Network effects on information acquisition by DeGroot updaters

Risco, Miguel Article — Published Version Network effects on information acquisition by DeGroot updaters Economic Theory Provided in Cooperation with: Springer Nature Suggested Citation: Risco, Miguel (2024) : Network effects on information acquisition by DeGroot updaters, Economic Theory, ISSN 1432-0479, Springer, Berlin, Heidelberg, Vol. 79, Iss. 1, pp. 201-234, https://doi.org/10.1007/s00199-024-01568-7 This Version is available at: https://hdl.handle.net/10419/318555 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Economic Theory (2025) 79:201–234 https://doi.org/10.1007/s00199-024-01568-7 RESEARCH ARTICLE Network effects on information acquisition by DeGroot updaters Miguel Risco1 Received: 27 May 2023 / Accepted: 9 March 2024 / Published online: 4 April 2024 © The Author(s) 2024 Abstract In today’s world, social networks have a significant impact on information processes, shaping individuals’ beliefs and influencing their decisions. This paper proposes a model to understand how boundedly rational (DeGroot) individuals behave when seeking information to make decisions in situations where both social communication and private learning take place. The model assumes that information is a local public good, and individuals must decide how much effort to invest in costly information sources to improve their knowledge of the state of the world. Depending on the network structure and agents’ positions, some individuals will invest in private learning, while others will free-ride on the social supply of information. The model shows that multiple equilibria can arise, and uniqueness is controlled by the lowest eigenvalue of a matrix determined by the network. The lowest eigenvalue roughly captures how two-sided a network is. Two-sided networks feature multiple equilibria. Under a utilitarian perspective, agents would be more informed than they are in equilibrium. Social welfare would be improved if influential agents increased their information acquisition levels. Keywords Information acquisition ·Learning ·Public goods ·Network effects · Information diffusion ·Bounded rationality JEL Classification D61 ·D83 ·D85 ·H41 Support by the German Research Foundation (DFG) through CRC TR 224 (Project B05) and funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant agreement 949465) are gratefully acknowledged. I thank Sven Rady, Francesc Dilmé, Alexander Frug, Alexander Winter, Simon Block, Justus Preusser, Axel Niemeyer, an anonymous referee and participants at the seminars at University of Bonn and Universitat Pompeu Fabra for their helpful comments and discussions. BMiguel Risco [email protected] 1Bonn Graduate School of Economics, University of Bonn, Bonn, Germany 123 202 M. Risco 1 Introduction Information is key to making decisions. Nowadays, social networks have a significant impact on information processes. We discuss various issues with family, friends, and colleagues, affecting their opinions and shaping our own. Random conversations about an upcoming election, the job market, or stock market performances can influence our beliefs. Breakthrough news spreads rapidly, and individuals are constantly updating their opinions. Apart from this social supply of information, individuals can learn privately, such as by searching on the internet or consulting a book. Therefore, it is essential to understand how individuals behave when they seek to acquire information to make decisions in situations where both social communication and private learning take place. To what extent do people exert effort themselves, and to what extent do they rely on others? In this paper, we propose a model of information acquisition in networks in which individuals are boundedly rational, behaving as mechanical updaters when it comes to learning. With this in hand, they decide how much to invest in a costly information source to improve their knowledge of the state of the world. Mechanical updating here consists of agents merely taking weighted averages of the signals received—the so-called DeGroot updating rule from DeGroot (1974). Despite considering boundedly-rational agents, we still apply the concept of Nash equilibrium at the stage where they determine their information acquisition. This is done in the spirit of an evolutionary concept of Nash equilibrium. An evolutionary model consists of a large population of boundedly-rational players playing some game repeatedly over time (Mailath 1998). Evolutionary theory shows that such players eventually learn to play Nash equilibrium,1even in the absence of perfect rationality. The crucial assumption is that more successful behaviors become more prevalent due to a combination of imitation and the failure of unsuccessful behaviors.2In our model, boundedly-rational agents that update mechanically face the problem of provision of a local public good. The task of gathering information for subsequent decision-making recurs numerous times throughout an individual’s life. In the spirit of evolutionary theory, we think of boundedly-rational agents who, despite their cognitive constraints, have learned to reach Nash equilibrium outcomes through their choices. To provide an intuition for the formal model, consider an agent who wishes to become more informed about a particular issue. We assume that she has some prior knowledge, and that informative conversations take place in her neighborhood—for example, at the office. There, each colleague exerts a different and fixed influence that shapes the agent’s final opinion. Anticipating this, she decides how much effort to spend on private learning. Given that learning tools are similar among neighbors, we assume a positive correlation when it comes to private learning signals. Hence, each 1In particular, the central notion in evolutionary game theory is that of evolutionary stable strategy and theory shows that any symmetric strict Nash equilibrium is indeed an evolutionary stable strategy. See Mailath (1998) or Samuelson (2002) for an overview. 2This is also discussed by Aumann (1997), asserting that ordinary people, in their daily activities, do not consciously adhere to rationality but evolve “rules of thumb” through evolution. If these rules prove effective, they proliferate and multiply, eventually reaching the equilibrium that strict rationality would have predicted. 123 Network effects on information acquisition by DeGroot… 203 agent faces a problem of information acquisition in which information is a local public good. Individuals have to decide how much to invest in private learning, knowing that free social learning will take place later. Depending on the substitutability between information acquired personally and information acquired by others, but also on the neighbors’ choices, agents will raise or lower their learning efforts. Free-ride behaviors will arise.3 This paper provides three main contributions. First, we analyze and characterize the equilibria arising in the model. Depending on the network structure and their positions, agents will contribute with some learning or completely free-ride. In principle, there are multiple equilibria, and all of them can be calculated. A sufficient condition for equilibrium uniqueness is our second contribution. If this condition does not hold, the equilibria computations run in exponential time. Equilibrium uniqueness is controlled by the lowest eigenvalue of a matrix given by the network. This eigenvalue essentially captures how two-sided the corresponding network is, that is, whether agents can be divided into two sets with many links between them but just a few within. In a game of strategic substitutabilities, when an agent increases her effort, her neighbors decrease theirs in response, so that the neighbors’ neighbors have to increase, and so on. When the network is two-sided, these direct effects accumulate and lead to several distinct equilibria. However, if the lowest eigenvalue is sufficiently large, the network will not be two-sided enough for the actions to rebound, and the equilibrium will be unique. Finally, we provide a welfare analysis. From a social (utilitarian) perspective, every agent would be more informed than she is in equilibrium. To satisfy this demand, at least the influential agents (those agents from which the others get the majority of information) have to increase their contribution. If the network is too unbalanced, this becomes a burden and the welfare of the influentials decreases. In general, the utilitarian optimum does not Pareto-dominate the equilibrium outcome. The choice of the updating rule, i.e., how individuals process and incorporate the information received, is a relevant decision when trying to model social learning. One has to decide whether to employ the fully rational Bayesian focus or the naive, boundedly-rational approach, mainly represented by the already mentioned DeGroot rule. Quoting Acemoglu and Ozdaglar (2011), “which type of approach is appropriate is likely to depend on the specific question being investigated”. We argue here that the DeGroot updating rule fits best within our context. Bayesian updating requires an unrealistic cognitive demand for learning in large networks. However, the DeGroot rule provides a convenient alternative, given its simplicity and lack of restrictive requirements. In a simultaneous setting, Bayesian agents who receive Gaussian private signals behave like DeGroot updaters when subject to persuasion bias, as shown by DeMarzo et al. (2003). In fact, if the game is one-shot (as it is in this paper), persuasion bias is not even necessary for such a result to hold. Still, their model deviates slightly from standard rational assumptions, as neighbors’ signal precision is unknown. The relationship between Bayesian and DeGroot rules, especially for one-shot games, supports our model and is further analyzed in the “Appendix”. Nonetheless, after the first period, a pure Bayesian (not suffering from persuasion bias) would adjust for the information buried in the network, while DeG3For an axiomatic characterization of costly information acquisition processes, see Duraj and Lin (2022). 123 204 M. Risco rootian agents would not.4The literature on networks has widely used the DeGroot rule in different settings. Golub and Jackson (2010) show that under some mild conditions on connectedness and influence, DeGroot agents converge to the belief that would result from the full aggregation of everyone’s signal. Both Golub and Jackson (2012), devoted to study homophily, and Acemoglu et al. (2010), which analyzes the tension between the spread of misinformation and information aggregation, also reflect how convenient DeGroot updating is for large networks analysis. However, the major drawback of the rule is that the choice of weights might seem arbitrary, particularly when communication lasts longer than one period. Furthermore, the assumption that everyone is informed at the outset may be too demanding. Banerjee et al. (2021) adapted the rule to sparse signals to address this issue. This having been said, the empirical evidence heavily supports DeGroot updating. Various papers confront it against Bayesian learning in an experimental setting, concluding that it approximates better people’s information aggregation rules (see Corazzini et al. 2012; Mueller-Frank and Neri 2013; Grimm and Mengel 2020; Chandrasekhar et al. 2020). Although there is no definitive approach, many recent papers tend to use a boundedly rational model for both sequential and simultaneous settings. For example, Dasaratha and He (2020) assume that agents neglect redundancies of information and then aggregate heuristically, and Mueller-Frank and Neri (2021) consider a large class of boundedly rational or quasi-Bayesian rules, respectively. Although modelling learning through a mechanical updating rule is overly simplistic, it allows us to isolate the network effects, which is the primary concern of this paper. Furthermore, we argue that assuming exogenous and fixed weights reflects human behavior. The influence that our neighbors exert on a concrete topic is almost predetermined. A wide range of factors such as past interactions, trustworthiness, and expected level of knowledge defines an influence level before communication occurs. Similarly, it seems sensible that agents can endogenously set the influence of their own private learning on their views: the more time devoted to researching, the more reliable the agent perceives it to be. Thus, the expenditure of costly attention will reduce player-specific noise. Galeotti and Goyal (2010) is a key paper in the literature on information acquisition in networks. In this paper, network-placed agents strategically select their links to access the information held by their neighbors. Every equilibrium displays the socalled “law of the few”: the majority of individuals tend to get most of their information from a tiny subset of the group, the influentials. Our model shows that this result holds true for networks where a subset of agents, the populars, has a significantly higher weight than the rest, such as the core-periphery network. In such networks, popular agents become influential and acquire most of the information, while the others free-ride. This finding contrasts with Banerjee et al. (2021), where the sparse-signals structure indicates that being popular alone is insufficient for being influential. However, two assumptions in Galeotti and Goyal (2010) differ from our model: links are endogenous, allowing an agent to reach any other individual in a potentially large network, and homogeneous, meaning perfect substitutability. Network effects on information acquisition have also been analyzed from a Bayesian perspective. In Myatt and 4SeeMolavietal.(2018) for an axiomatic foundation of the DeGroot rule under imperfect recall. 123 Network effects on information acquisition by DeGroot… 205 Wallace (2019), rational agents acquire information about the state of the world from sources that provide noisy signals. Paying costly attention reduces noise, and signals are possibly correlated across players, similar to our model. However, incentives differ as agents not only want to match the state of the world but also care about coordination asymmetrically. Furthermore, there is no communication stage. The player’s centrality (in the sense of Bonacich) and correlations determine information acquisition, but centrality entails less expenditure, in contrast to Galeotti and Goyal (2010) and our paper. Finally, Denti (2017) introduces the concept of entropy reduction to study how players endogenously acquire costly information to decrease their uncertainty about fundamentals. Network effects induce externalities in information acquisition and are a source of multiple equilibria. Regarding equilibrium analysis, our work closely follows that of Bramoullé et al. (2014). Following previous attempts in the literature to characterize conditions for equilibrium in linear games of strategic complements (cf. Ballester et al. 2006) and strategic substitutes (especially in public goods, cf. Bramoullé et al. 2007), the authors showed that equilibrium uniqueness and stability depend on the lowest eigenvalue of the network matrix.5This is dependent on the two-sidedness of the network. Although our paper differs in setting and motivation, the best response function derived from our model is similar to that of Bramoullé et al. (2014). Consequently, the result regarding the lowest eigenvalue characterizing equilibrium uniqueness is also similar. However, their model assumes that agents’ contributions are reciprocal and weighted equally, which differs from our assumptions. This has two consequences. First, the potential theory introduced in Monderer and Shapley (1996), on which Bramoullé et al. base their results, cannot be applied here; second, a wider range of networks can be analyzed. Nevertheless, if we restrict our setting to symmetric, homogeneous networks, an almost equivalent condition arises. Finally, our paper is also related to Bramoullé et al. (2007) model of pure public goods in exogenous networks, where again all contributions are weighted equally and there is perfect substitutability. In that model, the authors find that multiple equilibria typically exist, and there is always one in which some individuals contribute while others free-ride. This equilibrium is typically unique. The rest of the paper is organized as follows. Section2describes and analyzes our model, Sect.3studies the equilibria, provides a uniqueness condition and presents some examples, and Sect.4analyzes the model from a social planner perspective. Section5briefly introduces a dynamic version of the model, and Sect.6concludes. 2Model We consider a finite set of nagents interacting via a social network represented by an n×nmatrix G=(gij), which is predetermined and stochastic: the entries in each row are non-negative and sum to one. Interactions need not be symmetric or two-sided, so in general gij = gji and gij >0 does not imply gji >0. 5Using the lowest eigenvalue of the network matrix to determine the uniqueness of an outcome is a technique often employed in the social networks literature. See, for example, Melo (2022). 123 206 M. Risco Each agent holds a private signal siabout a common underlying state of the world μ∈R, drawn independently from a normal distribution with mean μand variance σ2>0. There are two learning resources available to improve this signal, presented in the order in which they become accessible to the agent: active private learning from a more informative but costly source, and social learning from neighbors. The first resource involves drawing a signal Iifrom a normal distribution with mean μand variance ˜σ2<σ 2, while the second resource involves aggregating the signals of the agent’s direct neighbors in the network. Both types of learning take the form of DeGroot updating of signals, following DeGroot (1974). Agents take a weighted average of their signals, i.e., they aggregate several indicators in just one. In the case of private learning, agent idecides the weights in the convex combination between siand Ii. The costly signal Iireceives weight xi∈[0,1]at linear cost xicwith c>0. Costly signals are positively correlated across agents, Cov(Ii,Ij)=α>0 for all i,j. The original private signals are independent across agents and independent of all costly signals. Regarding social learning, agent itakes the weighted average of her neighbors’ signals and her own. Weights are exogenously6given by the network matrix, and they represent influence or trust: agent ilistens to agent jprecisely at intensity gij. The mechanical updating process described can be viewed as active learning with attention costs for boundedly rational agents. In addition to normal signals, it can also be interpreted from a Bayesian perspective, as demonstrated in DeMarzo et al. (2003). Agents assign subjective precisions πij to each other, attempting to estimate the true precision of their signals. If the signals are independent and normal, Bayesian updating is equivalent to DeGroot updating, with weights given by πij n j=1πij for social learning.7 A similar argument applies to the active learning process; see the “Appendix” for a motivation of the present framework in terms of quasi-Bayesian updating as defined in DeMarzo et al. (2003). In the following, we use the term “beliefs” to refer to the most recently updated signal an agent holds about μ. A precise description of the learning process is as follows: The agent receives si∼N(μ, σ 2)and decides how much to spend on learning Ii∼N(μ, ˜σ2). Once xiis selected, the belief becomes pi=(1−xi)si+xiIiat cost xic. Finally, the social communication stage yields beliefs n  j=1 gij pj= n  j=1 gij (1−xj)sj+xjIj. Note that if iand jare not neighbors, gij =0, so summing over i’s neighbors is equivalent to summing over all nindividuals. At this point, only one communication stage is assumed, but considering tstages would imply the substitution of Gby Gt, as shown in Sect. 5. 6Rational learners might adjust the weights based on neighbors’ information levels, as discussed in Galeotti and Goyal (2010). However, in this case, we want to focus on situations where weights are pre-determined for a naïve learner due to past interactions, influence, or reputation, and cannot be modified. 7If gij =0, then πij =0. 123 Network effects on information acquisition by DeGroot… 207 Agent iaims to obtain the most precise belief about μat minimum cost, as deviations are penalized through a quadratic loss function. This is specified in the payoff function −⎛ ⎝μ− n  j=1 gij pj⎞ ⎠ 2 −xic=−⎛ ⎝μ− n  j=1 gij((1−xj)sj+xjIj)⎞ ⎠ 2 −xic. Although agent iis a naive, mechanical learner, we assume, based on evolutionary theory, that she is capable of reaching Nash equilibrium outcomes. Specifically, deciding how much to contribute to a public good is a typical example of a process in which boundedly-rational agents evolve toward Nash equilibrium outcomes in the long run (Mailath 1998). Hence, we allow agent ito form expectations and best respond to others’ choices, as if she were rational at this stage. She chooses the amount xithat maximizes her expected utility: max xi∈[0,1]⎧ ⎪ ⎨ ⎪ ⎩ E⎡ ⎢ ⎣−⎛ ⎝μ− n  j=1 gij((1−xj)sj+xjIj)⎞ ⎠ 2⎤ ⎥ ⎦−xic⎫ ⎪ ⎬ ⎪ ⎭ .(1) 3 Equilibrium The equilibrium concept used in this model is Nash equilibrium, where each agent i chooses her information level xiby best responding to others’ equilibrium choices. It is important to note that E[si]=E[Ii]=μfor all i. Additionally, every pair of signals is independent except for Iiand Ij. As a result, En j=1gij(xjIj+(1−xj)sj)=μ, while Var(xjIj+(1−xj)sj)=x2 j˜σ2+(1−xj)2σ2and Cov(xjIj+(1−xj)sj,xkIk+ (1−xk)sk)=αxjxk. These equalities, along with the payoff equation, imply that only second moments matter. In fact, E⎡ ⎢ ⎣−⎛ ⎝μ− n  j=1 gij(xjIj+(1−xj)sj)⎞ ⎠ 2⎤ ⎥ ⎦=−Var ⎡ ⎣ n  j=1 gij(xjIj+(1−xj)sj)⎤ ⎦. Using that for any sequence of random variables {˜ Xj}n j=1it holds that Var(n j=1˜ Xj)= n j=1Var(˜ Xj)+2n j=1j−1 k=1Cov(˜ Xj,˜ Xk), the maximization problem for agent i can be simplified as follows: max xi∈[0,1]⎧ ⎨ ⎩−˜σ2 n  j=1 g2 ijx2 j−σ2 n  j=1 g2 ij(1−xj)2−2α n  j=1 j−1  k=1 gijgikxjxk−cxi⎫ ⎬ ⎭ . (2) 123 208 M. Risco The objective is strongly concave in the choice variable. The first order condition for an interior solution yields xi=2σ2−c/g2 ii 2(˜σ2+σ2)−α gii(˜σ2+σ2) j=i gijxj. Note that this expression is bounded above by 1 but could be negative. As xi∈[0,1] by assumption, the optimal choice of active learning for agent igiven others’ choices x−iis x∗ i=max ⎧ ⎨ ⎩ 0,2σ2−c/g2 ii 2(˜σ2+σ2)−α gii(˜σ2+σ2) j=i gijxj⎫ ⎬ ⎭ . This best response function is similar to the one obtained when solving a maximization problem in a local public goods setting. Games of negative externalities or Cournot competition also yield similar forms. The only difference is that here, δis divided by gii, a parameter that varies across agents. In the other cases, the substitutability factor is the same for all agents. Note that only the weighted out-degree matters for information acquisition, but not the weighted in-degree.8In other words, agents care about who they are listening to (the gijs), but not who listens to them (the gjis). Furthermore, if gii =0, then x∗ i=0 trivially, as active learning is a waste of resources for someone who does not assign positive weight to herself. Therefore, without loss of generality we can assume gii >0. By setting ¯xi=2σ2−c/g2 ii 2(˜σ2+σ2), and δ=α (˜σ2+σ2), we obtain x∗ i=max ⎧ ⎨ ⎩ 0,¯xi−δ gii  j=i gijxj⎫ ⎬ ⎭ . Here, information refers to individuals’ costly learnt signals and is a local public good. Each agent benefits from others’ private learning via network communication. The quotient δ gii scales the benefit and indicates the substitutability between an agent’s and her neighbors’ information. Agent iseeks to reach at least the information target 8The out-degree of agent iis the total weight of links directed away from her. The in-degree is the total weight of links directed to her. 123 Network effects on information acquisition by DeGroot… 215 Fig. 8 Criminal network Fig. 9 Information acquisition for the criminal network As soon as δincreases, B-class agents take advantage of substitutability and free-ride on A and the C-class agents. Agent A has only B-class neighbors, so although she extracts information from them, she has to make up the difference. In contrast, C-class agents have some C-class neighbors, so the free-riding behavior of B-class agents does not affect them as severely. Figure9shows that x∗ Ais higher than x∗ Cfor all δ>0. 3.2 Equilibrium characterization Let us divide the agents into two groups: active (A) agents, who are active learners (x∗ i>0), and passive (P) agents. An equilibrium in which all agents belong to A is known as a distributed equilibrium, as effort is distributed among all agents. In contrast, a specialized equilibrium is such that only a few individuals (the specialists) learn, while the others free-ride. This part mainly follows Bramoullé et al. (2014). Without loss of generality, we can reorder the agents such that the first rare active and the last n−rare passive. As xj=0 for all j∈P, for any individual i,wehavej=igijxj=j∈A\{i}gijxj. 123 216 M. Risco Thus, for i∈1,...,r, an equilibrium requires that: x∗ i=¯xi−δ gii  j∈A\{i} gijx∗ j>0. For i∈{r+1,...,n}, an equilibrium requires that: ¯xi−δ gii  j∈A\{i} gijx∗ j≤0. Let ¯ xA=(¯x1,..., ¯xr)and ¯ xP=(¯xr+1,..., ¯xn). The diagonal of a matrix Ais denoted by dA.LetGAbe the r×rminor corresponding to the active agents of the network, while GPis the (n−r−1)×(n−r−1)minor of Gcorresponding to the passive agents. The (n−r−1)×rminor GP,Aof Gis given by (gij)where i∈P and j∈A. Rearranging the above expressions, we obtain the following result: Proposition 3.2 The profile of information levels x=(x∗ 1,...,x∗ n)=(xA,0)with xA=(x∗ 1,...,x∗ r)∈(0,1]rconstitutes an equilibrium if and only if dGA¯ xA=(1−δ)dGA+δGAxA, dGP¯ xP≤δGP,AxA.(3) Note that given nagents, there are 2npotential partitions. Obtaining all possible equilibria requires solving the system (3) for each partition. This can be done in two steps: (i) First, solve for xAin dGA¯ xA=(1−δ)dGA+δGAxA. The solution is unique if and only if det[(1−δ)dGA+δGA] = 0. (ii) Then, check whether all components of xAare strictly positive and dGP¯ xP≤ δGP,AxA. If the diagonal elements of Gare identical, i.e., gii =gjj for all i,j, the number of equilibria is weakly lower than 2nand can be computed in exponential time. In this case, the condition det[(1−δ)dGA+δGA]=0 simplifies to det −(δ−1)gii δId +GA=0, which holds if and only if GAhas an eigenvalue λ=(δ−1)gii δ. Consequently, for almost all δ, the equation has a unique solution. While Bramoullé et al. (2014) assume not only dG=dId but also matrix symmetry, we have shown that these assumptions are not necessary to obtain an explicit expression for equilibria. 3.3 Equilibrium uniqueness In general, there might be multiple equilibria in this model. We present two examples. The first example is a three-agent network that is incomplete and can also be visualized as a star. The weights of the connections between the agents are represented by the matrix G. Figure10 shows the graph corresponding to this network, with thicker arrows indicating larger weights. 123 Network effects on information acquisition by DeGroot… 217 Fig. 10 Incomplete three-agent network Fig. 11 Four-agents eye Since gAA =gBB =gCC,itfollowsthat ¯xA=¯xB=¯xC=¯x. Assuming δ=1 2, the best reply functions become: xA=max{0,¯x−xB+xC 2}, xB=max{0,¯x−xA}, xC=max{0,¯x−xA}. There are two distributed equilibria: (x∗ A,x∗ B,x∗ C)=(¯x 2,¯x 2,¯x 2)and (x∗ A,x∗ B,x∗ C)= (¯x 3,2¯x 3,2¯x 3). Additionally, there exist specialized equilibria where either agent A or both agents B and C purchase ¯xwhile the others free-ride. Another similar example holds for δ=1 kand a star with kagents, demonstrating that the multiplicity of equilibria does not depend on the extreme assumption that δ=1 2(which is extreme in the sense that it implies ˜σ2=0). The second example is a four-agents eye, shown in Fig. 11. Weights are given by the matrix G. Once again, we assume δ=1 2. Since ¯xi=¯xfor all i, the best replies are as follows: xA=max 0,¯x−xB+xC+xD 2, xB=max 0,¯x−3(xA+xD) 4, xC=max 0,¯x−3(xA+xD) 4, 123 218 M. Risco xD=max 0,¯x−xA+xB+xC 2. There are two specialized equilibria: 2 3¯x,0,0,2 3¯xand (0,¯x,¯x,0). The rough idea behind multiplicity is that agents can be divided into two distinct groups so that active learning contributions vary between them. When one group learns more, the other decreases its effort, and vice versa. We will discuss this in detail in Sect. 3.3.2. Next, we seek a structural condition on the network that guarantees uniqueness. It turns out that, given δ, the positive definiteness of a matrix that we denote Qensures equilibrium uniqueness. This matrix Qcan be determined from Gin a one-to-one correspondence once δis fixed. Recall that agent i’s expected payoffs are given by the following equation: ui(x1,...,xn)=E⎡ ⎢ ⎣−⎛ ⎝μ− n  j=1 gij((1−xj)sj+xjIj)⎞ ⎠ 2⎤ ⎥ ⎦−xic. Proposition 3.3 The profile of active learning choices x∗=(x∗ 1,...,x∗ n)is an equilibrium of the game if and only if (θ− ˆ Qx∗)T(x∗−x)≥0(4) for any x∈[0,1]n, with the matrix ˆ Q=(σ2+˜σ2)⎛ ⎜ ⎜ ⎜ ⎝ 2g2 11 2δg11g12 ···2δg11g1n 2δg22g21 2g2 22 ···2δg22g2n . . .. . ..... . . 2δgn1gnn 2δgn2gnn ··· 2g2 nn ⎞ ⎟ ⎟ ⎟ ⎠ and the vector θ=(2σ2g2 11 −c,...,2σ2g2 nn −c). Proof First, the following equivalence is established: the profile x∗is an equilibrium if and only if ∂ ∂xiui(x∗ i,x∗ −i)(x i−x∗ i)≤0 for all iand x i∈[0,1]. Fixing a profile x∗∈[0,1]nand an agent i, let us define g(t):= ui(x i+t(x∗ i−x i), x∗ −i) 123 Network effects on information acquisition by DeGroot… 219 for 0 ≤t≤1 and x i∈[0,1]. The derivative with respect to tis given by g(t)= ∂ ∂xi(ui(xi,x∗ −i))|xi=x i+(x∗ i−x i)(x∗ i−x i).Ifx∗is an equilibrium, g(t)has a maximum at t=1 and g(1)≥0. Hence, ∂ ∂xiui(x∗ i,x∗ −i)(x i−x∗ i)≤0. Now, let us show the converse. Concavity of gfollows from the concavity of ui,11 and then g(t)≤g(y)+g(y)(t−y)for any t,y∈[0,1]. Choosing t=0 and y=1, we see that g(0)≤g(1)−g(1). Moreover, g(1)≥0 by assumption, so that −g(1)≤0 and g(0)≤g(1). This inequality implies that ui(x∗ i,x∗ −i)≥ui(x i,x∗ −i) for all x i∈[0,1], and x∗is an equilibrium. Summing up with respect to all agent yields n  i=1 ∂ ∂xiui(x∗ i,x∗ −i)(x i−x∗ i)!≤0. Denoting by "∂ ∂xiui(x∗ i,x∗ −i)#ithe vector given by stacking up all ∂ui ∂xi, the previous inequality can be rewritten as ∂ ∂xi[ui(x∗ i,x∗ −i)]!T i (x−x∗)≤0. The profile of active learning choices x∗is an equilibrium if and only if this inequality holds for any x∈[0,1]n.12 It just remains to explicitely derive the vector components, which are given by ∂ ∂xi[ui(x∗ i,x∗ −i)]=2g2 iiσ2−2g2 iix∗ i(σ2+˜σ2)−2αgii  j=i gijx∗ j−c. 11 The function uiis clearly twice differentiable with respect to xiand ∂2ui ∂x2 i=−2(−gii(Ii−si))2<0. 12 If ˆ xis not an equilibrium, then there is some agent jsuch that ∂ ∂xjuj(ˆxj,ˆx−j)(xj−ˆxj)>0 for some xj∈[0,1]. Hence, defining the profile ˜ xas ˜xj=xjand ˜xi=ˆxifor i= j,  i ∂ ∂xi (ui(ˆxi,ˆx−i)!(˜xi−ˆxi)=∂ ∂xjuj(ˆxj,ˆx−j)(xj−ˆxj)>0. 123 220 M. Risco Finally, it is a mere verification to check that defining ˆ Qand θas above, x∗is an equilibrium if and only if (θ− ˆ Qx∗)T(x∗−x)≥0.  If ˆ Qis positive definite, there is just one vector of information levels x∗that satisfies (4). This is the sufficient condition for equilibrium uniqueness. Proposition 3.4 If the matrix ˆ Qis positive definite, the equilibrium is unique. Proof Suppose x∗ 1and x∗ 2are two different equilibria. Then, (θ− ˆ Qx∗ 1)T(x∗ 1−x∗ 2)≥0 and (θ− ˆ Qx∗ 2)T(x∗ 2−x∗ 1)≥0. Summing up both inequalities yields (θ− ˆ Qx∗ 1)T(x∗ 1− x∗ 2)+(θ− ˆ Qx∗ 2)T(x∗ 2−x∗ 1)≥0, which holds if and only if (x∗ 2−x∗ 1)Tˆ Q(x∗ 2−x∗ 1)≤0. But ˆ Qis positive definite, i.e. xTˆ Qx>0 for all x= 0. Consequently, x∗ 1=x∗ 2and the equilibrium is unique.  Dividing ˆ Qby σ2+˜σ2does not change its definiteness and simplifies the expression—recall that σ2+˜σ2>0.13 Thus, Qis given by Q=1 σ2+˜σ2 ˆ Q=⎛ ⎜ ⎜ ⎜ ⎝ 2g2 11 2δg11g12 ···2δg11g1n 2δg22g21 2g2 22 ···2δg22g2n . . .. . ..... . . 2δgn1gnn 2δgn2gnn ··· 2g2 nn ⎞ ⎟ ⎟ ⎟ ⎠ .(5) The following result shows that Qis completely determined by G, once δis fixed. Consequently, equilibrium uniqueness for this model depends solely on the influence network G. Proposition 3.5 Given δ, there is a one-to-one correspondence between Qand G. Proof Given δ, the matrix Qis defined element-wise from Gas in (5). Assume δis fixed and denote this transformation by φδ. Let us show that it is possible to recover Gfrom Q. Denoting by qij the elements in Q, let us define (element-wise) the transformation τδby τδ(qii)=√qii for all iand τδ(qij)=qij δ√qii for all i= j. It is trivial to check that τδ(φδ(G)) =Gand φδ(τδ(Q)) =Q. In general, the matrix Qis an n×nmatrix that need not be symmetric. Note that xTQx =1 2xT(Q+QT)x, and Qis positive definite if and only if A:= 1 2(Q+QT) is positive definite. Since Ais symmetric, we can use the characterization of positive definiteness in terms of eigenvalues: a symmetric matrix is positive definite if and only if all of its eigenvalues are positive. Let λ1(A)denote the lowest eigenvalue of A. 13 It would be possible to divide by 2(σ 2+˜σ2)instead, but keeping the factor 2 simplifies the expression for the matrix Alater. 123 Network effects on information acquisition by DeGroot… 221 Corollary 3.6 If λ1(A)>0, then the equilibrium is unique. The explicit expression for Ais given by: A=⎛ ⎜ ⎜ ⎜ ⎝ 2g2 11 δ(g12g11 +g21g22)···δ(g1ng11 +gn1gnn) δ(g21g22 +g12g11)2g2 22 ···δ(g2ng22 +gn2gnn) . . .. . ..... . . δ(gn1gnn +g1ng11)δ(gn2gnn +g2ng22)··· 2g2 nn ⎞ ⎟ ⎟ ⎟ ⎠ . Note that this sufficient condition is independent of the cost cof active learning but depends on the influences between agents and the substitutability of information acquisition. The scope of this condition is the focus of our subsequent analysis. We will make more restrictive assumptions on the model to explore particular cases of interest, which will eventually lead to a result similar to that of Bramoullé et al. (2014). Later, we will apply the sufficient condition to the examples in Sect.3.1. We first prove an auxiliary lemma. Lemma 3.7 Let Abe a symmetric matrix and β,δ > 0. The matrix βId+δAis positive definite if and only if λ1(A)≥−β δ. Proof The matrix βId +δAis positive definite if and only if all the solutions λ to det[λId −(β Id +δA)]=0 are strictly positive. The equation is equivalent to det[λ−β δId−A]=0. Note that the eigenvalues of Aare the solutions tto the equation det[tId−A]=0. Consequently, as t=λ−β δ, the condition λ>0 can be translated into all eigenvalues tof Averifying t>−β δ. This is precisely the condition λ1(A)>−β δ.  Next, we consider two particular cases that are worth exploring. Assuming that all agents pay the same attention to themselves, i.e., gii =gjj for all i,j, we can denote the diagonal terms of Gby β:= gii >0. We define ¯ Aas ¯ A=⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 0g12+g21 2... g1n+gn1 2 g21+g12 2 ...... g2n+gn2 2 . . .... .... . . gn1+g1n 2 gn2+g2n 2... 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ . Using Lemma 3.7, we see that λ1(A)>0 if and only if λ1(¯ A)>−β δ. Note that ¯ A is simply ¯ A=1 2(G+GT)−βId. Here, ¯ Areflects the average flow of information between a pair of networks, or the undirected network associated with G. Now, assume that the network displays reciprocal relations, i.e., gij =gji,in addition to same self-importance across agents. This means that the influence of agent ion agent jis the same as that of agent jon agent i, and so the matrix Gis symmetric and can be seen as undirected. Again, λ1(A)>0 if and only if λ1(¯ A)>−β δ,but ¯ A is now simply G−βId. The matrix ¯ A=G−βId can be seen as a generalization 123 222 M. Risco of the matrix Gin Bramoullé et al. (2014), where ¯aij ∈[0,1]instead of gij ∈{0,1}. The sufficient condition is equivalent to theirs. However, to derive such a result they use the potential theory developed by Monderer and Shapley (1996), which requires symmetry—this is why we cannot apply it to the general model. Proposition 3.8 The sufficient condition for the uniqueness of equilibrium can be specialized to two particular cases: •If self-importance is equal across agents (gii =gjj =βfor all i,j), the condition becomes λ1(¯ A)>−β δwith ¯ A=1 2(G+GT)−βId. •If on top of that the influences are reciprocal (gij =gji for all i,j), the condition becomes λ1(¯ A)>−β δwith ¯ A=G−βId. This proposition summarizes the results obtained so far, which show that the condition for the uniqueness of equilibrium can be specialized for two particular cases: when all agents have the same level of self-importance and when the network exhibits reciprocal relations between agents. In both cases, the condition involves the eigenvalue of a matrix ¯ A, which can be calculated based on the properties of the network. The precise definition of ¯ Ais given for each case. 3.3.1 Examples: uniqueness The networks analyzed in Sect.3.1 are reviewed again to apply the equilibrium uniqueness condition. First, we revisit the class of k-regular graphs with nagents that share their attention homogeneously. Proposition 3.8 applies, and the lowest eigenvalue of ¯ Ais λ1(¯ A)= −1 k. The equilibrium is unique if δ<1, which always holds. As an example of this class of networks, the matrix ¯ Aassociated with the complete graph is given by ¯ A=⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 01 n... 1 n 1 n0... . . . . . . ... ...1 n 1 n 1 n... 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ . Next, we consider the class of stars. Due to the asymmetry of Qand the different terms in the diagonal (self-importance is not equal across agents), only Corollary 3.6 applies. The equilibrium is unique if λ1(A)>0, which depends on both δand ε. Matrix Ais given here by: A=⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 2 n2δ((1−ε)ε +1 n2)...δ((1−ε)ε +1 n2) δ((1−ε)ε +1 n2)2(1−ε)2... 0 . . . ... .... . . δ((1−ε)ε +1 n2)0... 2(1−ε)2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ . 123 Network effects on information acquisition by DeGroot… 223 Fig. 12 The lowest eigenvalue of the star Figure12 shows the values for which a unique equilibrium is ensured—every pair (δ, ε) such that the blue surface is above the orange plane. A particular network structure belonging to the class of core-periphery networks was set in Fig.6. Here, A= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 2 9δ2 81 δ2 81 δ9 100 00 δ2 81 2 9δ2 81 0δ9 100 0 δ2 81 δ2 81 2 900δ9 100 δ9 100 002 100 00 0δ9 100 002 100 0 00δ9 100 002 100 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . and we apply Corollary 3.6. It turns out that λ1(A)>0 for all δ∈[0,1 2],sothe equilibrium is always unique.14 The criminal network from Ballester et al. (2006) was represented in Fig.8.Proceeding as before, we calculate the lowest eigenvalue of A.15 We find that λ1(A)>0 for all δ<0.45011, which guarantees a unique equilibrium for such values. Finally, we consider the incomplete network depicted in Fig.10. Recall that δ=1 2 and all diagonal terms are equal: β=1 3. To apply Proposition 3.8, we compute 14 The explicit expression for the lowest eigenvalue of Ais λ1(A)= 981−100δ−√670761−163800δ+541441δ2 8100 . We see that λ1(A)is a decreasing function of δin [0,1/2]. As it is strictly positive at δ=1/2, λ1(A)>0 for all δ∈0,1 2. 15 Let y1(δ),y2(δ) and y3(δ) be the three roots of −32400 −97200δ+259200δ3+(3096 +6192δ− 7200δ2)y+(−97 −97δ)y2+y3.Lety1(δ) be the smallest root in δ∈0,1 2. Then, λ1(A)=1 450 y1(δ) and λ1(A)>0⇔δ<0.45011. 123 224 M. Risco ¯ A=⎛ ⎝ 01 2 1 2 1 200 1 200 ⎞ ⎠. The uniqueness condition λ1(¯ A)>−β δis not satisfied because λ1(¯ A)=−1 √2< −2 3. This was expected, as we had already obtained two different equilibria for this particular network. 3.3.2 The lowest eigenvalue The present subsection explores the meaning of the uniqueness condition and provides an intuition. A network is bipartite if agents can be divided into two sets, say Rand S, such that if i∈R,iis not connected to any j∈Rexcept for herself. The network is completely bipartite if every i∈Ris connected to all j∈S. Bipartite networks represent disjoint or independent communities. An affiliation network is a classic example. Another bipartite network might be found when representing supervisorcandidate communication. A complete bipartite network represents one extreme of two-sidedness. The other extreme is the complete regular graph. In this subsection, we talk about two-sidedness as an intuitive measure of how close a network is to the complete bipartite graph. First, let us briefly characterize λ1(A).16 By definition, λ1(A)=min{λ∈R: ∃∈Rnsatisfying λ =A}. Assuming = 0, λ =Aimplies Tλ =TA, which leads to λT=TA, and finally to λ||||2=TA. So, if |||| = 1, then λ=TA. Hence, λ1(A)=min $λ∈R:λ=TAand |||| = 1%. Following Bramoullé et al. (2014), we can use an eigenvector associated to λ1(A) to separate the agents into two groups. If i≥0, agent ibelongs to set R. Otherwise, she belongs to set S. This leads to the decomposition λ1(A)=εTAε= >0 &'( )  i,j∈R ijqij + >0 &'( )  i,j∈S ijqij + <0 &'( ) 2 i∈R,j∈S ijqij . The greater the lowest eigenvalue, the more weight the network puts within sets and the less it puts between sets. Hence, the size of λ1(A)is related to the two-sidedness of the graph A. The closer the network is to the complete bipartite graph, the lower λ1(A). This is because transferring weight from links within Ror Sto links between both sets decreases λ1(A). Creating new links between sets or removing links within Ror Sbelong to that kind of weight transfer. Thus, making the graph more two-sided decreases the lowest eigenvalue. Let us show how the division of agents into the two groups is induced by agents’ listening structures. We have λ1(A)=tA=i,jqijijwith |||| = 1. Without loss of generality, let us assume that λ1(A)>0 (if not, a similar reasoning holds). Then, agent ibelongs to Rif and only if λ1(A)i>0. Since λ1(A)i=Ai,wesee 16 Remember that when Qis symmetric, then A=Q. 123 Network effects on information acquisition by DeGroot… 231 Fig. 14 Acquisition for the criminal network in the long-run In the long run, only in-degree matters, but not network position. Hence, agent A no longer has a distinct role and there are just two classes of agents: B-class and C-class (to which agent A belongs now). B-class agents have in-degree 6, and πj=6 59 , and C-class agents have in-degree five, so πj=5 59 . Best reply functions are given by: x∗ B=max 0,¯xB−δ59 6(3xB+7xC), x∗ C=max 0,¯xC−δ59 5(4xB+6xC). Similar to the one-period communication game, B-class agents acquire less information. In particular, for the configuration of parameters used in Sect. 3.1, we can observe in Fig.14 that B-class agents completely free-ride on C-class agents. This happens because B-class agents consider acquiring private information too costly, relying instead on the information obtained from the seven C-class agents. Finally, the examples from Sect.3.3 are trivial in the long run. The incomplete threeagent network converges to a matrix characterized by the limit vector π=1 2,1 4,1 4, while the four-agents eye converges to a matrix characterized by the limit vector π=3 10 ,1 5,1 5,3 10 . Both cases lead to unique equilibrium configurations. 6 Conclusion This paper has analyzed the behavior of DeGroot updaters in a networked environment and studied the impact of substitutability and network structure on information acquisition and welfare. We have shown that the substitutability of agents’ active learning efforts induces free-riding behavior and can lead to multiple equilibria. We have also provided a sufficient condition for equilibrium uniqueness in terms of the lowest eigenvalue of the matrix A, which is determined by Gand the parameter of 123 232 M. Risco substitutability δ. When this eigenvalue is positive, the equibrium is unique. Even if there are multiple equilibria, we have proposed a procedure for calculating them. In terms of welfare, we have found that the information target is lower in equilibria than under the utilitarian paradigm. This is significant since the target is precisely the level of information an agent will have at the end of the game. We have shown that it is socially desirable to increase the information level of every agent. While increasing agents’ active learning may seem like a solution, we show that in the one-shot game it is not. Not only the ranking in targets does not imply a ranking in acquisition levels, but the utilitarian optimum does not Pareto dominate the equilibrium allocation. Nevertheless, over the long run, neighborhood frictions are eliminated and the utilitarian allocation always exceeds the equilibrium allocation. An interesting avenue for further research would be the implementation problem of a planner trying to incentivize DeGroot updaters to move from equilibrium levels of active learning to the utilitarian optimum. Public information policies, such as subsidizing external information sources, rewarding learning contributions, or creating new links to foster communication, could also be explored. Funding Open Access funding enabled and organized by Projekt DEAL. Declarations Conflict of interest The authors declare that they have no conflict of interest. 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Appendix: Quasi-Bayesian foundation Regarding agents’ cognitive sophistication, this paper follows the boundedly rational approach, which assumes that agents have limited cognitive resources and do not possess precise knowledge of their environment. Nonetheless, it is useful to connect the assumptions of this paper to the standard Bayesian framework. In this “Appendix”, we provide a pure theoretical motivation for DeGroot updating in networks, following DeMarzo et al. (2003). DeGroot updating can be viewed as a Bayesian updating process for agents that receive normally distributed signals but do not know the true variances of their neighbors’ signals. Consider nagents who want to estimate some unknown parameter μ∈R. Agent i receives an independent signal x0 i∼N(μ, σ 2 i), and she assigns some precision πij = 1 Vari(x0 j)to agent j’s signal, which may or may not be the true precision. Note that this assumption does not align with the standard Bayesian approach, which assumes that agents have precise knowledge of the signal structure. Agents communicate according 123 Network effects on information acquisition by DeGroot… 233 to a social network ˜ G, which is a directed graph that indicates whether agent ilistens to agent j;˜gij =1 if agent ilistens to agent j, and ˜gij =0 otherwise. Each agent knows her own information, so ˜gii =1. Truthful reporting is assumed. Given normality and the assigned precisions, a sufficient statistic for the signals is their weighted average, with weights given by the precisions. DeMarzo et al. (2003) denote such a statistic by x1 i, and refer to it as agent i’s belief after communication: x1 i= n  j=1 ˜gijπij n j=1˜gijπij x0 j. The sufficiency of the statistic x1 icomes from the application of the Fisher-Neyman factorization theorem. Defining gij := n j=1˜gijπij n j=1˜gijπij , we obtain the stochastic matrix G=(gij). 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