Transitions between regimes in the bar attendance model

TRANSITIONS BETWEEN REGIMES IN THEBAR ATTENDANCE MODEL

Heymann, D. †, Perazzo, R.P.J. §, and Schuschny, A.‡

† CEPAL (Comision Economica para America Latina, Un. Nations) andFacultad de Ciencias Economicas, Universidad de Buenos Aires,

Paraguay 1178, (1057) Buenos Aires, Argentinae-mail: [email protected]

§Instituto Tecnologico de Buenos Aires, Dep. de Investigacion y DesarrolloAvda. Eduardo Madero 399, Buenos Aires,Argentina

e-mail: [email protected]

‡ CEPAL (Comision Economica para America Latina, Un. Nations)e-mail: [email protected]

Key Words: genetic algorithms, learning, contagion, selforganization, economic shocks, bankruns

Abstract. We develop several extensions of the “Bar Attendance Model” (BAM) describing theself organization of a collection of agents that coordinate their attendance schedules to avoidingan accepted level of “crowding”. We study the stylized features of the transitions triggered by anexogenous perturbation considering the interplay of information publicly available and anotherthat only concerns a neighborhood of each agent. We show that selforganization is still possiblein the presence of the two types of information. We also show that the effects produced bythe local information can give rise to a contagion cascade that resembles a “panic” effect. Wefinally study an extension in which the maximum acceptable attendance is determined by thesame organization process. In one appendix we discuss the connection of the BAM with one inwhich agents play mixed strategies. In a second appendix we use this framework to construct aschematic model to represent salient features of a real bank run.

Mecanica Computacional Vol. XXIIIG.Buscaglia, E.Dari, O.Zamonsky (Eds.)

Bariloche, Argentina, November 2004

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1 Introduction

In recent times considerable attention has been given to the description of self-organizationprocesses in multi agent models. These are geared to explain how (and when) individual decisionsgive rise to collective effects that show up in the behavior of the system as a whole. Such modelshave a wide number of interesting applications in a variety of situations ranging from the routingof messages in an information network to the analysis of the performance of an ensemble ofeconomic agents.

The main question posed in that line of research is how systems composed of multipleagents acting in a decentralized fashion reach states of “macroscopic order” in which individualdecisions are (roughly) consistent with one another. A canonical instance of this line of researchis the “Bar Attendance Model” (BAM) proposed by [Arthur, W. B. (1994)], which provides asimple illustration of a system where coordination is achieved through the inductive learning ofagents. In that model, a (large) number of individuals have to decide whether or not to go toa bar at a certain date. The physical capacity of the shop is taken as given. The prospectivecustomers share the perception that the bar becomes unacceptably crowded when the numberof people who are present exceeds a critical value, which is smaller than the total population.Otherwise, if attendance is below that threshold, they derive utility from going to the bar.The model is such that agents choose an action individually, based on information that can beconsidered to be global (i.e. it refers to aggregate values of the system) and public (i.e. it isavailable to all the agents of the system). This information is assumed to be the total attendancein the previous days. In the model agents are assumed to perform inductive reasoning to predictthe number of customers at the shop in the next day. The system self-organizes in such away that the attendance converges to the maximum acceptable level (plus or minus randomfluctuations) [Johnson, N.F. et al (1998)].

A basic input to this model is the tolerable attendance level. This parameter guides theself-organization process. The (stationary) state that is reached by the system involves agentswho behave differently. One can easily realize that a solution in which all agents act in thesame fashion cannot be stable. Conversely it is easy to see that there exist a set of strategiesby which agents just saturate the capacity of the bar every day. These states are sustained by adiversity in the behavior of individuals, who choose to go to the shop in different days, so thatcrowding is always avoided. There is a large multiplicity of such equilibria, none of which hassalient features.

The purpose of the present paper is to study several extensions of the BAM frameworkin order to analyze the dynamics that prevails when the internal order of a coordinated, stablesystem suddenly breaks down. We aim at constructing schematic, self-organization models thatare able to account for features that are typical of such transitional situations.

The BAM can be extended in several ways. A simple extension, but one with interestfrom an economic point of view, derives from assuming that there is a lower bound of attendancebelow which an individual agent finds that going to the shop is not convenient to him. Therefore,over a certain range there is “strategic complementarity” between the choices of individuals.This opens the possibility of obtaining different “macroscopic equilibria”, since when the totalattendance is too low agents have an incentive to abstain from visiting the shop. We show that“full attendance” states are often attained through the decentralized decisions of a collection of

D. Heymann, R. Perazzo, A. Schuschny

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adaptive agents whose learning is represented by a genetic algorithm. However, for some initialconditions, the system may get trapped in states where the shop remains “empty” in some days:these are analogous to equilibria where agents develop the self fulfilling expectation that totalattendance will not be sufficient to reach the lower threshold.

The actions of groups of agents sometimes appear to undergo sudden changes, as ifthere was a collective shift to a new mode of behavior. Such changes can display features suchas those of panic, suggesting the existence of initiation effects. In connection with this, westudy a modified representation of the learning procedure of agents. In practice, individualsoften have access not only to system-wide data, but also to information about the behavior ofother agents who are in some way “close to them”, and they can condition their actions on thatinformation, particularly when they find that their own strategies do not generate reliable oracceptable outcomes 1. We develop an extension of the framework of the BAM that allows for theintroduction of such “local” or “contagion” information channel that may influence individualdecisions.

We analyze the robustness of the self-organization process in this modified decisionprocedure and we also discuss the effects of the interplay of both information channels in thepresence of an external shock which makes attendance “fundamentally” inconvenient for theagent. The simulations that we perform suggest that, in such situations, the decline in thequantity of customers is gradual at first, as individuals, based on their own specific learningprocedures, start deserting the shop. This can be interpreted as the diffusion of an “awareness”of a change in conditions. After a few time periods, the number of clients starts to fall morequickly, while contagion becomes widespread. A situation of “panic” builds up. If the exogenousshock is reversed soon, the system rapidly returns to the original, undisturbed state. However,in the case where the new value of the lower threshold is maintained, a full “panic wave” buildsup, with a drastic fall in attendance. If now the exogenous parameter returns to its initial level,reversing the “fundamental” shift, a gradual reorganization of the system takes place. Thiscan happen in two ways. If the “bad fundamental state” has lasted for a long time, the modelgenerates the possibility that all individuals stay away from the shop, leading to an autarchicequilibrium. Alternatively, if the crisis is short-lived, the system again converges to a stationarystate with a high level of attendance.

The third extension that we consider, is to let the “capacity” of the shop vary endoge-nously as a response to the behavior of customers. With this purpose, we analyze a co-adaptationprocess in which the number of available positions at the bar is changed according to the at-tendance observed in the past. This system can actually be regarded to evolve in two timescales, one associated to the mean attendance values and the other to the adaptation processof individual agents. In the standard version of “El Farol”, the “size of the shop” is kept fixed.Therefore the traditional model falls into one particular case of the framework that we presentbecause it can be regarded to correspond to an extreme “adiabatic” picture in which the indi-vidual adaptations of the customers take place in times that are negligible compared to thoserequired to produce changes in the values of the acceptable attendance level. We study therelaxation to the self-organized stationary state, and we also mimic the effects of an exogenous

1This type of argument has been used in recent models of “herd behavior”; cf. [Banerjee, A. V. (1992)],[Bikchandani, S. , Hirshleifer, D. And Welch, I. (1991)]


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shock. This can be regarded as a sudden change in the expectations of the public or those ofthe owner of the bar. We study the transition dynamics that correspond to the response to thisexternal shock in the context of coadaptation and of the interplay between the global and thecontagion (local) information channels.

In Appendix 1 we discuss the connections of the present treatments with one in whichagents play mixed strategies. In Appendix 2 consider the possible uses and limitations of themodel as a “metaphor” to help in the analysis of the structure of events like banking crises.

2 Coordination through public information

2.1 The multiagent model

In this section we present a modification to the BAM to consider a situation in which it is commonbelief that the attendance should not exceed a given value for customers to be “comfortable”and, at the same time, it should not fall below a lower bound to make attendance convenient.The adaptation procedure is based upon the past attendance recorded at the bar, an informationthat is assumed to be available to all the agents of the system. The individual decisions makethus use only of public, global (aggregate) information. We model the relaxation to equilibriumusing genetic algorithms [Goldberg, D.E. (1989)].

The learning procedure acts by improving strategies taking into account the perfor-mance during a set of consecutive visits to the bar. Each agent has assigned a population ofNp “attendance plans” for the subsequent “week” of Nd days. Each strategy is encoded in a“genome” that specifies each day whether the agent goes to the bar or not. All the initial popu-lations of genomes are selected at random, and are updated by selection, crossover and mutation[Goldberg, D.E. (1989)], so that less successful strategies are progressively eliminated.

The formal steps of the algorithm are the following:

1. Each agent possesses Np attendance strategies. Each strategy consist of a chain of Nd bitswhere Nd stands for the number of days of the “week”. The “bit” p#t

k, for the t-th “day”of the p-th strategy, of the k-th agent, takes the value +1 (−1) in case she chooses to goto the bar (not to go) in that day 2.

2. Once each agent has selected one strategy (chosen on the basis of the data of the precedingweek), she implements daily the action of either going or not going to the bar. The barrecords the daily level of attendance. Let c be the current strategy of each agent. Thefraction Dt of the public who has attended the bar in the t-th day is:

Dt =1

Nag

Nag∑

k=1

θ(c#tk) (1)

where θ(x) = 1 if x > 0 and θ(x) = 0 otherwise.2The model only involves pure strategies, i.e. it does not consider the case in which agents attend the bar

with some, possibly time varying probability. The connection with the case of mixed strategies is discussed inAppendix 2


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3. Each agent calculates the “utility” obtained that day by comparing the attendance Dt withthe two bounds So and S1. The agents who have made a correct guess of the attendancelevel add a positive contribution to their weekly utility, Up

k (t + 1) = Upk (t) + (1 + C)

regardless of the fact that they have attended the bar or not. The agents who havemade a wrong guess of the attendance level bear a cost that lowers their weekly utility,Up

k (t + 1) = Upk (t) + (1−C). The cost is the same whether they have attended the bar or

not. The utility is recorded daily and during the period of Nd “days” of the “week”.

4. When a “week” finishes, the agent compares the performance of all her strategies 3. Thefitness of each strategy is determined by the weekly utility that could have been achievedwith the corresponding attendance policy. At the end of the week all the available strategiesare ranked according to their utility (fitness).

5. The population of strategies associated with each agent is updated according to the usualprocedure of the genetic algorithm. We have used an “elitist” version of the geneticalgorithm: those genomes ranked within the upper 50% survive to the next generation,while those in the lower 50% are replaced by new ones obtained by crossover of the survivingstrategies. All the genomes (strategies) are next subject to random mutations, i.e. a bit ischanged from 1 to 0 or vice versa with probability pmut. According to the common practicepmut is chosen to be very small (typically pmut ∼ .001) because otherwise the evolutionprocess does not converge to any stable state because successful strategies do not survive.

6. The fittest strategy is selected as the “current” attendance schedule for the next “week”and the whole procedure is iterated.

2.2 Self-organization

In the usual “El Farol” settings (with only an upper bound of tolerable attendance) the outcomeof processing the public, global information gives rise to an asymptotic coordinated state inwhich the agents self-organize to go to the bar as frequently as tolerated by the upper crowdingthreshold S1 (see figure 1). Its stability is due to the internal diversity of the system: the agentstend to use complementary strategies that avoid crowding. By contrast, if all the agents chosethe same schedules the resulting attendance level would fall outside the acceptable range. As aconsequence the system evolves a “population” of strategies by which different agents go to thebar in different days. Such configurations are dynamically stable. All agents continue to explorenew strategies that correspond to random mutations of their current plans and the system only“visits” configurations that are in the close neighborhood of the state that has been reached.

The case that we consider in the present paper involving an upper and a lower bound ofacceptable attendance, does not give rise to a unique coordinated state. In addition to the “fullattendance” case it may also happen that the ensemble of agents gets trapped in a configurationin which, during certain days, the bar receives no customers (figure 1). After few time steps ofevolution and simply due to the random fluctuations of the adaptation process, the attendance

3It is assumed that each individual change of attendance strategy causes a negligible change in the totalattendance of each day, otherwise the coadaptation process requires to test each possible combination of individualstrategies .e. a total of N

Npag combined situations


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Figure 1: Simulations without local interactions with 256 agents, C = −0.01, pmut = 0.005, So = 0.2,S1 = 0.8, Nd = 10 and Np = 20. The upper panel shows an optimum convergence while the lower oneshows how the selforganization results a in sub-optimal state. Each day 10% of the agents update theirstrategies through the genetic algorithm.

at one particular day turns out to be below the lowest threshold So. When this happens all theclients that choose to go to the bar on those days get a negative contribution to their utility.The choice of attending the bar is thus progressively discarded in the following adaptation stepsand therefore it becomes a common belief that the bar will be empty those days. The occurrenceof a configuration of this sort is an example of path dependence in the self organization process,because it becomes stable in an early stage of adaptation, conditioning all the future evolutionof the system.

When the system is confined into one of these configurations, the adaptation dynamicsallows a transition to the “macroscopic” ordered state of full attendance. Such transition is how-ever highly improbable because it requires that a large enough fraction of agents simultaneouslycoincide in changing their individually chosen actions for a particular day. In statistical mechan-ics, this situation is usually referred to as confination by “entropic barriers” [Palmer, R. (1989)].

In either of the cases presented above (“full attendance” or “empty days”) the system“visits” the neighborhood of a Nash equilibrium. No agent can improve her utility changingunilaterally her attendance strategy. There is a large multiplicity of equilibria: any two con-figurations in which a client of the bar exchanges her position with one who has not attended,are equivalent. One can check that the cumulative attendance of the agents for the whole weekfollows a binomial distribution. In the stationary state the (time) average of the fraction of theclients that go to the bar S1, is the same as the probability that a given agent can attend thebar. Thus the probability to reach a weekly cumulative attendance of Aw days is


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P (Aw) =

(Nd

Aw

)SAw

1 (1− S1)Nd−Aw (2)

3 Coordination with local contagion

3.1 The model for local contagion

This section incorporates into the model the effect of “local” information, in the sense that itreflects the situation of a small set of customers belonging to the neighborhood of any givenagent. This can be considered a “contagion” mechanism. Here every agent can choose betweentwo alternatives: (i) to continue with her individual attendance plan or (ii) to imitate thestrategy followed on the average by her nearest neighbors. The neighborhood is determined byplacing the agents in an square grid with periodic boundary conditions. The label k of eachagent is thus replaced by the row and column indices (i, j).

Each agent is allowed to abandon her current strategy whenever she finds that herperformance is not satisfactory. It may be noticed that, owing to this fact, one has to distinguishbetween the current action cat

(i,j) taken by the agent located in the site (i, j), on the t-th day,and her current plan for the same day c#t

(i,j). To decide a shift away from her plan, the agentdetermines the daily outcome obtained with her strategy and compares it with the one obtainedif she had imitated her nearest neighbors. For this purpose each agent calculates the “localmean action” (field):

h(i,j) =14

(cat

(i,j+1) + cat(i,j−1) + cat

(i+1,j) + cat(i−1,j)

)(3)

Once this is known, the agent can determine if she will follow her neighbors 4, although thealignment of each agent with the average local attendance policy is not entirely deterministic.We assume that there is some noise in the agent’s decision that is represented by a “thermalagitation” characterized by a temperature-like control parameter T = 1/β. This is introducedthrough a Glauber dynamics [Glauber, R. J. (1963)] in which the probability of either imitatingor ignoring the local policy of her neighborhood is 5:

4The arrangement of agents in a grid is borrowed from the Ising Model of ferromagnetism [Huang, K. (1987)].The option of either going to the bar or staying at home corresponds to the two possible orientations of the“magnetic” dipole (agent). The “imitation” process is similar to the alignment of the dipole moment with thelocal magnetic domain (neighborhood) acted upon by a ferromagnetic interaction.

5This corresponds to the physical picture of an Ising ferromagnet in contact with a thermal bath at finitetemperature. The thermal fluctuations cause that the spin of each site of the lattice updates its orientationthrough a stochastic dynamics with a probability that depends on the local field given by 4. In the absence ofthese fluctuations (T = 0), the transition probabilities are:

• If the local field h(i,j) 6= 0 then s(i,j)(t + δt) =sign(h(i,j)) with probability 1.

• If h(i,j) = 0 then s(i,j)(t + δt) = ±1 with probability 12.

The presence of a non vanishing temperature breaks down the degeneracy implicit in the last alternative.Thelowest energy state of a system of spins with attractive (ferromagnetic) interactions between nearest neighborshas all the spins aligned in the same direction. There are two (degenerate) ground states, one with si = +1∀iand the other with si = −1∀i. These states display a ferromagnetic ordering that is characterized by a global


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P (action = ±1) =1

1 + e∓2βh(i,j)(4)

Thus, each agent computes every day the utility that would have been obtained following thelocal mean action . If this is greater than the one computed with her current strategy, the agentchanges her attendance plan for the next day and uses (with the above probability) the actionprescribed by the local field. Otherwise, the agent sticks to her individual plan, using the nextbit of information of her current strategy.

3.2 Self-organization with local effects

The question that we address now is the robustness of the self-organization process. The resultthat is obtained is that this is indeed preserved in the presence of contagion effects. Thereare however several important modifications. The organization process involves a larger levelof fluctuations which are to be attributed not only to the random search performed in theindividual optimization, but also to the occasional contagion that is induced by the Glauberdynamics. The second relevant difference is discussed in the next section and relates to theabrupt changes that take place in the transitional dynamics involved in the change of regime.

In the first 300 time steps of figure 2 we show the evolution of the system when bothinformation channels are active. We have fixed the thermal fluctuations of the Glauber dynamicsat T = 0.1. For this value of T , in stationary conditions, approximately 15% of the agentsconstantly switch strategies, imitating their neighbors. From the genetic algorithm point ofview, that effect is analogous to an upward revision of the value of pmut. However, adaptationis still possible because the fluctuations are not large enough to destroy the information contentof the individual selection process. For even larger values of T all agents follow an increasinglyrandom attendance policy that amounts to deciding to go to the bar by tossing a coin. In thissituation learning becomes impossible and the system does not self-organize.

With moderate values of T , the presence of local contagion does not prevent the sys-tem from self-organizing. The learning procedure partially compensates the new uncertaintiesinduced by fluctuations by promoting attendance strategies with a somewhat higher attendancethan the highest tolerable level. The presence of larger fluctuations has on the other hand an im-portant consequence, namely the disappearance of metastable states: the buildup of the entropicbarriers (mentioned in the preceding section) is prevented because in the initial adaptation stepsthe dynamics forces the system to explore a wider space of possible strategies.

4 Transitions between regimes

The relevance of the interplay of both coordination mechanisms becomes noticeable in the eventof an external shock representing a sudden change in the global conditions. An exogenous

positive (large) magnetization. The fact that these states are also the attractors of the noiseless dynamics resultsfrom the interaction of the spins of the system. The ferromagnetic ordering is a cooperative phenomenon in thesense that the global magnetization scales with the size (number of spins) of the system. The evolution of thesystem correspond to a gradient flow along the directions that reduce the total energy of the system which ismonotonically reduced at every time step [Amit, D. J. (1989)]).


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Figure 2: The selforganization process and the effects of a crisis are shown when both informationchannels are active and with T = 0.1 The lower panel gives the effect of contagion i.e. the fraction of theagents that act according to the local field. The parameters are the same as those of the previous figure.Notice the “awareness”, “panic” and “recovery” periods

“crisis” can be induced by increasing the lower threshold level So to a value greater or equal toS1. This compels the whole set of agents to revise their decisions. The crisis can be finished byreturning So to its original value some time later.

In fig 3 we show the evolution during the crisis in the absence of contagion. After theshock, the behavior of the population does not switch abruptly; rather, agents gradually adapttheir strategies to the new situation as they learn that it is convenient to avoid going to the bar.The rate of adjustment of the average attendance is governed by the mutation probability of thegenetic algorithm. The evolution has an exponential time constant given by 1/pmut which is longcompared with the length of the business “week”. If the shift in S0 persists, agents eventuallylearn to desert the bar.

The main effect of the local contagion is to enhance collective effects, giving rise torapid changes in the aggregate behavior of the agents. Once the “crisis” begins, the collectiveperformance of the clients of the bar displays two features.

First, agents who were imitating their neighbors no longer do so, and choose to usetheir own strategies. The intuitive reason for this is that on the average agents make morewrong decisions and get poor utilities; therefore, contagion is hindered. For the settings chosenfor this simulations, this process may last for a period of the order of a couple of “weeks”.

Second, as the threshold S0 is maintained at its new (higher) level, the agents increas-ingly learn to avoid going to the bar. A moment is then reached in which a critical fractionof individuals acting in that way induce an avalanche in which all agents massively imitate thestrategy of not going to the bar, because acting in this way produces a good utility to theirneighbor. This has all the features of a panic. Thus, the period of “awareness” (based on the


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Figure 3: Evolution during a “exogenous crisis” without contagion. All the parameters are the same asbefore.

agents’s own learning) leads at some point to a “contagion cascade” or panic, with an abruptdecline in the attendance level (see figure 2). This is triggered only if a large enough fraction ofagents have decided “by themselves” to avoid going to the bar. When the panic emerges, thetransition takes place in a very short time 6

If the “crisis” is short, the coordinated state of high attendance remains essentiallyunaffected because the agents do not stop considering previously successful attendance strategies.Recovery in this case is also quite abrupt requiring little “re-training” of the agents. The reasonis that after few days in which the lower threshold has regained its previous value, most of theagents have not yet lost memory of old strategies and find that they are again suitable, startingto massively use them again “as if nothing had happened”. On the other hand, if the crisis islong enough to “forget” the once successful strategies, the new equilibrium corresponds to thebar with no clients 7.

In Appendix 2 we show how some of these qualitative features can be found in therecords of a bank run that took place in Argentina as a consequence of the devaluation of theMexican peso.

After a longer time the only successful overall strategy that survives is never goingto the bar. In this stage both (genetic) learning and contagion mechanisms act coherently togenerate the same behavior. In the model, once the attendance “collapses”, it is highly difficultto reconstruct it. The system finds itself trapped again between high entropic barriers. A veryimprobable circumstance has to occur to escape this situation, namely that many agents decideby chance and at the same time to go to the bar again. If this happens that strategy wouldproduce a high utility and stabilize again a regular attendance. As already observed, suchbarriers can only be overcome allowing a high level of fluctuations (setting for instance T to ahigh value), so as to induce the system to visit configurations which are far from the currentstate.

6Within the physical picture of the Ising model this corresponds to a ferromagnetic phase transition to a statewith a positive magnetization. In the thermodynamic limit that would correspond to an infinite number of agentsor when T = 0, the transition is instantaneous.

7Once the crisis is established and before zero attendance is reached, its value stays for a long time at a valuethat is close to 1− S1 (.2 in the case shown). This is so because all successful strategies required that all clients,at least 20% of the time, should not go to the bar. Thus at least 20% of the time the agents would follow theirstrategy (and mistakenly go to the bar) instead of imitating the neighbors. This effect is reinforced by the factthat the neighborhood strategy is followed only if it is strictly better that one’s own.


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5 A self-consistent coordination process.

The BAM and the modifications that we have considered in the previous sections contain externalparameters that guide the self-organization process: the threshold levels So and S1. If theseparameters are seen as the outcome of the actions taken by the same agents, the models thatwe have considered in the preceding sections, correspond to an extreme (“adiabatic”) limit inwhich the change of those (aggregate) values is infinitely slow, or to put it into different terms,all agents adapt themselves in negligible times compared to the one required to change thereference parameters.

In the present section we consider a situation in which the fraction S1 is time dependent.Its value is determined by previous attendance and the expectations of, say, the “owner” of thebar. The situation may be assimilated to a bar with an adjustable number of places. Everyweek the owner has to adjust the number of available chairs based upon previous attendanceand her own expectations. The clients evaluate their attendance policies using their individualexperience during the following days while the bar “owner” behaves adaptatively. We write:

S1(t) = λS1(t− 1) + (1− λ)Davg(t− 1) (5)

In 5 the time variable t counts the number of “weeks” (of Nd “days” each). The value S1(t) forthe t-th week is assumed to be the weighted average of S1(t − 1) and the average attendanceDavg(t − 1), recorded in the previous week. If the weighting parameter λ (0 < λ < 1) is setequal to 1 the reference attendance level becomes independent of the response of the public andwe fall into the models considered in the previous sections. If instead λ = 0, the capacity of thebar fully adjusts to the observed level of attendance. We therefore limit our discussions to thecase of small values of λ.

The first question that we address to is whether self-organization is robust under themore complex dynamics involved in this model. The time scale associated to the adaptationof the agents can be changed. Let Φ the fraction of the agents that are picked up at randomevery day to update their attendance policies. Φ is the probability per unit time that an agentupdates her attendance policy. The individual mean adaptation time is 1/Φ. We can introducethe “adiabaticity parameter” A = NdΦ. A ' 1 corresponds to the case in which (on the average)the whole ensemble of agents update their strategies during one week of Nd days. A regime ofrapid adaptation of agents corresponds to A > 1.

In the preceding analysis the actual value of the cost of a wrong decision and the rewardfor a good one were irrelevant. Those models were insensitive to the values of both because theywere irrelevant in the selection of good attendance schedules. This is not the case in the presentsituation because the self-organization process triggers a dynamics in which individual trials anderrors translate into costs or benefits that depend on how much time the agent spends exploringabove or below the (moving) reference value of S1(t).

An example of the self-organization is shown in the first few “weeks” of the evolutionshown in figures 4 and 5. The difference between the two experiments is in the value of Φ (Φ = .2and .02 respectively). The initial value S1(0) is the same in both cases and it is set equal to avery low value. The lower threshold has been set equal to 0. Initial attendance strategies aredrawn at random with a mean attendance of .5. A faster adaptation causes an initial dip thatcorresponds to the reaction of all the agents to a slowly growing value of S1(t). This reaction is


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Figure 4: Selfconsistent organization following eq.5 with 400 agents, pmut = .005;S1(t = 0) = .1 andλ = .3. The reward for a correct decision is 10 times larger than the cost for a wrong one. Each day 20%of the agents update their strategies. A crisis is produced in the day 400 (t = 40). Panel (a) withoutcontagion; panel (b) with contagion; panel (c) number of agents acting according to the local field. Noticethe gradual recovery as compared with the model presented in the preceeding section. Panels at the right:magnification of the first few days of the organization process. The effect of contagion is to increase noiseand to to produce a dip in the daily attendance (as compared with the upper panel) to meet the initiallow value of S1(0).

clearly absent in the case of Φ = .02 (or A = .2). Self-consistency is nevertheless rapidly reached(t ' 10 weeks) in either case. The effect of contagion is shown in the lower panels and reproducethe same qualitative features that have been pointed out in the previous models.

The second question that we address to, is the stability of the asymptotic state againstexternal shocks, as in the previous section. An external (negative) shock could correspond to(say) an accident that forces the bar to remain closed during some time and to return later tothe usual operating routine. This is simulated by changing (exogenously) the value of S1 holdingits value fixed in a very low value during a short period and later allowing it to change throughthe self-consistent evolution process. This happens (in both figures) from t = 40 to t = 60.

Contagion is seen to play the same role as in the examples of the previous section,giving rise to an abrupt change during the transition between regimes. With a higher value of Φthe “awareness” and “panic” periods can clearly be identified. An interesting additional resultis that the “recovery” period turns out to be gradual; this can be interpreted as a progressive


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Figure 5: Selfconsistent organization following eq.5. The parameters and conventions are the same as inthe preceeding figure.The fraction of the agents that update their strategies every day that is in this case2% instead of 20%. Notice that the crisis is not noticeable if contagion is excluded except for an increasein the fluctuations. In this case contagion (lower panel) produces a dramatic transition


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recovery of “confidence” of the customers in the services offered by the bar. The speed of recoveryis governed by the ratio of cost to reward. This is of course intuitive because customers tend toexplore wrong attendance schedules if the cost is not high. Fluctuations that act through thecontagion mechanism are also important in this recovery pattern. It is worth remarking that aslower adaptation causes a more abrupt transition to “panic”. A lower value of Φ enhances theeffects of contagion because this channel becomes the only possibility of updating the attendancestrategies thus improving utility.

6 Conclusions

We have considered several extensions of the BAM in which individual agents mutually adapttheir individual actions to reach a coordinated, macroscopically ordered configuration. In thesemodels agents learn to adjust their attendance schedules to avoid exceeding and staying belowcertain limits of attendance. The multiagent system displays several possible collective orderings.One is an equilibrium in which all agents coordinate their individual attendance schedules tosaturate the maximum accepted capacity of the bar. Another is a kind of “autarchic equilibrium”in which agents refuse to go to the shop. There is also a family of suboptimal, metastableconfigurations in which the bar is deserted in some specific isolated days. All these states can beconnected to the full attendance configuration, but moving from one to the other would requirethe highly improbable situation of a simultaneous change of behavior of a large collection ofagents. This is a qualitative feature that can be recognized in a wide number of everydaysituations in which “overcoming the barriers” amount in practice a global reorganization of thesystem.

We have also considered the use of information that concerns the immediate neigh-borhood of each agent. This provides a “local” or “private” perspective in the decisions of theagents. We found that the self-organization mechanism based only in public information (i.e.the aggregate attendance level) can be considered robust in the sense that addition of contagioneffects does not modify the end result of the adaptation process. However the presence of thetwo alternative information channels was found to play an essential role in shaping the transi-tions between regimes. The adaptation in the original version of the BAM can only give riseto gradual changes. The interplay between the two information channels is instead a way togenerate abrupt changes, which are commonly observed in real circumstances. The compositeeffect of the two competing mechanisms is to turn the ensemble of multiple agents into whatis known as an excitable system. These are complex systems found in a wide variety of fieldsthat encompass social, biological and physical situations displaying a special kind of boundedhomeostasis. They are stable against small perturbations, but if the shock is larger than a giventhreshold the system only turns back to its (equilibrium) original settings after performing a longexcursion away from that situation. We found that such excursion (or transitional dynamics) inthe present framework clearly displays differentiated moments that may be taken to correspondto “awareness”, “panic” and “recovery”.

It is important to remark that the same learning (adaptation) rules are capable ofdescribing the widely different situations that correspond to the self-organization leading to a(Nash) equilibrium on the one hand, and the out-of-equilibrium transition that follows exoge-


14

nous shocks of various intensities on the other. It is not necessary to make any other particularassumption about the behavior of the agents with reference to any particular global situationsthat may arise either before, during or after the crisis; neither is necessary to tailor the modelfor a specific duration or intensity of the shock. The rules that govern the adaptation andthe processing of information are the same before, during and after the transition episode andconstitute an essential part of the multiagent system.

Both the self-organization process and the general features of the transitional dynamicsprove to be robust when considering a self-consistent process. In this case the crowding limitsintroduced in the BAM are determined by the same organization process and are not treatedas external control parameters of the model. However new information is required, namelythe relative value of the utilities lost or gained by staying respectively above or below the(dynamically determined) crowding limit. The process is governed actually by the balancebetween the time in which the agent explores states above or below the (moving) referencecrowding limit. An important feature appears regarding the transitional dynamics and concernsthe recovery process after an external shock. That recovery is abrupt in the previous modelsand turns out to be gradual when self-consistency is considered. This can be interpreted as thegradual process by which the bar regains the “confidence” of customers.

In Appendix 2 we use the model to understand on qualitative grounds, a real situationof the transitional dynamics generated during the bank run that took place in Argentina as aconsequence of the devaluation of the Mexican peso. It is abundantly clear that the metaphorof the BAM should not be stretched: the model can certainly not be taken to describe theworkings of a financial system. However the ability of the model in reproducing stylized featuresof the empirical quantitative data of the transitional regime, and the enormous complexity of theinteractions that are involved in the real world dynamics, suggests that the transitions betweenregimes may be roughly described by simple schemes of self-organization in spite of the fact thatthe real life, detailed internal mechanisms may be very intricate.

In that application the emergent collective pattern of behavior can be explained witha simple structure that involves the interplay of only few degrees of freedom concerning public(global) information and the contagion of private attitudes. If considered as arising from alearning or adaptive mechanism, abrupt changes can emerge in a natural way by including thissecond information channel. On the other hand the interpretation of the gradual recovery afterthe crisis may have different interpretations; as either a “long term memory” of events thathappened before the crisis or are as the “self-consistent”, gradual recovery of confidence in thebanking system.


15

Appendix 1The case of mixed strategies8.

The dynamics of an ensemble of agents with mixed strategies have been extensivelystudied for the BAM and a modified version of it called the “Minority Game” (MG)(see refs.[Challet, D. and Zhang Y-C (1997)], [Burgos, E.; et.al] and references cited therein). Withinthe MG all members of the population have to make a binary choice (either go or not go tothe bar) and the minority group is the winner. In all instances of the MG the population isalways partitioned into two groups and only one (the minority) is the winner. This is not thecase within the BAM in which both groups can win if the group that goes to the bar is exactlyequal to the maximum tolerable attendance.

The MG has usually been investigated assuming a probabilistic relaxation dynamicsin which each agent updates her own attendance probability by keeping track of positive andnegative utilities gathered in previous rounds of the game. The result of these works prove thata stable equilibrium cannot be reached with a common mixed strategy for the whole ensembleof agents. That is, a system in which all the agents have the same probability p of going to thebar is in a Nash equilibrium. The agents tend to change their attendance probability to improvetheir individual utilities and the system thus relaxes to a stationary distribution of attendanceprobabilities in which the whole population is polarized into a fraction that always go to the bar(p ∼ 1) and the complementary group of agents who never go (p ∼ 0). Such configurations canvery well be approximated by a set of players with pure strategies. These results are consistentwith the way in which strategies are represented in the treatment of the BAM made in the text.

We now discuss analytically the determination of an equilibrium with mixed strategiesin the BAM. Let N be the number of agents and m = NS1 the maximum acceptable level ofattendance (m 6= N and m 6= 0) and p the probability of going to the bar at a given day. It isclear that p = 1 or p = 0 can not be an optimal strategy for every agent. Assume next that allagents except one (that we label N) have the common mixed strategy to attend the bar withprobability p. The probability that the N−th agent finds a party of less than m agents is:

S(N, m, p) =m−1∑

i=0

(N − 1

i

)pi(1− p)N−1−i (6)

It is simple to check that S(N, m, p) is a continuous, strictly monotonic, decreasing function ofp in the whole interval 0 ≤ p ≤ 1 and

S(N, m, 0) = 1 (7)S(N, m, 1) = 0 (8)

S(N,m1, p) < S(N, m2, p), for 0 < p < 1 and m1 < m2 (9)

As done in the paper we consider that the utility that is obtained for attending thebar if there are less than m clients is equal to the cost (normalized to 1) incurred by going

8The authors acknowledge helpful discussions with H. Ceva and E. Burgos concerning the points analyzed inthis appendix


16

if the number of clients is larger than m. It is easy to see that this assumption can be madewithout loss of generality and does not affect the present proof. If the N−th single differentagent chooses a mixed strategy with probability of attendance pN , she will have an expectedutility of9:

UN = pNS(N, m, p) + (1− pN )(1− S(N,m, p)− pN (1− S(N,m, p))− (1− pN )S(N,m, p)= (2pN − 1)(2S(N, m, p)− 1) (10)

The N−th agent has to choose pN in such a way as to maximize her utility UN given theattendance probabilities of the remaining N − 1 players. Since UN is linear in pN , its maximumis attained for pN = 0 if S(N,m, p) < 1/2 or for pN = 1 if S(N, m, p) > 1/2. This resultis completely intuitive: if the probability of joining a group of less than m agents is greaterthan 1/2 the N−th agent will maximize her utility by going every day to the bar, or doing theopposite if that probability is less than 1/2. Neither pN = 0 nor pN = 1 can be equal to theattendance probability of the remaining agents.

The only possible symmetric Nash equilibrium is when S(N, m, p) = 1/2. This equationhas always one root p = p∗ and pN can be taken to be equal to p∗ because UN = 0 and pN iscompletely indeterminate10. This corresponds to the case in which all agents have an expectedutility11 that is equal to 0.

However the equilibrium is unstable. To see this we assume that a small perturbationδp is introduced to the common strategy of the N−1 players. We we now search for a maximumof UN we obtain:

δpN

δp

)p=pN=p∗

= −(2p∗ − 1)S′(N,m, p∗)2S(N,m, p∗)− 1

(11)

The derivative S′ in the numerator of 11 is S′ 6= 0 ∀p because S(N, m, p) is strictly monotonic.However the denominator vanishes because at the equilibrium S(N,m, p∗) = 1/2. The ratio(δp∗/δp) therefore diverges thus indicating the instability of the equilibrium configuration. Thisis easy to understand since for any infinitesimal change δp the value of S varies from 1/2 andthe value for pN has to undergo a finite change from p∗ to 0 or 1 (depending on the sign of δp).

We thus conclude that there is not a common mixed strategy for all the populationof N agents leading to a stable equilibrium configuration. From an intuitive point of view,a distribution of individual attendance probabilities sharply peaked at some value of p (sayp ∼ S1) leads to the occurrence of attendance fractions which are close to S1 but not equal to

9Under the same assumptions, if the condition of strict minority is imposed as in the MG, the expected utilityis UN = pN (2S(N, m, p) − 1) + (pN − 1)(2S(N, m + 1, p) − 1). For large N , S(N, m, p) ' S(N, m + 1, p) exceptfor terms of O(1/N) and equation 10 is recovered.

10It turns out that the root is p∗ ' m/N = S1. This means that the average attendance (the average takenover of all the agents and over many instances of the game) is equal to the capacity of the bar.

11For the case in which strict minority is imposed, the expected utility UN = pN (2S(N, m, p) − 1) + (pN −1)(2S(N, m+1, p)−1) is strictly negative due to the property 9. This can not be an equilibrium situation becausethe N − th agent is then forced to change her attendance probability. The strict minority condition thereforeleads to a case in which there is no symmetric Nash equilibrium with mixed strategies.


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this value. Therefore the symmetric mixed strategy is less good than a set of strategies leadingto the formation of groups with exactly m agents at each day. This can be obtained with adensity distribution of attendance strategies having two sharp peaks at p ∼ 0 and p ∼ 1 withthe proper relative weights. It seems a noticeable results that in spite of the fact that there isno exchange of information between the agents, these self organize into two distinct parties withcomplementary attendance behaviors each day.


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Appendix 2An application to bank runs

Banking panics have been interpreted in two ways (c.f. [Diamond, D. et.al (1983)],[Jacklin, C. and Bhattacharya (1988)], [D’Amato, et. al. (1997)], [Temzelides, T. (1994)]): asevents based on the diffusion of information about “economic fundamentals”, which inducesa revision of beliefs about the prospects of the banks, or as pure coordination phenomena ina system with multiple potential equilibria, when some “extraneous” shock causes agents tochange the state on which they focus their expectations. Both explanations need not be mutu-ally exclusive. It may happen, for example, that some macroeconomic news, which modify theperceived return on deposits, induce a response which in turn triggers a self reinforcing avalancheof withdrawals, thus amplifying or accelerating the effect of the impulse. In this case, the panicis not simply generated by a “sunspot” (since without the news, agents would disregard themultiplicity of equilibria) but it does have a component whereby the observation that people aredrawing their assets from the banks leads to a cascade of more withdrawals.

Although the environment that we depict is highly schematic, it is tempting to use itto consider “runs” for which an actual quantitative record is available such as bank deposits.As we see below, the model can capture stylized features of the collective behavior that is foundin a crisis episode and its aftermath.

At a certain level, one can find intuitive analogies between the action of “going to thebar” in the BAM, and that of making a deposit in a bank. It seems clear that people spacetheir visits to banks in such a way that, in a stationary state, the aggregate “attendance level”remains more or less constant while the identity of the individuals who decide to go changesfrom day to day. This behavior could be rationalized, by analogy with the BAM, by defininga “maximum attendance level” beyond which, given the physical capacity of a bank, customerswould incur some “congestion costs”. However, that somewhat strained analogy would providea weak basis for a representation of the behavior of financial activities. In any case, our interestin the use of this self-organization model in a financial context has little to do with the way inwhich customers routinely choose the days in which they will be physically present at banks.Rather, what we want to consider is how the system reacts to a “fundamental shock” that createsdoubts about the solidity of banks. In this regard, the existence of strategic complementarityin the decisions to place resources in banks is a standard feature of the relevant literature.

On the other hand, it is not awkward to assume, as a simplifying hypothesis, that thereis a lower bound in the volume of bank deposits such that, if the aggregate value goes below thatbound, prospective depositors suffer losses due to the inability to fully recover their funds; thatlower limit would be a function of parameters such as the profitability of banks. Thus, “badnews” would correspond to an increase in that lower bound of deposits such that an individualdeposit is profitable. In the limit, the value of the bound would be so high that “well informed”agents would desert the bank. This is the scenario that we want to analyze. As stressed before,the self-organization in the timing of attendance (and thus, the nature of the upper bound on“attendance”) is of very secondary importance for our purposes: nothing would be changed ifwe just predetermined the days in which a given agent would go to the bank in case he wishedto make a deposit; the issue is whether he would want to “trust” the bank at all.

Our experiment consists in shocking the system so that making deposits in the bank


19

is “fundamentally” inconvenient, and investigating the dynamics of the process through whichagents learn to abstain from making deposits, and how (and if) they again “return” to the bankif the fundamental parameter goes back to its initial value. Within this picture the “attendanceschedule” that we refer to in the text, reflects the planned deposit schedule for the next Nd

“market days” of each agent, i.e. it would reflect the planned decision of making a depositin the bank or not doing it. It is implicitly assumed that at the end of each “day” the agentgets back her deposit, collects the corresponding interests and continues with her “investmentschedule” for the following day. The fact that the level fluctuates reflects the random eventsthat occasionally induce one agent to retain the money or make a deposit. When an agentchanges a 0 into a 1 in her attendance schedule, she switches its programmed action so as tomake a deposit that day, and the opposite when the change is from 1 to 0. In either case thereis produced a (small) change in the aggregate value of the deposits for that day.

The origin of the “crisis” may be attributed to the arrival of an exogenous informationthat casts doubts on the “fundamental” reliability of the financial system; the end of the dis-turbance can be associated to news (e.g. of government actions) which in some way determinean enhancement of the ability of the banks to honor deposit contracts.

A severe banking crisis took place in Argentina in the aftermath of the Mexican de-valuation by the end of 1994. In figure 6(a) we plot the total daily level of deposits (in currentaccounts, savings and time deposits, both in Argentine pesos and foreign currencies) during theperiod between November 1994 and October 1995. The picture also shows values generated withthe present model. The free parameters have been calibrated as explained in the figure caption.The vertical scale has been normalized in such a way that the upper level corresponds to thevolume of deposits at the beginning of the crisis and the lowest to the amount of funds at thebanks on the day when the government announced that deposits were guaranteed. Each day ofthe time series is equivalent to a computer “day” of the attendance model.

The phases of “normal performance”, “gradual withdrawal of deposits”, “panic” and“recovery” can be clearly identified in the figure. The simulated series describes quite well thefirst stages of the crisis, but not the recovery. This happens because agents have not forgotcompletely strategies that were successful prior to the crisis and start using them “as if nothinghas happened”. This can therefore be attributed to the fact that no memory is built into theartificial system to account for the asymmetry in the data which displays a sudden run and aslower return of deposits to the banks.

In order to include this kind of “long term memory”, we have elaborated a slightlymodified algorithm. We first choose a situation in which new initial strategies are drawn, lettingthe genetic algorithm start again with its adaptive work in a “fundamentally new” state of thesystem. This modification would correspond to the introduction of “new rules” of behavior.For the purpose of the simulation, this procedure was triggered whenever more than half of theagents act according to contagion. The second variant that we introduced is to assume that in acertain number of cases (say, 40 %), agents are “cautious”, i.e. they decline to follow contagionwhen the action suggested by the neighborhood is to trust the banking system and make adeposit. The evolution obtained by introducing these changes is presented in figure 6(b): it canbe observed that there is a reasonably good fit, both in the period where deposits declined andin the upswing.


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Figure 6: Normalized deposits during the Argentine banking crisis. The numerical experiment has thefollowing parameters: Nag = 256, Np = 20, T = 0.001, the mutation probability of the genetic algorithmis 0.005. The crisis begins in t = 30 (December 20th, 1994) and finishes in t = 120. We overlapped thenumerical experiment and the series of the total daily level of deposits. In the panel (b) we have includedin the experiment two threshold effects: 1)if the fraction of agents acting by contagion is greater than 50%, a new population of “predictors” is drawn again, and 2) 60 % of the agents are inhibited to depositwhen induced by contagion.

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Transitions between regimes in the bar attendance model

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