Observational Learning with Position Uncertainty

Transcription

1 Observational Learning with Position Uncertainty Ignacio Monzón and Michael Rapp December 009 Abstract Observational learning is typically examined when agents have precise information about their position in the sequence of play. We present a model in which agents are uncertain about their positions. Agents are allowed to have arbitrary ex-ante beliefs about their positions: they may observe their position perfectly, imperfectly, or not at all. Agents sample the decisions of past individuals and receive a private signal about the state of the world. We show that social learning is robust to position uncertainty. Under any sampling rule satisfying a stationarity assumption, learning is complete if signal strength is unbounded. In cases with bounded signal strength, we show that agents achieve what we define as constrained efficient learning: all individuals do at least as well as the most informed agent would do in isolation. JEL Classification: C7, D83, D85 We are grateful to Bill Sandholm for his advice, suggestions and encouragement. We also thank Dan Quint, Ricardo Serrano-Padial, Marzena Rostek, Marek Weretka, Ben Cowan, María Eugenia Garibotti, Danqing Hu, Sang Yoon Lee, Fernando Louge and David Rivers for valuable comments. All remaining errors are ours. Collegio Carlo Alberto, Via Real Collegio 30, 1004 Moncalieri O), Italy. ignacio@carloalberto.org, Department of Economics, University of Wisconsin Madison, 1180 Observatory Drive, Madison, WI 53706, USA. mrapp@wisc.edu,

2 1. Introduction In a wide range of economic situations, agents possess private information regarding some shared uncertainty. hese include choosing between competing technologies, deciding whether to invest in a new business, deciding whether to eat at a new restaurant in town, or selecting which new novel to read. If actions are observable but private information is not, one agent s behavior provides useful information to others. Consider the example of choosing between two MP3 players recently released to the market. One is inherently better for all individuals, but agents do not know which. 1 Individuals form a personal impression about the quality of each device based on information they receive privately. hey also observe the choices of others, which partly reveal the private information of those agents. An important characteristic of this environment has been overlooked: when choosing between competing alternatives, an agent may not know how many individuals have faced the same decision before him. Moreover, when an agent observes someone else s decision, he also may not know when that decision was made. For example, even when individuals know when the competing MP3 players were released, they may be unaware of exactly how many individuals have already chosen between the devices. In addition, when an agent observes someone else listening to an MP3 player on the bus, he does not know whether the person using the device bought it that morning or on the day it was released. We study settings with position uncertainty, and ask whether individuals eventually learn and choose the superior technology, by observing the behavior of others. he seminal contributions to the social learning literature are Banerjee 199] and Bikhchandani, Hirshleifer, and Welch 199]. In these papers, a set of rational agents choose sequentially between two technologies. he payoff from this decision is identical to all, but unknown. Agents know their position in the adoption sequence. In each period, one agent receives a signal and observes what all agents before him have chosen. Afterwards, this agent makes a once-and-for-all decision between the two technologies. Given that each agent knows that the signal he has received is no better than the signal other individuals have received, agents eventually follow the behavior of others and ignore their own signals. Consequently, Banerjee and Bikhchandani et al. 1 We study situations where network externalities do not play a salient role, including choosing between news sites, early search engines, computer software, MP3 players or computer brands. Our focus is on informational externalities. We are currently working on a setting with network externalities, for which we have partial results. 1

3 show that the optimal behavior of rational agents can prevent social learning. Smith and Sørensen 000] highlight that there is no information aggregation in the models of Banerjee 199] and Bikhchandani et al. 199] because these models assume signals are of bounded strength. In Smith and Sørensen 000], agents also know their positions in the adoption sequence and observe the behavior of all preceding agents. In contrast to Banerjee 199] and Bikhchandani et al. 199], agents receive signals of unbounded strength: signals can get arbitrarily close to being perfectly informative. In such a context, the conventional wisdom represented by a long sequence of agents making the same decision can always be overturned by the action of an agent with a strong enough signal. As a result, individuals never fully disregard their own information. In fact, Smith and Sørensen 000] show that if signals are of unbounded strength, social learning occurs. We take as our starting point the basic setup in the literature on social learning and introduce position uncertainty. In our setting, agents, exogenously ordered in a sequence, choose between two competing technologies. Agents receive a noisy private signal about the quality of each technology. We study cases of both bounded and unbounded signal strength. We depart from the literature by assuming that agents may not know 1) their position in the sequence or ) the position of those they observe. We refer to both phenomena as position uncertainty. We also depart from Banerjee 199], Bikhchandani et al. 199] and Smith and Sørensen 000] in that agents observe the behavior of only a sample of preceding agents. o understand the importance of position uncertainty, note that if an agent knows his position in the sequence, he can condition his behavior on three pieces of information: his private signal, his sample and his position in the sequence. In fact, he may weigh his sample and signal differently depending on his position. he typical story of social learning is one of learning over time. Early agents place a relatively larger weight on the signal than on the sample. As time progresses, information is aggregated and the behavior of agents becomes a better indicator of the true state of the world. Later agents place a relatively larger weight on the sample, which, in turn, can lead to precise information aggregation. In contrast, if agents have no information about their positions, such heterogeneity in play is impossible. Instead, agents place the same weight on the sample regardless of when they play. Heterogeneous play based on positions cannot drive social learning in this setting. Position uncertainty leads to an additional difficulty. Agents A and B do not know if A precedes B or vice versa. his adds a strategic component to the game: A s strategy affects B s payoffs inasmuch as B s strategy affects A s payoffs. hus, the game cannot be solved recursively and, in fact, there are cases with multiple equilibria.

4 his paper presents a flexible framework for studying observational learning under position uncertainty. Our framework is general in two ways. First, agents are placed in the adoption sequence according to an arbitrary distribution. For example, some agents may, ex-ante, be more likely to have an early position in the sequence, while others are more likely to be towards the end. Second, our setup allows for general specifications of the information agents obtain about their position, once it has been realized. Agents may observe their position, receive no information about it, or have imperfect information about it. For instance, agents may know if they are early or late adopters, even if they do not know their exact positions. We focus on stationary sampling, which allows for a rich class of natural sampling rules. Sampling rules are stationary if the probability that an agent samples a particular set of individuals is only a function of the distance between the agent and those he observes that is, how far back in time those decisions were made). his assumption implies that no individual plays a decisive role in everyone else s samples. We say social learning occurs if the probability that a randomly selected agent chooses the superior technology approaches one as the number of agents grows large. We find that learning is robust to the introduction of position uncertainty. We specify weak requirements on the information of agents: they observe a private signal and at least an unordered sample of past play. Agents need not have any information on their position in the sequence. Under unbounded signal strength, social learning occurs. his is due to the fact that as the number of agents grows large, individuals rely less on the signal. In cases with bounded signal strength, we show agents achieve what we define as constrained efficient learning: all individuals do at least as well as the most informed agent would do in isolation. In the present setting, social learning results from the combination of stationary sampling and a modified improvement principle. First introduced by Banerjee and Fudenberg 004], the improvement principle states that since agents can copy the decisions of others, they must do at least as well, in expectation, as those they observe. In addition, when an agent receives a very strong signal and follows it, he must do better than those he observes. As long as the agents observed choose the inferior technology with positive probability, the improvement is bounded away from zero. In Banerjee and Fudenberg s model, agents know their positions and observe others from the preceding period. If learning did not occur, this would mean that agents choose the inferior technology with positive probability in every period. hus, there would be an improvement between any two consecutive periods. his would lead to an infinite sequence of improvements, 3

5 which cannot occur, since utility is bounded. In that way, Banerjee and Fudenberg show social learning must occur in their setting. Acemoglu, Dahleh, Lobel, and Ozdaglar 008] and Smith and Sørensen 008] use similar arguments to show social learning in their models. We develop an ex-ante improvement principle, which allows for position uncertainty, and so places weaker requirements on the information that agents have. However, this ex-ante improvement principle does not guarantee learning by itself. Under position uncertainty, the improvement upon those observed is only true ex-ante i.e. before the positions are realized). Conditional on his position, an agent may actually do worse, on average, than those he observes. 3 Stationary sampling rules have the following implication: as the number of agents grows large, all agents become equally likely to be sampled. 4 his leads to a useful accounting identity. Note that the ex-ante expected) utility of an agent is the average over the utilities he would get in each possible position. As all agents become equally likely to be sampled, the expected utility of an observed agent approaches the average over all possible positions. hus, for all stationary sampling rules, the difference between the ex-ante utility and the expected utility of those observed must vanish as the number of agents grows large. o summarize, our ex-ante improvement principle states that the difference between the exante utility and the expected utility of an observed agent only goes to zero if, in the limit, agents cannot improve upon those observed. his, combined with stationary sampling rules, implies there is social learning if signal strength is unbounded and constrained efficient learning if signal strength is bounded. he result of constrained efficient learning is novel in the literature and is useful for two reasons. First, it describes a lower bound on information aggregation for all information structures. For any signal strength, agents do, in expectation, at least as well as if they were provided with the strongest signals available. Second, it highlights the continuity of social learning, or perfect information aggregation. Social learning becomes a limit result from constrained efficient learn- 3 o see this, consider the following example. here is a large number of agents in a sequence. Everyone knows their position exactly except for the agents in the second and last positions, who have an equal chance of being in each of these positions. If each agent observes the decision of the agent that preceded him, then, by the standard improvement principle, the penultimate agent in the sequence makes the correct choice with a probability approaching one. he agent who plays last, on the other hand, is not sure that he is playing last. He does not know if he is observing the penultimate agent, who is almost always right, or the first agent, who often makes mistakes. As a result, the agent who plays last relies on his signal too often, causing him to make mistakes more often than the agent he observes. 4 he following example illustrates this point. Let all agents be equally likely to be placed in any position in a finite sequence and let sampling follow a simple stationary rule: each agent observes the behavior of the preceding agent. An agent does not know his own position and wonders about the position of the individual he observes. Since the agent is equally likely to be in any position, the individual observed is also equally likely to be in any position except for the last position in the sequence). 4

6 ing: as we relax the bounds on the signal strength, the lower bound on information aggregation approaches perfect information aggregation. Before introducing the formal model, we describe the related literature on observational learning. Most of the literature has focused on cases in which agents know their own positions, and sample from past behavior. he aforementioned Banerjee and Fudenberg 004], Çelen and Kariv 004], Acemoglu, Dahleh, Lobel, and Ozdaglar 008] and Smith and Sørensen 008] are among them. Acemoglu et al. 008] present a model on social learning with stochastic sampling. In their model, agents know both their own position and the position of sampled individuals. Under mild conditions on sampling, social learning occurs. 5 Smith and Sørensen 008] is the only paper to allow for some uncertainty about positions. In their model, agents know their own position but do not know the positions of the individuals sampled. Since agents know their own positions, an improvement principle and a mild assumption on sampling ensures that social learning occurs. In the next section we explain the model and discuss the implications of stationary sampling. In Section 3, we present our results on social learning in the case that agents are ex-ante symmetric and play a symmetric equilibrium. Section 4 generalizes these findings to the asymmetric cases. We also extend our results to a non-stationary sampling rule: geometric sampling. Section 5 concludes.. Model here are players, indexed by i. Agents are exogenously ordered as follows. Let p : {1,..., } {1,..., } be a permutation and let P be the set of all possible permutations. hen, P is a random variable with realizations p P. he random variable P i) specifies the period in which player i is asked to play. We assume that every permutation has equal probability. In Section 4.1 we generalize the model to allow for arbitrary distributions over permutations and show that our results also hold in that case. Agents know the length of the sequence but may not know their position in it. We are interested in settings where is large. herefore we study the behavior of agents as approaches infinity. 5 o see why position uncertainty matters in a setup like the one described in Acemoglu et al. 008], consider the following example. Each agent observes the very first agent and the agent before him. However, they do not know which is which. Since the first agent may choose the inferior technology, his incorrect choice may carry over. his happens because individuals cannot identify who is the first agent in their sample. If agents knew the position of those observed, social learning would occur. 5

7 All agents choose one of two technologies: a A = {0, 1}. here are two states of the world: θ Θ = {0, 1}. he timing of the game is as follows. First, θ Θ and p P are chosen. he true state of the world and the agents order in the sequence are not directly revealed to agents. Instead, each individual i receives a noisy signal Z pi) about the true state of the world, a second signal S pi) that may include information about his position, and a sample ξ pi) of the decisions of agents before him. With these three sources of information, each agent decides between technologies 0 and 1, collects payoffs, and dies. hus, the decision of each individual is once-and-for-all..1 Payoffs We study situations with no payoff externalities. Let ua, θ) be the payoff from choosing technology a when the state of the world is θ. We assume that the optimal technology depends on the state of the world; that is, u0, 0) > u1, 0) and u1, 1) > u0, 1). We show in Appendix B.1 that under these assumptions, any model is equivalent to a case with Prθ = 1) = Prθ = 0) = 1 and payoffs as given in able 1, for some λ > 0. echnology a) rue State θ) λ able 1: Payoff Function. Minimum Information Structure We describe the information available to agents in two steps. First, we present some minimum restrictions on the private signal Z and the sample ξ that guarantee social learning. Next, we describe the additional signal S, which includes other information about the game that agents may have and that does not disrupt learning. In order to achieve social learning agents must be able both to learn from others through the sample and to add new idiosyncratic information through their private signal. he requirements we impose on the sample guarantee that agents can identify the choice of a randomly selected agent from their sample and therefore do at least as well as those observed. 6

8 ..1 Private Signals about the State of the World Agent i receives a private signal Z Pi), with realizations z Z. Conditional on the true state of the world, signals are i.i.d. across individuals and distributed according to µ 1 if θ = 1 or µ 0 if θ = 0. We assume µ 0 and µ 1 are mutually absolutely continuous. hen, no perfectly-revealing signals occur with positive probability, and the following likelihood ratio Radon-Nikodym derivative) exists: lz) dµ 1 dµ 0 z) An agent s behavior depends on the signal Z only through the likelihood ratio l. Given ξ and S, individuals that choose technology 1 are those that receive a likelihood ratio greater than some threshold. For this reason, it is of special interest to define a distribution function G θ for this likelihood ratio: G θ l) Pr lz) l θ). We define signal strength as follows. DEFINIION 1. UNBOUNDED SIGNAL SRENGH. Signal strength is unbounded if for all likelihood ratios l 0, ), and for θ {0, 1}, 0 < G θ l) < 1. Let suppg) be the support of both G 0 and G 1. If Definition 1 does not hold, we assume that ] the convex hull of suppg) is given by co suppg)) = l, l, with both l > 0 and l <, and we call this the case of bounded signal strength. We study this case in Section he Sample Let a t A be the behavior of the agent playing in period t. he history of past behavior at period t is defined by h t =..., a 1, a 0, a 1, a,..., a t 1 ). Actions a t for t 0 are not chosen strategically but are instead specified exogenously by an arbitrary distribution f h 1 θ) = Pr H 1 = h 1 θ) over a set H 1 of initial histories. 6 For example, the sample of the first agent in the sequence is always exogenously specified by this distribution. Conditional on θ, H 1 is independent of P and of S t for all t. he results we present in this paper do not place any other restrictions on the distribution of the actions of early agents. Let H t be the 6 Restaurants, hotels, clubs, and other establishments change hands occasionally but agents are not always aware of this. At the same time, changes in characteristics of the options may imply significant variations in consumers utility but may not always be observed. Consequently, the arbitrary distribution over initial histories may result from agents t 0 playing the same game when payoffs were different. 7

9 random) history at time t, with realizations h t H t. Agents observe the actions of a sample of individuals from previous periods. he random set O t, which takes realizations o t O t, lists the positions of the agents observed by the individual in position t. For example, if the individual in position t = 5 observes the actions of individuals in periods and 3 then o 5 = {, 3}. We place the following restrictions on O t. First, it should be nonempty, and independent of θ and P. Second, O t should be independent of its concurrent signal Z t. Lastly, the set of sampled individuals, O t, must be independent of the actual decisions of those individuals. o accomplish this, we assume that O t is independent of H 1, independent of Z τ and S τ for all τ < t, and independent of O τ for all τ = t. We also assume sampling to be stationary. We describe this assumption later. At a minimum, we assume that each agent observes an unordered sample of past play, fully determined by H t and O t. Formally, the sample ξ : O t H t Ξ = N is defined by ) ξo t, h t ) o t, a τ, where o t is the number of elements in o t. τ o t herefore, ξo t, h t ) specifies the sample size and the number of observed agents that chose technology 1. 7 Agents know the sample size and how many agents in the sample chose each technology. Choosing technology 1 with a probability equal to the proportion of agents choosing technology 1 in the sample is equivalent to picking an agent at random and following his behavior. Such a behavior leads, in expectation, to a utility equal to the expected utility of those observed. In other words, this minimum requirement on the information in the sample guarantees that agents can mimic those observed and obtain at least their average expected utility, in expectation. It is worth noting, however, that agents may have more information about the positions of observed agents. his information is included in the second signal S, as explained in the next subsection. We impose restrictions on the stochastic set of observed agents O t. o do so, we first define the ] expected weight of agent τ on agent t s sample by w t,τ E. hese weights play a role in 1{τ Ot } O t the present model because if agents indeed pick a random agent from the sample and follow his strategy, the ex-ante probability agent t picks agent τ is given by w t,τ. Now, different distributions for O t induce different weights w t,τ. We restrict the sampling rule by imposing restrictions directly 7 For example, if t = 5, h 5 =..., a 1 = 1, a = 1, a 3 = 0, a 4 = 0) and o 5 = {, 3} then ξ 5 o 5, h 5 ) =, 1); that is, the agent in position 5 knows he observed two agents and that only one of them chose technology 1. 8

10 on the weights. We suppose that weights only depend on relative positions. DEFINIION. SAIONARY SAMPLING. A sampling rule is stationary if the weight of agent τ on agent t s sample depends only on the distance between these agents: w t,τ = w t τ) for all t and all τ. he weights w t,τ depend only on the distance between agents, and so no individual plays a prevalent role in the game. A rich family of such rules can be obtained by assuming that for all t, t and t 1, t,..., t n ), the following condition holds: Pr O t = { t t 1, t t,..., t } ) ) t n = Pr O t = {t t 1, t t,..., t t n }. 1) his condition imposes a stronger restriction than w t,τ relative positions determine the likelihood of samples. 8 rules include: = w t τ) since it requires that only Some examples of stationary sampling 1. Agent t observes the set of his M predecessors.. Agent t observes a uniformly random agent from the set of his M predecessors. 3. Agent t observes a uniformly random subset K of the set of his M predecessors. Section 4. studies an example of non-stationary sampling rules, geometric sampling, which includes the case when agents draw uniformly from all those who have already played..3 Additional Information: he Signal S he previous subsections presented the minimum information requirements of our model. ogether with stationary sampling, the assumptions on the signal Z and the sample ξ guarantee social learning, as we show later. he random variables Z and ξ are observable, whereas P, O, H and of course θ are not. Our model, however, allows agents to possess more information about the game they play. We add a second observable signal S, with realizations s S, to allow for more general information frameworks. Signal S may provide information about the unobserved 8 he following example shows a stationary sampling rule that violates equation 1). Every individual in an even position t observes two agents, those in positions t 1 and t. Every individual in an odd position t observes one of two agents, either the agent in position t 1 or that in position t. Weights are w t,τ = 1 if τ {t 1, t } and zero otherwise, so sampling is stationary. 9

11 random variables. Our only additional assumption is that conditional on θ, S t be independent of Z τ for all τ. For technical reasons, we also assume that S is finite for any length of the sequence. o illustrate how S can provide additional information, assume that agents know their position. In such a case, S = {1,..., } and S t = t. his setting corresponds to the model of Smith and Sørensen 008] where agents know their position and observe an unordered sample of past behavior. Our setup can also accommodate cases in which agents have no information about their position, but have some information about the sample, which can be ordered. 9 If agents have imperfect information about their position in the sequence, then S specifies a distribution over positions. Moreover, S may provide information on several unobservable variables at the same time. In Section 4 we show that S plays a critical role when allowing for general position beliefs. o summarize, the signal S represents all information agents have that is not captured by Z or ξ. he results we present in this paper do not depend on the distribution of S. In particular, social learning does not depend on the information agents possess about their position..4 Strategies and Equilibrium All information available to an agent is summarized by I = {z, ξ, s}, which is an element of I = Z Ξ S. Agent i s strategy is a function σ i : I 0, 1] that specifies a probability σ i I) for choosing technology 1 given the information available. Let σ i be the strategies for all players other than i. hen the profile of play is given by σ = σ i, σ i ). he random variables in this model are Ω) = ) θ, P, {O t }, {Z t}, {S t}, H 1. hese random variables, along with the strategy profile, determine the history of past play. Actions a t for t 0 are arbitrarily specified by the distribution f h 1 θ) = Pr H 1 = h 1 θ), as stated earlier. For all t 1 each h t leads to one of two possible values h t+1, depending on the action a t. Let h t and h t+1 list the same actions up to period t 1. We say that h t+1 = h t 1 if a t = 1 and that h t+1 = h t 0 if a t = 0. We define the distribution f h t θ, p, σ) Pr H t = h t θ, p, σ) recursively 9 ake the example presented in the previous subsection, with t = 5, h 5 =..., a 1 = 1, a = 1, a 3 = 0, a 4 = 0) and o 5 = {, 3}, which leads to ξ 5 o 5, h 5 ) =, 1). If samples are ordered, then s 5 = 1, 0), denoting that the first agent in the sample chose technology 1 and the second one chose technology 0. 10

12 by Pr a t = 1 θ, p, σ, h t ) f h t θ, p, σ) if h t+1 = h t 1 f h t+1 θ, p, σ) = Pr a t = 0 θ, p, σ, h t ) f h t θ, p, σ) if h t+1 = h t 0 0 otherwise With the distribution over histories, we can write the utility of agent i, conditional on a particular order of agents, p, as follows u i σ, p) = 1 θ h pi) H pi) f ) ) h pi) θ, p, σ Pr a pi) = θ θ, p, σ, h pi) λ 1 θ) + θ). We focus on properties of the game before both the order of agents and the state of the world are determined. We define the ex-ante utility as the expected utility of a player. DEFINIION 3. EX-ANE UILIY. he ex-ante utility ū i is given by ū i σ) = 1! u i σ, p). p P Profile σ = σ i, σ i) is a Bayes-Nash equilibrium of the game if: ū i σ i, σ i) ūi σi, σ i ) for all σ i and for all i. Agents are ex-ante homogeneous; the information they obtain depends on their position P i) but not on their identity i. hat is why we focus on strategies where the agent in a given position behaves in the same way regardless of his identity. A profile of play is symmetric if σ i = σ for all i. A symmetric equilibrium is one in which the profiles of play are symmetric. Using a fixed point argument, we show that symmetric equilibria exist. PROPOSIION 1. EXISENCE. For each there exists a symmetric equilibrium σ ). See Appendix B.3 for the proof. In the present model, multiple equilibria may arise, and some may be asymmetric. Appendix A.1 presents an example of such equilibria. In the remainder of this section, however, we focus on symmetric equilibria. In Section 4.1 we allow for asymmetric ones and show that the results we present in this section hold for all equilibria. Under symmetric profiles of play, an agent s utility is not affected by the order of other agents. 11

13 In other words, if σ is symmetric, the utility of an agent facing a game with positions p is given by u t σ) = u i σ, p) for any p such that p i) = t. Only the uncertainty over one s own position matters, and so the ex-ante utility of every player can be expressed as follows: ū i σ) = 1 u t σ). his simplifies the analysis of the game, and so we drop the index i and focus on the position t in the sequence. For social learning, we require that the expected proportion of agents choosing the right technology approaches 1 as the number of players grows large. We can infer this expected proportion through the average utility of the agents. DEFINIION 4. AVERAGE UILIY. he average utility ū is given by ū σ) = 1 ū i σ). i=1 he expected proportion of agents choosing the right technology approaches 1 if and only if the average utility reaches its maximum possible value, λ+1. In principle, there can be multiple equilibria for each length of the game. We say social learning occurs in a particular sequence of equilibria, {σ )} =1, when the average utility approaches its maximum value. DEFINIION 5. SOCIAL LEARNING. Social learning occurs in a sequence {σ )} =1 of equilibria if lim ū σ )) = λ + 1. Our definition of social learning also applies to the asymmetric cases studied in Section 4.1. In the main part of the paper we focus on the symmetric case all players are ex-ante identical and play symmetric strategies) and so all agents have identical ex-ante utilities. As a result, the average utility is equal to each agent s ex-ante utility, ū i σ) = ūσ). In this symmetric setting, social learning occurs if and only if every individual s ex-ante utility converges to its maximum value. 1

14 .5 Average Utility of hose Observed We present an ex-ante improvement principle in Section 3.1: as long as those observed choose the inferior technology with positive probability, an agent s utility can always be strictly higher than the average utility of those observed. o accomplish this, we need to define the average utility ũ i of those that agent i observes. Let ξ t denote the action of a randomly chosen agent from those sampled by agent i when in position t. hen, we define ũ i as follows. DEFINIION denoted ũ i σ i ), is given by 6. AVERAGE UILIY OF HOSE OBSERVED. he average utility of those observed, ũ i σ i ) = 1 θ 1 Pr ] ) ξ t = θ θ, σ i λ 1 θ) + θ). he average utility ũ i of those that agent i observes is the expected utility of a randomly chosen agent from agent i s sample. When i is in position t the weight w t,τ represents the probability that agent τ is selected at random from agent i s sample. hus, ũ i can be reexpressed as follows. PROPOSIION. he average utility of those observed can be rewritten as 10 ũ i σ i ) = 1 τ:τ<t w t,τ u τ σ i ). See Appendix B.4 for the proof. From now on, we utilize the expression for ũ i given by Proposition. Finally, we also use ũσ) to denote the average of ũ i σ i ) taken across individuals. Notice that since players are ex-ante symmetric, when the strategy profile σ is symmetric, ũσ) = ũ i σ i )..6 Consequences of Stationary Sampling Stationary sampling has the following useful implication: the weights that one agent places on previous individuals can be translated into weights that subsequent individuals place on him. he graph on the left side of Figure 1 represents the weights w t,τ agents t place on agents τ. For example, w,4 represents the weight agent 4 places on agent and w 0, represents the weight agent places on agent 0. hese weights are highlighted in the left-hand graph of Figure 1. Since the 10 We allow agents to observe individuals from the initial history, so τ can be less than 1. 13

15 distance between agents and 4 is equal to the distance between agents and 0, then these weights must be equal: w,4 = w 0, = w). he graph on the right side of Figure 1 shows the sampling weights only as a function of the distance between agents. τ τ w 6,5 w 1) w 5,4 w 6,4 w 1) w ) w 4,3 w 5,3 w 6,3 w 1) w ) w 3) w 3, w 4, w 5, w 6, w 1) w ) w 3) w 4) w,1 w 3,1 w 4,1 w 5,1 w 6,1 w 1) w ) w 3) w 4) w 5) w 1,0 w,0 w 3,0 w 4,0 w 5,0 w 6,0 t w 1) w ) w 3) w 4) w 5) w 6) t w 1, 1 w, 1 w 3, 1 w 4, 1 w 5, 1 w 6, 1 w 1, w, w 3, w 4, w 5, w 6, w ) w 3) w 4) w 5) w 6) w 7) w 3) w 4) w 5) w 6) w 7) w 8) Figure 1: Stationary Sampling he horizontal ellipse in the right-hand side of Figure 1 includes the weights subsequent agents place on agent. he vertical dashed ellipse shows the weights agent places on previous agents. Since weights are only determined by the distance between agents, the sum of the horizontal weights must equal the sum of the vertical weights. 11 By definition, 1 the vertical sum of weights is given by τ:τ<t w t,τ = 1. hus, for any agent far enough from the end of the sequence, the sum of horizontal weights those subsequent agents place on him is arbitrarily close to 1. Intuitively, this means that all individuals in the sequence are equally important in the samples. Since the weights subsequent agents place on any one agent add up to 1, the average observed utility ultimately weights all agents equally, and the proportion of the total weight placed on any fixed group of agents vanishes. hen, as we show next, the homogeneous role of agents 11 Algebraically, for any J > 0, t 1 τ=t J w t,τ = τ=t J t 1 wt τ) = J j=1 wj) = t+j τ=t+1 w t,τ 1 1{τ Ot ] ]] ] } τ:τ<t w t,τ = τ:τ<t E = E 1 O t O t E τ:τ<t 1 {τ Ot } O t = Ot E = 1. O t 14

16 under stationary sampling imposes an accounting identity: average utilities and average observed utilities get closer as the number of agents grows large. his is not an equilibrium result but a direct consequence of stationary sampling, as the following proposition shows. PROPOSIION 3. Let sampling be stationary, and, for each, let { y t } t= with 0 y t λ+1. Let ȳ) = 1 y t and ỹ) = 1 τ:τ<t w t,τ y τ. hen, be an arbitrary sequence, lim sup {yt } t= ȳ) ỹ) = 0. See Appendix B.5 for the proof. We use arbitrarily specified sequences to highlight that Proposition 3 holds for all possible payoff sequences. urning to the case of interest, let {σ)} =1 be a sequence of symmetric strategy profiles that induces average utility ū) = ūσ)) and average utility of those observed ũ) = ũσ)). Define the ex-ante improvement by v) ū) ũ) to be the difference between the average utility and the average utility of those observed. Proposition 3 has the following immediate corollary. COROLLARY 1. If sampling is stationary and the strategy profile is symmetric, then lim v) = 0. he ex-ante improvement v) represents how much an agent in an unknown position expects to improve upon an agent he selects at random from his sample. Corollary 1 shows that under stationary sampling rules, the ex-ante improvement v) vanishes as the number of agents in the sequence grows larger. 3. Social Learning In this section, we present an improvement principle for settings in which agents do not know their position in the sequence. he improvement principle states that agents can do better, on average, than those they observe. Moreover, if the average utility of those observed is bounded away from the maximum possible utility, then the improvement upon their utility must be bounded away from zero. Banerjee and Fudenberg 004] are the first to use an improvement principle to study social learning. Smith and Sørensen 008] and Acemoglu et al. 008] adopt a similar approach. In those papers, agents do better than those they observe, and so, over time, agents utilities increase. Our argument is similar in the sense that agents do better than those they observe. It is different, 15

17 though, in that our improvement principle is stated ex-ante; agents do better than those they observe, unconditional on their position. In what follows, we first explain why an ex-ante improvement principle holds in the present setting. hen, we show how this ex-ante improvement principle can be used to study social learning in this setting. 3.1 he Ex-ante Improvement Principle We restrict agents information in a particular way and show that agents can improve upon those they observe. Consequently, with less restricted information they can still improve upon those observed. o see this, pick an agent at random from those agent i observes and disregard the rest of the information from his sample. Let ξ denote the action of the selected agent. Also, disregard all information contained in S. After restricting the information in this way, what remains is Ĩ = { z, ξ } with Ĩ Ĩ = Z {0, 1}. he average utility ũ i of those observed by agent i depends on how likely they are to choose the superior technology in each state of the world. Define π 0 and π 1 to be the likelihoods that a randomly picked agent from the sample chooses the superior technology in states θ = 0 and θ = 1, respectively. hat is, define π 0 Pr ξ = 0 θ = 0 ) and π 1 Pr ξ = 1 θ = 1 ). hen, the average utility of those observed by agent i can be expressed as ũ i = 1 π 0λ + π 1 ). A simple strategy, copy a random agent from the sample, guarantees the average utility ũ i of those observed. o show that agents do strictly better than those they observe, we show that there is always an alternative that does better than this simple strategy. he likelihood ratio L : {0, 1} 0, ) implied by the information from the randomly chosen agent is given by L ξ ) Pr θ = 1 ξ, σ i ) Pr θ = 0 ξ, σ i ). hen, the information provided by ξ is captured by L1) = π 1 1 π 0 and L0) = 1 π 1. Now, if the information provided by the signal is more powerful than the information reflected in these likelihood ratios, agents are better off following the signal than following the sample. We use this to define a smarter strategy σ i that utilizes not only information from the sample, but also information from the signal. If signal strength is unbounded and observed agents do not always choose the superior technology, the smarter strategy does strictly better than simply copying a π 0 16

18 random agent from the sample. In fact, fix a level U for the utility of observed agents. he thick line in Figure corresponds to combinations of π 0 and π 1 such that ũ i σ i ) = U, and the shaded area represents combinations such that ũ i σ i ) U. In the shaded area, the improvement upon those observed is bounded below by a positive-valued function CU). π 1 ũ i σ i ) = U π 0 Figure : Ex-ante Improvement Principle with Unbounded Signals PROPOSIION 4. EX-ANE IMPROVEMEN PRINCIPLE. If signal strength is unbounded, then there exists a strategy σ i and a positive-valued function C such that for all σ i See Appendix B.6 for the proof. ) ū i σ i, σ i ũi σ i ) C U) > 0 if ũ i σ i ) U < λ + 1. Proposition 4 presents a strategy that allows the agent to improve over those observed. hen, in any equilibrium σ ), he must improve at least that much. In addition, since we focus on symmetric equilibria, ū i σ )) = ū σ )) and ũ i σ i ) ) = ũ σ )). With this two facts, we present the following corollary. COROLLARY. If signal strength is unbounded, then in any symmetric equilibrium σ ), ū σ )) ũ σ )) C U) > 0 if ũ σ )) U < λ + 1. o study social learning, we focus on the behavior of the average utility ū σ )) as the number of players approaches infinity. As Corollary 1 shows, stationary sampling implies that v) = ū σ )) ũ σ )) must vanish as the number of players grows large. If v) van- 17

19 ishes, Corollary implies that ū σ )) approaches the maximum possible utility; that is, social learning occurs. PROPOSIION 5. SOCIAL LEARNING WIH UNBOUNDED SIGNALS. If signal strength is unbounded and sampling is stationary, then social learning occurs in any sequence of symmetric equilibria. See Appendix B.7 for the proof. 3. Bounded Signals: Constrained Efficient Learning So far, we have studied social learning when signal strength is unbounded. However, one can imagine settings in which this assumption may be too strong. If signal strength is bounded, social learning cannot be guaranteed. We can define, however, a constrained efficient learning concept that is satisfied in this setting. o study constrained efficient learning, imagine there is an agent in isolation who only receives the strongest signals, those yielding likelihoods l and l. Under such a signal structure the likelihood ratios determine how often each signal occurs in each state. Let u cl denote the maximum utility in isolation, that is the expected utility this agent would obtain by following his signal. We show that in the limit, agents utility cannot be lower than u cl. 13 In order to show that constrained efficient learning occurs, we present an ex-ante improvement principle when signal strength is bounded. In this setting, as long as the information from private signals is sometimes more precise than that obtained from the sample, agents can still improve upon those observed. he signal can always be used if either of the following inequalities are satisfied: L1)l > λ or L0)l < λ. he dotted lines in Figure 3 correspond to combinations of π 0 and π 1 such that the previous expressions hold with equality. Consequently, outside of the shaded area in Figure 3, signals can be used to improve upon those observed, as Proposition 6 shows. PROPOSIION 6. EX-ANE IMPROVEMEN PRINCIPLE UNDER BOUNDED SIGNALS. If signal strength is bounded, then there exists a strategy σ i and a positive-valued function C such that for all σ i ū i σ i, σ i) ũi σ i ) C U) > 0 if ũ i σ i ) U < u cl. See Appendix B.8 for the proof. We say that a model of observational learning satisfies constrained efficient learning if agents 13 A policy implication may be that if a regulator is no more informed than the most informed agents in the sequence, she would not be able to outperform the result of a social learning algorithm. 18

20 π 1 1 L0)l = λ ũ = u cl L1)l = λ 1 π 0 Figure 3: Ex-ante Improvement Principle with Bounded Signals receive at least u cl in the limit. PROPOSIION 7. CONSRAINED EFFICIEN LEARNING. If signal strength is bounded and sampling is stationary, then constrained efficient learning occurs in any sequence of symmetric equilibria. See Appendix B.9 for the proof. 4. General Setup and Extensions his section is still preliminary. 4.1 General Position Beliefs and Asymmetric Equilibria In this section we show that our results of social learning with signals of unbounded strength and constrained efficient learning with signals of bounded strength hold for arbitrary distributions over permutations and for all equilibria, including asymmetric ones. When agents are not placed in the sequence according to a uniform distribution or when the equilibrium played is not symmetric, agents are not identical ex-ante. As a result, the ex-ante utility of agents depends on their identity, ũ i need not be constant over agents anymore. o show our results for games with agents that are not identical ex-ante, we construct an analogous game in that agents exhibit the same behavior, but they are ex-ante identical. Since we have established the result for the symmetric case, the intuition behind the general model is simple. For any game with an arbitrary distribution over permutations we can construct an analogous one in 19

21 which players are ex-ante symmetric but have interim beliefs corresponding to the arbitrary distribution over permutations. his is done by adding information through the additional signal. Since we have established that the additional signal does not prevent learning, we are done. Likewise, when agents are allowed to have extra information, the assumption of a symmetric equilibrium is without loss of generality. o see this, suppose that some players do not react in the same way to the same information. hen, we can simply say that they are different types. Since their type is part of the information they receive, now they do react the same way to the same information. hat is, before agents are assigned their type, they act symmetrically. he game played under arbitrary beliefs, which we call Γ 1, only requires an alternate specification of the random permutation and the additional signal. Everything else remains as stated in the main model. If beliefs satisfy the common prior assumption, there is a commonly understood prior and signaling structure. his is modeled by allowing Q P to be a random permutation with an arbitrary distribution. Also, let S i S i be an additional signal that agent i receives from a finite set, and S S be the signal profile with an arbitrary distribution. We assume that, conditional on θ, S is independent of Z. We also assume that S Q 1 τ) is independent of O t for all τ < t. he pair Q, S) can induce any beliefs over positions which satisfy the common prior assumption. Finally, fix a possibly asymmetric) Nash equilibrium, and let σ i denote the strategy for player i. PROPOSIION 8. If signal strength is unbounded then social learning occurs in any sequence of equilibria of Γ 1. Next, we present the equivalent corollary to Proposition 7 for bounded signals. PROPOSIION equilibria of Γ If signal strength is bounded then limited social learning occurs in any sequence of We proceed by defining an auxiliary game, Γ. On the one hand, players in Γ 1 have arbitrary beliefs about positions. On the other hand, the game Γ satisfies all the assumptions of the model in the main section. We then show that for each equilibrium of Γ 1 there is a symmetric equilibrium of Γ in which the distribution over histories is the same. Our results do not immediately apply to arbitrary permutations because agents in Γ 1 are allowed to have non-identical ex-ante beliefs about positions. o remedy this, we create the game Γ, where agents are first uniformly shuffled to become the players in Γ 1. As a result, the ex-ante belief of each agent is that he has an equal chance of playing in any time period. After being shuffled, players in Γ are informed about which player they are in Γ 1 through the extra signal S. 0

22 hus, the game Γ 1 is informationally equivalent to the auxiliary game Γ after agents know which player in Γ 1 they are. Moreover, Γ satisfies all the requirements of our model. he auxiliary game Γ, is constructed as follows. Let P P be a uniformly-distributed random permutation. Agents are assign to time periods by the composed permutation P Q, which is itself a uniformly distributed random permutation. he additional signal the agents receive is defined by S t Q 1 t), S Q 1 t)). One can think of this game as follows. he permutation P is a mapping from agents playing in Γ to positions in Γ 1. Next, we tell each agent which position of Γ 1 he occupies, and pass along the information that the agent in Γ 1 receives in that position. Since Q is an injective function, the inverse Q 1 t) indicates which position of Γ 1 is asked to play in period t. And S Q 1 t) is the extra signal received by the agent in position Q 1 t). So the agent of Γ that plays in period t knows which player in Γ 1 plays in period t as well as all the extra information of that player. So the Γ player has all of the information of the Γ 1 player. Additionally, since P is a uniform random permutation, the fact that the player in Γ knows his identity provides him no useful information. In both games the players playing in period t have the same information. he game Γ satisfies the assumptions given in Section. As we have already mentioned the mapping from agents to time periods, P Q, is uniformly random. Given our assumptions on S, the signal S τ is independent of O t for all τ < t. Last, we can restrict attention to symmetric equilibria of Γ since the strategy profile with σ i s t, z t, ξ t ) = σ q 1 t) s q 1 t), z t, ξ t ) is a symmetric strategy profile and, as we see next, an equilibrium of Γ. In order to show that these two games induce the same distribution over histories, it is useful to realize the two games on the same probability space. o do so, we need players in each position to use the same random variable to employ a mixed strategy. An agent playing in period t uses B t, a random variable distributed uniformly on 0, 1] and independent of all other random variables. In this case, the histories of Γ 1, written H 1 t σ ), and the histories of Γ, written H t σ ), are the same in every period. We can show this by induction on t. By definition, H1 1 = H 1. If the histories are equivalent up to period t, H 1 t σ ) = H t σ ) = h t, then the agents in both games playing in period t receive the same sample, ξo t, h t ). In Γ 1, the agent chooses technology one ) when B t σ Q 1 t) S Q 1 t), Z t, ξo t, h t ). Similarly, in Γ, the agent chooses technology one when B t σ P 1 Q 1 t)) S t, Z t, ξo t, h t )). hese are the same, by the definition of σ i s t, z t, ξ t ). Since the agents have the same information in each game, σ is a symmetric equilibrium of Γ. Proposition 8 is now a straightforward corollary of the Proposition 5. 1