Table Of ContentEPJ manuscript No.
(will be inserted by the editor)
9
9
9
1
Competition, efficiency and collective behavior in the “El Farol”
n
a bar model
J
2
M. A. R. de Cara, O. Pla, and F. Guinea
1
e-mail: [email protected]
]
h Institutode Ciencia deMateriales
c Consejo Superior deInvestigaciones Cient´ıficas
e 28049 Madrid SPAIN
m
- February 7, 2008
t
a
t
s Abstract. TheElFarolbarmodel,proposedtostudythedynamicsofcompetitionofagentsinavarietyof
. contexts(W.B.Arthur,Amer.Econ.Assoc.Pap.andProc.84,406(1994))isstudied.Wecharacterizein
t
a detailthethreeregionsofthephasediagram(efficient,betterthanrandomandinefficient)ofthesimplest
m version of the model (D. Challet and Y.-C. Zhang, Physica A, 246, 407 (1997)). The efficient region is
- showntohavearichstructure,whichisinvestigatedinsomedetail.Changesinthepayofffunctionenhance
d further thetendencyof themodel towards a wasteful distribution of resources.
n
o
PACS. 02.50.-r Probability theory, stochastic processes, and statistics – 02.50.Ga Markov processes –
c
05.40.+j Fluctuation phenomena, random processes, and Brownian motion
[
2
v 1 Introduction ble strategies. Strategies use the full information of the
2
m previous outcomes to decide the next move. As there
6 m
Inrecentyearstherehasbeenagrowinginterestinunder- are 2 possible combinations of past events, the number
11 standing the dynamics of systems of interacting individ- of strategies is 22m. After each event, the agents update
1 uals with competing goals (frustration). Simple rules for the score of their set of strategies. The gain made by the
8 the behavior of the individuals may lead to unexpected successful strategies can either be a fixed constant,or de-
9 propertiesinthebehaviorofthecollectivity.Theserather pendonthe sizeofthe groupformedatthattime step.In
/ generalpremises canapply to problems indifferent fields, thesimplestversionofthe model,onepointisassignedto
t
a like economy [1], ecology [2] or physics [3]. each successful strategy. When an agent has two or more
m ToillustratethesefactsBrianArthurintroducedwhat strategies with the same score, one of them is picked at
- he called “El Farol” bar problem (EFBP) [4]. N individ- random.Thischoiceofpayoffistheonediscussedindetail
d uals decide, ateachtime step,to goto a bar orto stayat below. The model is defined by the three parameters: N,
n
home.Thebarisenjoyableonlyiftheattendancedoesnot the number of agents, m, the number of time steps used
o
c surpass some critical number, that can be thought of as by each strategy in determining the next best move, and
: some kind of comfort capacity. But each individual does s,thenumberofstrategiesavailabletoeachagent.Exten-
v
notknowbeforehandwhatisgoingto happen.Tobe able sionstootherpayoffschemes,similartothoseusedin[5–7]
i
X tomakethedecisionforthenexttimesteptheindividuals are also mentioned. Note that the original work [4] used
r (which we will call agents in the following, as in previous a much less constrained set of strategies and a different
a literature of this model) are provided each one with a set comfort capacity (60%).
ofstrategies.Using these strategies,andthe knowledgeof The model, with the set of rules described above, was
whathashappenedintheportionofthehistorythatthey investigated in [6,8]. The authors analyze the mean size
can recall, the agents take decisions. and the fluctuations of the groups taking each of the two
D. Challet and Y.-C. Zhang [5] have given a precise choicesavailable.Itisarguedthatthe modelcanbe char-
m
set of rules which determine the model. The two possible acterized in terms only of the combination ρ = 2 /N.
N
choices, going to the bar or staying at home, are repre- The average group size is . The distribution of sizes is
2
sented by 0 and 1. A choice is successful if the agents symmetricalaroundthis value. The meanquadratic devi-
which make it are in the minority (comfort capacity = ation from the average, σ, is a measure of the number of
50%). The outcome of a given simulation is represented pointsaccumulatedbyalltheagents.Thisnumberismax-
by a seriesof0’sand1’s whichcharacterizethe successful imumwhenthetwogroupsarealmostequal,inwhichcase
choices at each time step. Each agent uses a fixed set of σ O(1). As function of σ2/N and ρ three regimes can
∼
s strategies, taken at random from the pool of all possi- bedistinguished,asfunctionofthetotalnumberofstrate-
2 M. A.R. deCara et al.: Competition, efficiency and collective behaviorin the“El Farol” bar model
gies at play [8]: i) When ρ 1, the number of strategies
≫
available to the agents is small, and the value of σ ap-
proachesthe limit expectedwhenthe agentstakerandom
decisions, σ2/N = 1/4. ii) If ρ 1, almost all possible
≪
strategies are in possession of the agents, and their per-
formanceisworstthanrandom,as σ2 > 1.iii)Finally,for
N 4
ρ 1theagentsperformstatisticallybetterthanrandom.
∼ σ2 100
The curve of versus ρ shows a minimum. The authors
N
(b) (c)
define regime i) as inefficient, as the agents have little in-
formation, and regime ii) as efficient, as agents have all
available information at their disposal. 10
Insection2,weanalyzethemodeldefinedabove,with
emphasis on the structure shown in the efficient region.
Section 3 presents an interpretation of the results. Then, 1 (a)
section 4, we discuss results obtained by varying the pay-
offfunctionwhichdeterminesthechoiceofstrategies.Sec-
tion 5 analyzesa seemingly trivialvariationof the model:
0.1
the majority game, when it becomes preferable to be in
the majority. The final section presents the conclusions. 0.01 0.1 1 10 100
2 Minority game. Fig. 1. Different phases found in the EFBP. The lower part
2
shows the evolution of σ /N as function of ρ, circles are for
s=2 and stars are for s=6. The insets show histograms of the
The transition discussed in [8] is displayed in fig. 1 for
attendance number in the different phases, with N = 101: a)
s=2 and s=6. The difference between the efficient and
efficient,m=2,s=2;b)betterthanrandomm=6,s=2;andc)
inefficient regimes is sharper for small values of s. Each
inefficient m=10, s=2. The top figure shows the difference in
simulation of the model starts from a history of length punctuation between the maximum scored and the minimum
m+3 to initialize the scoresof the strategies.The results scoredstrategiesinthesethreecases:dottedlinefor(a),dashed
shown in the paper are averages over the 2m+3 possible line for (b),and continuousline for case (c).
initial conditions defined in this way. In almost all cases,
the system evolves towards a steady state which is inde-
pendent of the initial conditions. thatthedistributionofattendancesshowsthreeseparated
The peaks in the size distributions are alwayswell ap- peaks.
proximatedbyGaussianfunctions.Thelargevalueofσ in We have completed the study the evolution of the dif-
the efficient region is due to the formation of new peaks ferent peaks by analyzing their evolution after an initial
awayfrom N/2.A pictorialview of this effect is shownin seriesofrandomchoices.Inthetimeseriesshowninfig.5,
fig. 2, where the different regimes are studied by varying the agents make choices randomly, although their strate-
m and s. The attendances have been normalized to one gieskeepupdatingthescores.Atagiventimestep(2048),
in the interval [ 1,1]. In the range of values of ρ where the agents start to use the strategies at their disposal.
−
threepeaks canbe clearlyresolved,the weightofthe cen- The peak structure is robust, and develops immedi-
tral peak is one half of the total, and the other two peaks ately. As shown in fig. 5, the peaks split from the central
include one fourth of the recorded attendances. The cen- peak and move to their positions in the steady state dis-
tral peak is always well approximated by a Gaussian of cussed earlier.
width √N/2 (see also fig. 3), which corresponds to ran-
dom choices by the agents.
Asoneleavestheefficientregion,thepeaksmergewith
3 Interpretation.
the central one, whose width decreases first and then in-
creases, to reach the random value for large values of ρ.
For small values of ρ, the peak structure is very rich, and The results presented in the previous section allow us to
seems self similar, as shown in fig. 3. gain some understanding of the complex dynamics of the
As pointed out in [8], it is somewhat unexpected the efficient regime. In this region, no strategy can stay with
poorperformanceoftheagentswhenalargeamountofin- the highest score for long. The repeated use of a given
formationisavailable.Itis evenmoreremarkablethe rich strategy by a significant number of agents leads to the
structureshowninfig.3,whichshowsthattheevolutionis raise of other strategies, preferably those more anticorre-
farfromrandom.Thisbehaviorisalsoconsistentwiththe lated with the one at play. As a result of this, the most
existenceofnontrivialpatternsinthetimeseries,beyond punctuated strategy (the best considered by the agents)
the reach of the agents [8]. hasmanychancesofmakingitsuserstoloose.And,even-
A plotof the attendances at successivetimes is shown tually,theagentssegregateintoanticorrelatedgroupswhen
in fig. 4. We have chosen the parameters in such a way some degree of evolution is incorporated [9].
M. A.R. deCara et al.: Competition, efficiency and collective behaviorin the“El Farol” bar model 3
10 800
5
0
10
5 600
0
10
5
0 400
10
5
0
10 200
5
0
-1 0 1-1 0 1-1 0 1-1 0 1-1 0 1 200 400 600 800
s=2 s=4 s=6 s=8 s=10
Fig. 2. Attendancenumbersdistributions for N =1001 Fig.4.Attendanceinagivengroupattwosuccessiveintervals.
Theparameters used are s=2,m=2 and N =1001.
(b) (c)
0.0002
0.0001
0
0.001 (a)
0.0005
0.0004
0.0003
0.0002
0.0001
0
0
Fig. 3. Attendancenumbersdistributionfor N=100001, s=4,
and m=4, normalized in the interval [0,100001]. (a) Full dis- Fig. 5. Attendancenumberversustimeforthegameinwhich
tribution where the y-axis has been truncated in order to ap- atransition isforced from arandom gametoaminority game
preciate the spreading of the lateral peaks. (b) Magnification (seetext).Theparametersoftheminoritygameare:N=1001,
oftheregionmarkedin(a)withdashedlines.(c)Pointsinthe s=4,and m=4(6) for the bottom (top) graphs.
central peak. The continuous line is a Gaussian, centered at
N/2, with weight half of the total distribution and deviation
√N/2.
of these nrandom agents can be taken to be at random,as
they are unable to recognize the series which give rise to
the high scores of x and x¯.
For simplicity, we now assume that there are two an- Whenstrategyxhasthehighestscore,thetwogroups
ticorrelated strategies, x and x¯ which have the highest willhavesizesclosetonrandom/2+ncorrelandnrandom/2
−
scores most of the time. Let us denote nx and nx¯ the ncorrel, respectively. This outcome will give no points to
number of agents which have strategy x and x¯. We can x, while strategy x¯, which would have lead to the most
take nx nx¯ = ncorrel. We now denote as nrandom the favorablechoice,gainsonepoint.Ifthescoreofx¯remains
≈
numberofagentswhichhaveneitherxnorx¯.Thechoices belowthatofx,theprocessrepeatsitself.Asteadystateis
4 M. A.R. deCara et al.: Competition, efficiency and collective behaviorin the“El Farol” bar model
reachedwhenthe scoresofx andx¯ differ by,atmost,one
point.Then,anoutcomewithtwounequalgroupsofsizes 80
nrandom/2+ ncorrel and nrandom/2 ncorrel is followed
− 40
by the formation of two groups of similar size, N/2.
≈
The fact that there are nrandom agents acting at random
impliesthatthesevaluesaretheaverageofGaussianpeaks
of similar width.
6
We can estimate the value of ncorrel from the analysis
in[7].Weclassifythe 22m strategiesinto2m mutuallyun-
correlated,maximallycorrelatedoranticorrelatedclasses.
m
Then, nx N/2 =1/ρ. 4
≈
The previous analysis gives a plausible explanation
of the three peaks observed throughout most of the ef-
ficient region of parameter space. It can be extended, in
a straightforward way, to the case when the dominant 2
strategies are more than two. The main new ingredient
is that there are situations in which two, or more, dom-
inant strategies can have the same score. Let us imagine
that the strategies with the highest scores are x1,x2,x¯1 0 -1 -0.5 0 0.5 1
and x¯2. Then, at a given instant, the strategy with the
highestscorecanbe x1,x2 ...,butalsox1 andx2 (orsim-
Fig. 6. AttendancedistributionsforN=1001, m=4,ands=4.
ilar combinations) simultaneously. If, in addition, x1 and
Dashed line is for the step payoff, continuous line for ∆p=a,
x2 lead to the same outcome, the majority group will be
and dotted line for ∆p=N/2 a.
of size nrandom/2+nx1 +nx2. This combination will be, −
probably, less likely, leading to lower peaks further away
fromtheaverage,inagreementwiththe findingsreported
here. strategies.Outcomeswithnearlyequalgroupsgiveriseto
We havecheckedthatthere is atrivialcasewhere this large changes in the scores of the strategies. Thus, long
analysis reproduces the observed evolution: m = 2 and living cycles, of the type described in the previous sec-
s = 16, where all agents have all strategies. The atten- tion, cannot form. The highest ranking strategy changes
dance histograms show two sharp peaks at 1 and N, and rapidly.As all strategies are in play,groups of many sizes
a Gaussian peak with half the weight of the total distri- aregenerated,despite the fact thatthe payofffavorssizes
bution at N/2, and deviation √N/2. close to N/2.
In the opposite case, with payoff function equal to
N/2 a,we ascribethe largepeakatN/2tofrequentsit-
−
4 Varying the rewards for the winners. uations when many strategies have the same score. This
situationis selfsustaining,as,whenthe twogroupsareof
Wenowlooktotheeffectofchangingthewayinwhichthe sizesN/2 andN/2+1,there is no changein the scoresof
different strategies are updated after each outcome. The thestrategies.Thisiswhathappensinhalfofthepossible
simplest modification is to relate the change in the score 2m+3 initialconditions,andcorrespondstothedeltapeak
to the size of the minority group [5,6]. In the following, in fig. 4. The rest of the distribution is a good average of
we assume that the payoff, ∆p, depends linearly with the what happens in the other half of the initial conditions.
size, a. If the score is incremented by a, strategies which The shift of the payoff by a constant described earlier re-
leadto groupswith attendances close to N/2are favored. duces the probability of tie-ups, and leads to a double
If,ontheotherhand,thescoreisincrementedbyN/2 a, peaked distribution. These peaks displaced from the cen-
−
the tendency is the opposite, and strategieswhich leadto terseem,inthis case,relatedtothe twopeaksinthe step
very small groups are favored. payoff case. It is likely that the evolution of the model is
Thedistributionsgeneratedbythesetwopayoffchoices governedby cycles with a few dominating strategies.
are plotted in figure 4. The distribution obtained by the
steppayoffdiscussedintheprevioussectionisalsoplotted,
for comparison. 5 Majority game.
Contrarytointuition,thetwodistributionsseemtogo
intheoppositedirectiontowhatthechoiceofpayoffleads We have also studied the majority game, in which the
tothink.Itmustbenotedthat,whenthesecondchoiceof agents prefer to be in a overcrowded bar or leave the
payofffunctionisshiftedbyaconstant,∆p=N/2 a+k, bar empty. The methodology is the same as in the mi-
−
the central peak tends to disappear, and it is replaced nority game, in which the different initial conditions tend
by two peaks at the sides. This result is similar to other to give similar results. Here, initial conditions may make
findings with a payoff which also favors small groups [6]. big changes in the attendance distributions.
We interpret the broad structure for the payoff func- Results are trivial (the full majority is attained at
tion a as due to the swift shuffle of the highest ranking all time steps) only when all agents have all strategies
M. A.R. deCara et al.: Competition, efficiency and collective behaviorin the“El Farol” bar model 5
(s=22m). Evenin this case,and depending onthe initial 6 Conclusions.
conditions, the group (0 or 1) which obtains the majority
may oscillate in time. As we have seen, the El Farol bar problem has a rich
The obtained distributions for different values of m structure.We have focused mostly on the behavior in the
and, consequently, ρ, are plotted in fig. 5. efficient regime, where most of the strategies are at the
disposal of the agents. As already remarked in [8], the
model has many features in common with frustrated sys-
6 tems in statistical mechanics. In particular, most initial
conditions lead to a poor performance of the system as a
whole.The model seems unable to select a pool of strate-
4
giessuchthatthe globalgainby the agentsis maximized.
Inparticular,thoseagentswhichhaveaccesstothestrate-
2 gies with the highest scores at a given moment perform
worse than those which do not. The latter play basically
atrandom,andprofitfromtheunproductivecoordination
0 of the players using the nominally best strategies.
-1 -0.5 0 0.5 1-1 -0.5 0 0.5 1-1 -0.5 0 0.5 1 Thiseffectseemstoremainwhenthepayofftothedif-
ferentstrategiesisvaried.Itisalsoremarkablethatthein-
trinsic frustrationofthe modelshowsup whenthe agents
10
try to be in the majority. Most initial conditions lead
to evolutions where the agents fail to coordinate among
themselves.
1
Financial support from DGCYT through project no.
PB96/0875and the European Union through project
ERB4061PL970910is gratefully acknowledged.
0.1
0.1 1 10
We would like to thank the helpful comments of N. Garc´ıa,
Fig. 7. The analogous diagram of figure 1 for the majority EnriqueLouis, Pedro Tarazona and Yi-ChengZhang.
s
rule.HereN=101 and s=4.Thedashedlineis for σm =N/2
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in [8].
