfare implications of our results?

  • In cases where asymptotic learning holds, how fast do agents learn and consequently buy the superior product?

5.1 General Signals

Throughout the analysis we have restricted our attention to signal structures that satisfy Assumption 1. In many applications this assumption fails to hold. In particular, Assumption 1 does not hold when the set of signals is countable or finite. It is therefore important to understand whether our condition can be stated more generally to capture all signal distributions.

Fortunately, it turns out that such a general condition does exist. Let be a general signal distribution and let be the CDFs as defined in Definition 3. Define as follows:

Obviously, are both well defined. Note that is defined using the limit from the left () whereas uses the limit from the right ().

We can now state the more general condition for vanishing likelihood as follows.

Definition 8.

The signal structure satisfies vanishing likelihood if .

Note that if satisfies Assumption 1, then the condition in Definition 8 coincides with the condition in Definition 5. Moreover, note that for finite signal distribution we have , and thus vanishing likelihood fails. Our results hold verbatim under the more general definition of vanishing likelihood for when firms are myopic.

We omit the proofs for the general setting but note that the underlying ideas for the proofs are similar while their exposition becomes more cumbersome. The primary reason for this is that with an arbitrary signal structure the consumer can be indifferent between two options (e.g., indifferent between the two products or between a product and exiting) with positive probability. Therefore, given a price pair, the consumer may have more than one best reply. In addition, it is not necessarily the case that any such best reply induces a two-player game between the firms that possesses an equilibrium. The underlying reason is that the consumer strategy may lead to discontinuity in firms’ payoffs as a function of prices. Under Assumption 1 the consumer has a unique best reply with probability one and such discontinuity can be ignored.999The omitted proofs are available from the authors upon request. Although we have not written a rigorous proof for the case where firms are farsighted, we believe that the results carry through.

An additional challenge that results from the aforementioned discontinuity pertains to the mere existence of an equilibrium in Absent this equilibrium, our results become vacuous. Fortunately, we can use the result of Reny [30] to overcome this.

Consider the following specific best-reply consumer strategy: whenever a consumer is indifferent between buying from one firm and the outside option he always chooses to buy from the firm. Whenever a consumer is indifferent between buying from Firm and Firm and his expected utility from purchasing a product is at least zero, he chooses the firm that is a priori favorable. That is, in this case he chooses Firm whenever and Firm whenever . In all other cases he strictly prefers one alternative and therefore chooses this alternative.

With this consumer strategy, game satisfies Reny’s better-reply secure condition [30] for any Theorem 3.1 in [30] thus guarantees the existence of a mixed subgame perfect equilibrium in

5.2 Monopolistic Market

A natural question to ask is what are the necessary and sufficient conditions for asymptotic learning when there is a single firm that competes against an outside option. It turns out that the vanishing likelihood condition plays a crucial role in the monopolistic case as well.

More precisely, consider a monopolistic model with a single firm and a binary state space. In state the firm offers a high-quality product (the common value is one) whereas in the other state () the common value is zero. At each stage the firm sets a price and a new consumer arrives. The consumer receives a private signal and chooses whether or not to buy the good (the outside option is valued at zero). The methodology and techniques discussed in this paper may be used to show that in the monopolistic setting, vanishing likelihood guarantees asymptotic learning and the lack of vanishing likelihood (at both ends) implies that asymptotic learning fails.101010Formally, the notion of vanishing likelihood is related to two conditions on the signal structure, that correspond to the two extreme values: the highest and lowest possible signals. Indeed, if one of these conditions fails then learning fails in the duopolistic setting. By contrast, in the monopolistic setting failure of learning is guaranteed only when both conditions are not satisfied.

5.3 Social Welfare

The main goal of this paper is to identify the informational structure that guarantees learning. As we now turn to show, in our model, learning guarantees that eventually all agents purchase the superior product. This implies that vanishing likelihood is a necessary and sufficient condition for the market to be (asymptotically) efficient.111111In some variants of the herding model, such as those studied in by Mueller-Frank [26, 27], learning does not entail market efficiency.

Corollary 2.

Let be a myopic Bayesian Nash equilibrium. If asymptotic learning holds, then conditional on state ,

The corollary follows from the proof of our main theorem and its proof is relegated to Appendix D.

5.4 Rate of Convergence

A natural question to ask is, in cases where asymptotic learning holds, how fast does the market reach a situation where the consumer buys the superior product with high probability. In the herding model, this question has only recently received attention (see, e.g., Rosenberg and Vieille [31] and Hann-Caruthers et al. [20]). In particular, Rosenberg and Vieille [31] demonstrate a sharp negative result showing that for “reasonable” unbounded signals, the expectation of the first correct decision by a consumer, is infinite. In our setting we conjecture that under vanishing likelihood breaking bad herds is faster and thus the rate of convergence is higher. We leave this interesting question for future research.

6 Case Studies

An intriguing application of our theoretical results relates the outcome of the evolutionary process of Schumpeterian growth. One particular interest (and concern) is a corollary of our main theorem. Our main theorem asserts that whenever one firm is sufficiently a-priori advantageous (i.e. the public belief from the outset is that it has the superior product with high probability), it can use predatory pricing to obstruct the entry of entrants yielding innovative substitute products and technologies. In fact, whenever the demand side does not exhibit the vanishing likelihood property, predatory pricing will eventually become the incumbent’s optimal strategy. Recent technological advances had led to an increase in the number and complexity of proposed innovations [29]. The rise of the Internet increased the exposure to new products and the pace at which adoption decisions are made. Social networks inform potential consumers over the action taken by their predecessors, therefore we suspect that such an advantage is quite typical.

2 Social Learning and Myopic Pricing

Our model comprises a countably infinite number of consumers, indexed by , and two firms: Firm and Firm . There are two states of nature . In state , firm produces the superior product. We normalize the value of the superior product to and the value of the inferior product to . In every time period the two firms first set (non-negative) prices for their products. Then consumer receives a private signal and must decide whether to buy product , product , or neither product. Formally, the action set of every consumer is , where the action corresponds to the decision to buy from firm and the action corresponds to the decision to exit the market and not to buy either product. The payoff of every consumer

, given the price vector

as a function of the realized state , is

(1)

For simplicity we assume that both firms have no marginal cost of production. Hence, firm ’s stage payoff, given a price vector , can be described as a function of the consumer’s decision as follows:

(2)

We assume that the state is drawn at stage according to a commonly known prior distribution, such that . The state is unknown to both the firms and the consumers. Each consumer forms a belief on the state using two sources of information: the history of prices and actions, , and a private signal (where is some abstract measurable signal space).222An alternative model is to assume that consumers do not observe prices. In this alternative formulation our results go through when restricting attention to pure equilibria. This follows from the observation that the consumption history determines the corresponding equilibrium prices at each stage. The firms observe only the realized history at every time and receive no private information. Conditional on the state , signals are independently drawn according to a probability measure . We refer to the tuple as a signal structure. We assume throughout that and are mutually absolutely continuous with respect to each other.333 and are mutually absolutely continuous whenever for any measurable set

. Note that with this assumption the probability of a fully revealing signal, for which the posterior probability is either

or , is zero. The prior and the functions and are common knowledge among consumers and firms.

We let be the set of all finite histories and let be the set of all infinite histories. We let be the set of decision rules for the consumer; i.e., is the set of all measurable functions that map pairs consisting of a price vector and a signal to a consumption decision. A (pure) strategy for consumer is a measurable function