Table Of ContentDarwinian Selection and Non-existence of Nash Equilibria.
Daniel Eriksson1 and Henrik Jeldtoft Jensen1,∗
1Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2BZ, U.K.
WestudyselectionactingonphenotypeinacollectionofagentsplayinglocalgameslackingNash
3
equilibria. Aftereachcycleoneoftheagentslosingmostgamesisreplacedbyanewagentwithnew
0
random strategy and game partner. The network generated can be considered critical in the sense
0
thatthelifetimesoftheagentsispowerlawdistributed. Thelongestsurvivingagentsarethosewith
2
thelowest absolute score pertime step. The emergent ecology is characterized bya broad range of
n behaviors. Nevertheless, theagents tend to besimilar to theiropponents in terms of performance.
a
J
PACSnumbers: 87.10.+4,87.23.-n,02.50.Le,05.65.+b
4
1
Ithasbeenarguedthatbiologicalevolutionisdrivenby Collective adaptive agents striving to be among the mi-
] a combination of natural selection and self-organization nority were studied in mean-field models by Arthur [5]
h
c [1, 2]. Mutations act at random at the level of genotype and Challet and Zhang [6]. Aspects of Darwinian evo-
e while selection acts at the phenotype. Darwinian selec- lution were added to the minority game by Zhang [7].
m tion is expected to occur because some phenotypes are ThislineofapproachwerefurtherdevelopedbyPaczuski,
- more viable than others [3]. The viability or fitness of a Bassler and Corral (PBC) who considered agents play-
at given phenotype is, of course, not an absolute quantity ing the minority game [8]. The goal of agents is to be
t but depends oncontext and the environmentthe pheno- among the global minority though they are linked only
s
. typeisexposedto[4]. Theenvironment(physicalaswell to a subset of all the agents. The strategy of the worst
t
a as biotic) is to a large extent produced by co-existing performing agent is randomly changed after each cycle
m phenotypes, and hence the environment and the corre- while the links of the network remain unchanged [8].
- sponding fitness are self-organized emergent properties Here we modify the model introduced by PBC to a
d of the ecology. Moreover,fitness or vigor is necessarily a
Local Darwinian Network model (LDN) in which we al-
n
relative quantity. An organism can become so vigorous low agents to play competitive local games and to adapt
o
thatitremovesitsownfoundationofexistence;abalance
c theirconnectionsaswellastheirstrategies. Viewedfrom
[ very relevant e.g. to host-parasite systems. For organ- the perspective of evolutionary ecology it appears to be
isms to be able to coexist for extended periods of time more natural to consider agents performing local games
1
they need to develop phenotypes which are in a kind of
v (organismsorspecies entangledin a webofmutual com-
restrained poise.
6 petitions). The renewal of the properties of the worst
4 Here we analyze Darwinian selection acting on phe- performing agent should be thought of as representing
2 notype in a model consisting of an adaptive network of either the “mutation” of an individual or, if agents are
1 agentsofdifferentstrategiesmutuallycompetinginlocal thought of as representing species, as a species being su-
0 zero-sumgameslackingNashequilibria. Weletselection perseded by an invading species with different affinities.
3
actbyremovingtheagentswiththesmallestscore(most Ineithercaseitisnaturaltoupdatethesetofinteraction
0
/ negative) after a round of games. Thus the viability of links. Further, from a statistical mechanics or complex
at an agent is a function of his phenotypical behavior in a systems perspective we expect systems coupled globally
m specific environment. When the local games are of op- tomorereadilyenteracriticalstate,henceitisofinterest
posing interestwithonlyasinglewinnerwefindthatthe to explore the criticality of purely local systems.
-
d agents organize into a system where agents are homo-
The LDN net is critical for K = 2 as is the original
n geneous in terms of success but heterogeneous in terms
Kauffman net [1]. The PBC net is critical for K =3 [8].
o of activity. The agents achieve longevity by organizing
c Here K is the number of independent arguments of the
themselves in a way that minimize the absolute value of
: agents’ Boolean strategy functions.
v their score. By this strategy they avoid to be amongst
i the worst performers themselves; moreover low absolute The homogeneity parameter P of an agent measures
X
score allows the agents to retain well tuned partners by the fraction of 0’s and 1’s – whichever is in majority –
ar notforcingmanylostgamesontotheiropponents,which in the Boolean output assigned to the K2 distinct input
would lead to the elimination of the partner. states of the agent. For K = 2, the homogeneity pa-
rameter is constrained to one of the values 1/2, 3/4 or 1
Ourmodel is closelyrelatedto previousnetworkmod-
where in the last case the agent will not switch output
els. KauffmanintroducedBooleannetworkstostudythe
signalatall. IncontrasttoPBC,wefixthe homogeneity
relationship between evolution and self-organization [1].
parameter of each agent to 3/4 ensuring that the net-
work is logically entirely homogeneous with respect to
the output ofthe agents. It followsthatanydiscrepancy
∗Authortowhomcorrespondence shouldbeaddressed: in the switching behavior of two competing agents is of
[email protected];URL:http://www.ma.ic.ac.uk/~hjjens/ dynamical origin.
2
The model – Consider N agents, each assigned a b) The Darwinian update is done in the following
Boolean signal S = 0 or 1, for i = 1,...,N. The dy- way. After testing the performance of the agents, the
i
namics consists of two types of updates: a) the agents worstperformingagentisreplacedbyanewagentwitha
playing rounds of games; b) and the act of Darwinian new Boolean function (i.e., genotype) chosen completely
selection/mutation. at random. (In case the worst performance is shared
a) The games are performed in the following way. by more then one agent, then one of them is chosen at
Each agent is defined as aggressor (A) of one other ran- random for replacement.) The new agent is randomly
domlyassignedagent(thusagentsparticipateonaverage assigned an opponent and a set of K source-agents. In-
in two games) who acts as opponent (O) in a zero sum tuitively, this means that agents belonging to ecological
game. IneachtimestepAandOcomparetheirBoolean nichesthatno longerexistareremovedfromthe system,
signals SA and SO. If SA = SO the aggressor A scores whereas new candidate niches are sampled fortuitously
+1 and the opponent scores –1. If SA 6= SO the aggres- byagentswhothemselveshaverandomproperties. How-
sor A scores –1 and the opponent scores +1. The game ever, the wiring of those agents that attack an agent be-
isidenticaltothesimple”coinguessing”gamewhereone ingreplacedisnotchanged. Theseagentsnowattackthe
of two persons is to pick the hand – left or right - that new agent as we proceed with a new round of games, or
is holding the coin. The game has the property that, test cycle, as described under a).
whenconsideringonlypure(non-randomized)strategies, In this letter, we use a Non-Darwinian version of the
there exists no Nash equilibrium. No Nash equilibrium model as benchmark allowing us to isolate the effect of
existsinthesensethatitwillalwaysbepossibleforeither the Darwinian updating. In the Non-Darwinian model,
playertobenefitbychanginghisstrategyiftheopponent at the end of eachtest cycle, we randomly pick an agent
sticks to his strategy. In our simulations, two agents are for replacement. Agents are in this situation not pun-
never allowed to act as mutual aggressors in which case ished for performing poorly and are all subject to the
both agents could receive a zero score. Neither do we same probability of being replaced.
allow agents to act as aggressorfor themselves. The fact
Results-Theinitialconfigurationconsistsofrandomly
thatagentscannotcollaborateinordertoachieveneutral
assigned input links, Boolean functions and game part-
(zero) scores will be referred to as an opposing interests
ners. As the sequence of test cycles and Darwinian re-
property of the network.
placements are repeated the model organizes gradually
Thesignalsusedforthezero-sumgamesaregenerated
into a stationary state. The critical properties of the
in the following way. The output signal at a given time
network are indicated in Fig. 1 where we exhibit the
stepS (t)ofagenti isdetermineddeterministicallyfrom
i probability of the length of transients p(t) and of the
theoutputattheprevioustimestepfromK othersource
length of periodic attractors p(a) respectively. The dis-
agents S (t−1),S (t−1),...,S (t−1)
i1 i2 iK tributions do exhibit scale free power law like behavior,
though an accurate determination of the precise func-
S (t)=f (S (t−1),S (t−1),...,S (t−1)), (1)
i i i1 i2 iK tional form (and precise values of the exponents) is un-
where f is a randomly chosen Boolean function associ- fortunately not possible. The behavior of p(t) for K =2
i
atedwithagenti. ForK =2thefunctionissuchthatfor for large N appears to be consistent with p(t)∼t−α for
exactly three of the functions four input configurations α ≃ 1.4. The behavior of p(a) for K = 2 for large N
({0,0}, {0,1}, {1,0}, {1,1}) the same Boolean output appears to consistent with p(a) ∼ a−β for β ≃ 1.2. The
is assigned, whereas the signal is different for the fourth behavior of the Non-Darwinian version of the network
configuration. This ensures P =3/4. is similar as shown in the figures. Hence the Darwinian
One can think of a specific choice for the functional moveisnotcrucialfortheexistenceofthepowerlawlike
form of f as the genotype of agent number i. The K behavior of p(t) and p(a). Nevertheless we shall see be-
i
source-agentsassociatedwith icaninclude theopponent low that the Darwinian selective move has a significant
of i but will not include agent i. The source-agents and effect on the characteristics of the agents. A transient
the opponents are to be thought of as the environment and corresponding first occurrence of periodic attractor
of an agent. make up a test cycle. We find that the distribution of
For fixed assignments of source-agents as well as ag- testcyclesalsoappearstobepowerlawlike. Noticethat
gressor and opponents Eq. 1 is now parallel updated ∀i. in order to find the largest observations in Fig. 1 we it-
Since the state space of the net is finite and the dynam- erated the network more than the maximum number of
icsisdeterministic,periodicorbitswillalwaysbereached updates of a test cycle and then reentered the state at
(thoughthemaximumlengthofaperiodishuge2N). We 104 updates to perform the Darwinian update.
measure the average score per time step of the agents We define the lifetime l of an agent as the number of
overa roundofgamesconsistingofeither simulating the updates survived at the moment the agent is removed
transient plus one periodic cycle or by performing 104 fromthe system. As exhibitedin Fig. 2 we find thatun-
parallel updates which ever is the shortest. We denote der Darwinian updating the network develops what ap-
this sequence of deterministic updates a test cycle. The pears to be a power law like distribution of lifetimes p(l)
score gained by an agent characterizes the success of his with increasing system size, consistent with p(l) ∼ l−γ
phenotypical behavior. for γ ≃ 0.9. Notably the magnitude of the power law
3
100 x 106
N=64 Darwinian 3.5
N=128 Darwinian
10−1 NN==11002244 DNaornw−iDniaarnwinian 3
10−2 2.5
Frequency10−3 Lifetime1.25
10−4
1
10−5
0.5
10−6
100 101 102 103 104 −04 −3 −2 −1 0 1 2 3 4
Transient Length Lifetime Yield
100
N=64 Darwinian
N=128 Darwinian FIG.3: Thecorrelationbetweenlifetimeandlifetimeyieldfor
10−1 NN==11002244 DNaornw−iDniaarnwinian the Darwinian model (stars) and the Non-Darwinian model
(circles). N =1024.
10−2
Frequency10−3 of an agent the score averaged over the lifetime of the
10−4 agent. The yield of an agent characterizes the efficiency
of the agent’s phenotypical behavior. Fig. 3 shows that
10−5 longevity is directly related to near zero values of the
agent’s lifetime yield. Thus the selective pressure on the
10−1600 101 102 103 104 agentsinthemodelleadstoafinelytunedecology,where
Attractor Period agents tend to avoid winning too much, which might in-
duce annihilation of their partners. Instead agents man-
FIG.1: Theprobabilityofthelengthoftransients(topcurve)
and attractors (bottom curve)for different systemssizes N. age to keep their partners and live long by entering into
near neutral patterns.
Fig. 4 is a histogram of yield accumulated over en-
N=64 Darwinian tiretestciclesoftheDarwinianandbenchmarkecologies
N=128 Darwinian
10−2 NN==11002244 DNaornw−iDniaarnwinian respectively. The accumulated yield of an agent is the
average score of the agent over the length of his life so
far. (Noticethatagentswerenotbornatthesametime.)
10−3
Frequency10−4 lnTaehtteewdsotyrakine.lddWariesdfiadnemdvieaaatssiuomnraeδlloYefrotδhfYetheienqdutiahslteirtDiybauinrtwiosiunnciocafensasecccionulmotghuye-
for all considered system sizes N =128, 256 and 1024.
10−5 Indeed, one would expect that removal of the worst
performingagentsshouldreducetheabundanceofpoorly
10−6 performing agents with large negative average scores.
100 101 102 103 104 105
Lifetime But we observe that the abundance of well performing
agents with large positive scores is also somewhat re-
FIG. 2: The probability of the number of updates agents duced. IntheDarwinianmodelagentsarethusclustered
survivefor different system sizes N.
around the zero score. Notice that given opposing in-
terests of agents, the collected environment of an agent
receives the same score as the agent but with opposite
exponent is smaller than one. Since test cycles are nu- sign. In the Darwinian model, agents are therefore sim-
mericallytruncatedatafinite lengthof104 updates, the ilar to their respective environments in terms of success.
distributionoflifetimesissomewhatdistortedforlengths Aninterestingpropertyofe.g. theMinorityGame(MG)
longerthan the numericalcutoff. Because of logarithmic modelisthatanyfavorableconfigurationforaparticular
binning the effect is not visible in Fig. 2. We find that agentisinherentlyunstableasotheragentsacttoreverse
for all considered values of N = 64,128 and 1024 the the situation [6]. A similar situation holds for LDN as
lifetime distribution for the Non-Darwinian model (the well but unlike MG the environments of the agents are
circles in Fig. 2) are approximately linear as function of heterogeneous.
the logarithm of the argument, i.e. not at all described Letusnowdescribeinmoredetailhowtheagentsman-
by a power law. age to achieve low absolute scores and thereby survive
Next we address the behavior of the agents in the sta- many test cycles. For the benchmark and Darwinian
tionary critical state. We denote by the lifetime yield model respectively, we have investigated the switching
4
tough still, perhaps, power law distributed. This version
0.2 of the model is difficult to handle numerically.
As pointed out by PBC, an interesting question is –
whatgamesleadtocomplex,scalefreestatesunderDar-
0.15
Frequency 0.1 whaleilnroewianintnlygseptlerhceetvieloonnst?inaggWeaengtssepnfertcosumtlaoatceimhtihpeavrotinvtgehmethgueatiumrauelngsfhaaoivnuolwrdahbiinllee-
situation by a change of behavior.
0.05 We suggestthat the ”coin-guessing”game of the LDN
model is a suitable representation of such situations
where agents tend to become equal in strength to their
0
−3 −2 −1 0 1 2 3
Accumulated Yield opponents implying their exact ”task” becomes irrele-
vant. Forexample,thegameissimilartomanysituations
FIG. 4: Histogram of accumulated yields for the Darwinian
in the business world, e.g. the trade of a stock involv-
model (stars) and the Non-Darwinian model (circles). N =
ing one seller and one buyer. From a dynamical point
1024.
of view, it is significant that either the buyer or seller in
a trade will necessarily become a winner and the other
agent a loser as the price of the stock diverges from its
activity of the agents. By a switch we mean the change
value when the deal was made. (If the price of the stock
from sending signal 0 to 1 or vice versa. By a distinct
goesupthesellerwouldhavebeenbetteroffontheother
switching activity we mean a certain observed number
sideofthetrade). AsintheLDNmodel,thedeterminant
of switches of one or more agents over a test cycle. The
of who is the winner is of complex dynamical origin.
number W(N,t+a) of observed distinct activities can-
not be larger than the number of agents in the ecology The score distribution of the Darwinian model might
N northe lengthofthetestcycle(t+a). We haveinves- be compared to e.g. the distribution of performances
tigated the dependence of W on (t+a) for system sizes of fund managers. Interestingly, extensive investigations
N =128,256and1024. Wefindthatforallcyclelengths havegivenlittleevidencethatfundmanagerscanconsis-
2≤(t+a)<104,theDarwinianecologyexhibitsalarger tently beat relevant market indexes (see p. 368 in [9]).
ratio W/(t+a). This may suggest that the Darwinian Similarly,the LDN model has the propertythat success-
updating of the network produces a greater variety of ful agents tend to be short-lived.
phenotypes. We have presented a Local Darwinian Network model
We now discuss the robustness of the model. Increas- which exhibits critical behavior without global interac-
ing the number of opponents per agent does not appear tions. The model demonstrates that – given that agents
to change the test cycle distribution. We do not observe actunderopposinginterestwiththeirrespectiveenviron-
criticality in case the input wiring of the agents is un- ments – Darwinian phenotypical selection of the worst
changed as the agents are replaced. We point out that, performer together with genotype mutations can lead
ingeneral,testingthestationarityofthemodelisnumeri- to an ecology where the distribution of lifetimes lacks
callychallenging. Onecanconsideranalternativeversion a characteristic scale. The agents of the emergent ecol-
of the network where the homogeneity parameter is not ogy are heterogeneous with respect to their activity but
fixed. Inthis situation,PBCfindthat whenagentscom- in terms of success they are nearly equal to their respec-
pete globally, the homogeneity parameter self-organizes tive environments. Longevity is achieved by adapting to
to a distinct value as the network enters a critical state a near neutral yield which ensures the stability of the
[8]. In contrast, for the LDN model we find that the ho- fellow partners.
mogeneity parameter decreases from the initial unbiased Helpful initial inspiration from M. Paczuski, P. Bak
valuemakingthetransientsandperiodicorbitsverylong, and K. Nagel are gratefully acknowledgedby DE.
[1] S. Kauffman, At Home in The Universe (Oxford Univer- [5] W. B. Arthur,Am.Econ. Rev.84, 406 (1994).
sity Press, 1993). [6] D.ChalletandY.-C.Zhang,Physica(Amsterdam)246A,
[2] J. M. Smith and E. Szathm´ary, the origins of life (Chap- 407 (1997).
man and Hall, Oxford, 1999). [7] Y.-C. Zhang, Europhys.News 29, 51 (1998).
[3] C. Darwin, On the Origin of Species by Means of Natural [8] M.Paczuski,K.E.Bassler,andA.Corral,Phys.Rev.Lett
Selection.(JohnMurray(thised.1951,OxfordUniversity 84, 3185 (2000).
Press)., Oxford, 1859). [9] Z. Bodie, A. Kane, and A. J. Marcus, Investments
[4] L. Pagie and P. Hogeweg, Evolutionary Computation 5, (McGraw-Hill, NewYork,2002), 5th ed.
401 (1998).
