Information revelation through bunching

Games and Economic Behavior 102 (2017) 568–582

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Games and Economic Behavior www.elsevier.com/locate/geb

Information revelation through bunching Tao Wang a,b,∗ a b

School of Economics, Zhejiang University, Hangzhou, Zhejiang, 310027, China Economics Department, Queen’s University, Kingston, Ontario K7L 3N6, Canada

a r t i c l e

i n f o

Article history: Received 27 July 2016 Available online 9 March 2017 JEL classification: C23 D62 D82 Keywords: Bunching Herding Endogenous timing Asymmetric equilibrium Information externality

a b s t r a c t In this paper, we analyze a dynamic game of pure information externality. Each player receives a private signal and chooses whether and when to invest. Bunching occurs when a subgroup of the players make decisions contingent on their signals, while the rest of the players wait regardless of their signals. We focus on asymmetric pure strategy equilibria, where players’ private information is revealed gradually through bunching. When players become patient enough, the most efficient equilibrium contains no herding of investing, while the least efficient equilibrium resembles the outcomes in an exogenous timing model. When players’ discount factors differ, less patient players will bunch earlier than more patient players. © 2017 Elsevier Inc. All rights reserved.

1. Introduction Frequently in life, firms or individuals need to make Yes or No decisions under uncertainty. For example, one needs to decide on whether or not to purchase a newly-released cell phone, whether or not to go to the cinema to watch a new movie, or whether or not to make an investment. These decisions involve risk, since the outcomes are uncertain. In this kind of environment, an agent could benefit from the observations of other agents’ actions, as those actions could reveal information which is useful to the agent. The existing literature has focused on herd behavior; when later investors observe many investments (or no-investments) made by earlier investors, they ignore their own private information and follow suit. An important setup in a typical herding model is that players make their choices according to an exogenous order; they must act in their exogenously determined time slot, no sooner and no later. Therefore, the amount of information revealed through players’ actions in each period is exogenously determined. We categorize this type of models as the exogenous timing models.1 In this paper, we analyze an endogenous timing model. In the model, players choose whether and when to invest. Every player possesses private information (good signal or bad signal) which could be revealed to other players through his action. A player must balance the benefit to wait for more information to be revealed with the discounting of payoffs due to the wait. We focus on the pure strategy equilibria in our model. Under general conditions, these equilibria are asymmetric. In the first period and maybe some periods that follow, only a subgroup of players make decisions contingent on their private

* 1

Correspondence to: School of Economics, Zhejiang University, Hangzhou, Zhejiang, 310027, China. E-mail address: [email protected]. See Banerjee (1992) and Bikhchandani et al. (1992) for the pioneer work on the exogenous timing models.

http://dx.doi.org/10.1016/j.geb.2017.02.017 0899-8256/© 2017 Elsevier Inc. All rights reserved.

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signals; that is, to invest if the signal is good and to wait if the signal is bad. The rest of the players just wait and observe the actions of these players. We term this subgroup of players a “bunch”. Being in a bunch does not require a joining action. In an arbitrary period, a bunch is just a collection of players whose strategy in that period is to invest if the signal is good and to wait if the signal is bad. Therefore, in an equilibrium, a bunch member’s private information is completely revealed in that period. In contrast, in the same period, a non-member’s private information is not revealed; his strategy is to wait regardless of his private signal. Therefore, information revelation in the game is done only through the actions of the bunch members, and the number of signals revealed in a period is exactly equal to the bunch size. A player waiting in one period reveals a bad signal if and only if he belongs to the bunch in that period. We obtain many properties in our analysis. First, the bunch sizes and the amount of information to be revealed in each period is endogenously determined, instead of assumed exogenously. Second, when players are enough patient, the least efficient asymmetric equilibrium in our model produces outcomes resembling those in the one-per-period exogenous timing model. In the equilibrium, only one player makes a decision in each period (i.e., the bunch size is one in every period) until herding occurs. This one-per-period decision making is an equilibrium behavior in our model, while it is assumed in an exogenous timing model. Meanwhile, in the most efficient equilibrium in our model, bunching occurs most often and herding occurs least often. In this equilibrium, as long as herding on waiting does not occur, players make their investment decisions one by one according to their private signal, and herding on investing completely disappears. The information revelation is maximized in this way, and the equilibrium outcome is more efficient than the one obtained from the one-per-period exogenous timing model. Third, when players are enough impatient, the bunch sizes in our equilibria become very large. The aggregate information revealed through this very large amount of signals becomes very precise. In this case, the equilibria in our model are more efficient than any exogenous timing model. Finally, when players differ in their discount factors, impatient players will be in earlier bunches, while patient players will be in later bunches. This resolves the multiplicity issue of asymmetric equilibria in the symmetric model where players are ex ante identical. The literature on herding in economics was started by Banerjee (1992) and Bikhchandani et al. (1992), with the latter referring to herd behavior as information cascade, as people no longer reveal their own private information. In a herding equilibrium, the equilibrium outcome depends crucially on the private information of the first few individuals, and only these first few players’ private signals are revealed in the equilibrium, leading to socially inefficient outcomes.2 Note that these are exogenous timing models. Chamley and Gale (1994) are the first to employ an endogenous timing model to investigate the timing of investment issue. They examine a model where an unknown number of investment options are held by a certain number of players, and the expected payoff of each individual depends on the total number of options. They focus on the symmetric mixed strategy equilibrium of that model. As players use mixed strategies, the amount of information to be revealed becomes uncertain.3 The linkage to the exogenous timing models is not established as the latter are asymmetric in nature. Later on, several other researchers have also used different endogenous timing models to investigate various herding and related issues. Chari and Kehoe (2004) use endogenous timing to regenerate herding in a continuous investment model. In Rosenberg et al. (2007), players play a one-armed bandit and continuously receive private signals. Zhang (2009) generalizes Chamley and Gale (1994) to both continuous signals and multiple discrete signals.4 Chamley (2004a) addresses the nonexistence of symmetric pure strategy equilibrium in Chamley and Gale (1994) by allowing players to have different beliefs drawn from a continuous distribution, and establishes the existence of multiple symmetric pure strategy equilibria. He examines the issue of large economies and finds that the amount of information revealed converges to the model in Chamley and Gale (1994) if the belief distribution is bounded. Levin and Peck (2008) also establish the existence of symmetric pure strategy equilibrium by adding a second signal (cost of investment) to Chamley and Gale’s model. Zhang (1997) presents another endogenous timing model where players with different precision of signals choose their optimal timing to act in continuous time. His paper shows that the player with the highest precision invests first and then there is an investment surge leading to an information cascade. Similar results are also found in Aoyagi (1998) in a model with a different setup. All of these papers focus on symmetric equilibria. Both Gul and Lundholm (1995) and Chamley (2004b, p. 122) briefly analyze the asymmetric equilibria in the two-player case with different settings. The phenomenon of bunching requires more

2 Herding and information cascades have been investigated by many researchers. See Bowden (2013) for a recent survey and Chamley (2004b) for a well-organized and detailed analysis of herding and its applications in economics. Cao et al. (2011) show that even if players can communicate with each other regarding their payoffs from past choices, herding can still occur, but cannot last forever. Khanna and Mathews (2011) study a model in which players’ actions may affect each other’s payoffs. They find that a leader may spend more to get more precise information to avoid the adverse effect of actions being copied, and this information gain can dominate the information loss from herding. 3 Chamley (2004a) generalizes Chamley and Gale (1994) by allowing players to have different beliefs (which is equivalent to different signals) from a continuous distribution. It is an endogenous timing model similarly to ours, and it focuses on symmetric pure strategy equilibria. We have a discrete signal distribution and we focus on asymmetric pure strategy equilibria. The comparison of information revelation between the two models is not straightforward as the signal distributions are different. 4 Herd behavior in models of endogenous timing has been confirmed in experiments. (See Sgroi, 2003, for a review.) In an empirical study, Moretti (2011) examines the behavior of movie goers and finds that peer effect (in the form of information externality) plays an important role in their movie watching decisions.

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than two players in the game. In this paper, we focus on asymmetric pure strategy equilibria and highlight their different properties. This paper is organized as follows. In Section 2, we describe the model and its assumptions. In Section 3, we perform some preliminary analysis, including defining the myopic best response function, defining the patterns of equilibrium behavior, and determining the bunch sizes. In Section 4, we characterize the asymmetric pure strategy perfect Bayesian equilibria (PBE). We also perform comparative static analysis regarding the discount factor. In Section 5, we conclude. Finally, Appendix A contains all of the proofs, and Appendix B provides an equilibrium analysis when the main assumptions of the model are not satisfied. 2. The model Consider a model with n players and T periods. In these T periods, a player can invest only once. Once he has invested in a period, he has no further actions to play in all subsequent periods. In any period that a player has not yet previously invested, his action is given by a ∈ { I , W }, where I stands for investing, and W stands for waiting (i.e., not investing in that period). For example, player i can choose I or W in period 1. If he chooses W , then he can choose I or W in period 2; if he chooses I , then he has no further actions to take from period 2 onward. Assume that T ≥ n + 1. This ensures that all players will have sufficient time to make their investment decisions.5 The outcome of a player’s investment depends on the state of the economy. The state of the economy is denoted as S ∈ {G , B }, where G denotes the Good state and B denotes the Bad state. For any player, the payoff from investment is πG when the state is G, and π B when the state is B, where πG > 0 > π B . Anyone who waits in all periods (i.e., never invests) earns 0. Ex ante, with probability q the state is G, and with probability 1 − q the state is B. The state of the economy remains the same throughout the game. In any period t, each player can observe other players’ actions (but not the payoffs) in all previous periods (from periods 1 to t − 1). Players are impatient and their payoffs are discounted by a common discount factor δ ∈ [0, 1]. Before period 1, each player receives a private signal s ∈ { g , b}, which is correlated with the state of the economy. A player’s action may reveal his private signal, which is then useful to other players in predicting the state. However, one player’s action does not affect another player’s payoff directly. Therefore, this is a game of pure information externality. Conditional on the state of the economy, a player’s signal is distributed independently and identically across all the players. Assume that Pr( g |G ) = Pr(b| B ) = p > 12 . Here, we assume that the signal precision (i.e., p) is the same in state G or B for simplicity. Because of the symmetry in signal precision, Pr( S | g , b) = Pr( S ), where S = G , B; that is, a good signal and a bad signal cancel out each other’s effect.6 Therefore, the difference between the total number of good signals and the total number of bad signals known by a player becomes the sufficient statistics for the signals for a player. Denote by dt the number of net public good signals at the beginning of period t, which is the same for all players and could be positive, zero, or negative. This number is based on the total number of good and bad signals inferred from the actions of players from periods 1 to t − 1. Furthermore, denote d˜ t as a player’s information including his own private signal. Therefore, d˜ t is equal to dt + 1 or dt − 1 depending on whether the player has a good signal or a bad signal. This d˜ t may be different from player to player. We shall omit the index for the players since it will not cause any confusion in later analysis. The probability of a good state conditional on d˜ t net good signals is denoted by Pr( S |d˜ t ), and the expected payoff for a player playing action a in period t given d˜ t is written as π (a|d˜ t ). We focus on pure strategy equilibria in the analysis. Consider any period. A player who has not yet invested has four strategies to choose from: to invest if his signal is g and to wait if it is b; to wait if it is g and to invest if it is b; to invest regardless of his signal; and to wait regardless of his signal. Obviously, the actions of this player may reveal his signal: the former two strategies completely reveal his signal, while the latter two strategies do not reveal his signal. In this paper, we will focus on the following assumptions:

π ( I |1) = Pr(G |1)πG + Pr( B |1)π B > 0,

(A1)

π ( I |0) = Pr(G |0)πG + Pr( B |0)π B < 0.

(A2)

and

With these assumptions, without learning any other information, it is profitable for a player to invest if his private signal is g and not to invest if he has no private signal. Note that (A2) implies that

π ( I | − 1) = Pr(G | − 1)πG + Pr( B | − 1)π B < 0.

(1)

If a player will not invest when he has no signal, he will definitely not invest when he has a bad signal. The above assumptions are made to simplify the notation and the analysis. The qualitative results in this paper does not depend on them. In Appendix B, we analyze the cases when (A1) or (A2) does not hold. As long as profitability requires one more good signal than unprofitability, which is generally true, our results remain valid qualitatively.

5 If T ≥ n + 1, the players’ strategies do not depend on T as no investment will occur after n + 1th period in the equilibria in our model. However, if T < n + 1, their equilibrium strategies may depend on T . 6 When the precisions are different, the Bayesian updating is more complicated, but the analysis is similar.

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3. Preliminary analysis 3.1. Myopic best response In period t, for a player, say player i, who has waited in all previous periods, with his information on signals d˜ t , define B R (d˜ t ) ∈ { I , W } as his myopic best response function, that is, B R (d˜ t ) is the best response function for player i on the hypothesis that period t is the last period that he can invest. This myopic best response function plays an important role in the characterization of the equilibria in the analysis. Given Assumptions (A1) and (A2),

B R (d˜ t ) =



I, W,

if d˜ t ≥ 1; otherwise.

(2)

Of course, a player may or may not always follow this myopic best response function, as period t may not be the last period, and waiting in period t may not mean that he will get zero profit (as he still has the chance to invest in a later period). However, as we will show in the proof of Theorem 1, for any player deviated from his equilibrium action, it is optimal for him to make an immediate decision according to this myopic best response function in the period right after his deviation. Given player i’s information d˜ t at the beginning of period t, suppose that there are kt new signals to be revealed in period t. Then player i’s expected payoff from using the myopic best response function in period t + 1 is

R (kt |d˜ t ) =

kt 

π ( B R (d˜ t + r − (kt − r ))|d˜ t + r − (kt − r )) Pr(rg , (kt − r )b|d˜ t ),

r =0

where r is the number of additional good signals among the kt new signals and Pr(rg , (kt − r )b|d˜ t ) is the conditional probability that there are r additional good signals and kt − r additional bad signals. This R function captures player i’s expected payoff when he waits one period (expecting kt additional signals to be revealed) and then uses his myopic best response. We have the following lemma. Lemma 1. R (kt |d˜ t ) is strictly increasing in kt and converges to R (∞|d˜ t ) ≡ Pr(G |d˜ t )πG . This lemma illustrates two properties of the myopic best response function. The first property is this function is increasing in the number of expected signals, that is, the more signals to be revealed in the next period, the higher a player’s payoff from waiting (as he can make a more precise decision in the next period). The second property is that this function has an upper bound. When there are infinitely many signals to be revealed, by the Law of Large Numbers, the proportions of good signals and bad signals will be, respectively, exactly the same as the probabilities of receiving a good signal and a bad signal conditional on the true state. Therefore, by examining the proportions of good signals and bad signals, a player can infer exactly which state is the true underlying state. If the underlining state is revealed to be G, the player will invest, and if the underlining state is revealed to be B, the player will wait forever (i.e., not invest). 3.2. Patterns of equilibrium behavior As will become clear later, only a few patterns of behavior will occur in a given period in a pure strategy PBE of this game. Specifically, in a given period, it could happen that only a subgroup of players make differential decisions contingent on their private signals, i.e., invest if the signal is good and wait if the signal is bad. Meanwhile, all other players wait regardless of their signals. In this case, we call this subgroup a bunch and this phenomenon bunching. Note that a bunch member’s waiting reveals a bad signal, while a non-member’s waiting reveals nothing. Alternatively, in a given period, players could also herd. When herding occurs, every player who has not invested takes the same action regardless of his own signal.7 Note that bunching is not an action of the game, and neither is herding. If a player’s strategy in a particular period is to invest if his signal is g and wait if his signal is b, then he is automatically classified as a bunch member for that period, alongside with all other players who has the same contingent strategy in that period. The following definitions define all possible patterns of behavior in a pure strategy PBE. Definition 1. In a given period, Herding of Investing occurs when every player (regardless of his signal) who has waited in all previous periods invests now.

7 Different from the herd behavior that leads to information cascades in our model, Scharfstein and Stein (1990) examine reputational herding, where smart types receive perfectly correlated signals and dumb types receive random signals. Players herd so as to build a reputation for being a smart type. Gul and Lundholm (1995) define the concept of clustering, a behavior that appears similar to an information cascade, but is in fact different. When clustering, players make similar decisions at almost (but not exactly) the same time, but players’ private information is completely revealed. The definition for herding is slightly different in financial markets literature as it involves price movements. It is used by Park and Sabourian (2011) to investigate informational herding and contrarianism in financial markets.

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Definition 2. In a given period, Herding of Waiting occurs when every player (regardless of his signal) who has waited in all previous periods still waits. Definition 3. In a given period, Bunching of Size k occurs when a group (i.e., a bunch) of k players among those who did not belong to a previous bunch have the same contingent strategy – invest if the signal is g and wait if it is b. All other players who have not previously invested just wait. Definition 4. In a given period, the Last Bunching occurs when there are not enough players left for a bunch of the desired size to form. All players who do not belong to any previous bunch have the same strategy – invest if the signal is g and to be determined if the signal is b. In the analysis, a contingent strategy is usually to invest if his signal is g and to wait if his signal is b. (It is never optimal for a player to do the opposite; that is, invest if his signal is b and wait if his signal is g.) In a period, every bunch member uses this contingent strategy, and his information is revealed. As we shall see, either bunching or herding occurs in any particular period, and they can never occur at the same time. In the definition for the last bunching above, the decisions made by the bunch members could be different from those made by a player in a regular bunch. More specifically, a player with a bad signal in a regular bunch waits. But a player with a bad signal in the last bunch may invest given that it is optimal to do so. (Refer to Lemma 6 in the next subsection for their behavior.) Meanwhile, a player with signal g always invests in a bunch, whether he is in a regular bunch or the last bunch. Note that the period when the last bunching occurs may not be the last period of actions. Players with signal b from earlier bunches may invest in the period after that if the information revealed is favorable enough. In the asymmetric equilibria we will characterize, either Herding of Investing, Herding of Waiting, or (regular or last) Bunching may occur in a particular period. Bunching will always occur in period 1, and may occur again in later periods if bunching occurred in all preceding periods. 3.3. Bunch size determination In any period, given any history of the game, a player who has not yet invested needs to decide whether or not to invest. He trades off the benefit of investing right away with the benefit of waiting. Some players wait regardless of their signals, while others decide according to their signals (i.e., bunching and thus effectively forming a bunch). Bunching occurs frequently in our equilibria and the size of bunching is the focus of this subsection. Recall that in a given period, bunching of size k occurs when k players among those who did not belong to any previous bunch make the same contingent decision – invest if the signal is g and wait if it is b. Suppose that at the beginning of a given period, say, period t, dt = d. This d is the number of net good signals revealed publicly in previous periods. When d is negative, bad signals outnumber good signals by |d|, and no player will invest. In this case, further bunching can never occur. Therefore, we will focus on a non-negative d in the following analysis, and we interpret this d as the degree of public optimism. Suppose that in equilibrium a bunch of size kd∗ is to be formed in a given period. Then kd∗ must satisfy two conditions. The first condition is that a player (with either signal) inside the bunch does not want to be a non-bunch person. This means that if the player has signal g, he prefers to invest now instead of waiting one period; if the player has signal b, he prefers to wait one period instead of investing now. We have

π ( I |d + 1) ≥ δ R (kd∗ − 1|d + 1), 



∗

π ( I |d − 1) ≤ δ R kd − 1|d − 1 .

(IC1) (IC2)

The second condition is that a player (who has not previously invested, with either signal) outside the bunch does not want to deviate. This implies that, regardless of his signal, the player prefers to wait one period instead of investing now. We have





π ( I |d + 1) ≤ δ R kd∗ |d + 1 , 

∗



π ( I |d − 1) ≤ δ R kd |d − 1 .

(IC3) (IC4)

The following lemmas will help us to establish the existence and the properties of kd∗ . We first have Lemma 2. For any given k, π ( I |d) − δ R (k|d) is increasing in d. This lemma implies that the benefit of additional k signals is decreasing in the number of net good signals already obtained. When there are more net good signals, the additional k signals become less valuable. Therefore, if a player with more net good signals finds it optimal to wait for k additional signals, it must be optimal for a player with fewer net good signals to wait as well. The next lemma consolidates the incentive compatibility constraints. From Lemma 1, we have

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Fig. 1. Determination of kd∗ .

Lemma 3. (IC2) implies (IC4); (IC1) and (IC3) imply (IC2). This lemma means that if it is profitable for a player with signal b to wait for kd∗ − 1 signals, then it must be profitable for him to wait for kd∗ signals. Furthermore, if it is optimal for a player with signal g to wait for kd∗ signals but not for kd∗ − 1 signals, then it is optimal for a player with signal b to wait for kd∗ − 1 signals. This lemma reduces (IC1), (IC2), (IC3) and (IC4) to (IC1) and (IC3). Suppose that π ( I |d + 1) ≥ δ R (∞|d + 1) = δ Pr (G |d + 1) πG . Then even if there are infinitely many signals to be revealed in this period, a player with signal g does not find it optimal to wait for these signals. In this case, any player with signal g will invest right away. It happens when players discount future payoffs a lot (i.e., δ is very small) or when signals are already very precise (i.e., p is very close to 1). Now suppose that π ( I |d + 1) < δ R (∞|d + 1). In this case, as we will show in the following lemma, there exists a unique and finite kd∗ satisfying (IC1) and (IC3). Note that (IC1) and (IC3) cannot be binding at the same time. Lemma 4. (a) Suppose that π ( I |d + 1) g ) < δ Pr(G |d + 1)πG . Then for each d ≥ 0, there exists a unique kd∗ satisfying (IC1) and (IC3); that is, there exists a unique solution to

δ R (kd∗ − 1|d + 1) ≤ π ( I |d + 1) ≤ δ R (kd∗ |d + 1). (b) Suppose that π ( I |d + 1) ≥ δ Pr(G |d + 1)πG . Then there exists no finite kd∗ satisfying (IC1) and (IC3). In this case, we define kd∗

(3)

= ∞.

The kd∗ uniquely determined in the above lemma is increasing in d, the number of net good signals at the time, as is shown in Fig. 1. kd∗ is the smallest k such that δ R (k|d + 1) exceeds π ( I |d + 1). Intuitively, given d, kd∗ is the smallest bunch size that is worthwhile for a player (with signal g) outside the bunch to wait for the signals from the bunch. The following lemma shows that when there are more net good signals, the bunch size will be larger. We have Lemma 5. k∗0 ≤ k∗1 ≤ k∗2 ≤ k∗3 ≤ · · ·. One way to interpret this lemma is that the situation is more optimistic when there are more net good signals. Therefore, players have more incentive to invest now. In order for the players outside the bunch to wait, the bunch size (and thus the number of signals to be revealed) in this case must be larger than the bunch size when the number of net good signals is small. From the proof of this lemma in the appendix, we can see that when d is larger, (IC1) is not affected; however, (IC3) becomes more difficult to satisfy. When (IC3) is violated for some k, one must increase k to meet the requirement of (IC3). The above kd∗ is the number of additional signals such that a player inside the bunch does not wait while a player outside the bunch prefers to wait. However, in the later periods of the game, the number of players not belonging to any previous bunches may be smaller than this kd∗ . Let n˜ t denote at the beginning of period t the number of players who did not belong to any previous bunches in the game. The determination of the bunch size above assumes implicitly that n˜ t ≥ kd∗ . When t n˜ t < kd∗ , that is, when the number of players not belonging to any previous bunch is not large enough to form a new bunch t of the desired size kd∗ , these players still form a (last) bunch. t The following lemma characterizes the behavior of players when last bunching occurs. Lemma 6. Suppose that the Last Bunching occurs in period t. In that period, a player with signal g inside the last bunch will invest. Meanwhile, a player (inside or outside the last bunch) with signal b will act as follows. (a) If π ( I |dt − 1) > δ R (˜nt − 1|dt − 1), then all players with signal b will invest in period t.

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(b) If π ( I |dt ) ≤ δ R (˜nt |dt ), then all players with signal b will wait in period t. Every signal is thus revealed at the end of period t. All players with signal b will invest in period t + 1 if and only if dt +1 ≥ 1. (c) If π ( I |dt ) > δ R (˜nt |dt ) and π ( I |dt − 1) ≤ δ R (˜nt − 1|dt − 1), all players with signal b belonging to any previous bunch will invest in period t, and all players with signal b in the last bunch will wait. Every signal is thus revealed at the end of period t. All remaining players with signal b will invest in period t + 1 if and only if dt +1 ≥ 1. Note that π ( I |dt − 1) > δ R (˜nt − 1|dt − 1) implies π ( I |dt ) > δ R (˜nt |dt ). Note also that when the last bunching occurs in period t, the difference between the a player with signal b in an earlier bunch and in the last bunch is that the former has already revealed his private signal by waiting (since the strategy of a player in a regular bunch is fully separating), while the latter has not yet revealed his private signal (since his earlier actions are fully pooling). Therefore, the latter has one more bad signal in his signal count than the former. If the latter prefers to invest in period t (than to wait), then the former prefers to invest in period t. This is Lemma 6(a). Conversely, if the former prefers to wait in period t, then the latter prefers to wait in period t as well. This is Lemma 6(b). It is also possible that the former prefers to invest in period t while the latter waits in period t. This is Lemma 6(c). However, it is never possible that the latter prefers to invest in period t while the former does not. If dt = 0 or 1, the condition in (a) never holds, since π ( I |dt − 1) < 0 and δ R (˜nt |dt − 1) ≥ 0. Therefore, the players in the last bunch act according to their signals; i.e., a player with signal g invests and a player with signal b waits. If dt ≥ 2, then it is possible that a player with signal b in the last bunch will invest right away. This is because investing right away brings in positive expected payoff, while waiting one period for just a few extra signals does not bring enough gain to offset the discounting. This happens when n˜ t is relatively small. 4. Equilibrium analysis In this paper, we focus on the analysis of asymmetric pure strategy equilibria. In our model, symmetric and asymmetric pure strategy equilibria do not co-exist. When

π ( I |1) ≥ δ R (n − 1|1)

(4)

holds, there exists a unique equilibrium. This equilibrium is in the form of symmetric pure strategies. Any player with signal g will invest in period 1; conditional on the number of investments in period 1, a player with signal b will invest in period 2 only if it is profitable to do so. No further investments will occur after period 2. Inequality (4) is equivalent to n ≤ k∗0 ; that is, the total number of players does not exceed the size of the supposed first bunch. Inequality (4) holds when n is very small or when δ is very small. In the rest of this section, we will focus on the situation where (4) is violated. 4.1. Asymmetric pure strategy PBE In this subsection, we characterize an asymmetric pure strategy equilibrium when n > k∗0 , or equivalently, when

π ( I |1) < δ R (n − 1|1).

(5)

We have the following theorem. Theorem 1. Suppose that (A1), (A2) and (5) hold. There exists an asymmetric pure strategy PBE, which has the following recursive properties. Let d1 = 0. In period t, t = 1, 2, ... (a) if dt > 1, Herding of Investing occurs, and all investment actions end; (b) if dt < 0, Herding of Waiting occurs, and all investment actions end; (c) if dt = 0 and n˜ t ≥ k∗0 , then Bunching of size k∗0 occurs; (d) if dt = 1 and n˜ t ≥ k∗1 , then Bunching of size k∗1 occurs; (e) if dt = 0 and n˜ t < k∗0 , or if dt = 1 and n˜ t < k∗1 , then the Last Bunching of size n˜ t occurs. (f) players with signal b in each bunch wait until the Last Bunching occurs, and at that time they behave according to Lemma 6. Note that in the above theorem (cf. Fig. 2), in any period t, dt is updated according to the actions of the players in the bunch of period t − 1. Any off-equilibrium behavior by players outside that bunch will be ignored. This off-equilibrium belief (and, in fact, any off-equilibrium belief) will support the equilibria in the theorem, since there is no action externality in our model. When a player deviates from “wait” to “invest”, what happens after the deviation becomes irrelevant to him. In equilibrium, Herding of Investing occurs when the actions in previous periods reveal that good signals outnumber bad signals by a sufficiently large number. Herding of Waiting occurs when the opposite happens, i.e., the actions in previous periods reveal that bad signals sufficiently outnumber good signals. In short, herding occurs when the probability of either state (G or B) becomes sufficiently large. Bunching occurs when the situation is not as certain. Note that the actions of the players in a bunch (except maybe the last one) will reveal their signals completely. Meanwhile, herding will reveal none of the players’ signals.

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Fig. 2. Flow Chart for Theorem 1.

In (c), (d), and (e) of the above theorem where bunching occurs, players with signal b wait. In (e), the final bunch is formed, and it is undersized. From this theorem, we know that in the first period, Bunching of size k∗0 forms since d1 = 0. In the second period, if d2 > 1, that is, d2 ≥ 2, then even a player with signal b will find it profitable to invest since his belief is at least one net good signal. Therefore, by Assumption (A1), the expected payoff from investing with (at least) one good signal is positive. If d2 < 0, that is, d2 ≤ −1, then even a player with a good signal waits since his belief is at most zero net good signals. From Assumption (A2), a player’s payoff is negative with a belief of zero net good signals. From this theorem, we can conclude that information revelation in this dynamic game is done through blocks. A block of signals are revealed in the first period. Depending on the difference between the two types of signals, another block of signals may be revealed in the second period, and so on and so forth, until herding occurs and thus no further signals are revealed. Note that the pace of information revelation does not depend on the amount of information to be revealed (which, holding the precision of each signal fixed, can be measured by the total number of private signals, n). The equilibrium in the above theorem has the following property. In any period, players outside the bunch do not need to know the identities of the players in previous bunches in order to infer the total numbers of good signals and bad signals in those bunches. The total number of players who have previously invested gives them the total number of good signals among the previous bunches; by subtracting this number from the total size of all previous bunches, they can then infer the total number of bad signals. For example, suppose that the size of the first bunch k∗0 = 5 and 3 players invested in the first period. Then a player need not know who these 3 players are; he can infer right away that there are 3 good signals and the rest of the bunch (i.e., 2 players) have bad signals. The equilibrium bunch sizes characterized by Theorem 1 exhibit strong patterns. Suppose that k∗0 is even. Then if there is bunching in a period, the bunch size is k∗0 . Suppose that k∗0 is odd and k∗1 is even. Then the bunch size is k∗0 in the first period, and k∗1 in all subsequent periods. Suppose that both k∗0 and k∗1 are odd. Then the bunch size is k∗0 in any odd period, and k∗1 in any even period. Note that the determination of k∗0 and k∗1 is independent of the total number of players n. The results in Theorem 1 are robust to the signal profitability requirements (A1) and (A2), which assume that investment is not profitable given no signal, but it is profitable given one good signal. These assumptions can be relaxed to the more general case, where investment is not profitable given m + 1 bad signals but becomes profitable given m bad signals. In Appendix B, we show that results similar to those in Theorem 1 continue to hold in this general case. The results in Theorem 1 are also robust to small perturbations in signal acquisition costs. The equilibria characterized in the paper are still valid if private signals are costly and players have the choice of not acquiring their signals. The equilibrium strategies in our model are also robust to small perturbations in players’ discount factors, as well as to introducing some degree of player risk aversion. We can rank the asymmetric equilibria according to the social welfare they generate in these cases, and in each case there is a unique asymmetric equilibrium that maximizes the social welfare. For example, in the case of costly signal acquisition, players with lower costs will form the earlier bunches (and acquire the signals), while players with higher costs will form the later bunches. In the case of unequal discount factors, players with lower discount factors (i.e., less patient players) will form earlier bunches and reveal their information first, while players with higher discount factors (i.e., more patient players) will form later bunches. Similarly, when players are risk averse, players who are less risk averse will form earlier bunches, while players who are more risk averse will form later bunches. The differentiations in players’ characteristics serve as a natural coordinator for the players.8 In the next subsection, we formally characterize the asymmetric pure strategy equilibria when players have different discount factors.

8 In a different context, Leduc et al. (2016) show that agents with low degree (few friends) have incentives to adopt a new technology early, while agents with high degree have incentives to free ride in a network model.

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ˆ Fig. 3. Determination of bunch size k.

4.2. Coordination when players have different discount factors When players have different discount factors, their incentives to wait for more revealed signals are different from player to player. This creates a natural and effective device to sort the players into different bunches, resolving the coordination issue in the asymmetric equilibria of a symmetric game. Suppose that we rank the players from the least patient to the most, and without loss of generality, let δ1 ≤ δ2 ≤ · · · ≤ δn . For player i with discount factor δi and for each public information d, there exists a unique bunch size kd∗ (δi ), determined similarly to (3) by the following:

    δi R kd∗ (δi ) − 1|d + 1 ≤ π ( I |d + 1) ≤ δi R kd∗ (δi )|d + 1 .

(6)

We have the following lemma. Lemma 7. kd∗ (δ1 ) ≥ kd∗ (δ2 ) ≥ · · · ≥ kd∗ (δn ). The above lemma shows that the cutoff bunch size is larger for a less patient player. This is very intuitive, since a less patient player needs more revealed signals to be persuaded to wait until the next period. With this lemma, we now show that less patient players will be in earlier bunches. We have the following proposition. Proposition 1. Suppose that players’ discount factors may be different. Then the equilibrium characterized in Theorem 1 is still a PBE, with equilibrium bunch size kˆ (for a given d) determined by the following: (1) Rank all (say, k) players who did not belong to any previous bunch by their discount factors:

δn1 ≤ δn2 ≤ · · · ≤ δnk . (2) Determine the bunch size kˆ by

kd∗ (δn ˆ ) ≥ kˆ , k

and kd∗ (δn ˆ

k +1

) ≤ kˆ .

(7)

ˆ will be in the bunch, some of the players at the cutoff (with k∗(δn ) = k) ˆ will be in the bunch, Less patient players (with kd∗ (δni ) > k) i d ˆ will not be in the bunch. and more patient players (with kd∗ (δni ) < k)

Note that when some players’ discount factors are close to each other, there could multiple players with kd∗ (δni ) at

ˆ In this case, there could be multiple equilibria. Every player with k∗ (δn ) > kˆ is definitely in the bunch. The the cutoff k. i d

ˆ For example, suppose that there remaining bunch players are to be selected from those marginal players with kd∗ (δni ) = k. are still 7 players not belonging to previous bunches, with kd∗ (δn1 ) = 6, kd∗ (δn2 ) = 5, kd∗ (δn3 ) = 5, kd∗ (δn4 ) = 4, kd∗ (δn5 ) = 4,

kd∗ (δn6 ) = 4, and kd∗ (δn7 ) = 3. In this case, kˆ = 4. Players n1 , n2 , and n3 are definitely in the bunch, and exactly one of the players n4 , n5 , and n6 is to be in the bunch as well (cf. Fig. 3). In this example (and in any situation with multiple players ˆ the coordination issue for asymmetric equilibria does not go away completely. Nevertheless, except satisfying kd∗ (δni ) = k), for those players at the cutoff, all other players are sorted out fully.

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Fig. 4. Flow chart for Theorem 2.

In our main model where every player has identical discount factor, kd∗ (δni ) is the same for every player. In this case, the

bunch size kˆ is equal to kd∗ (δni ), and any kˆ of the players are to be in the bunch. 4.3. Other asymmetric pure strategy PBE

The equilibrium characterized in Theorem 1 is not the only asymmetric pure strategy equilibrium of this game. There are other equilibria, which take similar forms but with more bunching. It turns out that when the difference between the numbers of good signals and bad signals revealed before period t is at least 2, i.e., dt ≥ 2, herding is not the unique equilibrium behavior; bunching may also occur in this situation. The equilibrium characterized in Theorem 1 has the least bunching; all other equilibria we will characterize in this subsection will have more bunching. We first have the following lemma.

¯ and k∗ is infinite when dt > d. ¯ Lemma 8. For any fixed δ ∈ [0, 1), there exists an integer d¯ > 0, such that kd∗ is finite when dt ≤ d, d We can now characterize the additional asymmetric equilibria (Fig. 4). Theorem 2. Suppose that (A1), (A2) and (5) hold. Then the following recursive properties characterize an asymmetric pure strategy PBE. Let d1 = 0. In period t, t = 1, 2, ..., (a) if dt < 0, Herding of Waiting occurs, and all investment actions end; (b) if dt = 0, 1, and n˜ t ≥ kd∗ , then Bunching of size kd∗ occurs; t

t

¯ and n˜ t ≥ k∗ , then Bunching of size k∗ occurs, or, alternatively, Herding of Investing occurs; (c) if dt = 2, ..., d, dt dt (d) if dt = 0, 1, and n˜ t < kd∗ , then the Last Bunching of size n˜ t occurs; t (e) if dt ≥ 2, and n˜ t < kd∗ , then the Last Bunching of size n˜ t occurs, or, alternatively, Herding of Investing occurs. t (f) players with signal b in each bunch waits until the Last Bunching occurs, and at that time they behave according to Lemma 6.

Again, the off-equilibrium beliefs will be assigned similarly to Theorem 1. When dt < 2, the strategies in this equilibrium are the same as those in Theorem 1. When dt ≥ 2, there are multiple equilibria in this “subgame”. Bunching is one equilibrium, while herding is another equilibrium. When bunching occurs, it reveals the private signals of the players in the bunch. Meanwhile, herding reveals no private information. Therefore, the outcomes of the equilibria in Theorem 2 where bunching occurs in certain “subgames” are more informative than those in Theorem 1 where herding occurs in the same “subgames.” Again, the information revelation in these equilibria is done through blocks. 4.4. Comparative static analysis on δ In this subsection, we investigate the effect of the discount factor δ on the asymmetric equilibrium strategies. We also compare our asymmetric equilibria with the equilibrium in the following one-per-period exogenous timing model. In this model, the set up is the same as in Section 2 except the timing is exogenous: only one pre-specified player makes a decision on investment in any given period. Each player has only one chance to invest. If a player waits in his specified period, he will have no other chance to invest later on.9 We first have the following lemma.

9

See Bikhchandani et al. (1992), Banerjee (1992) and Wang (2016).

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Lemma 9. Suppose that (A1), (A2) and (5) hold. For a given d, if δ increases, the bunch size kd∗ decreases. An increase in δ means that the players become more patient and waiting is less costly. So in order for a more patient player in the bunch to invest right away (instead of waiting one period for more signals to be revealed), the number of signals to be revealed needs to be smaller. This can be seen from (IC1). Furthermore, (IC3) becomes easier to satisfy when δ increases. π ( I |n−2) We now discuss the case of large δ , where players are sufficiently patient. Define δˆ ≡ R (1|n−2) < 1. As long as δ ≥ δˆ , the bunch size for any d ≥ 0 is always 1. This is because δ ≥ δˆ guarantees kn∗−2 = 1, and kd∗ is increasing in d. We have the following proposition.

ˆ 1]. Then k∗ = 1 for any d ≥ 0. In this case, The outcomes Proposition 2. Suppose that (A1), (A2) and (5) hold. Suppose also that δ ∈ [δ, d of the asymmetric pure strategy equilibria described in Theorem 1 coincide with the outcomes of the equilibrium in the one-per-period exogenous timing model. When δ ≥ δˆ , the cost of waiting for more revealed signals is small. It is not an equilibrium that more than one player is in the bunch, since one of the bunch members would wait.10 Players reveal their signals at most one per period until herding occurs. Of course, there are still other asymmetric pure strategy equilibria. This is because when d ≥ 2 in a given period, bunching with a bunch size of 1 is one equilibrium behavior, and herding is another equilibrium behavior. Theorem 1 and Theorem 2 characterize all of the asymmetric pure strategy equilibria. The equilibrium with the least bunching nests the equilibrium outcome in the one-per-period exogenous timing model. Since d1 = 0, bunching of size k∗0 = 1 occurs in period 1. If the bunch member waits and reveals that his signal is b, then d2 = −1 and herding of waiting occurs and no more investing will happen. If he invests and reveals that his signal is g, then d2 = 1. In this case, k∗1 = 1 and bunching of size 1 occurs in period 2. In period 2, if the revealed signal is b, then d3 = 0 and bunching of size 1 occurs again in period 3. If the revealed signal is g, then d3 = 2 and herding occurs in this equilibrium. This outcome corresponds to the one-per-period exogenous timing model as in Banerjee (1992) or Bikhchandani et al. (1992). In such a model, players make decisions sequentially according to an exogenous order, and only one player can invest in one period. Players herd when d = −1 or when d = 2 in the equilibrium. Our asymmetric pure strategy equilibria naturally include this equilibrium outcome from the one-per-period exogenous timing model. The above equilibrium has the least bunching (and thus the most herding) among all co-existing asymmetric pure strategy equilibria. Because the information revealed is the least, this equilibrium is the least efficient. We now consider the equilibrium with the most bunching. When dt ≤ 1, players’ behavior is exactly the same as in the above equilibrium. However, when dt ≥ 2, bunching of size 1 occurs (instead of herding) and private signals are still being revealed. Herding of investing never occurs. The total amount of information revealed by this equilibrium is maximized. The above discussions are for the case of large enough δ . When δ becomes smaller, the bunch size for any d becomes π ( I |1) larger (cf. Lemma 9). In this case, the asymmetric equilibria in our model become even more efficient. When δ ≤ Pr(G |1)π , G the size of the first bunch (as well as any subsequent bunch) is equal to infinity (cf. Lemma 4). In this case, when there are infinitely many players, the true state of the economy is learned after the first period. This must be more efficient than any N-player-per-period exogenous timing model. This conclusion is also true when δ is slightly larger (and thus the first bunch is finite but large). 5. Conclusion In this paper, we characterize the asymmetric pure strategy equilibria when players with private information decide not only whether or not to make an investment in a risky environment, but also when to make it. In these equilibria, players either herd or bunch in any given period. Bunching can occur only when there has been bunching in all previous periods. After herding occurs, the investment action ends. In these equilibria, some players make decisions earlier and some later. Later players learn information from the earlier decision makers and can make more informed decisions, but their payoffs are discounted. We focus on bunching in this paper. When bunching occurs, a bunch of players make decisions while the rest of the players do nothing regardless of their signals. When players are very patient, the least efficient asymmetric equilibrium resembles the one-per-period exogenous timing model, while in the most efficient equilibrium, the bunch size is always equal to 1 and herding of investing disappears. When players are less patient, the bunch sizes become larger, and more information is revealed through bunching. When players have different discount factors, we show that less patient players will join earlier bunches, while more patient players will join later bunches. The analysis in this paper has many applications. It can be applied to the adoption of new products, the release of new movies, the issuing of IPOs, the investments in the financial markets, the group purchase of coupons on websites such as

10 When δ = 1, waiting is costless, and (5) always holds. In this case, it is an equilibrium behavior that every player waits in some periods, even though every player waiting forever is not an equilibrium.

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Groupon.com, etc. When people make decisions in a dynamic setting, the asymmetric equilibria in our model predict that they are likely to make decisions in groups. This and other predictions can be tested empirically in future research. Acknowledgments We would like to thank the editor, an advisory editor and a referee for their very helpful and constructive comments. We would also like to thank Archishman Chakraborty, Christophe Chamley, Ettore Damiano, Douglas Gale, Rene Kirkegaard, Thorsten Koeppl, Jonathan Levin, Li Hao, Wei Li, Sumon Majumdar, Maxwell Pak, Jacob Sagi, Shouyong Shi, Veikko Thiele, Ruqu Wang, Jan Zabojnik, Charles Zheng, and seminar participants at the SIOMS at SWUFE, CEA, and Queen’s University for their helpful comments. Financial support from the Young Scholar Creative Research Grant from the Central Government in China is gratefully acknowledged. Appendix A. Proofs

Proof for Lemma 1.

R (kt |d˜ t )

=

kt 

π ( B R (d˜ t + r − (kt − r ))|d˜ t + r − (kt − r )) Pr(rg , (kt − r )b|d˜ t )

r =0

=

kt  

π ( B R (d˜ t + r − (kt − r ))|d˜ t + r + s − (kt − r − s)) Pr((r + s) g , (kt − r − s)b|d˜ t )

s=0,1 r =0

<

kt  

π ( B R (d˜ t + r + s − (kt − r − s))|d˜ t + r + s − (kt − r − s)) Pr((r + s) g , (kt − r − s)b|d˜ t )

s=0,1 r =0

= R (kt + 1|d˜ t ) This completes the proof for the first part of the lemma. When kt goes to infinity, there will be infinite number of signals to be revealed in period t. Given the underlying state, the signals are i.i.d. According to the Law of Large Numbers, the proportion of good signals converges to its expected value with probability one, that is, it converges to p in state G, and 1 − p in state B. Note that p > 12 . Therefore, in period t + 1, when the state is G, the total number of good signals will exceed the total number of bad signals by any given finite number with probability one. Seeing this, the best response will always be a = I , that is, to invest. Likewise, in period t + 1, when the state is B, the total number of bad signals will exceed the total number of good signals by any given finite number with probability one. Seeing this, the best response will always be a = W , that is, to wait. Therefore, the expected payoff is Pr(G |d˜ t )πG + Pr( B |d˜ t )0. This completes the proof for the second part of the lemma. 2 Proof for Lemma 2.

π ( I |d) − R (k|d) ⎛

=⎝

k 

−

r =0

=





⎞ ⎠ π ( I |d + r − (k − r )) Pr(rg , (k − r )b|d)

d+r >k−r

π ( I |d + r − (k − r )) Pr(rg , (k − r )b|d).

d+r ≤k−r

π ( I |d + r − (k − r )) is increasing in d and each π ( I |d + r − (k − r )) is negative when d + r ≤ k − r. Also, Pr(rg , (k − r )b|d)

= Pr(rg , (k − r )b|G ) Pr(G |d) + Pr(rg , (k − r )b| B ) Pr( B |d)

 = pk−r (1 − p )r − pr (1 − p )k−r Pr( B |d) + pr (1 − p )k−r . When d + r ≤ k − r and p > 12 , pk−r (1 − p )r − p r (1 − p )k−r > 0. Recall that Pr( B |d) is decreasing in d. Thus, Pr(rg , (k − r )b|d) is decreasing in d and it is positive. Thus, each term after the summation sign is negative and decreasing in d in absolute value. Also, the range for r such that d + r ≤ k − r is shrinking as d increases. Therefore, π ( I |d) − R (k|d) is increasing in d.

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Since

π ( I |d) − δ R (k|d) = (1 − δ)π ( I |d) + δ[π ( I |d) − R (k|d)], and

π ( I |d) is increasing in d, we conclude that π ( I |d) − δ R (k|d) is increasing in d. 2

Proof for Lemma 3. From Lemma 1, it is immediate that (IC2) implies (IC4). By Lemma 2 and (IC3), we have

0 ≥ π ( I |d + 1) − δ R (kd∗ |d + 1) > π ( I |d) − δ R (kd∗ |d). Also,

π ( I |d) = π ( I |d − 1) Pr(b|d) + π ( I |d + 1) Pr( g |d), and













R kd∗ |d = R kd∗ − 1|d − 1 Pr(b|d) + R kd∗ − 1|d + 1 Pr( g |d).     From (IC1) and π ( I |d) < δ R kd∗ |d , it must be that π ( I |d − 1) < R kd∗ − 1|d − 1 . This completes the proof. 2 Proof for Lemma 4. From Lemma 1, R (k|d + 1) is increasing in k. Since π ( I |d + 1) < δ R (∞|d + 1), the curve of δ R (k|d + 1) intersects the line π ( I |d + 1) only once and at k ≥ 1. If π ( I |d + 1) ≥ δ R (∞|d + 1) = δ Pr (G |d + 1) πG , then no k can satisfy (IC1) and (IC3). In this case, a player with signal g would not wait for any finite number of signals to be revealed. In this case, we define kd∗ = ∞. 2 Proof for Lemma 5. Since π ( I |d) − δ R (k|d) is decreasing in k, and by Lemma 2, an increase in d shifts the curve π ( I |d) − δ R (k|d) upwards, resulting in a bigger k. Note here the inequalities are not strict because of the discreteness of k. 2 Proof for Theorem 1. Consider any period t when a regular bunch is to form. For a type b player either in the bunch or outside the bunch, there is no incentive for him to deviate by investing right away because (IC2) and (IC4) hold as long as (IC1) and (IC3) hold. For a type g player outside the bunch, he does not have incentive to deviate by investing right away because of (IC3). Now we prove that if a type g player inside the bunch deviates by waiting in period t, he will use his myopic best response in period t + 1 instead of waiting further. If a type g player inside the bunch deviate by waiting in period t, and at the beginning of period t + 1, the public information is updated as d good signals. Then it is actually d + 2 good signals to him. If herding occurs in period t + 1, no information will be revealed in this period. Thus, this player will wait no further and use his myopic best response. If bunching of size kd∗ occurs in period t + 1, this player will also use his best response if



π ( I |d + 2) ≥ δ R kd∗ |d + 2



(A.1)

holds. We have

π ( I |d + 2) = π ( I |d + 3) Pr( g |d + 2) + π ( I |d + 1) Pr(b|d + 2), and

















R kd∗ |d + 2 = R kd∗ − 1|d + 3 Pr( g |d + 2) + R kd∗ − 1|d + 1 Pr(b|d + 2). By (IC1), we have

π ( I |d + 1) ≥ δ R kd∗ − 1|d + 1 . By Lemma 2, it must be





π ( I |d + 3) ≥ δ R kd∗ − 1|d + 3 . Therefore, inequality (A.1) holds. For the equilibrium strategies characterized in Theorem 1, d takes the value of either 0 or 1. The above analysis shows that no player would deviate. This completes the proof for Theorem 1. 2

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Proof for Lemma 8. Given δ , kd∗ is finite only if

π ( I |dt + 1) < δ R (∞|dt + 1) , that is, when

Pr (G |dt + 1) πG + Pr ( B |dt + 1) π B < δ Pr (G |dt + 1) πG . Rearranging, we have

πG + Since

Pr ( B |dt + 1) Pr (G |dt + 1)

Pr( B |dt +1) Pr(G |dt +1)

π B < δ πG .

is decreasing in dt and

π B is negative, the second term on the LHS is increasing and converging to 0. Thus,

¯ there exists an upper bound for dt for the above inequality to hold. Define this upper bound as d. ∗ When the above inequality does not hold, kd = ∞ (cf. Lemma 4). This concludes the proof of this lemma.

2

Proof for Theorem 2. For d = 0 or 1, the proof follows Theorem 1. For d ≥ 2, herding could occur. If every player who does not belong to an earlier bunch herds, then no more private information is revealed. In this case, there is no incentive to wait. Therefore, everyone herds is one equilibrium, as is characterized in Theorem 1. Meanwhile, for d ≥ 2, kd∗ exists. Therefore, it is also an equilibrium that a bunch of size kd∗ is formed. The proof in Theorem 1 still goes through. Therefore, there are two different types of equilibria for every d ≥ 2. This completes the proof of Theorem 2. 2 Proof for Lemma 9. When δ increases, δ R (k|d + 1) shifts upward for any given k. Therefore, k has to decrease so that (3) is still satisfied.

2

Appendix B. Equilibrium characterization when (A1) or (A2) does not hold In the analysis in the text, we assumed that (A1) and (A2) hold simultaneously. In this appendix, we will analyze the situations where (A1) or (A2) does not hold. There are two cases. Case 1: π ( I |1) < 0. In this case, the unique equilibrium is a trivial one, where every player waits in the first period and thus also in all subsequent periods. This result is obvious, because in the first period even a player with signal g finds it not profitable to invest and therefore players with either signal wait in the first period. Since there is no information revelation in the first period, the situation repeats itself in the second period, and in every period afterwards. Every player waits in all periods, and there is no investment by any player in any period. Case 2: For some m > 0,

π ( I |1 − m ) > 0

(A1 )

π ( I | − m ) < 0.

(A2 )

and

Assumptions (A1 ) and (A2 ) imply that it is profitable to invest when conditional on m − 1 net bad signals, but not when conditional on m net bad signals. Note that when m = 0, Assumptions (A1 ) and (A2 ) become Assumptions (A1) and (A2). In this case, we can also characterize the asymmetric pure strategy equilibria similarly to Theorem 2. Theorem 3. Suppose that (A1 ), (A2 ) and (5) hold. Then all asymmetric pure strategy PBE have the following recursive properties. Let d1 = 0. In period t, t = 1, 2, ..., (a) if dt < −m, Herding of Waiting occurs, and all investment actions end; (b) if dt = −m, −m + 1, and n˜ t ≥ kd∗ , then Bunching of size kd∗ occurs; t

t

¯ and n˜ t ≥ k∗ , then Bunching of size k∗ is one equilibrium, and Herding of Investing is another equilibrium; (c) if dt = −m + 2, ..., d, dt dt (d) if dt = −m, −m + 1, and n˜ t < kd∗ , then the Last Bunching of size n˜ t occurs; t (e) if dt ≥ −m + 2, and n˜ t < kd∗ , then the Last Bunching of size n˜ t is one equilibrium, and Herding of Investing is another equilibrium. t (f) players with signal b in each bunch waits until the Last Bunching occurs, and at that time they make their investment decisions similarly to Lemma 6.

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The proof is analogous to the proof of Theorem 2. The new assumptions change the threshold between profitability and unprofitability. Now a player with signal g can invest profitably even if he infers that among other players there are m or fewer net bad signals. However, as long as the bunch is at the right size, it provides the correct incentive for some players to bunch and the rest to wait in a certain period. Theorem 3 and Theorem 2 differ in what may happen in the first period. In Theorem 2, bunching of size k∗0 is the unique equilibrium outcome. However, when m ≥ 2, it is profitable for a player with signal b to invest in the first period. Therefore, both herding of investing and bunching of size k∗0 are possible equilibrium outcomes in the first period. In fact, for a given m, the situation facing the players in the first period under (A1 ) and (A2 ) is similar to the situation facing the players when d = m in some period under (A1) and (A2). Therefore, the strategies in Theorem 3 can be viewed as the strategies in the “subgames” of Theorem 2. References Aoyagi, M., 1998. Equilibrium delay in a simple dynamic model of investment. Econ. Theory 12 (1), 123–146. Banerjee, A.V., 1992. A simple model of herd behavior. Quart. J. Econ. 107 (3), 797–817. Bikhchandani, S., Hirshleifer, D., Welch, I., 1992. A theory of fads, fashion, custom, and cultural change as informational cascades. J. Polit. Economy 100 (5), 992–1026. Bowden, M.P., 2013. Social learning, herd behavior and information cascades: a review of the recent developments in relation to their criticisms. J. Econ. Res. 18, 205–236. Cao, H.H., Han, B., Hirshleifer, D., 2011. Taking the road less traveled by: does conversation eradicate pernicious cascades? J. Econ. Theory 146, 1418–1436. Chamley, C., 2004a. Delays and equilibria with large and small information in social learning. Europ. Econ. Rev. 48, 477–501. Chamley, C., 2004b. Rational Herds: Economic Models of Social Learning. Cambridge University Press, Cambridge. Chamley, C., Gale, D., 1994. Information revelation and strategic delay in a model of investment. Econometrica 62 (5), 1065–1085. Chari, V.V., Kehoe, P.J., 2004. Financial crises as herds: overturning the critiques. J. Econ. Theory 119, 128–150. Gul, F., Lundholm, R., 1995. Endogenous timing and the clustering of agents’ decisions. J. Polit. Economy 103 (5), 1039–1066. Khanna, N., Mathews, R.D., 2011. Can herding improve investment decisions? RAND J. Econ. 42 (1), 150–174. Leduc, M.V., Jackson, M.O., Johari, R., 2016. Pricing and Referrals in Diffusion on Networks. Working Paper. Stanford University. Levin, D., Peck, J., 2008. Investment dynamics with common and private values. J. Econ. Theory 143, 114–139. Moretti, E., 2011. Social learning and peer effects in consumption: evidence from movie sales. Rev. Econ. Stud. 78, 356–393. Park, A., Sabourian, H., 2011. Herding and contrarian behaviour in financial markets. Econometrica 79 (4), 973–1026. Rosenberg, D., Solan, E., Vieille, N., 2007. Social learning in one-arm bandit problems. Econometrica 75 (6), 1591–1611. Scharfstein, D., Stein, J.C., 1990. Herd behavior and investment. Amer. Econ. Rev. 80 (3), 465–479. Sgroi, D., 2003. The right choice at the right time: a herding experiment in endogenous time. Exper. Econ. 6 (2), 159–180. Wang, T., 2016. Performance Measurement and Herding. Working Paper. Queen’s University. Zhang, J., 1997. Strategic delay and the onset of investment cascades. RAND J. Econ. 28 (1), 188–205. Zhang, Y., 2009. Robust Information Cascade with Endogenous Ordering. Working Paper. Singapore Management University.