Social Interactions under Incomplete Information with Heterogeneous Expectations

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1 Social Interactions under Incomplete Information with Heterogeneous Expectations Chao Yang and Lung-fei Lee Ohio State University February 15, 2014 A Very Preliminary Version Abstract We build a framework to analyze social interactions under various types of information structures when exogenous characteristics may or may not be observed to all agents in a group. We consider two types of heterogeneity in conditional expectations about group members behaviors. First, due to variation in individual features, expectations on behaviors of two different agents may differ. Second, because of asymmetry in private information, two agents expectations on the behavior of a third person may not be the same. We relate our model to a simultaneous move game with incomplete information and adopt the solution concept of Bayesian Nash Equilibrium. This extends current literature, where due to the rational expectation assumption, the second type of heterogeneity is missing. Hence, our model is capable of investigating social interactions with a variety of assumptions about asymmetric private information. Additionally, it can be used to inspect influences of information structures on intensity of social interactions. There is a one-to-one correspondence between a Bayesian Nash Equilibrium and the conditional expectation of group members behaviors. As the latter is a vector-valued function of privately known group members characteristics, we use functional contraction mapping to derive a sufficient condition for equilibrium existence and uniqueness. We show how to solve equilibrium under some special information structures. Moreover, we find that, by Monte Carlo experiments, nested fixed point maximum likelihood estimation performs well for linear model, binary choice model and Tobit model. Extensions to some more complicated cases of incomplete information, such as unobserved group random effects and private information about social relations, are also discussed. 1 Introduction We develop a framework which analyzes social interactions under incomplete information. By allowing asymmetry of both personal exogenous characteristics and individual private information sets, we extend current literature by allowing the expectation of an agent on 1

2 behavior of socially related agents to be heterogeneous in two aspects: the agent whose behavior expectation is taken about and the set of private information on which prediction is based. The framework can incorporate many frequently used models, such as linear model, binary choice model, and Tobit model. Additionally, our analysis applies to general form of incomplete information. Therefore, our model makes it possible to investigate interactions of various types of behaviors for socially related agents under different information structures. It is natural to believe that in a social group, like a class, some information on group members is shared by everybody but some information is known only to a subgroup of members. An agent may not know the features of all the other agents in her group. For example, when analyzing the interactions of college students academic performances, individual IQ and /or SAT scores are often used as exogenous covariates. As we know, it is not likely that a student knows the IQ and/or SAT score of all the other students in her class. She might only know that information about her close friends. Moreover, since different students make different friends, one student s knowledge about students IQ s in her class differs from that of another student in general. As a result, their predictions on math score of a third student will not necessarily be equal. With different expectations on other classmates performances in math class, those two students may make different study plans, even when all of their other personal traits are the same. Therefore, heterogeneity in private information can influence social effects. Different information structures can result in different intensities of social interactions. However, this type of heterogeneity has not been analyzed in existing literature. In previous models, after rational expectation assumption is imposed, an individual forms that same expectation over other agents behaviors as econometricians do. Therefore, two agents form the same expectation on a third individual in the group. The rational expectation assumption largely simplifies calculation of equilibrium. Additionally, it is suitable for the case when all group and personal characteristics are public information or when individuals make the same type of decisions repeatedly so that a stable status is formed. However, when there is asymmetric private information about some variables, such as individual s IQ, or when some decisions are made just once, it is not realistic to impose that assumption any more. So we try to investigate social interactions with information asymmetry without making that assumption. To achieve our goal, we build a behavioral model. We assume that an agent s decision on a certain economic action is related to her expectation of the behaviors of other agents she is associated with, conditional on her own information about publicly known group and individual characteristics, social relationships between group members, and personal traits of a subset of agents in her group. That framework is related to a simultaneous move game with incomplete information. In such a game, agents in a group make decisions at the same time. The potential payoff of an agent is affected by actions of agents with whom she is associated. Then she will make a prediction on their actions conditional on her private information and choose her best response in order to maximize her expected 2

3 payoff. Moreover, when taking expectations, a rational agent will take into account that her rival will make decisions in the same way. Therefore, an agent s strategy is to take the best response given others strategy. That is, a Bayesian Nash Equilibrium (BNE) will be derived when rational agents play the game. In our model, when we view agents behaviors as best responses, observed outcome is a realization of a BNE. One advantage of adopting BNE as a solution concept, instead of rational expectation, is that BNE allows the information on which expectation is based to be different, which makes it possible to bring in information heterogeneity. We show that we can solve equilibrium via conditional expectation on group members behaviors. First, it is shown that for an arbitrarily chosen individual, conditional expectation on her behavior is a function of possible vectors composed of privately known personal characteristics of some group members. Equilibrium conditional expectations, as a result, is a vector-valued function. Second, we prove that there is a one-to-one correspondence between a BNE and a conditional expectation function satisfying the consistency condition. Therefore, we can analyze existence and uniqueness of BNE by functional analysis. Embedding conditional expectation in a complete normed function space, we find that a conditional expectation function satisfies consistency condition if it is a fixed point of an operator. Utilizing Banach fixed point theorem, we derive that that operator is a contraction mapping when intensity of social interactions is within a reasonable range. Owing to contraction mapping, we get not only a sufficient condition for existence and uniqueness of an equilibrium, but also a possible method to solve equilibrium, namely, contraction mapping iterations. Moreover, it is possible to solve equilibrium analytically in some special cases. We illustrate solution of equilibrium by some concrete examples of information structures. If privately known personal features are discrete, we can represent an equilibrium conditional expectation as a finite-dimension vector. Coordinates of that vector correspond to function values on points in the finite support. Then we can employ contraction mapping to solve it. If privately known personal features are continuously distributed, we can either discretize the sample space and use the previous method, or use the projection method, introduced by Juud (1998)[11]. The key idea in this method is to approximate conditional expectation function as a linear combination of a finite number of basis functions. We choose the coefficients such that consistency condition is satisfied. Our framework is general. It can incorporate a variety of models frequently used in empirical studies, such as linear model, binary choice model, and Tobit model with social interactions. It also allows general information structures, including the case when all information is public, some features are self-known and some characteristics are shared by friends. However, there are still some restrictions about information structure in the basic framework. First, it is assumed that all social relations are public information to group members. Second, idiosyncratic shocks are not used to make predictions. Additionally, all exogenous characteristics used to make predictions can be observed by econometricians. Those assumptions are later relaxed. First of all, we show that owing to independence between idiosyncratic shocks and exogenous variables as well as independence of idiosyncratic 3

4 shocks between agents, including idiosyncratic shocks in the private information used by agents to make predictions does not change conditional expectation. So there is not a loss of generality in our basic framework where we assume that idiosyncratic shocks do not play a role in individual prediction. We then extend our model to include group characteristics that are observed by group members and not by econometricians. We treat them as group random effects. Equilibrium can be solved in the same way. However, when estimating the model, we need to simulate the unobserved group random effects. A more challenging extension case is that social relationships is privately known. That is, the complete social relationship in a group is not necessarily known to everyone in the group. In that case, when an agent is predicting the behavior of one agent she is associated with, it is necessary of her to consider not only possible realizations of private personal features in that agent s information set but also the subset of social relations that agent knows. We show that theoretically we can analyze the model in a way similar to that is used in the basic framework. But computational burden will be largely increased. However, in some cases, computation is tractable. Model parameters can be estimated by nesting solution of conditional expectation in an inner loop of maximum likelihood estimation. Since the conditional expectation function is a fixed point of a function operator, the method can be called nested fixed point (NFXP) MLE. That method is frequently used to estimate discrete choices in dynamic models, such as Rust[16]. Moreover, when there is unobserved group random effects, we need to simulate group random effects. In that case, the model is estimated by nesting fixed point maximum simulated likelihood estimation (NFXPMSL). Monte Carlo experiment results are present to illustrate computational aspects and finite sample properties of proposed estimators. We focus on three popular models, linear model, binary choice model and Tobit model; and two information structures, one with publicly known characteristics and another with self-known characteristics. We see that generally those estimation methods perform well. One interesting thing about our experiment is that we estimate each model under two distinct information structures, one consistent with data generating process and one not. We compare estimation bias, standard deviation, as well as maximized sample likelihood. It is found that when data is generated in the situation that all characteristics are known to every group member, estimation under the assumption that personal characteristics are self-known does not bring in additional bias (but has a smaller value of maximized sample log likelihood on average). In contrast, when data is generated in the case that personal characteristics are self-known, but all of them were regarded to be publicly known by an empirical researcher for estimation, there can be additional bias, which can not be reduced even if sample size is increased. Note that after imposing rational expectation assumption, for any arbitrarily chosen agent, conditional expectation about her behavior will be constant. As a result, in terms of model estimation, to assume public information is equivalent to assume rational expectation. Therefore, as it is shown by our Monte Carlo experiments, we need to be cautious when following previous literature and assuming rational expectation. Information structures do influence our estimation of social effects. 4

5 The linear model is related to the linear-in-means model of social interactions by Manski (1993)[14]. In that model, he assumes that agents in a group make predictions on group average behaviors conditional on publicly known group features. Hence, in his model, expectation on others behavior is a scalar. In their micro-based model, Brock and Durlauf (2007)[5], model an individual makes predictions on other agents outcomes. However, after imposing symmetry and rational expectation, expectation reduces to a scalar. Relaxing symmetry in exogenous characteristics, Lee, Li and Lin (2013)[13] allow expectation on one agent s behavior to be different from expectation on another agent s behavior. That is, there is one type of heterogeneity in their model, the heterogeneity due to difference in personal features. But by imposing rational expectation assumption, any two agents will have the same expectation on actions of a third agent. So the second type of heterogeneity is missing there. The paper proceeds as follows. In Section 2, we introduce the model framework, as well as the relationship between the model and a simultaneous move game with incomplete information. In Section 3, we first show the one-to-one correspondence between a BNE and a conditional expectation function and then discuss existence and uniqueness of equilibrium. Equilibrium solution and model estimation are respectively discussed in Section 4 and Section 5. After three possible extensions in Section 6, some Monte Carlo experiment results are present in Section 7. Section 8 concludes. Some interesting qualitative analysis of equilibrium conditional expectations is put in the AppendixA. 2 Models 2.1 A Model Framework Consider a group of socially related agents 1 The size of the group is n. We use i to index an individual in that group. The n n matrix, W, is used to represent their (exogenous) social relations. For example, it can be used to represent friendship relations or competitor relations. In that case, for any i j, W i,j = 1 if i is associated with j; and W i,j = 0 otherwise. For any i, W i,i = 0. More generally, W i,j can depend on geographic, economic or social distance between i and j. We only require that elements of W are non-negative, W i,j 0 for any i, j; and W i,i = 0 for any i. So it can show not only whether i and j are associated but also the strength of relations. Note that W may be asymmetric or symmetric. To incorporate possible discrete choices in additional to continuous outcomes, the model is presented with a latent variable and a relationship between observed outcomes and latent variables. The observed outcome, or observed agent s behavior, y i, depends on the latent 1 In empirical studies, we may encounter many independent groups. In that case, interactions within each group can be analyzed in the same way. So theoretically, we can analyze social interactions in a single group. As a result, in Section 2 and Section 3, we just look at one group. When we need to discuss different groups, we will use subscript, g, to denote variables for group g. 5

6 variable, y i. y i = h i (y i ), (2.1) where h i ( ) : R R is a real-valued function, which can be linear, piecewisely linear, or nonlinear. From (2.1), for generality, we may allow the function form to vary across different agents. the latent variable for i is interacted with her expectations about other agents outcomes in her group. y i = u(x i ) + λ j i W i,j E[y j X p J i, Z] ɛ i. (2.2) From Equation (2.2), we can see that yi is determined by three parts of factors. The first part, u(x i ), represents the direct effect of exogenous covariates. X i = (X g, Xi c, Xp i ) is composed of three types of exogenous characteristics, the group features, X g ; some commonly known individual characteristics, Xi c, as well as some privately known information, X p i. The third part, ɛ i, is the idiosyncratic shock. It is assumed that ɛ i s are i.i.d. and independent of all the exogenous characteristics and social relations. Its distribution is characterized by a pdf function, f ɛ ( ), with its CDF function, F ɛ ( ). Assume that ɛ i is observed by individual i herself, but not other agents, or econometricians. Innovations of our model reside in the second part. It shows two features of social interactions. First, i s latent variable is related to that of agent j only if i is associated with j, i.e., W i,j 0. Second, instead of assuming that latent variables are directly socially interacted, in this model, j influences i indirectly through i s expectation on the true outcome of j s behavior. This assumption is quite intuitive in the situation where agents in the same group make decisions simultaneously when there are some privately known personal traits. In that case, when an individual, i, is making a decision, she does not know either the true realizations of other agents payoffs or the actions they will take. But it is the actions actually taken by other agents that influence her payoff. As a result, she makes predictions about her payoff on the basis of her expectations of other agents actions (or behaviors). For example, when counties in a state are setting a tax rate, they will make a trade off between increasing average tax they gained from each person and a loss of tax base, for residents or business may move to neighboring counties who choose lower tax rates. As a result, a county in tax competition will make plans based on its expectation on tax rates set by its neighbors. Later, we will see that this assumption is consistent with Bayesian Nash Equilibrium in a simultaneous move non-cooperative game under incomplete information. Moreover, in this model, expectation is based on i s information about exogenous characteristics, social relations as well as information structure in the group. Since group common features, X g, and some individual characteristics, Xj c s, are publicly known in a group, they are known to i. We assume that all social relations are known to each group member. So W is also known to i 2. However, information about another part of personal 2 Here, W represents the vector formed by all its elements, (W 1,1,, W 1,n,, W n,1,, W n,n), which 6

7 features, X p j s, is not public. Therefore, except for Xp i, i may only know part of the rest characteristics. We can see that, in fact, the information structure in a group can be fully described by specifying for every agent the subset of agents whose X p s are known to her. Therefore, for each i, we define a n 1 vector, J i, in the following way: J i (j) = 1, if i knows X p j ; (2.3) and J i (j) = 0, otherwise, for each 1 j n. Hence, information structure in a group can be represented by an n 2 1 vector, J = (J 1,, J n). (2.4) For every i, we define by X p J i, the vector formed by the random variables, X p j s, which are known to i. X p J i = (X p j : J i (j) = 1). (2.5) Suppose that X p j is of dimension k p. Then the dimension of X p J i is M i = ( n j=1 J i(j))k p. For instance, if 1 only knows X p 1, Xp 2 and Xp 3, J 1(j) = 1 for j = 1, 2, 3 and J 1 (j) = 0 for j > 3. We also have that X p J 1 = (X p 1, Xp 2, Xp 3 ). We assume that information structure is common knowledge. So i knows J. This assumption can be interpreted in this way. Under our assumption, which type of information is known to i is publicly known. But the realization of those random variables are private information. For example, it is publicly known that agent 1 knows her own feature and that of agents 2 and 3. But the realizations of X p 1, Xp 2 and Xp 3 are unknown to agent 4. To simplify notation, we summarize all the publicly known variables in one vector, Z = (X g, X c 1,, X c n, W 1,1,, W 1,n,, W n,1,, W n,n, J 1,, J n). (2.6) Agent i is private information is represented by X p J i. Therefore, i s expectations about other agents behaviors are based on two vectors, one about private information, X p J i ; and another about public information, Z. Variables in those two vectors are random variables. When a random variable is included in X p J i or Z, i knows its realization. If the same random variables are included in X p J i and X p J j, i.e., J i = J j, their realizations must be the same. In that case, those two agents have the same private information. Although it is natural to assume that an agent s prediction is also based on the realization of her idiosyncratic shocks, there is no loss of generality if we exclude them from private information X p J i. We will show in Section 6 that due to independence between idiosyncratic shocks, {ɛ i } s, and exogenous characteristics, X i s, as well as independence among idiosyncratic shocks, an agent s conditional expectations do not change if ɛ i is also used to make predictions. In Section 6, we also consider the case that there is private shows all social relations in the group. 7

8 information about W. In that case, W is not included in Z. Instead, some of its elements may be private information. It is shown that we can use similar method to analyze the model in that case. But there will be additional computational burden. We allow X p J i to vary across agents. As a result, there are two types of heterogeneity in expected outcomes, E[y j,g X p J i, Z] s. On one hand, since agent j and j may be different in terms of personal characteristics, i.e., X j X j, their expected outcome may differ in i s eyes. That is, in general, E[y j X p J i, Z] E[y j X p J i, Z], for i j, i j, and j j. On the other, because two agents may have different private information, their expectation on the behavior of the same third person might differ. Namely, E[y j X p J i, Z] E[y j X p J k, Z] when J i J k in general. Those two types of heterogeneity distinguishes the current model from previous research. For instance, in the classical model about social interactions by Manski (1993)[14], expectations are taken on the average behavior of the whole group based on commonly known group features. So in that model, everyone has the same expectation on group mean. There is no heterogeneity in expected outcomes in a group. In the discrete choice model by Brock and Durlauf [5], agents make predictions on others choices. However, due to symmetry among agents, after imposing rational expectation assumption, every agent has the same expectation on any other agent s choice in her group. Their work is extended by Lee, Li and Lin [13], who introduce asymmetry in personal characteristics and allow expectation on choices of different agents to differ. However, with rational expectation, any two agents have the same expectation on the choice of a third agent. That is, there is the first type of heterogeneity in expectation, but not the second type. Therefore, our model extends existing research in social interactions with incomplete information by incorporating two types of heterogeneity in conditional expectations about group members behaviors. In the discussion below, we will see this extension largely changes solution of equilibrium. Moreover, it makes it possible to analyze social interactions under various information structures. We suppose that all the exogenous characteristics, X g, Xi c s, Xp i s, W and J are observed by econometricians. Therefore, any variables, except idiosyncratic risks, used by agents to make predictions are known to econometricians. That assumption is reasonable in many cases. That is because econometricians are capable of gathering much information about members of the whole group by survey. Of course, there can be some variables known to agents in a group but are not reflected by data set. One case in point is quality of services, which is hard to measure objectively. So we relax that assumption in Section 6, where unobserved group random effects are added to the model. The intensity of social interaction effects is denoted by the parameter, λ. It is assumed to be homogeneous across all the agents. If λ > 0, the social interaction effect is positive. If λ < 0, outcomes are negatively related. The case of λ = 0 represents absence of social interactions. 8

9 2.2 Models and Information Structures Simple as it is, the model, (2.1) and (2.2), provides with a framework broad enough to incorporate many interesting cases. First, by specifying the dependence of observed outcomes on latent variable, we consider several specific models important in empirical studies. 1. (Linear Model) h i (z) = h(z) = z for any z R. That is, the latent variable can be perfectly observed. y i = y i. (2.7) 2. (Binary Choice) h i ( ) = h( ) for any i. h( ) can take only two values, 0 and 1. Namely, observed outcome is a binary variable, whose value depends on sign of the latent variable. y i = I(y i > 0). (2.8) 3. (Ordered Multinomial Choice) More generally, observed outcome can take a finite number of integer values. Suppose that h i ( ) = h( ) for any i, which is a step function. Let S be a finite integer, y i {0, 1,, S}. S+1 y i = I(c s 1 < yi c s )(s 1), (2.9) s=1 where c s for s = 1,, S are S cutoff points and c 0 =, c S+1 = (Tobit Model) The latent variable is right-censored. The cutoff value of the censoring rule is homogeneous and normalized to be equal to 0. That is, yi is observed if and only if yi > 0. In this case, h i( ) = h( ) and h( ) is piecewisely linear with a kink of 0. y i = I(yi > 0)yi. (2.10) 5. (Two-sided Censored Data) The latent variable is double-sided censored. The two cutoff points of the censoring rule are the same for all the agents. y i = I(y i < c 1 )c 1 + I(c 1 y i < c 2 )y i + I(y i c 2 )c 2. (2.11) Specifications on X p J i s reflect information structures. Some examples are cited below. They will be elaborated later. 1. (Publicly Known Characteristics) All exogenous group and individual characteristics, social relations and information structure is publicly known. In that case, X p J i include all X p j s. X p J i = (X p 1,, Xp n ). (2.12) 9

10 2. (Self-Known Characteristics) X p i is observed by i but not by any other group members. That is, X p J i = X p i. (2.13) 3. (Socially-Known Characteristics) i knows X p j if i is associated with j, i.e., W i,j Game Theoretical Explanation X p J i = (X p j ) j:1 j n,j=i, or W i,j 0. (2.14) It can be shown that the model, (2.1) and (2.2), can be related to a simultaneous move game with incomplete information. There are n agents in a group. The association among agents in the group is represented by the matrix, W, as before. Agents in the same group are making decisions at the same time. As a result, when i is making her decision, she does not know which actions other group members will take. Hence, she will make predictions on them. Assume that there is incomplete information and different agents may have different private information. Suppose that i makes predictions on the basis of public information Z and private information X p J i. Let s focus on two important cases, continuous choice and binary choice. 1. (Continuous Choice) Denote the action taken by i as y i. Let X p J i be any feasible action in i s set of actions. The payoff i gets when other agents take actions, y i is as follows: r(x p J i, y i ) = s(x i ) + (u(x i ) ɛ i )a i + λ j i W i,j y j a i 1 2 a2 i. Therefore, her expected payoff is E[r(X p J i, y i ) X p J i, Z] = s(x i ) + (u(x i ) ɛ i )a i + λ j i W i,j E[y j X p J i, Z]a i 1 2 a2 i. This is a quadratic function of X p J i. It is easy to derive the optimal strategy, y i = u(x i ) + λ j i W i,j E[y j X p J i, Z] ɛ i. Observed outcomes are realizations of optimal strategies. y i = yi. Therefore, we derive the following relationships of optimal strategies(also for observed outcomes): y i = u(x i ) + λ j i W i,j E[y j X p J i, Z] ɛ i. (2.15) for all i. When agents are rational, they will perceive that their rivals choose optimal strategies in the same way. Hence, Equation (2.15) defines a Bayesian Nash Equilibrium. 10

11 2. (Binary Choice) Consider the case that agent i s action, y i can only take two values, 0 and 1. The payoff associated with action 0 is normalized to be equal to 0. The payoff associated with y i = 1 is t i = u(x i ) + λ j i W i,j y j ɛ i. Thus, her expected payoff when taking action 1 is y i = u(x i ) + λ j i W i,j E[y j X p J i, Z] ɛ i. Therefore, i s optimal strategy is y i = 1 if and only if u(x i ) + λ j i W i,j E[y j X p J i, Z] > ɛ i ; and y i = 0 otherwise. Therefore, the following equation system, y i = I(u(X i ) + λ j i W i,j E[y j X p J i, Z] ɛ i > 0), (2.16) defines a BNE of the model when rational agents expect their rivals to choose strategies in the above way and take best response accordingly. From the above two examples, we can see that our model, (2.1) and (2.2), is consistent with a simultaneous move game under incomplete information where a (reduced-form) BNE {y i,g } is characterized by the following equation system: y i = h i (u(x i ) + λ j i W i,j E[y j X p J i, Z] ɛ i ). (2.17) Observed outcomes are just realizations of an equilibrium, y i. 3 3 Equilibrium Analysis 3.1 Equilibrium and Expectations Given that observed outcomes are realizations of an equilibrium satisfying Equation { (2.17), } we need to solve equilibrium strategies via solving the expectation terms, E[y j X p J i, Z]. Pick any i and k such that i k. By consistency, we get that E[y i X p J k, Z] = E[h i (u(x i ) + λ j i W i,j E[y j X p J i, Z] ɛ i ) X p J k, Z]. (3.1) 3 A strategy is a decision rule. It is a function of exogenous variables and idiosyncratic shocks. So it is a random variable. Observed action (or, outcome) is its realization, hence, a scalar. To simplify notation, however, I use y i to denote both a strategy and its realization. 11

12 As we defined in last section, Z is a vector representing public information, representing publicly known group and individual features, social relations, and information structures. X p J i is a vector representing private information, composed of a part of privately known personal characteristics observed by i. According to our assumption, k only knows which type of information is in X p J i. She does not know the realizations of those random variables. From her eye, X p J i, which contains privately known characteristics, is a random vector; and E[y j X p J i, Z] is a random variable. As a result, she has to integrate over all possible realizations of X p J i conditioning on realizations of her own information, X p J k, Z. Conditional on public information in the group, Z, her expectation about i s behavior might change when she knows X p s of a different subgroup of agents, i.e., the random variables in X p J k differ. Therefore, given public information, Z, conditional expectation, E[y i X p J k, Z], is a function of random vector, X p J i. Conditional expectations for behaviors of all group members, (E[y 1, Z],, E[y n, Z]), is a vector of such types of functions. That distinguishes our model from previous research by Manski [14], Brock and Durlauf [5], and Lee, Li and Lin [13]. Manski [14] and Brock and Durlauf [5] solve just one expectation, expected group average behaviors or expected behavior of symmetric group members, as a scalar. Lee, Li and Lin [13] assume rational expectation. So expectation formed by an individual agent is the same as expectation formed by econometricians. For each j, they only need to solve E[y j ] as a scalar, given all observable characteristics of members in the group and group features. When j runs over all group members, they solve expected behaviors in a group as a vector. Nonetheless, adopting BNE, with heterogeneity in private information, we have to know how expectation of y j vary with private information used to make predictions. Since an agent only predicts the behaviors of agents with whom she is associated, for each i, we only need to define conditional expectation on random vectors known to associated agents. Conditional on Z = z, let (Ω, F, P ) denote the probability space on which the random variables, X p j s are defined. For each i, for any j such that W j,i 0, by (2.3), the type of private information of k is described by J j. X p J j is a random vector composed of some X p k s. Therefore, when the dimension of each Xp k is k p, X p J j is a measurable function from (Ω, F) to (R M j, B M j R ), where M j = ( n j=1 J k(j))k p is the dimension of the vector X p J j, B M j R is the Borel set. Then for each J j, we define a measurable function, ψi,j e j, from (R M j, B M j R ) to (R, B R) such that ψ e i,j j (x) = E[y i X p J j = x, z], (3.2) for any x R M k. Then the composite, ψ e i,j j (X p J j ) : (Ω, F, P ) (R, B R, m R ) is a random variable with ψ e i,j j (X p J j ) = E[y i X p J j, z]. (3.3) Therefore, given J j, the random vector used to make predictions is fixed. Then the function ψ i,jj shows how conditional expectation on y i varies with realization of that vector. So 12

13 we can characterize how conditional expectation changes with the random vector on which expectation is based by defining a function on random vector. Let s begin with the domain. { } A i = X p J j : W j,i 0. (3.4) for any 1 i n. For A i A i, there is j, such that W j,i 0 and A i = X p J j. Denote the set of all random variables on (Ω, F, P ) by C, define ψ e : n i=1 A i C n, such that for any A = (A 1,, A n ) n i=1 A i, ψ e (A) i = ψ e i (A i ) = ψ e i,j j (X p J j ) = E[y i X p J j, Z = z], (3.5) where A i = X p J j for some j = 1,, n. For simplicity, we will write E[y i X p J j, z] instead of E[y i X p J j, Z = z] in the discussions below. We now investigate the relationship between a BNE and expectation functions. Proposition 3.1 Conditional on public information Z = z, if a profile of strategies, n y = (y 1,, y n ) : A i R n, is a BNE in the model, i.e., they satisfy (2.17), the vector function, ψ e : n i=1 A i C n, defined by (3.5), for any i = 1, 2,, n, and A = (A 1,, A n ) n i=1 A i, satisfies the consistency condition: ψ e (A) i = ψ e i (A i ) i=1 = E[h i (u(x i ) + λ j i W i,j ψ e j (X p J i ) ɛ i ) A i ], (3.6) for any i. On the reverse, if ψ e : n i=1 A i C n satisfies ψ e (A) i = ψ e i (A i ), (3.7) for any i = 1, 2,, n, and A = (A 1,, A n ) n i=1 A i, and condition (3.6), (3.5) holds. Moreover, the strategy profile, (y 1,, y n ), defined by y i = h i (u(x i ) + λ j i W i,j ψ e j (X p J i ) ɛ i ) (3.8) is a BNE, i.e., it satisfied (2.17). Proof. The first part follows straightforwardly from the definition of ψ e and consistency condition of expectations in equilibrium, (3.1). For the second part, from (3.8), we have that E[y i A i, z] =E[h i (u(x i ) + λ j i W i,j ψ e j (X p J i ) ɛ i ) A i, z] =ψ(a) i, 13

14 for any i = 1, 2,, n, and A i A i. Therefore, we have that y i = h i (u(x i ) + λ j i W i,j E[y j X p J i, z] ɛ i ). The second part is proved. Hence, there is a one-to-one correspondence between a Bayesian Nash Equilibrium and expectation function satisfying (3.6). We can analyze equilibrium via analyzing the functional equation, (3.6). 3.2 Existence and Uniqueness From above discussion, we can see that there is a BNE in our model if and only if there is a expectation function, ψg e : n g i=1 B i,g C n, satisfying (3.7) and (3.6). We investigate possible existence and uniqueness of such a function by functional analysis. First, impose the following assumptions on model primitives. Assumption 3.1 f ɛ (ɛ) > 0 for all < ɛ <. That is, the support for all the ɛ i s is R. Assumption 3.2 Both h i ( ) for each i = 1,, n and u( ) are continuous. Assumption 3.3 For any real number x, E[ h i (x ɛ) ] <. Furthermore, H i (x) = E[h i (x ɛ)] is differentiable with respect to x. Assumption 3.4 The derivative of H i ( ) is uniformly bounded. That is, max 1 i n dh i(x) dx <. Let Ψ be a set of functions, fixing Z = z, any ψ Ψ is a function mapping a profile of sets A = (A 1,, A n ) n i=1 A i to a random vector in C n, such that (3.7) holds for any i and A. For each i and A i A i, there is a j such that W j,i 0 and A i = X p J j. Suppose that conditional on Z = z, the joint distribution of X p = (X p 1,, Xp n) is F p ( Z = z) 4, which is denoted as F p for simplicity. Functions of Ψ satisfy the following integrability condition: max max ψ i (X p J j ) df p (X p ) <, (3.9) 1 i n {j:w j,i 0} where X p J i is a random vector composed of X p j s. That is, ψ is integrable with respect to the distribution of private personal characteristics, conditional on public information in the group, Z = z. From discussions above, we can see that conditional expectation in 4 We allow correlation between publicly information, Z, and privately known characteristics, (X p 1,, Xp n). When they are independent, conditional distribution will be the same as unconditional distribution. 14

15 equilibrium, ψ e, is a mapping from n i=1 A i to C n and satisfies (3.7). We assume that it meets the requirement, 3.9. So ψ e Ψ. 5 Define sum and scalar product operation of functions in Ψ in the conventional way. That is, for any ψ, ψ Ψ, α R 1, and (A 1,, A n ) n i=1 A i, (ψ + ψ )(A 1,, A n ) = ψ(a 1,, A n ) + ψ (A 1,, A n ); (αψ)(a 1,, A n ) = αψ(a 1,, A n ). Zero is a function which is equal to 0 almost everywhere with respect to F p ( ). It is easy to see that under the above definition, Ψ is a linear space. In addition, define ψ = max max (ψ) i (X p J j ) df p (X p ), (3.10) 1 i n {j:w j,i 0} which is shown to be a norm is the following lemma. Lemma 3.1 is a well-defined norm on Ψ. Proof. By the definition of Ψ, for any ψ Ψ, ψ is non-negative. Moreover, ψ = 0 if and only if ψ i (A i ) = 0 a.e. for any i and A i A i, i.e., ψ = 0. For any real number α, αψ = max max (αψ) i (A i ) df p (X p ) 1 i n {j:w j,i 0} = α max max ψ i (A i ) df p (X p ) 1 i n {j:w j,i 0} = α ψ. Additionally, for any ψ, ψ Ψ, ψ + ψ = max max (ψ + ψ ) i (A i ) df p (X p ) 1 i n {j:w j,i 0} max max ψ i (A i ) df p (X p ) + max 1 i n {j:w j,i 0} = ψ + ψ. max 1 i n {j:w j,i 0} psi i(a i ) df p (X p ) By Lemma 3.1, (Ψ, ) is a linear normed space. In addition, it is complete as the following lemma shows. 5 We use ψ e to represent exclusively equilibrium conditional expectation. ψ is used to denote any arbitrarily chosen point in the function space Ψ. 15

16 Lemma 3.2 The linear normed space (Ψ, ) is complete. Hence, it is a Banach space. Proof. Suppose {ψ m } is a Cauchy sequence in Ψ. m 1 m 2, such that (ψ mk+1 ) i (A i ) (ψ mk ) i (A i ) df p (X p ) ψmk+1 ψ mk < 1 2 k, for any i and A i A i. For every k, define φ k such that and function φ such that (φ k ) i (A i ) = (ψ m1 ) i (A i ) + (φ) i (A i ) = (ψ m1 ) i (A i ) + k (ψ ml+1 ) i (A i ) (ψ ml ) i (A i ). l=1 (ψ ml+1 ) i (A i ) (ψ ml ) i (A i ). l=1 It is easy to see that {j:w j,i 0} φ k ψ m Because φ k φ, by monotonicity convergence, max max (φ) i (X p i J j ) df p (X p ) = max max lim i {j:w j,i 0} k ψ m (φ k ) i (X p J j ) df p (X p ) Therefore, φ < a.e.. So the series ψ m1 + k=1 (ψ m k+1 ψ mk ) converges a.e.. ψ = lim k ψ mk is well-defined and ψ φ. Therefore, ψ Ψ. By Fatou s Lemma, max max (ψ m ) i (X p i J {j:w j,i 0} j ) (ψ) i (X p J j ) df p (X p ) max max lim inf (ψ m ) i (X p i {j:w j,i 0} k J j ) (ψ mk ) i (X p J j ) df p (X p ) lim inf max max (ψ m ) i (X p k i J j ) (ψ mk ) i (X p J j ) df p (X p ). {j:w j,i 0} By definition of the Cauchy sequence, lim n ψ m ψ = 0. Define an operator, T, on Ψ, such that (T (ψ)) i (A 1,, A n ) =(T (ψ)) i (A i ) =E[H i (u(x i ) + λ j i W i,j ψ j (X p J i )) A i, z]. (3.11) 16

17 where H i (x) = h i (x ɛ)f ɛ (ɛ)dɛ. (3.12) We can see that if H i (u(x i ) + λ j i W i,jψ j (X p J i )) is integrable with respect to F p ( ), T (ψ) Ψ for any ψ Ψ. If ψ e Ψ, it is a fixed point of T. Additionally, if function ψ Ψ is a fixed point of T, it satisfies consistency condition. So it is an equilibrium conditional expectation function. That is to say, if we focus on functions in (Ψ, ), there is a one-to-one correspondence between equilibrium conditional expectation functions and fixed points of operator T. The proposition that follows shows that T can be a contraction mapping. Proposition 3.2 Under the condition that λ W D < 1, (3.13) where D = max i sup x dh i(x) dx is well-defined under assumption 3.4, T : (Ψ, ) (Ψ, ) is a contraction mapping. Therefore, by the Banach fixed point theorem, T has one and only one fixed point. Thus, there is one and only one BNE in the model. Proof. Pick any two functions, ψ, ψ (Ψ, ), T (ψ) T (ψ ) = max max E[(H i (u(x i ) + λ W i,j ψ j (X p J i )) j i 1 i n {k:w k,i 0} H i (u(x i ) + λ j i W i,j ψ j(x p J i ))) X p J k, Z = z] df p (X p ) max max 1 i n {k:w k,i 0} = max max 1 i n {k:w k,i 0} λ D W i,j j i = max max 1 i n {k:w k,i 0} λ D W i,j j i λ W D ψ ψ. E[D λ j i W i,j ψ j (X p J i ) ψ j(x p J i ) X p J k, z]df p (X p ) E[ ψ j (X p J i ) ψ j(x p J i ) X p J k, z]df p (X p ) ψ j (X p J i ) ψ j(x p J i ) df p (X p ) It is conventional to row-normalize W such that W = 1. As a result, how stringent the condition (3.13) will be depends on the upper bound of dh i(x) dx. We take some popular models as examples below, linear model, binary choice model and Tobit model. 17

18 1. (Linear Model) For the case of (2.7), H i (x) = (x ɛ)f ɛ (ɛ)dɛ = x E[ɛ]. Thus, sup x dh i(x) dx = (Binary Choice) For the case of (2.8), H i (x) = F ɛ (x). Thus, sup x dh i(x) dx = sup x f ɛ (x). In this case, the uniform boundedness of dh i(x) dx is due to boundedness of ɛ s density. In statistics, many density functions are bounded. Cases in point are the standard normal distribution where sup x f ɛ (x) 1 2π, and the logistic distribution where sup x f ɛ (x) (Tobit model) For the case, (2.10), H i (x) = xf ɛ (x) Thus, dh i (x) dx ɛ<x = F ɛ (x). ɛf ɛ (ɛ)dɛ. As a result, max i sup x dh i(x) dx = sup x F ɛ (x) 1. That is, the absolute value of the derivative of H i (x) is uniformly bounded by 1. 4 Equilibrium Solution and Calculation Proposition 3.2 is the key result for discussion of general model framework. It not only provides with a sufficient condition for equilibrium existence and uniqueness, but also suggests a numerical method to solve equilibrium. That is because the fixed point of a contraction mapping can be derived by recursive iteration beginning with an arbitrary initial guess. In this section, we choose three different information structures and illustrate how equilibrium can be solved in each case. We show that in some special cases, it is possible to solve equilibrium conditional expectations analytically. For all the analysis that follows, we assume that condition (3.13) is satisfied. So there is one and only one BNE. 18

19 4.1 Publicly Known Characteristics When all personal characteristics are publicly known, X p J i = (X p 1,, Xp n). An agent has the same expectation as that of econometricians, for any i and j, E[y i X p J j, z] = E[y i X p 1,, Xp n, z]. We can see that the equilibrium conditional expectation is a vector, ψ = (E[y 1 X p 1,, Xp n, z],, E[y n X p 1,, Xp n, z]). Hence, consistency condition for equilibrium expectation, (3.6), reduces to ψ e i = E[h i (u(x i ) + λ j i W i,j ψ e j ɛ i ) X p 1,, Xp n, z], (4.1) for any i. It can be solved by contraction mapping iteration. This is the case in Lee, Li and Lin [13]. In general, for the linear model, h i (x) = h(x) = x for any i 6, given that E[ɛ i ] = 0 for any i, we have that ψ e i = u(x i ) + λ j i W i,j ψ e i, (4.2) which corresponds to the spatial autoregressive model (SAR) in spatial econometrics. For linear model, with W row-normalized (i.e., sum of each row of W is 1), condition (3.13) reduces to λ < 1. We have the equilibrium: ψ e = ψ e 1. ψ e n 4.2 Self-known Characteristics u(x 1 ) = (I λw ) 1. u(x n ) In this case, X p i is known only to i. We have that X p J i = X p i and A i = { } X p j : W j,i 0.. (4.3) Since X p i,g s are the same type of characteristics of agents in the same group, it is natural to assume that they have the same support. Furthermore, we simplify discussion by imposing an exchangeable property on their joint distribution conditional on group public information, Z = z. Assumption 4.1 Conditional on public information, Z = z, X p i s have the same support, S p. Moreover, their conditional joint distribution, f p (X p 1,, Xp n; Z = z), is exchangeable 6 Note that in this part, we only require that h i( ) is the identity function. We do not impose linearity on u( ) here. 19

20 with respect to the identities of the agents. That is, for any permutation, s : {1,, n} {1,, n}, f p (X p 1,, Xp n, Z = z) = f p (X p s(1),, Xp s(n), Z = z). (4.4) Under Assumption 4.1, f(x p i Xp k = x, Z = z) = f(xp i Xp k and k i. When realizations X p k = Xp k = x, = x, Z = z), for any k i ψ e i,j k (x) =E[y i X p k = x, z] = H i (u(x g, Xi c, y) + λ W i,j ψj e (y))f p (y Xk P j i = = x, Z = z)dy H i (u(x g, Xi c, y) + λ W i,j ψj e (y))f p (y X P = x, Z = z)dy k j i (4.5) =E[y i X p k = x, z] =ψ e i,j k (x). where H i (x) = h(x ɛ)f ɛ (ɛ)dɛ. Thus, with Assumption 4.1, the identity of private characteristics used to make predictions does not matter. It is its realization that influences conditional expectation. Previously, we define conditional expectations in two steps. We first describe how conditional expectation varies with the random vector used to make predictions. We then characterize how conditional expectation changes with realization of a given random vector. Owing to the fact that conditional expectation is neutral to the identity of the random private characteristics, now we can directly define ψi e as a mapping from R kp (k p is the dimension of X p j s) to R1. To be specific, for any i, pick a k i, define ψ e i (x) = ψ e i,j k (x) = E[y i X p k = x, z] (4.6) By (4.5), ψ e i is well-defined. Then define ψe : R nkp R n by ψ e (x 1,, x n ) i = ψ e i (x i ), (4.7) similar to what we did in previous sections. Because this is a special case of the basic framework, our previous analysis about equilibrium existence and uniqueness applies here. In this case, consistency condition (3.6) reduces to ψi e (x) = H i (u(x g, Xi c, y) + λ W i,j ψj e (y))f p (y x)dy, (4.8) j i where f p (y x) is a simplified notation for the density of X p i conditional on X p j = x and Z = z. By exchangeability, f p (X p j = y Xp i = x, z) = f p(x p k = y Xp i = x, z), 20

21 for any j i and j i. So this simplification is reasonable. We investigate condition (4.8) in three different cases that follow Independent Characteristics Given Z = z, if the characteristics X p j s are independent of each other, conditional distribution f p (y x) = f p (X p k = y Xp i = x, z) is the same as f p (y) = f p (X p k = y z). So we have that ψi e (x) = H i (u(x g, Xi c, y) + λ W i,j ψj e (y))f p (y)dy. j i So ψi e(x) is a constant function of x, whose value can be denoted as ψe i. Specifically, it satisfies the following equations: ψi e = H i (u(x g, Xi c, y) + λ W i,j ψj e )f p (y)dy. (4.9) j i for i = 1,, n. The vector, ψ e = (ψ1 e,, ψe n) can be solved by contraction mapping iteration as in Lee, Li and Lin[13]. For the linear model, (4.9) is a linear equation system and we can derive an analytical solution: ψ e 1 E[u(X g, X1 c ψ e =. = (I λw ) 1, y)].. (4.10) E[u(X g, Xn, c y)] ψ e n Compared with (4.3), instead of true realizations, expectation is used in (4.10). This is because X p i is not observed by any agent other than i in the group Correlated Discrete Characteristics In general, X p i s may be correlated with each other among group members7. In this subsection, we focus on the case that X p i has a finite support. As a result, any expectation, ψi e ( ), is a function defined on a finite set. To solve it is to obtain its value at every point in that finite set. Thus, for each i, we can view ψi e ( ) as a vector. Suppose that the support of X p i is { x l : 1 l m }, where x l is a vector with specific values and m is the number of support points of X p i for all i = 1,, n. Under exchangeability, conditional distribution is fully captured by the matrix, p 11 p 12 p 1m P =......, p m1 p m2 p mm 7 This is possible, for example, even conditional on observed group variables, there might still be unobserved group variables which may influence members in a group equally (so exchangeable), and those unobserved group variables are random across different groups. The notion of exchangeability has been characterized in this way by De Finetti (1975). 21

22 where p ll = prob(x p i = x l X p k = xl ), for k i, which is invariant with respect to k by xchangeability. Using P, we have that ψ e i (x l ) = m l =1 p ll H i (u(x g, X c i, x l ) + λ j i W ij,g ψ e j (x l )). (4.11) Stacking all possible values of those conditional expectations in a vector as ψ e = (ψ e 1(x 1 ),, ψ e n(x 1 ),, ψ e 1(x m ),, ψ e n(x m )), we may solve it by contraction mapping iteration. In particular, for the linear model, the equilibrium conditional expectation, ψ e, which is an nm 1 vector, is characterized by ψ e i (x l ) = m l =1 p ll (u(x g, X c i, x l ) + λ j i and can be solved analytically. To be specific, define W i,j ψ e j (x l )), (4.12) u = (u(x g, X c 1, x 1 ),, u(x g, X c n, x 1 ),, u(x g, X c 1, x m ),, u(x g, X c n, x m )). Then equilibrium condition can be written as ψ e = (P I n )u + λ(p W )ψ, where I n is the n n-dimension identity matrix. As P W P W = W, under (3.13), λ(p W ) < 1. So I λ(p W ) is invertible. Thus, we have that Correlated Continuous Characteristics ψ e = (I λ(p W )) 1 (P I n )u. (4.13) Now we consider the case where X p i s have a continuous pdf conditional on Z = z. Since the support is no longer finite, solution of model equilibrium is more complicated. However, there is still a analytical solution for linear model when conditional on X p j and Z, expectation of X p i is linear in X p j. To be specific, we impose the following assumptions. Assumption 4.2 h i (x) = x for any z and i. Namely, the model is linear. Assumption 4.3 E[ɛ i ] = 0 for any i. 22