Contagious Synchronization and Endogenous Network Formation in Financial Networks
CContagious Synchronization and Endogenous
Network Formation in Financial Networks ∗ Christoph Aymanns † , Co-Pierre Georg ‡ July 1, 2014
Abstract
When banks choose similar investment strategies the financial sys-tem becomes vulnerable to common shocks. We model a simple fi-nancial system in which banks decide about their investment strategybased on a private belief about the state of the world and a socialbelief formed from observing the actions of peers. Observing a largergroup of peers conveys more information and thus leads to a strongersocial belief. Extending the standard model of Bayesian updating insocial networks, we show that the probability that banks synchronizetheir investment strategy on a state non-matching action criticallydepends on the weighting between private and social belief. Thiseffect is alleviated when banks choose their peers endogenously ina network formation process, internalizing the externalities arisingfrom social learning. ∗ We would like to thank Toni Ahnert, Jean-Edouard Colliard, Jens Krause, TarikRoukny, three anonymous referees, seminar participants at ECB, Bundesbank, as wellas the 2013 INET Plenary Conference in Hong Kong, and the VIII Financial StabilitySeminar organized by the Banco Central do Brazil for helpful discussions and comments.This paper has been prepared under the Lamfalussy Fellowship Program sponsored bythe ECB whose support is gratefully ackonwledged. The views expressed in this paperdo not necessarily reflect the views of Deutsche Bundesbank, the ECB, or the ESCB. † Mathematical Institute, University of Oxford and Institute for New Economic Think-ing at the Oxford Martin School. E-Mail: [email protected] . ‡ Deutsche Bundesbank and University of Cape Town Graduate School of Business.E-Mail: [email protected] . a r X i v : . [ q -f i n . E C ] A ug eywords: social learning, endogenous financial networks, multi-agent simulations, systemic risk JEL Classification:
G21, C73, D53, D85 Introduction
When a large number of financial intermediaries choose the same invest-ment strategy (i.e. their portfolios are very similar) the financial system asa whole becomes vulnerable to common shocks. A case at hand is the fi-nancial crisis of 2007/2008 when many banks invested into mortgage backedsecurities in anticipation that the underlying mortgages–many of which be-ing US subprime mortgages–would not simultaneously depreciate in value.This assumption turned out to be incorrect resulting in one of the largestfinancial crises since the great depression. How could so many banks choosea non-optimal investment strategy despite the fact that they carefully mon-itor both economic fundamentals and the actions of other banks?This paper presents a simple agent-based model in which financial inter-mediaries synchronize their investment strategy on a state non-matchingaction despite informative private signals about the state of the world. In acountable number of time-steps N agents, representing financial intermedi-aries (banks for short), choose one of two actions. There are two states ofthe world which are revealed at the end of the simulation. A bank’s actionis either state-matching, in which case the bank receives a positive payoffif the state is revealed, or it is state-non-matching in which case the bankreceives zero. Banks are connected to a set of peers in a financial networkof mutual lines of credit resembling the interbank market. They receive aprivate signal about the state of the world and observe the previous actionsof banks with whom they are connected via a mutual line of credit, but notof other banks. Based on both signals banks form a belief about the stateof the world and choose their action accordingly.Our model differs from the existing literature along two dimensions. Firstand foremost we develop an agent-based model of the financial system withstrategic interaction amongst agents. This differs from existing models (see,3or example, Poledna et al. (2014), Bluhm et al. (2013), Georg (2013), andLadley (2013)) where agent behaviour is myopic. Agents in myopic modelsreact to the state of the world but when choosing an optimal action they donot take into account how other agents will react to their choice. Thus, thenotion of equilibrium in myopic models is a rather mechanical. Strategic in-teraction amongst agents arises in our model from the fact that agents learnabout their neighbors’ actions, i.e. via the social belief. All the aforemen-tioned papers furthermore use an exogenous network structure as startingpoint for the agent-based simulation, while our model uses an endogenousnetwork formation process to arrive at a pairwise stable network structurethat maximizes expected utility from social learning.Second, while our model is mildly boundedly rational it shares a number ofassumptions with the literature on Bayesian learning in social networks. Themain difference to this literature (see, for example, Acemoglu et al. (2011),Gale and Kariv (2003)) is that we model an externality that is not presentin the standard model of Bayesian learning in social networks. We assumethat banks receive more information about the actions of other banks thanthey can computationally use. This assumption seems natural in a financialsystem that is increasingly complex. The underlying assumption is thatbanks cannot adjust their actions (i.e. their investment strategy) as fast asthey receive information from their peers and thus have to aggregate overpotentially large amounts of information. The social belief in our modelis formed not just from observing one neighbor at a time, but rather fromobserving a set of neighbors simultaneously. It is thus reasonable to as-sume that receipt of more information, i.e. observing the actions of a largersubset of agents, will create a stronger social belief than the receipt of lessinformation. We model this by allowing for different weights of the social See, for example, Haldane and Madouros (2012) for a discussion of increasing com-plexity in financial regulation. This assumption renders the agents boundedly rational, albeit mildly so, as for ex-ample DeMarzo et al. (2003) argue. socialclique ) can communicate at low costs, while others communicate at highcost. The main difference to our model is that we endogenously obtain adecreasing marginal value of additional links. We obtain a resulting endoge-nous network structure which is pairwise stable in the sense of Jackson andWolinsky (1996).We obtain two sets of results, one for networks with exogenous networkstructure, and one for endogenously formed networks. First, we analyze dif-ferent ways of weighting private and social belief. In particular, we comparethe standard equal weighting scenario in which agents place equal weightson their private and social belief, with two scenarios where agents placemore weight on the social belief when they have a larger neighborhood. Inthe neighborhood size scenario the social belief is weighted with the size ofthe neighborhood, i.e. the private signal is weighted equal to every ob-served neighbor action. In the relative neighborhood scenario agents putmore weight on the social belief when the neighborhood constitutes a largershare of the overall network. For completely uninformative signals thereis no difference between these weighting functions. For informative sig-nals, however, the weighting function has an impact on the probability thatagents synchronize their investment decisions on a state non-matching ac-5ion, i.e. for the probability that choosing a state-non-matching action iscontagious. Contagion is, very generally, understood as the transmissionof adverse effects from one agent to another and is more likely if agentsplace greater weight on their social belief and depends on the density of theunderlying exogenous network structure. The probability of contagion in-creases by a factor of in the neighborhood size scenario compared to theequal weighting scenario, which highlights the importance of understandingthe learning dynamics when agents place different weight on their socialbelief, depending on the size of their neighborhood.We show that contagious synchronization occurs even if private signals areinformative and if agents are initialized with an action that is on averagestate matching. The probability of contagion depends non-monotonously onthe density of the network. For small network densities ρ (cid:46) . the prob-ability of contagion increases sharply and then decreases slowly for largernetwork densities. We confirm the robustness of our results by conducting , independent simulations where we observe the average final action asa function of the average initial action with varying network densities.This result is of particular interest for policy makers as it relates two sourcesof systemic risk: common shocks and interbank market freezes. When thenetwork density is too small, for example in the aftermath of an interbankmarket freeze, banks are unable to fully incorporate the information abouttheir peers’ actions. This effect is empirically tested by Caballero (2012),who documents a higher correlation amongst various asset classes in theworld in the aftermath of the Lehman insolvency, i.e. during times of ex-treme stress on interbank markets and heightened uncertainty about thestate of the world. This can be understood as a contagious synchronization Such informational cascades are a well-documented empirical phenomenon. See, forexample, Alevy et al. (2007), Bernhardt et al. (2006), Chang et al. (2000), Chiang andZheng (2010), and Cipriani and Guarino (2014). For a more thorough discussion of the different forms of contagion, see for exampleBandt et al. (2009).
6f bank’s investment strategies for which our model provides a simple ra-tionale.Second, turning to the extension of endogenously formed networks, we showthat endogenous link formation in the first stage of our model can signif-icantly improve the speed of learning and reduce the probability of conta-gious synchronization relative to random networks. When private signalsare less informative, the additional utility from forming a link is smaller andthe endogenously formed network is less dense. This in turn can increase theprobability of contagious synchronization in the second stage of the model.Heightened uncertainty about the state of the world, i.e. a less informativesignal, does therefore not only directly increase the probability of contagioussynchronization, but also indirectly because agents have less incentives toendogenously form links. If agents are heterogenous in the informativenessof their private signals, i.e. if some agents receive signals with higher preci-sion than others, we show that the resulting endogenous network structureis of a core-periphery type. The structure of real-world interbank marketsis often of this particular type, as for example Craig and von Peter (2014)show. Naturally, these endogenously formed networks transfer informationmore effectively from highly informed agents to less informed agents thansimple random networks.This paper relates to three strands of literatures. First and foremost, thepaper develops a financial multi-agent simulation in which agents learn notonly from private signals, but also via endogenously formed interbank links.This is in contrast with existing multi-agent models of the financial systemwhich include Nier et al. (2007) and Iori et al. (2006) who take a fixednetwork and static balance sheet structure. Slight deviations from these Closely related is the literature on financial networks. See, for example, Allen andGale (2000), and Freixas et al. (2000) for an early model of financial networks. Thevast majority of models in this literature consider a fixed network structure only (see,amongst various others, Battiston et al. (2012)).
There is a countable number of dates t = 0 , , . . . , T and a fixed number i = 1 , . . . , N of agents A i which represent financial institutions and arecalled banks for short. By a slight abuse of notation the model parameter θ is sometimes called the state of the world and we assume it can take twovalues θ ∈ { , } . The probability that the world is in state θ is denotedas P ( θ ) and we assume that each state of the world is obtained with equalprobability . At each point in time t bank i chooses one of two investmentstrategies x it ∈ { , } which yields a positive return if the state of the worldis revealed and matches the investment strategy chosen, and nothing oth-erwise. Agents take an action by choosing a certain investment strategy.Taking an action and switching between actions is costless. The utility ofbank i from investing is given as: u i ( x i , θ ) = if x i = θ else (1)The state of the world is unknown ex-ante and revealed at time T . Thissetup captures a situation where the state of the world is revealed less often(e.g. quarterly) than banks take investment decisions (e.g. daily). In an alternative setup the state of the world is fixed throughout and an agent collectsinformation and takes an irreversible decision at time t , but receives a payoff that isdiscounted by a factor e − κt . Both formulations incentivize agents to take a decision infinite time instead of collecting information until all uncertainty is eliminated. N = { , , . . . , n } and the set of banks to which bank i is directly connected is denoted K i ⊆ N . Bank i thus has k i = | K i | directconnections called neighbors. This implements the notion of a network ofbanks g which is defined as the set of banks together with a set of unorderedpairs of banks called (undirected) links L = ∪ ni =1 { ( i, j ) : j ∈ K i } . A link isundirected since lines of credit are mutual and captured in the symmetricadjacency matrix g of the network. Whenever a bank i and j have a link,the corresponding entry g ij = 1 , otherwise g ij = 0 . When there is no riskof confusion in notation, the network g is identified by its adjacency matrix g . For the remainder of this section, I assume that the network structureis exogenously fixed and does not change over time. I assume that banksmonitor each other continuously when granting a credit line and thus ob-serve their respective actions.In this section, the network g is exogenously fixed throughout the simula-tion. In t = 0 there is no previous decision of agents. Thus, each bankdecides on its action in autarky. Banks receive a signal about the stateof the world and form a private belief upon which they decide about theirinvestment strategy x it =0 . The private signal received at time t is denoted s it ∈ S where S is a Euclidean space. Signals are independently generatedaccording to a probability measure F θ that depends on the state of the world θ . The signal structure of the model is thus given by ( F , F ) . I assume that F and F are not identical and absolutely continuous with respect to eachother. Throughout this paper I will assume that F and F represent Gaus-sian distributions with mean and standard deviation ( µ , σ ) and ( µ , σ ) respectively.In t = 1 , . . . bank i again receives a signal s it but now also observes the t − actions x jt − of its neighbors j ∈ K i . The model outlined in this section isimplemented in a multi-agent simulation where banks are the agents. Date10 = 0 in the model timeline is the initialization period. Subsequent dates t = 1 , . . . , T are the update steps which are repeated until the state of theworld is being revealed in state T . Once the state is revealed, returns arerealized and measured. In the simulation results discussed in Section 2.2 thestate of the world was revealed at the end of the simulation after the systemhas reached a steady state in which agents do not change their actions anymore. Banks form a private belief at time t based on their privately observed signal s it and a social belief based on the observed actions x jt − their neighboringbanks took in the previous period. The first time banks choose an actionis a special case of the update step with no previous decisions being taken.The information set I it of a bank i at time t is given by the private signal s it , the set of banks connected to bank i in t − , K it − , and the actions x jt − of connected banks j ∈ K it − . Formally: I it = (cid:8) s it , K it − , x jt − ∀ j ∈ K it − (cid:9) (2)The set of all possible information sets of bank i is denoted by I i . A strat-egy for bank i selects an action for each possible information set. Formally,a strategy for bank i is a mapping σ i : I i → x i = { , } . The notation σ − i = { σ , . . . , σ i − , σ i +1 , σ n } is used to denote the strategies of all banksother than i .A strategy profile σ = { σ i } i ∈ ,...,n is a pure strategy equilibrium of thisgame of social learning for a bank i ’s investment, if σ i maximizes the bank’sexpected pay-off, given the strategies of all other banks σ − i . Acemoglu et al. In practice this is ensured by having many more update steps than it takes the systemto reach a steady state. i , x it = σ i ( I it ) is given as: x i = if P σ ( θ = 1 | s it ) + P σ ( θ = 1 | x jt − , j ∈ K it − ) > if P σ ( θ = 1 | s it ) (cid:124) (cid:123)(cid:122) (cid:125) private belief p + P σ ( θ = 1 | x jt − , j ∈ K it − ) (cid:124) (cid:123)(cid:122) (cid:125) social belief q < (3)and x i ∈ { , } otherwise. The first term on the right-hand side of Equation3 is the private belief, the second term is the social belief, and the thresholdis fixed to . This equation can be generalized when introducing weights onthe private and social belief. In its most general form, it can be written as: x i = if t ( p i , q i ) > if t ( p i , q i ) < (4)where t ( p i , q i ) is a weighting function depending on the private and socialbelief. A simple weighting function t ( p i , q i ) = ( p i + q i ) if | K i | > ,p i | K i | = 0 , (5)implements the model of Acemoglu et al. (2011) where agents place equalweight on their private and social belief (we denote this weighting functionas the equal weighting scenario).The simple weighting function is appropriate in a setting where agents re-ceive two signals at a time only: their private signal and the signal of theirdirect predecessor in the social network. In our setting, however, banksreceive multiple signals, only one of which is their private signal. It is thusnatural to allow for more general weighting functions which, however, haveto satisfy two conditions: (i) In the case with no social learning ( K it − = ∅ ),the weighting should reduce to the simple case t ( p i , q i ) = p i in which theagent will select action x i = 1 whenever it is more likely that the state of12he world is θ = 1 and zero otherwise; and (ii) With social learning theweighting should depend on the number of neighboring signals, i.e. the sizeof the neighborhood k it − = | K it − | . The underlying assumption is that theagent will place a higher weight on the social belief when the neighborhoodis larger. We consider two different scenarios for the weighting function: (i)The private signal and each observed action are equally weighted (calledthe neighborhood size scenario): t ( p i , q i ) = (cid:18) k it − + 1 (cid:19) p i + (cid:18) k it − k it − + 1 (cid:19) q i (6)And (ii) Observed actions are weighted with the relative size of the neigh-borhood (called the relative neighborhood scenario): t ( p i , q i ) = (cid:18) − k it − N − (cid:19) p i + (cid:18) k it − N − (cid:19) q i (7)The private belief of bank i is denoted p i = P ( θ = 1 | s i ) and can easily beobtained using Bayes’ rule. It is given as: p i = (cid:18) d F d F ( s it ) (cid:19) − = (cid:18) f ( s it ) f ( s it ) (cid:19) − (8)where f and f are the densities of F and F respectively. Bank i isassumed to form a social belief q i by simply averaging over the actions ofall neighbors j ∈ K it − : q i = P σ ( θ = 1 | K it , x j , j ∈ K it − ) = 1 /k it − (cid:88) j ∈ K it − x jt − (9)Given these private and social beliefs, agents choose an action according toequation (4).Averaging over the actions of neighbors is a special case of DeGroot (1974)who introduces a model where a population of N agents is endowed withinitial opinions p (0) . Agents are connected to each other but with varying13evels of trust, i.e. their interconnectedness is captured in a weighted di-rected n × n matrix T . A vector of beliefs p is updated such that p ( t ) = T p ( t −
1) = T t p (0) . DeMarzo et al. (2003) point out that this processis a boundedly rational approximation of a much more complicated infer-ence problem where agents keep track of each bit of information to avoid apersuasion bias (effectively double-counting the same piece of information).Therefore, the model this paper develops is also boundedly rational. The model in this section can be formulated as an agent-based model. Banksare agents a i who choose one of two actions x i ∈ { , } . Variables deter-mined in the model internally are given by the private and social belief ofagent i at time t , p it , q it and the only exogenously given parameter is thestate of the world θ which is identical for all agents i . Each agent has aninformation set I i given by Equation (2). Agents receive utility (1) anddecide on their optimal strategy given in Equation (3). The interaction ofagents is captured in a network structure g , encapsulated in an agent i ’s in-formation set. Equations (8) and (9) specify how agents take their decisionsand choose an optimal strategy. Our interest is to understand under which conditions agents in the modelwith an exogenously fixed network structure coordinate on a state non-matching action. Before analyzing the full model, we build some intuition bydiscussing useful benchmark cases. Let the state of the world be θ = 0 andassume that f = mf . For m = 1 the signal is completely uninformative.For m > the signal is informative and more so the larger m is. In theequal weighting scenario, equation (3) together with equations (8) and (9) This bounded rationality can be motivated analogously to DeMarzo et al. (2003) whoargue that the amount of information agents have to keep track of increases exponentiallywith the number of agents and increasing time, making it computationally impossible toprocess all available information.
12 (1 + m ) − + 12 1 k (cid:88) j ∈ K it − x jt − > ⇔ (cid:88) j ∈ K it − x jt − > k (cid:20) − m ) (cid:21) (10)For completely uninformative signals, m = 1 , equation (10) implies that anagent will always follow the majority of her neighbors. For very informativesignals, m (cid:29) , equation (10) implies that an agent will only ignore herprivate signal if she receives a strong social signal. The required strengthof the social signal increases with the precision of the private signal. Nowconsider the neighborhood size scenario. The equation analogous to (10)for this scenario reads: (cid:18) k it − + 1 (cid:19) (1 + m ) − + (cid:18) k it − k it − + 1 (cid:19) k it − (cid:88) j ∈ K it − x jt − > (11) ⇔ (cid:88) j ∈ K it − x jt − > k it − (cid:20) −
11 + m (cid:21) (12)For completely uninformative signals m = 1 this condition reduces to theequal weighting scenario. For highly informative signals, m (cid:29) , however,the agent is almost as willing to follow her neighbors as in the uninforma-tive equal weighting scenario. The neighborhood scenario thus captures thesituation where the agent is aware not only of her own private signal infor-mativeness, but also of that of her neighbors. In the relative neighborhoodscenario equation (10) reads: (cid:18) − k it − N − (cid:19) (1 + m ) − + (cid:18) k it − N − (cid:19) k it − (cid:88) j ∈ K it − x jt − > (13) ⇔ (cid:88) j ∈ K it − x jt − > ( N − (cid:20) −
11 + m (cid:21) + k it − m (14)Again, for completely uninformative signals this equation reduces to theequal weighting case. For highly informative signals, m (cid:29) , this equa-tion reduces to (cid:80) j ∈ K it − x jt − > ( N − if the neighborhood is sufficiently15mall m (cid:29) k it − . An agent will thus ignore her private signal and fol-low the majority of her neighbors, if her neighborhood is larger than halfof the network. The central node in a star network will thus always followthe majority of the spokes and spokes will always follow their private signal.We can gain further insights into the model dynamics by resorting to amean-field approximation in which we consider the simplified action dynam-ics of a representative agent. Given the adjacency matrix g of a network g ,the social belief q of the representative agent is given as q = g x /k where x is the vector of all agent’s actions. The social belief in the mean-fieldapproximation is simply the average action of the population: q = Pr( x = 0 | q = q ) · x = 1 | q = q ) · x = 1 | q = q ) (15)which yields a self-consistency relation for the social belief and hence forthe average action of the population. The equilibrium average action q ∗ isimplicitely given by the solution to the self-consistency condition: q ∗ = Pr( x = 1 | q = q ∗ ) = (cid:90) − q ∗ p P ( p | θ = 0) dp. (16)Note that Pr( x = 1 | q = 0) = 0 and Pr( x = 1 | q = 1) = 1 . Since Pr( x = 1 | q = q ∗ ) is the cumulative distribution function of the bell shapedprivate belief it will have a sigmoid shape. Therefore the self consistencyequation will have three solutions: q = 0 , q = 1 and some q ∈ (0 , . Weillustrate this in figure 4. q , q are stable fixed points while q is unstable(this can be seen graphically in figure 4 and is a direct result from the spec-ified learning dynamics). q defines the “critical” social belief beyond whichthe system synchronizes on the state non-matching action.These exercises show that the impact of the weighting function on agents’strategies is easily understood in the case of completely uninformative andfully informative signals or in a mean field approximation. But what hap-16ens in the more realistic interim region? Are densely connected networksmore conducive for agents to coordinate on state non-matching actions orsparse networks? And how does the fraction of nodes that coordinate ona state non-matching action depend on the initial conditions? We addressthese questions in an agent-based simulation for three cases. In the case(I) of informed agents the distance between the mean of the two signals µ − µ = 0 . − . . while in the case (U) of uninformed agents thedistance between the mean of the two signals is µ − µ = 0 . − .
49 = 0 . .In both cases we use a standard deviation of σ , = √ . . We conduct oursimulations with N = 100 agents and update T = 100 times. To analyzethe impact of the network structure on the probability of coordination ona state non-matching action, we vary the network density ρ of a random(Erdös-Rényi) graph within ρ = [0 . , . in 20 steps. Each simulation isrepeated S = 1 , times to account for stochasticity. For all simulationswe assume that the state of the world is θ = 0 . An overview of the param-eters used can be found in Table 1.Figure (1) shows the average final action of the system after T = 100 up-date steps for a random graph with varying densities in the informed anduninformed case for the equal weighting, neighborhood size, and relativeneighborhood scenario. When the network density is very low, agents effec-tively act on the basis of their private signal only and the fraction of agentsthat choose a state non-matching action is proportional to the signal in-formativeness. With increasing network density, social learning sets in andthe fraction of agents with a state non-matching action is reduced. Withan informative signal and equal weighting there is almost no agent thatchooses a state non-matching action after T = 100 update steps. In theneighborhod size scenario agents are more likely to follow their neighborsthan in the equal weighting scenario. The initial conditions of the simu- A larger standard deviation would make the signal less informative, without changingour results qualitatively. i will only choose action x i = 1 if more than half of her neighbors chosethat action. With increasing network density the dependence on the ini-tial conditions will become more important, which explains the increase inthe fraction of agents choosing a state non-matching action with increasingnetwork density in the center-left panel of Figure (1). Note that this effectis not present for uninformative signals as can be seen in the right panel ofFigure (1). Finally, agent i in the relative neighborhood scenario chooses x i = 1 only if a relatively large fraction of her neighbors also chooses thisaction. The threshold for this is independent of the size of the neighborhoodand only depends on the total network size.The initial conditions become more important in the case of informativesignals for the neighborhood size scenario. In order to understand howexactly our results depend on the initial conditions, Figure (2) shows theprobability that a large fraction ( > ) of agents coordinate on a statenon-matching action for three cases: (1) For a full sample of S = 1 , sim-ulations; (2) conditional on agents starting on average with a state matchingaction: ˆ x = (cid:80) i x i (0) /N < ; (3) conditional on agents starting on averagewith a state non-matching action: ˆ x > . As expected, the probability thatagents coordinate on a state non-matching action drastically increases whenagents initially start with a state non-matching action. However, less so inthe equal weighting scenario because agents place less weight on their socialbeliefs and private signals are informative. A comparison of the left-centerwith the left-top panel in Figure (2) shows that the probability of conta-gion increases by a factor of roughly in the neighborhood size scenariocompared to the equal weighting scenario. Again, for uninformative signalsthis effect is not present. 18hile Figure (2) shows the existence of contagion even for initial actionsthat are state matching, the relationship between average initial and averagefinal action is not yet quantified. Therefore, in Figure (3) we plot theaverage final action versus the average initial action in a density plot. Weconducted a total of S = 1 , × simulations and show the resultingpair of average initial and average final action as a dot with the respectivecoordinates. The left side of Figure (3) is for an informative signal and wedraw initial actions according to the (informative) private signal. The meanof initial distributions is thus ˆ x I < . , i.e. informative on average. In theright panel we show the same results for an uninformative signal and themean of initial distributions is thus much closer to ˆ x I = . Contagionis shown in the upper right quadrant of each subfigure. For the equalweighting scenario (top), only very few simulations yield a final averageaction that is state non-matching. A similar picture can be seen for therelative neighborhood (bottom) scenario. In the neighborhood size (center)scenario, however, a substantial number of simulations with an initially statematching average action yield a final state non-matching average action,confirming the existence of a contagious regime. Banks form interbank networks endogenously. The decision whether or nottwo banks engage in interbank lending, e.g. in the form of agreeing on amutual line of credit, depends in reality on many factors, including liquidityneeds and counterparty risk. In the previous section we analyzed how onebank can learn about an underlying state of the world by observing theaction of another bank to which it has issued an interbank credit. Thisadditional information can create a benefit for the loan-issuing bank thatconstitutes a, possibly positive, externality for the lending decision. Whenbanks coordinate on a state non-matching information, however, “learning”19bout a neighboring bank’s action constitutes a negative externality. Thenet effect of both externalities determines whether two banks are willing toengage in interbank lending. We compute the value of an additional link inthree steps. First, we compute the probability that an agent chooses a statematching action, given her signal structure, private beliefs and neighbors’actions. Given this probability, we compute, second, an agent’s expectedutility conditional on her social belief which depends on her strategic choiceto establish a link. Once an agent’s expected utility with and without alink is computed, we can use the concept of pairwise stable networks todetermine the equilibrium network structure.
Based on the signal structure, the update of private beliefs, and neighbors’action choice we can derive the probability that an agent i chooses a statematching action. To see this, we first derive the distribution of privatebeliefs. The private signal structure is given as: f ( s ) = 1 √ πσ exp (cid:18) − ( s − µ ) σ (cid:19) f ( s ) = 1 √ πσ exp (cid:18) − ( s − µ ) σ (cid:19) (17)and we assume that σ = σ = σ . Denote the probability distribution ofagent i ’s private belief p i as f p ( p i ) . We can then state the following: Proposition 1
For θ ∈ { , } the probability that agent i chooses a statematching action x i = θ , given a social belief q i = q and private belief p i , isgiven by: Pr( x i = θ | q i = q ) = (cid:90) − q f p ( p i | θ = 0) dp i = (cid:90) − q f p ( p i | θ = 1) dp i , (18)20 here the distribution f p ( p i | θ = 0) of agent i ’s private belief is given as: f p ( p i | θ = 0) = (cid:16)(cid:16) − − pp − p (cid:17) pσ (cid:17) exp (cid:18) − ( ( µ − µ ) − σ log ( p − )) σ ( µ − µ ) (cid:19)(cid:0) √ πσ (cid:1) ((1 − p )( µ − µ )) . (19) and similarly for f p ( p i | θ = 1) . Proof, see Appendix (C). (cid:4)
In the absence of maturity, liquidity and counterparty risk, the value of aninterbank loan, is proportional to the probability that the newly connectedneighbor chooses a state-matching action and thus proportional to (18).
Recall that an agent i ’s utility u i is given as: u i ( x i ) = if x i = θ if x i (cid:54) = θ (20)The expected utility of agent i conditional on her social belief q i is thus: u i ( x i | q i ) = Pr( x i = θ | q i = q ) . (21)The value of a link is given by the marginal utility from establishing alink, which in turn depends on the change in the social belief q . An agentcan thus influence her social belief by strategically choosing neighbors. Theprobability that a neighbor takes a state matching action depends in turn onthe social belief that this neighbor forms about her neighbors, which leadsto complex higher-order effects which we neglect in this paper. Rather,we assume that an agent i has constant beliefs about her neighbors’ socialbeliefs q (cid:48) . An agent who ignores the effect of second-nearest neighbors (i.e.neighbors of neighbors) on the social beliefs of nearest neighbors will simplyassume that her neigbhors’ social belief is q (cid:48) = . This amounts to assuming21hat the neighbors’ actions are independent of each other. The expectedutility of agent i conditional on a given q (cid:48) and neighborhood K i is given as: u i ( q (cid:48) , K i ) = (cid:88) a ∈ Q i Pr( q i = a | q (cid:48) ) Pr( x i = θ | q i = a ) , (22)where Q i is the set of all possible values of the social belief of agent i . Thefirst term on the right-hand side of equation (22) is the probability thatagent i has a certain social belief given the social belief of her neighbors.The second term is the probability of choosing a state matching action giventhat social belief and given by equation (18). For a given size of the agent’sneighborhood k i , Q i is simply Q i = { n/k i | n ∈ Z , ≤ n ≤ k i } . Theprobability of a particular social belief can be computed by summing overthe probabilities of combinations of actions chosen by the neighbors of agent i . Define the set of feasible action vectors of i ’s neighbors conditional onagent i having a social belief q i = a : X ai = { x | (cid:88) j x j = a, x j ∈ { , } , j ∈ K i } . (23)i.e. X ai is the set of all action vectors that are compatible with a socialbelief q i = a . Then the probability of agent i having a private belief of q i = a given the social beliefs of all i ’s neighbors, q (cid:48) , is given as: Pr( q i = a | q (cid:48) ) = (cid:88) y ∈ X ai (cid:89) j ∈ K i Pr( x j = y j | q j = q (cid:48) ) . (24)Now define the probability that neighbor j chooses a state matching actionas: z j = Pr( x j = θ | q j = q (cid:48) ) = (cid:90) − q j f p ( p j | θ = 0) dp (25)Note, that f p ( p j | θ = 0) depends on the signal structure of neighbor j .We can write for the probability of agent i having a private belief of q i = a
22n equation (24):
Pr( x j = y j | q j = q (cid:48) ) = z j if y j = 01 − z j if y j = 1 (26)If z j = z ∀ j the distribution in 24 would be a simple binomial distribution.In general, however, this is not the case and we need to resort to numericalmethods to compute the equilibrium network structures. We now have all necessary ingredients to compute the expected utility of anagent i given her neighborhood K i and expectations about her neighbors’social belief q (cid:48) . In the endogenous network formation process the agent willseek to maximize her utility by changing her neighborhood while holding q (cid:48) fixed. In this section we outline an algorithm for endogenous networkformation that ensures a pairwise stable network in the sense of Jacksonand Wolinsky (1996): Definition 1
A network defined by an adjacency matrix g is called pairwisestable if(i) For all banks i and j directly connected by a link, l ij ∈ L : u i ( g ) ≥ u i ( g − l ij ) and u j ( g ) ≥ u j ( g − l ij ) (ii) For all banks i and j not directly connected by a link, l ij (cid:51) L : u i ( g + l ij ) < u i ( g ) and u j ( g + l ij ) < u j ( g ) where the notation g + l ij denotes the network g with the added link l ij and g − l ij the network with the link l ij removed. When maintaining a link iscostly, there will be some network density that depends on the cost c > per link. The marginal utility of an additional link decreases with the num-ber of links because the expected utility is bounded by (the pay-off is and the probability of choosing the correct action is less than, or equal to, ).23he algorithm to ensure a pairwise stable equilibrium starts by choosing arandom agent i from the set of agents N . Then, choose a second agent j from the set of agents N \ K i that are not yet neighbors of i . Agents arechosen with the following probability: w j = exp( βE j ) /Z, (27)where Z = (cid:80) k w k is a normalization constant and E j = | − µ j | is a proxyfor agent j ’s signal strength. For β = 0 i chooses the new agent with equalprobability. While this makes it more likely that agent i considers forminga link with agent j when j has a higher signal strength, it does not implythat such a link is actually formed. This decision is solely based on theutility that both i and j obtain from establishing the link.Now, let K (cid:48) i = K i ∪ j , i.e. the neigborhood of agent i after adding addingagent j , and similarly K (cid:48) j = K j ∪ i . The marginal utilities of adding j and i to the respective neighborhoods are then: ∆¯ u i ( q (cid:48) , K (cid:48) i , K i ) = ¯ u i ( q (cid:48) , K (cid:48) i ) − ¯ u i ( q (cid:48) , K i )∆¯ u j ( q (cid:48) , K (cid:48) j , K j ) = ¯ u j ( q (cid:48) , K (cid:48) j ) − ¯ u j ( q (cid:48) , K i ) (28)Given the marginal utilities of agents i and j and their cost of maintain-ing link c i and c j the agents will form a link if ∆¯ u i ( q (cid:48) , K (cid:48) i , K i ) > c i and ∆¯ u j ( q (cid:48) , K (cid:48) j , K j ) > c j . If ∆¯ u i ( q (cid:48) , K (cid:48) i , K i ) > c i and ∆¯ u j ( q (cid:48) , K (cid:48) j , K j ) < c j ,the algorithm selects the least informative agent in the neighborhood of j : l = argmin m ∈ K j E m . (29)Now, define K (cid:48)(cid:48) j = K (cid:48) j \ k . If ∆¯ u i ( q (cid:48) , K (cid:48) i , K i ) > c i and ∆¯ u j ( q (cid:48) , K (cid:48)(cid:48) j , K j ) >c j , form the link l ij and remove the link l jk (and similarly for i → j and j → i ). Otherwise, don’t form the link. If ∆¯ u i ( q (cid:48) , K (cid:48) i , K i ) < c i and ∆¯ u j ( q (cid:48) , K (cid:48) j , K j ) < c j repeat the previous step, i.e. consider removing the24east informative neighbor and re-evaluate the utilities. The endogenously formed network in an economy with identically informedagents and positive cost c of maintaining a link is a simple Erdös-Rényinetwork with a network density depending on the signal structure and linkcost. To analyze more realistic situations, we can harvest the strengths ofmulti-agent simulations. In the following we therefore assume that agentsare heterogenously informed about the underlying state of the world: a few“informed” agents have relatively precise signals and low costs of maintain-ing a link, while many “uninformed” agents have relatively imprecise signalsand higher cost of maintaining a link. Table 2 summarizes the parameterswe are using for the rest of this section.In order to compare the dynamics on the endogenous networks to the ERnetworks we run the following simulations. We first create , networksusing the network formation algorithm described above. Then, we run thesocial learning algorithm described in Section 2 while holding the networkstructure constant throughout. The underlying assumption is that banksare updating their investment decisions faster than the network structurechanges. This can be empirically corroborated by looking at the term struc-ture of interbank lending. While of the turnover in interbank marketsis overnight, about of exposures between banks stems from the termsegments.We also run the social learning algorithm with an initialization bias in whichwe set the initial action of all agents to some pre-defined value. To assessthe efficiency of the endogenous network formation, we compare the perfor-mance of the endogenously formed networks to the performance of Erdös-Rényi networks. All simulations in this section are conducted using theequal weighting scenario. An example of a resulting network structure can25e found in Figure 5 and the degree distribution of the endogenously formednetworks in , is shown in Figure 6. The degree distribution of the en-dogenously formed networks is clearly bimodal. One peak corresponds tothe uninformed nodes with small degree while the second peak correspondsto the informed nodes with high degree. Figure 7 shows the distribution ofthe final action for the , simulations conducted. A clear improvementover the Erdös-Rényi networks can be seen, highlighting the importance ofcore banks with more precise private signals. Since core banks are highlyinterconnected, there is a higher chance that they are in the neighborhoodof a peripheral bank (as opposed to a peripheral bank being in the neighbor-hood of another peripheral bank) which increases the precision of peripheralbanks’ social belief.To further understand the difference between endogenously formed and ran-dom networks, we analyze the time it takes learning to converge. We assumethe learning has converged at time t if: (cid:88) i | x i ( t ) − x i ( t + ∆ t ) | /N < . , where ∆ t = 15 . (30)It can be seen from Figure (8) that, except for very long convergence times,the system always converges faster in the endogenous network case thanin the Erdös-Rényi case. Note, that this simulation was conducted with-out initialization bias, i.e. with average initial action of . Finally, theprobability of contagion as a function of an initialization bias, i.e. as afunction of average initial action is shown in Figure (9). Again, the pictureis unanimously showing that the probability of contagion, i.e. the probabil-ity that more than of agents coordinate on a state non-matching actionis significantly smaller in the endogenous network case than in the case ofa random Erdös-Rényi graph. 26 Conclusion
This paper develops a model of contagious synchronization of bank’s in-vestment strategies. Banks are connected via mutual lines of credit andendogenously choose an optimal network structure. They receive a privatesignal about the state of the world and observe the strategies of their coun-terparties. When banks observe the actions of more peers they put moreweight on their social belief. We compare three scenarios of weighting func-tions. First, in the equal weighting scenario, agents place equal weightson their private and social belief. Second, in the neighborhood scenarioagents place proportionately more weight on the social signal when the sizeof the neighborhood increases. Third, in the relative neighborhood scenarioagents place more weight on the social belief if their neighborhood con-stitutes a larger fraction of the overall network. Social learning increasesthe probability of choosing a state matching action and thus agents’ utility.When agents strategically choose their neighbors they take the additionalutility from learning into account. The more neighbors a given agent has,the lower is the marginal utility from another link and the network endoge-nously reaches an equilibrium configuration.We obtain two results which are policy relevant. First, in a complex fi-nancial system where agents cannot take the action of all their peers intoaccount when taking an investment decision, the probability of contagioussynchronization depends on two things: (i) the weighting between the pri-vate and social belief; and (ii) the density of the financial network. Ourmodel thus relates two empirically relevant sources of systemic risk: com-mon shocks interbank market freezes. Second, the probability of contagioussynchronization is substantially reduced when agents internalize the posi-tive effects of social learning in a strategic decision with whom to form alink. The benefit from learning is reduced when the private signals aboutthe state of the world are less informative.27he model has a number of interesting extensions. One example is thecase with two different regions that can feature differing states of the world.Such an application could capture a situation in which banks in two coun-tries (one in a boom, the other in a bust) can engage in interbank lendingwithin the country and across borders. This would provide an interestingmodel for the current situation within the Eurozone. The model so far fea-tures social learning but not individual learning. Another possible extensionwould be to introduce individual learning and characterize the conditionsunder which the contagious regime exists. Finally, the model can be ap-plied to real-world interbank network and balance sheet data to test for theinterplay of contagious synchronization and endogenous network structure.One drawback of the model is that there is no closed-form analytical solutionfor the benefit a bank obtains through learning from a peer that takes intoaccount higher order effects. This benefit will depend on whether or not aneighboring bank chose a state matching on state non-macthing action inthe previous period and thus on the social belief of neighboring banks. Inthe former case, the benefit will be positive, while in the latter case it will benegative. Agents have ex ante no way of knowing what action a neighboringbank selected until the state of the world is revealed ex post. Finding such aclosed-form solution is beyond the scope of the present paper which focuseson the application in an agent-based model, but would provide a fruitfulexercise for future research. 28 eferences
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Tables
Variable Description Value1 - (I) 2 - (U) 3 - (H) N Number of agents
100 100 100 µ Average signal for θ = 0 0 . .
49 0 . µ Average signal for θ = 1 0 . .
51 0 . σ Standard deviation of signal for θ = 0 √ . √ . √ . σ Standard deviation of signal for θ = 1 √ . √ . √ . T Number of iterations of updater
100 100 100 ρ Density of ER network [0 , .
95] [0 , .
95] 0 . p Probability of being informed NA NA [0 . , . S Table 1: Parameters for runs for the case of informed (I), uninformed (U),and heterogeneous (H) agents. iotation Description Value N Number of agents N I Number of informed agents N U Number of unformed agents µ I Average signal for θ = 0 for informed agents . µ I Average signal for θ = 1 for informed agents . σ I Standard deviation of signal for θ = 0 for informed agents √ . σ I Standard deviation of signal for θ = 1 for informed agents √ . µ U Average signal for θ = 0 for uninformed agents . µ U Average signal for θ = 1 for uninformed agents . σ U Standard deviation of signal for θ = 0 for uninformed agents √ . σ U Standard deviation of signal for θ = 1 for uninformed agents √ . c I Cost per link for informed agent c U Cost per link for informed agent . T C Number of iterations of network algorithm q (cid:48) Agent’s belief of average action of neighbors of neighbors . β Intensity of choice in agent selection T Number of iterations of action updater n Number of endogenously formed networks used for simulation S Number of simulations per parameter configuration ρ Average density of ER networks . Table 2: Parameters for network formation and runs with endogenous net-works. ii
Figures
Average action at t = 100
Average action at t = 100
Average action at t = 100
Average action at t = 100
Average action at t = 100
Average action at t = 100
Figure 1: Average final action of agents as a function of the network den-sity rho of a random graph for the informed (left) and uninformed (right)case. Top: Equal weighting scenario; Center: Neighborhood size scenario;Bottom: relative neighborhood scenario.iii .0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.00000.00050.00100.00150.00200.00250.00300.00350.00400.0045
Probability of contagion unconditionalxi <= 0.5 unconditionalxi > 0.5
Probability of contagion unconditionalxi <= 0.5 unconditionalxi > 0.5
Probability of contagion unconditionalxi <= 0.5 unconditionalxi > 0.5
Probability of contagion unconditionalxi <= 0.5 unconditionalxi > 0.5
Probability of contagion unconditionalxi <= 0.5 unconditionalxi > 0.5
Probability of contagion unconditionalxi <= 0.5 unconditionalxi > 0.5
Figure 2: Fraction of simulations per parameter configuration S (1000) inwhich agents synchronize on the state non matching action (more than 80%of agents choose state non-matching action) as a function of network den-sity rho of a random graph for the informed (left) and uninformed (right)case. Top: Equal weighting scenario; Center: Neighborhood size scenario;Bottom: relative neighborhood scenario. We distinguish three cases: (1)unconditional: we compute the fraction based on the full sample S . (2)conditional ˆ x ≤ : we compute the fraction based on the sub-set of simu-lations in which the average initial action ˆ x = (cid:80) i x i (0) /N ≤ , i.e. whenthe agents start with a state matching action. (3) conditional ˆ x > : wecompute the fraction based on the sub-set of simulations when the agentsstart with a state non matching action.iv .0 0.2 0.4 0.6 0.8 1.0average initial action0.00.20.40.60.81.0 a v e r a g e f i n a l a c t i o n Dependence on initial condition l o g o f f r e q u e n c y a v e r a g e f i n a l a c t i o n Dependence on initial condition l o g o f f r e q u e n c y a v e r a g e f i n a l a c t i o n Dependence on initial condition l o g o f f r e q u e n c y a v e r a g e f i n a l a c t i o n Dependence on initial condition l o g o f f r e q u e n c y a v e r a g e f i n a l a c t i o n Dependence on initial condition l o g o f f r e q u e n c y a v e r a g e f i n a l a c t i o n Dependence on initial condition l o g o f f r e q u e n c y Figure 3: Average final action ˆ x F = (cid:80) i x i ( T ) /N versus the average ini-tial action ˆ x I = (cid:80) i x i (0) /N for the informed (left) and uninformed (right)case. Top: Equal weighting scenario; Center: Neighborhood size scenario;Bottom: relative neighborhood scenario. Data points are averages over S = 1000 simulations and all network densities ρ (20 values equally dis-tributed over the interval [0 , . ). The color code indicates the frequencywith which a point occurs in the sample (total size × ), the scale ofthe color code is logarithmic of base 10.v .0 0.2 0.4 0.6 0.8 1.0Social belief0.00.20.40.60.81.0 A v e r a g e b e li e f Pr( x =1 | q = q ) q Figure 4: Probability of choosing state matching action (equivalent to aver-age social belief and average action in mean field) given that the state of theworld is θ = 0 , f p ( p i | θ = 0) for µ = 0 . . Furthermore we use µ = 1 − µ and σ = 0 . . The intersections between the probability function and thediagonal mark the fixed points of the dynamical system.Figure 5: Example network from ensemble of n = 1000 endogenously formednetworks. Black nodes are “informed” agents, while white nodes are “unin-formed”. vi -5 -4 -3 -2 -1 P r o b a b ili t y Average network density: 0.08 ± endogenousER Figure 6: Degree distribution of ensemble of n = 1000 endogenously formednetworks compared to ER networks with same average density. -4 -3 -2 -1 P r o b a b ili t y d e n s i t y endogenousER: Figure 7: Distribution of final action in ER networks vs. endogenous net-works. This is without bias, i.e. the initial action is random based on theprivate belief only. vii
20 40 60 80 100Convergence time10 -7 -6 -5 -4 -3 -2 -1 P r o b a b ili t y d e n s i t y endogenousER: Figure 8: Distribution of convergence time ER networks vs. endogenousnetworks. This is without bias, i.e. the initial action is random based onthe private belief only. -5 -4 -3 -2 -1 P r o b a b ili t y o f c o n t a g i o n endogenousER: Figure 9: Probability of contagion vs. initialization bias. Missing valuescorrespond to zero frequency. viii
Proofs
Proof of Proposition (1) . We use the notation f P ( p i | θ = 0) to indicatethat the functional form of the probability distribution of the private belief p i has be derived assuming that θ = 0 . Now, let s ( p i ) be the inverse of theprivate belief: s ( p i ) = µ i − µ + 2 σ log (cid:16) − p i p i (cid:17) µ − µ ) (31)The distribution of the private belief can be computed as follows: f p ( p i ) = ∂s ( p i ) ∂p i f s ( s ( p i )) . (32)where the probability density function for signal s is given as: f s ( s ) = 1 √ πσ exp (cid:18) − ( s − µ ) σ (cid:19) (33)and the private belief p i ( s ) is given by Equation (8). Substituting in theexpression for s ( p i ) and computing the partial derivative we obtain: f p ( p i | θ = 0) = (cid:16)(cid:16) − p i p i − (cid:17) σ (cid:17) exp (cid:32) − (cid:16) ( µ − µ ) − σ log (cid:16) pi − (cid:17)(cid:17) σ ( µ − µ ) (cid:33)(cid:0) √ πσ (cid:1) ((1 − p i )( µ − µ )) (34)Example distributions of the private belief are shown in Figure (10). Wehave µ = 1 − µ and σ = 0 . . Note, that the majority of the probabilitydensity of the private belief is to the left of . in all cases. Therefore, theprivate signal tends to produce private beliefs that yield the state matchingaction. If we increase the | µ − . | the distribution becomes more skewedtowards the actual state of the world. Hence the private belief becomesmore informative.Now that we have defined the pdf of the private belief we can compute theix .0 0.2 0.4 0.6 0.8 1.0private belief02468101214 P r o b a b ili t y d e n s i t y µ =0 . µ =0 . µ =0 . Figure 10: Probability density function of the private belief given thatthe state of the world is θ = 0 f P ( p | θ = 0) for three values of µ ∈ { . , . , . } .probability that the agent chooses x i = 0 given some social belief q = q i as: Pr( x i = 0 | q i = q ) = (cid:90) − q f p ( p i | θ = 0) dp i , (35)where we use the notation f p ( p i | θ = 0) to indicate that the functionalform of f p has be derived assuming that θ = 0 . This result generalizes toall θ due to the symmetry of the signal structure. It can be shown that: Pr( x i = θ | q i = q ) = (cid:90) − q f p ( p i | θ = 0) dp i = (cid:90) − q f p ( p i | θ = 1) dp i ..