Collective Behaviour and Diversity in Economic Communities: Some Insights from an Evolutionary Game

Abstract
Many complex adaptive systems contain a large diversity of specialized components. The specialization at the level of the microscopic degrees of freedom, and diversity at the level of the system as a whole are phenomena that appear during the course of evolution of the system. We present a mathematical model to describe these evolutionary phenomena in economic communities. The model is a generalization of the replicator equation. The economic motivation for the model and its relationship with some other game theoretic models applied to ecology and sociobiology is discussed. Some results about the attractors of this dynamical system are described. We argue that while the microscopic variables -- the agents comprising the community -- act locally and independently, time evolution produces a collective behaviour in the system characterized by individual specialization of the agents as well as global diversity in the community. This occurs for generic values of the parameters and initial conditions provided the community is sufficiently large, and can be viewed as a kind of self-organization in the system. The context dependence of acceptable innovations in the community appears naturally in this framework.

arXiv:adap-org/9804003v1 30 Apr 1998
COLLECTIVE BEHA VIOUR AND DIVERSITY IN ECONOMIC COMMUNITIES:
SOME INSIGHTS FROM AN EVOLUTIONAR Y GAME
VIVEK S. BORKAR
Department of Computer Science and Automation
Indian Institute of Science
Bangalore 560 012, India
SANJAY JAIN
Centre for Theoretical Studies
Indian Institute of Science
Bangalore 560 012, India
AND
GOVINDAN RANGARAJAN
Department of Mathematics and Centre for Theoretical Studies
Indian Institute of Science
Bangalore 560 012, India
Abstract. Many complex adaptive systems contain a large diversity of s pecialized
components. The specialization at the level of the microsco pic degrees of freedom,
and diversity at the level of the system as a whole are phenome na that appear
during the course of evolution of the system. We present a mat hematical model to
describe these evolutionary phenomena in economic communi ties. The model is a
generalization of the replicator equation. The economic mo tivation for the model
and its relationship with some other game theoretic models a pplied to ecology and
sociobiology is discussed. Some results about the attracto rs of this dynamical system
are described. We argue that while the microscopic variable s – the agents comprising
the community – act locally and independently, time evoluti on produces a collective
behaviour in the system characterized by individual specia lization of the agents
as well as global diversity in the community. This occurs for generic values of the
parameters and initial conditions provided the community i s suﬃciently large, and
can be viewed as a kind of self-organization in the system. Th e context dependence
of acceptable innovations in the community appears natural ly in this framework.
Preprint No. IISc-CTS-4/98
To appear in the proceedings of the Workshop on Econophysics held at Budapest,
Hungary, July 21-27, 1997.

2 BORKAR, JAIN AND RANGARAJAN
1. Introduction
Several complex adaptive systems in the course of their evol ution exhibit the phe-
nomenon that the individual components comprising the syst em evolve to perform
highly specialized tasks whereas the system as a whole evolv es towards greater diver-
sity in terms of the kinds of components it contains or the tas ks that are performed
in it. Here are some examples:
1. Living systems are made of cells, which in turn are made of m olecules. Among
the various types of molecules are the proteins. Each type of protein molecule
has evolved to perform a very speciﬁc task, e.g., catalyse a s peciﬁc reaction in the
cell. At the same time, during the course of evolution, diver se kinds of protein
molecules have appeared – the range of specialized tasks bei ng performed by
protein molecules has increased.
2. In an ecology, species with highly specialized traits app ear (e.g., butterﬂies with
a speciﬁc pattern of spots on their wings). Simultaneously, the ecology evolves
to support a diverse variety of specialized species.
3. Many early human societies (such as hunter-gatherer soci eties) were perhaps
characterized by the fact that there were relatively few cho res (e.g., hunting,
gathering, defending, raising shelter) to be performed, an d everyone in the com-
munity performed almost all the chores. These societies evo lved to have special-
ist hunters, tool makers, farmers, carpenters, etc. Indivi duals specialized, and
simultaneously a diverse set of specialists appeared.
4. In an economic web, we ﬁnd ﬁrms exploring and occupying inc reasingly spe-
cialized niches, while the web as a whole supports an increas ingly diverse set of
specialists.
In the examples above the systems and their underlying dynam ics are quite diﬀer-
ent. But they all share the twin evolutionary phenomena of in dividual specialization
and global diversiﬁcation. In all these systems, the nonlin ear interaction among the
components seems to play a crucial role in the manifestation of this type of be-
haviour. For example, in an ecology, the development of high ly specialized traits in
a species is a result of its interaction with and feedback fro m other species. In an
economic community, each agent’s choices depend upon feedb ack from exchanges
(of goods, money, etc.) with other agents. Moreover, there i s no purposeful global
organizing agency which directs the behaviour of individua l components and ordains
them to be specialists. The phenomenon happens ‘spontaneou sly’, arising from the
local moves made at the level of individual components. Simi larly diversity also arises
as individuals capitalize on innovations – mutations, tech nological innovations, etc.
– which suit them in the existing context.
In this article, we describe a mathematical model which seem s to exhibit the
above twin evolutionary phenomena. The (idealized) behavi our of agents in eco-
nomic communities provides the basic motivation of the mode l. The model consists
of a set of coupled nonlinear ordinary diﬀerential equations describing the time evo-
lution of the activities of individual agents. In the next se ction we motivate and
present the model and place it in the perspective of existing work. In section 3 we
deﬁne more precisely the notions of specialization and dive rsity in the context of
the model and outline what type of behaviour we are looking fo r. Essentially we

are seeking attractors of the dynamical model that have the p roperty of individual
specialization and global diversity. Section 4 states cert ain theorems and numerical
results for the attractors of the system and discusses their consequences. The results
imply that under certain conditions that do not destroy gene ricity in parameter
space, the desired attractors (in which the system exhibits individual specialization
and global diversity) exist and have basins of attraction th at cover the entire conﬁg-
uration space. Thus the evolutionary phenomena mentioned a bove occur generically
in the model. In this section we also discuss self-organizat ion and the emergence of
innovations in the model. Finally, section 5 contains a brie f summary.
2. The model
The system is a community of N agents labeled by the index α = 1, 2, . . . , N . Each
agent can perform s strategies or activities labelled by i ∈ S = {1, 2, . . . , s }. At time
t, agent α performs strategy i with a probability pα
i (t), ∑ s
i=1 pα
i (t) = 1. The vector
pα (t) = ( pα
1 (t), p α
2 (t), . . . , p α
s (t)) is the mixed strategy proﬁle of agent α at time t. In
particular, if pα
i (t) = δij, then the agent α is said to pursue the pure strategy j or
to have specialized in strategy j.
The vectors pα (t) constitute the basic dynamical variables of the model. The
equation governing their evolution is taken to be
˙pα
i (t) = pα
i (t)[
∑
β ̸=α
s∑
j=1
aijpβ
j (t) −
∑
β ̸=α
s∑
i,j =1
pα
i (t)aijpβ
j (t)]. (1)
Here aij denotes the ijth element of the s-dimensional payoﬀ matrix A.
This dynamics is motivated as follows. Each agent is interac ting pairwise with
all other agents and receiving a payoﬀ at every interaction t hat depends on the
strategy pursued by each of the agents during that interacti on. Each agent updates
her strategy proﬁle based on the payoﬀs received, so as to incr ease her payoﬀ at
subsequent interactions. In a time ∆ t, agent α has a total of m∆ t interactions with
every agent ( m assumed constant). If in a particular interaction with agen t β , agent
α plays pure strategy k and β plays pure strategy j, then the payoﬀ to α is akj
(by the deﬁnition of payoﬀ matrix elements). Since α plays the strategy k with
probability pα
k and β plays the strategy j with probability pβ
j , the average payoﬀ to
α from the m∆ t interactions with β is
m∆ t
∑
k,j
pα
k (t)akj pβ
j (t).
The average payoﬀ to α from the whole community is
m∆ t
∑
β ̸=α
∑
k,j
pα
k (t)akj pβ
j (t).
This is the second term in the [ ] in Eq. (1). We have assumed tha t ∆ t is large
enough for there to be a statistically suﬃcient number of int eractions m∆ t so that
averages make sense. Yet it is small enough compared to the ti me scale at which
agents update their strategies so that pα
k can be considered constant during ∆ t, i.e.,

4 BORKAR, JAIN AND RANGARAJAN
there is a separation of time scales between the individual i nteractions of agents
(which happen on a short time scale) and the time scale over wh ich agents update
their strategy proﬁle (a long time scale).
If agent α were to pursue not the mixed strategy proﬁle pα during this interval
but instead the pure strategy i, then the payoﬀ received during this period would
have been
m∆ t
∑
β ̸=α
∑
j
aijpβ
j (t).
This is the ﬁrst term in the [ ] in Eq. (1). This quantity depend s on i and for some i
will be greater than the average payoﬀ and for some it will be l ess than the average
payoﬀ. At the end of period ∆ t, the agent α updates her strategy proﬁle pα
i to
pα
i + ∆ pα
i , adding a positive weight ∆ pα
i to those strategies i that do better than
the average and a negative weight to those doing worse than th e average. ∆ pα
i /p α
i
is chosen to be proportional to the amount by which the pure st rategy payoﬀ diﬀers
from the average payoﬀ:
∆ pα
i
pα
i
= cm∆ t[
∑
β ̸=α
∑
j
aijpβ
j (t) −
∑
β ̸=α
∑
i,j
pα
i (t)aijpβ
j (t)]. (2)
Taking the limit ∆ t → 0 and rescaling t by the factor cm, we recover Eq. (1).
Therefore the equation embodies the statement that at all ti mes, all agents update
their individual mixed strategy proﬁles so as to increase th eir own payoﬀs in the
current environment of the strategy proﬁles of other agents .
The reason why ∆ pα
i /p α
i and not just ∆ pα
i appears in the l.h.s. of (2) is that the
dynamics must respect the probability interpretation of pα
i . If two pure strategies i
and i′ provide the same payoﬀ to agent α , she must increment them in proportion
to their current strength in her proﬁle. This is needed to ens ure that pα (t) remains
normalized at all times, ∑ s
i=1 pα (t) = 1. If we start with normalized pα , the pro-
portionality factor pα
i on the r.h.s of (1) ensures that it remains normalized, since∑ s
i=1 ˙pα
i (t) = 0.
Thus, we have a community of N interacting agents, each responding to the rest
of the environment by updating their own proﬁle according to the above dynamical
equation. This is a “non-cooperative game”. Agents act on th eir own (not in concert,
per se ) and are selﬁsh – their actions are designed to increase thei r own payoﬀ,
without consideration for others or the community as a whole . Agents also exhibit
“bounded rationality” – they do not anticipate other agents ’ future strategies, but
merely respond to the aggregate of the other agents’ current strategies. There is
no global organizing agency at work, the community evolves j ust through these
individual actions of the agents.
Nevertheless, we will argue that the community does exhibit a kind of global
organization under certain circumstances. If the communit y starts with some arbi-
trary initial condition in which each pα at t = 0 is speciﬁed (each agent starts with
some mixed strategy proﬁle which could be diﬀerent for diﬀeren t agents) and evolves
according to Eq. (1), it will settle down to some attractor of the dynamics. The orga-
nization referred to above is in the nature of the attractors . When the payoﬀ matrix
elements satisfy certain inequalities, and when the size of the community is larger
than a certain ﬁnite bound that depends on the payoﬀ matrix (i .e., N is suﬃciently

large), then we ﬁnd that these attractors are characterized by each individual agent
having specialized to some pure strategy or the other, and at the same time the com-
munity as a whole retaining its full diversity of strategies , i.e., every pure strategy is
pursued by some agent or the other in the attractor conﬁgurat ion. Such attractors
seem, generically, to be the only stable attractors of the sy stem under the above
conditions. Most of the time, we will consider the system wit h a ﬁxed set of pure
strategies. At the end, we will mention applications of our r esults for the innovation
of new strategies. The instability of attractors in which th e community does not
have the full diversity of available strategies provides a m echanism by which new
pure strategies, or innovations, can invade the system.
Before proceeding further, we would like to place this model in the perspective of
existing work in the subject. Consider the “homogeneous sec tor”, where all agents
have the same (but in general mixed) strategy proﬁle: pα = x ∀ α . Then Eq. (1)
reduces to
˙xi(t) = xi(t)[
∑
j
aijxj(t) −
∑
k,j
xk(t)akj xj(t)](N − 1). (3)
The overall factor of N − 1 can be absorbed in a rescaling of time. This equa-
tion is the well known replicator equation [1]. It has applic ations in diverse ﬁelds
such as economics and sociobiology (where it models evoluti onary games) macro-
molecular evolution (describing evolution of autocatalyt ic networks, in particular the
hypercyclic feedback), mathematical ecology (Eq. (3) maps onto the Lotka-Volterra
equation) and population genetics (where it is the continuo us counterpart of the
discrete selection equation). This system exhibits a great diversity of solutions in-
cluding ﬁxed points, limit cycles, heteroclinic cycles, et c. For more details, see [2]. In
these applications i labels strategies or species of molecules or organisms, dep ending
upon the application. xi represents the fraction of individuals of type i in a large
population, and Eq. (3) models the change of the composition of the population with
time.
Eq. (1) has been considered as a multi-population generaliz ation of the replicator
dynamics and has been studied as such in the literature [2] [3 ]. The index α now labels
populations, e.g., α = 1 might correspond to a population of frogs, and α = 2 to a
population of insects. The idea here is to model the co-evolu tion of the populations of
frogs and insects in interaction with each other. p1
i now stands for the fraction of the
frog population with genotype i and p2
j for the fraction of the insect population with
genotype j. The indices i and j run over values s1 and s2 respectively which need
not be equal, and now there are two payoﬀ matrices, one for the frogs, A1 = ( a1
ij),
whose matrix element a1
ij equals the payoﬀ to a frog of type i in an encounter with
an insect of type j, and another for the insects, A2 = ( a2
ji) whose matrix element
a2
ji equals the payoﬀ to the insect in the same encounter. The dyna mics for the two
populations is now given by
˙p1
i (t) = p1
i (t)[
s2∑
j=1
a1
ijp2
j (t) −
s1∑
k=1
s2∑
j=1
p1
k(t)akj p2
j (t)],
˙p2
j (t) = p2
j (t)[
s1∑
i=1
a2
jip1
i (t) −
s2∑
k=1
s1∑
i=1
p2
k(t)akip1
i (t)]. (4)

6 BORKAR, JAIN AND RANGARAJAN
This is a so called ‘bimatrix game’ and reduces to Eq. (1) with N = 2, when
s1 = s2 = s and A1 = A2 = A.
Our interpretation of Eq. (1), presented earlier, is diﬀeren t from the “multi-
population” interpretation. The index α labels individuals and not populations. In
the multipopulation interpretation, individuals in each p opulation are hard-wired to
be of some speciﬁc genotype, while the composition of the pop ulation is plastic and
subject to selection. For us, the composition of the mixed st rategy proﬁle of each
individual is subject to selection. In the multipopulation interpretation there is no
reason for A1 and A2 to be equal; frogs and insects are quite diﬀerent. However a
single payoﬀ matrix A is natural in the present context if the community consists
of N identical agents (identical in that the payoﬀs to agents in any interact ion de-
pends on the strategies played in that interaction and not on the identity of the
agents). This allows us to study large communities (large N ) without the simulta-
neous proliferation of parameters. To our knowledge, the in terpretation of Eq. (1)
as modelling not N populations but a single community of N identical individuals
is new. While we make use of existing mathematical results fo r Eq. (1), the new
interpretation prompts us to investigate certain other mat hematical properties of
the model which have not received attention. Since Eq. (1) is a generalization of
the replicator dynamics, we will refer to it as the generaliz ed replicator dynamics
(GRD) whereas Eq. (3) will be referred to as pure replicator d ynamics (PRD).
Note that in (4) frogs receive payoﬀs only from insects, not fr om other frogs,
and insects only from frogs, not from other insects. This is b ecause the competition
among the diﬀerent genotypes of frogs happens not directly, b ut indirectly via their
interactions and competition with insects: the more succes sful genotypes among frogs
might be the ones (depending upon the payoﬀ matrix) which do b etter at capturing
insects. Similarly insects do not compete with each other di rectly but only with frogs;
the insect population proﬁle evolves because some insect ge notypes do better than
others at, say, evading frogs. A similar justiﬁcation might be provided for agents
in the present context. A single isolated agent has no compet ition and hence no
motivation to change her strategy proﬁle. There is no direct competition among
the weights of diﬀerent pure strategies within the strategy p roﬁle of a single agent;
this competition and consequent evolution arises indirect ly because of the external
pressure on the agent from the other agents. A ﬁrm that produc es a number of
goods in the economy need not change its production proﬁle if there are no other
producers. But if other producers enter the fray, the ﬁrm may need to change (say,
specialize in the production of a few items), in order to comp ete eﬀectively. This
feature is captured in the model by the exclusion of the β = α term on the r.h.s. of
(1) – agents don’t compete with themselves but with other age nts. We will see later
that this property is important for the emergence of special ization in the model.
A well known example is the “Hawk-dove game” [4], in which the re are two
pure strategies, “hawk” ( i = 1) and “dove” ( i = 2). The payoﬀ matrix elements are
a11 = (g − c)/ 2, a12 = g, a21 = 0, and a22 = g/ 2, with (typically) c > g > 0. In this
game individuals interact pairwise and every interaction i s a competition for some
resource. In an interaction, a hawk always escalates and ﬁgh ts, irrespective of what
the opponent does. A dove “displays”, but retreats if the opp onent escalates. Thus
when hawk meets dove, the dove always retreats and gets zero p ayoﬀ, while the hawk

gains a payoﬀ g from the resource. When dove meets dove, both have equal chan ce of
getting the resource or retreating, hence the average payoﬀ to each party in such an
encounter is g/ 2. When hawk meets hawk, there is a ﬁght, and with equal probab ility
one wins without injury and gains g, while the other retreats with an injury resulting
in a cost c. The average payoﬀ in hawk-hawk encounters to each party is t herefore
(g − c)/ 2. It is instructive to contrast the treatment of this game in PRD and GRD.
In the former, there is a large population of individuals, ea ch hardwired to be pure
hawk or pure dove in every encounter. The fraction of the popu lation that is hawk,
x1, and the fraction that is dove, x2 = 1 − x1, evolves according to (3) in response
to selection pressure and birth/death processes. The point (x1, x 2) = ( g, c − g)/c is
a stable equilibrium of (3), and generically, the populatio n ends up in this attractor,
i.e., with a ratio of hawks to doves being g/ (g−c). In GRD, one would have N agents,
each allowed to play both hawk and dove strategy in an encount er with the respective
probabilities pα
1 and pα
2 . It is not obvious that each agent will end up specializing in
a pure hawk or pure dove strategy, but that is what does happen . A consequence
of one of the theorems to be described later is that the only st able attractor of this
GRD is a conﬁguration where agents tend to distribute themse lves in a pure hawk
or pure dove strategy roughly in the ratio g/ (c − g) for ﬁnite N , and exactly in
this ratio as N → ∞. Thus individual specialization, which was true by assumpt ion
in PRD, is a dynamical outcome in GRD. Moreover, while indivi duals specialize in
their self interest to some pure strategy or the other depend ing upon their initial
conditions, collectively the community seems to obey some g lobal constraints.
3. Deﬁnitions and Notation
Consider
J = {x = (x1, x 2, . . . , x s) ∈ Rs|
s∑
i=1
xi = 1, x i ≥ 0},
which is the simplex of s-dimensional probability vectors. J is the full conﬁguration
space of PRD dynamics and is invariant under it. The conﬁgura tion space for GRD
is J N , the N -fold product. A generic point of J N is p = ( p1, p2, . . . , pN ), each pα
being an s-dimensional probability vector pα (t) = ( pα
1 (t), p α
2 (t), . . . , p α
s (t)) belonging
to J (α ) (the latter being a copy of the simplex J corresponding to agent α ).
A point x ∈ J such that xi = δij for some j is called the jth corner of J. If
an agent α has specialized to the pure strategy j, then pα
i = δij, i.e., pα has gone
to the jth corner of J (α ). If every agent has specialized to some strategy or the
other, the corresponding point in J N will be called a corner of J N and we say that
the community is fully specialized. Note that every corner of J N is an equilibrium
point of GRD, since the r.h.s. of (1) vanishes. Hence we refer to corners as corner
equilibrium points (CEPs).
A CEP can be characterized by an s-vector of non-negative integers n = (n1, n 2, . . . , n s)
where ni denotes the number of agents pursuing the pure strategy i at CEP. There
can be many CEPs with the same n vector. These would diﬀer only in the identity
of the agents at various corners. In this article we will igno re the diﬀerences between
such corners and characterize a CEP by its n-vector alone, since the agents are
identical and diﬀer only in their strategy proﬁle.

8 BORKAR, JAIN AND RANGARAJAN
Consider the following subset of J N : Fk ≡ { p ∈ J N |pα
k = 0 ∀ α } for some ﬁxed
k ∈ S. By deﬁnition, at a point in Fk, every agent has opted out of strategy k. Fk
is also invariant under (1), i.e., if pα
k is zero at some time, it remains zero. At the
“face” Fk, strategy k therefore becomes extinct from the population, and we say
that the full diversity of strategies is lost. As long as the s ystem is not in some Fk,
we say that the community exhibits the “full diversity” of strategies. Note that the
word ‘diversity’, as used here, does not stand for variabili ty among agents, but to
indicate that all strategies are supported. For example we c an have no variation but
full diversity at points p where pα = c ∀ α and none of the components of c are
zero. This is a “homogeneous” point, since all agents are doi ng the same thing.
If a CEP is such that ni ̸= 0 for all i, i.e., each strategy is played by at least one
agent at the CEP, we will refer to it as a fully diversiﬁed CEP or FDCEP. If one
or more ni is zero, the full diversity of strategies is lost and such CEP s are called
non-FDCEPs.
We are interested in studying the circumstances under which FDCEPs are the
preferred attractors of the dynamics, for then, individual specialization and global
diversity will arise dynamically in the community. If it hap pens that the FDCEPs
are attractors and their basins of attraction cover most of J N (all of J N except a
set of lower dimension), then for generic initial condition s the community is bound
to end up in an FDCEP, which means that it will exhibit individ ual specialization
and as well as global diversity.
4. Results
In this section we discuss some results concerning attracto rs of GRD. The proofs of
the theorems are omitted here; these and further results can be found in [5, 6]. We
will discuss the signiﬁcance of these results for specializ ation and diversity in GRD.
4.1. INTERIOR EQUILIBRIUM POINTS
An equilibrium point of GRD is called an interior equilibriu m point (IEP) if none
of the pα
i is zero. We have the following theorem:
Theorem 1: There is at most one isolated IEP. If there is one, it is homoge neous
and is given by pα
i = xi ∀ α, i where xi ≡ ui/ detB, ui is the cofactor of B0i, and B
is the ( s + 1) × (s + 1) matrix (whose rows and columns are labelled by the indice s
0, 1, 2, . . . , s )
B ≡








0 1 1 · · · 1
−1
−1 A
.
.
.
−1








. (5)
A necessary and suﬃcient condition for an isolated IEP to exi st is given by
A1: ui ̸= 0 ∀ i, and all ui have the same sign.
PRD also has an isolated IEP if condition A1 holds, which is then unique and
given by the same xi as given above for GRD. Further, note that at the IEP in GRD,
the system exhibits full diversity since no strategy is opte d out of by any agent.

However, there is no specialization. The above formula for t he IEP in particular
yields the point ( x1, x 2) = ( g, c − g)/c for the hawk-dove game.
4.2. SPECIALIZATION
For generic payoﬀ matrices A, generic initial conditions, and suﬃciently large N ,
we ﬁnd that the system ﬂows into a corner of J N . Thus specialization is a generic
outcome of the dynamics. This observation is based on the fol lowing facts:
1. Theorem 2: [2, 3] Any compact set in the interior of J N or the relative interior
of any face cannot be asymptotically stable. An equilibrium point is asymptot-
ically stable if and only if it is a strict Nash equilibrium. ( A strict Nash equilib-
rium is a point p such that at this point if any single agent unilaterally chan ges
her strategy – unilaterally means that all other agents rema in where they are –
then her payoﬀ strictly decreases.)
2. Theorem 3: Every asymptotically stable attractor must contain at leas t one
corner equilibrium.
3. Numerical W ork: The GRD equation for s = 3 was numerically integrated
using Runge-Kutta method of fourth order. We randomly gener ated ten 3 × 3
payoﬀ matrices and numerically integrated the GRD equation s for long times for
each payoﬀ matrix with ten randomly chosen initial conditio ns. When this was
done with N = 5, in 90 out of the 100 cases the dynamics converged to a corne r.
The remaining 10 cases (all corresponding to a single payoﬀ m atrix) converged
to a heteroclinic cycle. (In these 10 cases the system cycled between regions close
to a few corners, moving rapidly between these regions, and a t every successive
cycle spending increasing amounts of time near the corners a nd coming closer
to them.) When N was increased to 10 for the same ten payoﬀ matrices studied
above, all 100 cases converged to a corner. This suggests tha t the typically,
the stable attractors are corners or heteroclinic cycles, w ith corners becoming
overwhelmingly more likely at larger N .
Why are corners the preferred attractors of this dynamics? W e give here an
intuitive argument, which, though not rigorous or complete , provides some insight.
(For rigorous arguments, refer to the proofs of the above men tioned theorems.) Recall
that according to the dynamics each agent updates her strate gy proﬁle to increase
her payoﬀ in the current environment. Pick an agent α . Her payoﬀ at any point is
P α = ∑
k pα
k cα
k where cα
k ≡ ∑
β ̸=α akj pβ
j . Given a set of s numbers cα
k for a ﬁxed
α , generically one of them will be the largest. Let the largest one be cα
l (for some
particular l). Then it is clear that since the payoﬀ P α is linear in pα
k , the choice
pα
k = δkl will maximize it. Thus, as long as the index of the largest of t he cα
k remains
k = l, the agent α will move towards the pure strategy l. This argument can be
made for any agent. Thus every agent is, at any time, moving to wards some pure
strategy. In this argument it is crucial that cα
k is independent of pα (which it is
because of the exclusion of the β = α term in the payoﬀ to α ). If it were not, then
the nonlinear dependence of P α on pα would have invalidated the argument (as is
the case in PRD, where the analogous quantity ∑
k,j xkakj xj is quadratic in the xi,
and corners are not the generic attractors).

10 BORKAR, JAIN AND RANGARAJAN
This argument also sheds some light on why heteroclinic cycl es could be attrac-
tors. The point is that cα
k are not constants, but depend upon the strategy proﬁles
of agents other than α . If the change in these proﬁles causes some other cα
k (for some
k = l′, diﬀerent from l) to overtake cα
l , then from that time onwards, agent α will
have to change track and move towards pure strategy l′ rather than l.
In particular the above theorems mean that the IEP is always u nstable in GRD.
4.3. COLLECTIVE BEHA VIOUR
We have seen above that the GRD ﬂows to corners generically. T he next step is to
determine which corners the dynamics ﬂows to. For the moment we restrict ours elves
to FDCEPs. Characterizing an FDCEP by its n vector (described in the previous
section), we have the following result:
Theorem 4: Let n and n′ be any two asymptotically stable FDCEPs with N ≥ s.
If condition A1 holds then all components of n − n′ are bounded by a function of
the payoﬀ matrix A alone (and not of N ). Further,
lim
N →∞
ni
N = xi.
Thus, out of a large number (of order N s−1) of FDCEPs all of which are equi-
libria for GRD, only a few are stable. Further, even though th e agents are acting
individually and selﬁshly and going to corners (specializi ng), the system as a whole
retains a memory of the unique interior equilibrium point (w hich is guaranteed to
exist under the conditions of the above theorem) and tunes th e ratios ni/N such
that they are close to the IEP xi values.
One can give a physical or “economic” interpretation of this collective behaviour.
A stable equilibrium is by theorem 2 a strict Nash equilibriu m. Thus it cannot be
advantageous for any agent to switch her pure strategy unila terally. This means that
all agents must receive more or less the same payoﬀ. More preci sely, since a switch
of strategy by a single agent causes changes of O(1) in the payoﬀs to other agents,
at a strict Nash equilibrium it must be the case that diﬀerence s of payoﬀs among
agents could not be larger than O(1), since otherwise it would be possible for some
agent to make an advantageous switch without aﬀecting others . Thus stability is
achieved only at those equilibria at which diﬀerences in payo ﬀ among agents are a
very small fraction of the total payoﬀ to any agent, which is O(N ). This requirement
of “near-equality” of payoﬀs narrows down the set of stable eq uilibria considerably.
As to why the ratio ni/N gets tuned to be close to the IEP, we remark that in both
PRD and GRD, the IEP is characterized by exactly equal payoﬀs t o all strategies.
The above remarks also help explain some of our numerical res ults. When we
increased the number of agents from 5 to 10, we found that all c ases converged
to corners. This is because as N increases, the ratios ni/N can reproduce the xi
values corresponding to IEP more accurately and thereby ach ieve the near-equality
of payoﬀs required for the existence of a strict Nash equilibr ium.
4.4. DIVERSITY AND SELF-ORGANIZATION
We have seen above that among the FDCEPs, only a very small sub set can be
asymptotically stable. Now we consider non-FDCEPs. It coul d happen that along

with an FDCEP, some non-FDCEPs are also stable. In that case, if the community
starts in the basin of attraction of a non-FDCEP, it would eve ntually lose its diver-
sity. We would like to eliminate such attractors of the dynam ics. It turns out that
this can be achieved by imposing certain inequalities on the payoﬀ matrix elements.
Consider the following condition:
A2: The payoﬀ matrix is diagonally subdominant, i.e., aii < a ji∀j ̸= i.
Theorem 5: For s = 2, if A2 holds, then all non-FDCEPs are unstable for
N ≥ 2. For s = 3, if A1, A2 hold, then there exists a positive number N0 depending
on A, such that all non-FDCEPs are unstable for N > N 0.
The condition A2 means that each pure strategy gives more payoﬀ to other
strategies than itself, clearly a tendency that would suppo rt diversity. It is interesting
that the model also has the desirable feature that larger com munities favour diversity.
It will be useful to have conditions for higher s also which make all the non-
FDCEPs unstable. Partial results in this direction are cont ained in [6]. We expect
that for higher s the condition of suﬃciently large N and further inequalities on the
payoﬀ matrix elements would ensure the instability of non-F DCEP.
We now discuss the behaviour of the system when such conditio ns hold. Notice
that since these conditions are inequalities on the payoﬀ ma trix elements (and not
equalities), the behaviour of the system is structurally st able or generic, i.e., is not
destroyed by a small perturbation of the parameters. From th e evidence presented
in section 5.2, the system is expected to go to a corner with ge neric initial condi-
tions. By theorem 5 (and its generalizations to higher s), this corner cannot be a
non-FDCEP. Hence it must be an FDCEP. But then theorem 4 appli es and tells us
that it must be a very speciﬁc corner. At this corner the numbe r of agents ni pursu-
ing the pure strategy i is ﬁne tuned to a value close to N xi with xi determined by
the payoﬀ matrix via Theorem 1, and all agents receive the sam e O(N ) payoﬀ upto
diﬀerences of O(1). The ﬁnal state is ﬁne-tuned but robust in that it arises w ithout
ﬁne-tuning the parameters or the initial state. In this sens e (of spontaneous dynam-
ical ﬁne-tuning) the system exhibits self-organization, a lbeit without any obvious
critical behaviour. Furthermore, this self organization i s of the kind that we were
originally seeking, namely, in which the community exhibit s individual specialization
and global diversity.
4.5. INNOV ATIONS
So far we have considered strategy spaces of a ﬁxed size s. However, the growth of
diversity in the systems mentioned in the introduction has t o do with the appearance
of new strategies and disappearance of some old strategies. We now discuss how the
above considerations of the instability of non-FDCEPs are r elevant for the generation
of innovations. As an example consider a community of N agents which has initially
only two strategies and a payoﬀ matrix A satisfying condition A2. Then all non-
FDCEPs are unstable and only those FDCEPs with n = (n1, n 2) such that the ratios
ni/N close to xi of the IEP are stable. Let us assume that the system has settle d
into such a state. Now, assume that a new, third strategy aris es which enlarges the
2 × 2 matrix A into a 3 × 3 matrix A′ that contains A as a submatrix. The third
row and column of A′ represent the relationship of the new strategy with respect
to the old – how much payoﬀ it gives to them and receives from th em. At the time

12 BORKAR, JAIN AND RANGARAJAN
this strategy arises, the system is in the state ( n1, n 2, 0) since the third strategy
(being new) is as yet unpopulated. Note that this state is a no n-FDCEP for s = 3.
Now if A′ is such that it satisﬁes the conditions A1, A2 in Theorem 5, and N is
suﬃciently large, then the above state is unstable. Therefo re, any small perturbation
in which the agents start exploring the new strategy ever so s lightly will destroy the
old state and take the system to a new stable state which must b e an FDCEP with
3 strategies. By Theorem 4 this state will be ( n′
1, n ′
2, n ′
3) with
N (n′
1, n ′
2, n ′
3) ≈ (x′
1, x ′
2, x ′
3)
where x′
i are the components of the IEP corrsponding to the payoﬀ matri x A′ and are
all non-zero since A′ satisﬁes A1. Thus the innovation has destabilized the previous
state of the system and brought it to a new state where a ﬁnite f raction of the
population has adopted strategy 3. In such a case, we say that the innovation has
been “accepted” by the community. Note that the only require ments for this to
happen is that the elements of the new row and column in the pay oﬀ matrix satisfy
certain inequalities with respect to the existing matrix el ements (contained in the
conditions A1, A2 to be satiﬁed by A′) and that the community be suﬃciently
large. This tells us what properties a new strategy should ha ve in the context of
already existing activities in order for it to be “accepted” by the community. Thus
the model suggests a natural mechanism for the emergence of c ontext dependent
innovations in the community.
5. Conclusions
To summarize, Generalized Replicator Dynamics, eq. (1), is a nonlinear dynami-
cal model of learning for a community of N mutually interacting agents with the
following features:
1. Each agent is selﬁsh and exhibits bounded rationality.
2. This is a non-cooperative game and there is no global organ izing agency at
work. It is in general a non-hamiltonian system.
3. Specialization of individual agents to pure strategies i s a generic outcome of the
dynamics.
4. Under certain generic conditions on the payoﬀ matrix para meters the agents ex-
hibit a collective behaviour, and for suﬃciently large N , the community exhibits
diversity and self-organization.
5. A ‘good’ innovation (one that satisﬁes conditions A1, A2 , etc., with respect to
the exisiting strategies) makes the society unstable and ev olve until the inno-
vation is accepted.
It is noteworthy that in this dynamical system, order is gene rated at large N
(unlike the systems where increasing the number of degrees o f freedom makes the
system less orderly, in some sense). This order is not the usu al statistical mechanical
kind of order, the order of appropriately deﬁned macroscopi c variables, but an order
in the original dynamical variables themselves. However, i n another sense, this order
is also statistical since we do not know which pure strategy a n individual agent
follows. We only know about the fraction of agents pursuing a given strategy.

Acknowledgement: SJ acknowledges support from the Jawaharlal Nehru Centre
for Advanced Scientiﬁc Research, Bangalore, and the Associ ateship of the Interna-
tional Centre for Theoretical Physics, Trieste.
Email addresses :
borkar@csa.iisc.ernet.in, jain@cts.iisc.ernet.in, ran garaj@math.iisc.ernet.in
References
1. Taylor, P. and Jonker, L. (1978) Evolutionarily stable st rategies and game dynamics, Math. Bio-
sciences, 40, 145-156; Hofbauer, J., Schuster, P. and Sigmund, K. (1979) A note on evolutionarily
stable strategies and game dynamics, J. Theor. Biol. , 81, 609-612; Gadgil, S., Nanjundiah, V.
and Gadgil, M. (1980) On evolutionarily stable composition s of populations of interesting geno-
types, J. Theor. Biol. , 84, 737-759; Schuster, P. and Sigmund, K. (1983) Replicator dy namics,
J. Theor. Biol. , 100, 533-538.
2. Hofbauer, J. and Sigmund, K. (1988) The Theory of Evolution and Dynamical Systems . Cam-
bridge University Press, Cambridge.
3. Weibull, J. W. (1995) Evolutionary Game Theory . MIT Press, Cambridge.
4. Maynard-Smith, J. (1972) Game theory and the evolution of ﬁghting, in Maynard-Smith, J. On
Evolution. Edinburgh University Press.
5. Borkar, V. S., Jain, S. and Rangarajan, G. (1998) Dynamics of individual specialization and
global diversiﬁcation in communities, Complexity, Vol. 3, No. 3, 50-56.
6. Borkar, V. S., Jain, S. and Rangarajan, G. (1997) Generali zed replicator dynamics as a model
of specialization and diversity in societies, preprint num ber IISc-CTS-11/97.


Dynamics of Individual Specialization and Global Diversification in Communities

Abstract
We discuss a model of an economic community consisting of $N$ interacting agents. The state of each agent at any time is characterized, in general, by a mixed strategy profile drawn from a space of $s$ pure strategies. The community evolves as agents update their strategy profiles in response to payoffs received from other agents. The evolution equation is a generalization of the replicator equation. We argue that when $N$ is sufficiently large and the payoff matrix elements satisfy suitable inequalities, the community evolves to retain the full diversity of available strategies even as individual agents specialize to pure strategies.

arXiv:adap-org/9804004v1 30 Apr 1998
DYNAMICS OF INDIVIDUAL SPECIALIZATION AND GLOBAL
DIVERSIFICATION IN COMMUNITIES ∗
Vivek S. Borkar †
Department of Computer Science and Automation,
Indian Institute of Science, Bangalore 560 012, India
Sanjay Jain ‡
Centre for Theoretical Studies,
Indian Institute of Science, Bangalore 560 012, India
Govindan Rangarajan §
Department of Mathematics and Centre for Theoretical Studi es,
Indian Institute of Science, Bangalore 560 012, India
Abstract
We discuss a model of an economic community consisting of N interacting agents.
The state of each agent at any time is characterized, in general, by a mixed strategy
proﬁle drawn from a space of s pure strategies. The community evolves as agents
update their strategy proﬁles in response to payoﬀs received fro m other agents. The
evolution equation is a generalization of the replicator equation. We a rgue that when
N is suﬃciently large and the payoﬀ matrix elements satisfy suitable ineq ualities, the
community evolves to retain the full diversity of available strategies even as individual
agents specialize to pure strategies.
∗Journal Ref: Complexity, Vol. 3, No. 3, 50-56, (1998).
†E-mail: borkar@csa.iisc.ernet.in
‡Also at Jawaharlal Nehru Centre for Advanced Scientiﬁc Rese arch, Bangalore 560 064, and
Associate Member of ICTP, Trieste. E-mail: jain@cts.iisc.ernet.in
§E-mail: rangaraj@math.iisc.ernet.in

I. INTRODUCTION
One of the striking phenomena exhibited by a wide variety of complex a daptive systems
is that individual agents or components of the system evolve to per form highly specialized
tasks, and at the same time the system as a whole evolves towards a greater diversity in terms
of the kinds of individual agents or components it contains or the ta sks that are performed in
it. Some examples of this include living systems which have evolved incre asingly specialized
and diverse kinds of interacting protein molecules, ecologies which de velop diverse species
with specialized traits, early human societies which evolve from a stat e where everyone
shares in a small number of chores to a state with many more activitie s performed largely
by specialists, and ﬁrms in an economic web that explore and occupy in creasingly specialized
and diverse niches.
In this paper we study a mathematical model of economic communitie s that exhibits these
twin evolutionary phenomena of specialization and diversity. The sys tem is a community of
N (say, human) agents. There are s strategies or activities each agent can perform labelled
by i ∈ S ≡ {1, 2, . . . , s }, and at the time t the agent α (α = 1, . . . , N ) performs the activity
i with a probability pα
i (t) (thus
∑ s
i=1 pα
i (t) = 1 ∀ α, t ). The vector pα (t) = ( pα
1 (t), . . . , p α
s (t))
is called the mixed strategy proﬁle of agent α at time t. If pα
i (t) = δij for some j ∈ S then
the agent α is said to pursue the pure strategy j or to have ‘specialized’ in the strategy j at
time t. The set of vectors pα , α = 1 , . . . , N constitute the basic degrees of freedom of the
model. The dynamics is deﬁned by the equation
˙pα
i (t) = pα
i (t)

 ∑
β ̸=α
∑
j
aijpβ
j (t) −
∑
β ̸=α
∑
k,j
pα
k (t)akjpβ
j (t)

 , 1 ≤ α ≤ N, 1 ≤ i ≤ s, (1.1)
which determines the rate of change of an individual pα in terms of the current mixed
strategy proﬁles of all the agents and the payoﬀ matrix A = [[aij]].
We motivate this model as follows: Agents interact with each other o n a short time scale,
receive payoﬀs based on each other’s activity, and update their ind ividual strategy proﬁles
on a longer time scale so as to increase their payoﬀs. As is usual in gam e theory, aij denotes
the payoﬀ received by an agent pursuing a pure strategy i in a single interaction with an
agent pursuing the pure strategy j. Then the average payoﬀ received by the agent α from the
rest of the community in the period t to t+ ∆ t is proportional to ∆ t ∑
β ̸=α
∑
k,j pα
k (t)akjpβ
j (t).
This assumes that every agent interacts equally often with all othe r agents and that there
is a separation of time scales: ∆ t can be chosen long enough for there to be a statistically
suﬃcient number of interactions during the period, yet short enou gh that the change in the
strategy proﬁles during this period can be ignored in the computatio n of the average payoﬀ.
If α had played the pure strategy i in this period, she would have received an average payoﬀ
proportional to ∆ t
∑
β ̸=α
∑
j aijpβ
j (t). The agent α increments pα
i by an amount proportional
to pα
i as well as to the diﬀerence between the average payoﬀ she would ha ve got in this
interval if she had pursued a pure strategy i and the average payoﬀ she actually received:
∆ pα
i = c∆ tpα
i [
∑
β ̸=α
∑
j aijpβ
j (t) −
∑
β ̸=α
∑
k,j pα
k (t)akjpβ
j (t)], where c is a constant. Equation
(1.1) follows upon dividing by ∆ t, taking the limit, and rescaling time by a factor c. By
construction, each agent makes a positive change in the weight of s trategy i in her own
strategy proﬁle if she perceives that the pure strategy i would give a higher payoﬀ in the

current environment than her current strategy proﬁle, and a ne gative change if it were to
give a lower payoﬀ.
Eq. (1.1) is nothing but the ‘multipopulation replicator equation’ discu ssed in [1] (and
references therein). There each α represents a population, and pα
i represents the fraction
of individuals in the population α pursuing the strategy i. Since for us each α represents
an individual and not a population, we refer to dynamics speciﬁed by ( 1.1) as simply the
generalized replicator dynamics (GRD). By contrast the replicator dynamics (which we
hereafter refer to as the ‘pure replicator dynamics’ (PRD)) is give n by (see [2])
˙xi(t) = xi(t)[
∑
j
aijxj(t) −
∑
k,j
xk(t)akj xj(t)], i = 1, . . . , s. (1.2)
This is a standard model in evolutionary biology describing the growth and decay of s species
under selection pressure with xi representing the fraction of the ith species in the population.
PRD and its variants are also extensively studied in economics in game t heory as models
for dynamical selection of equilibria (see, e.g., [3] [4]). Its generalizat ions have also been
studied in the context of the emergence of organizations in complex adaptive systems (see
[5] and references therein). For extensive accounts of more rec ent contributions to PRD and
further references, see the recent books [6,7].
We view GRD as a model of learning in a community of N interacting agents. The
agents are identical in that each is capable of pursuing the same set of strategies with the
same payoﬀs. This is a non-cooperative game in which the agents act selﬁshly (each is
concerned with increasing her own payoﬀ without consideration of im pact on others or the
community), and exhibit bounded rationality (no anticipation of othe rs’ strategy, merely a
response to the current aggregate behaviour of others). Ther e is no global organizing agency
at work, individual actions alone are responsible for the evolution of the system.
Nevertheless, we shall argue that the community as a whole seems t o exhibit a kind of
global organization under certain circumstances. Individual agen ts tend to specialize, while
the community as a whole retains its diversity, i.e., each pure strateg y is pursued by some
agent or the other. We attempt to ﬁnd conditions on the paramete rs of the model (the size
N of the community and the s × s payoﬀ matrix A) such that this behaviour occurs. While
most of the time we work with a strategy space of a ﬁxed size (and re fer to diversity as the
maintenance of all strategies in this ﬁxed size space) the results als o have bearing on the
conditions under which new strategies can enter the community.
Section 2 sets the notation and discusses some relationships betwe en PRD and GRD. In
section 3 we identify conditions under which attractors of GRD can e xhibit simultaneously
specialization and diversity, and characterize these attractors q uantitatively. Section 4 sum-
marizes the results, discusses their possible signiﬁcance and outline s some open questions.
Due to constraints on space, proofs for some of the results have not been included in this
paper. These and other generalizations of our results are the sub ject of a detailed follow-up
paper [8].
II. RELA TIONSHIPS BETWEEN GRD AND PRD; INTERIOR EQUILIBRIA

A. GRD preliminaries
Notation, deﬁnition of specialization and diversity
Let J denote the simplex of s-dimensional probability vectors:
J = {x = (x1, · · · , x s)T ∈ Rs|
s∑
i=1
xi = 1, x i ≥ 0}. (2.1)
J is the full conﬁguration space of PRD, and is invariant under it.
The conﬁguration space of GRD will be denoted J N = Π N
α =1J (α ) where J (α ) is a copy
of J for the α th agent. A point of J N will be denoted p = ( p1, p2, . . . , pN ), where pα =
(pα
1 , p α
2 , . . . , p α
s ) ∈ J (α ). J N is invariant under GRD, as the norm of every pα is preserved
under (1.1).
A point of J N at which every agent has specialized to some strategy or the other will
be referred to as a corner of J N , and at such a point we say that the community is ‘fully
specialized’. It is evident that a corner is an equilibrium point of (1.1) sin ce ˙pα
i vanishes if pα
i
does and (1.1) preserves norm, hence we often refer to a corner as a ‘corner equilibrium point’
or CEP. A CEP can be characterized by an s-vector of non-negative integers n = (n1, . . . , n s)
where ni is the number of agents pursuing the pure strategy i at the CEP, 1 ≤ i ≤ s (thus∑
i ni = N). Two CEPs with the same associated n vector are interchangeable, since they
diﬀer only in the identity of the agents, irrelevant for our purposes .
The set Fk ≡ { p ∈ J N |pα
k = 0 ∀ α } for any k ∈ S is the subset of the boundary of J N
where all agents have opted out of strategy k. At the ‘face’ Fk, strategy k becomes extinct
from the population and the full diversity of strategies is lost. The c ommunity will be said
to exhibit ‘diversity’ at all points that do not belong to some Fk. Note that we use the
word ‘diversity’ not to signify the variation between individual agent s, but to indicate that
all strategies are supported. Indeed we can have no variation but full diversity if all agents
pursue the same mixed strategy: for all α , pα = c ∈ J ◦. (The superscript ◦ for any set
denotes its relative interior.) When pα is independent of α , the community is completely
‘homogeneous’ since all agents are doing the same thing. The commu nity can be fully
specialized and diversiﬁed at the same time: each agent chooses a pu re strategy and every
strategy is chosen by some agent or the other. This corresponds to CEP with n such that
each ni is nonzero, which will be called a ‘fully diversiﬁed’ CEP or FDCEP. By cont rast,
CEP where one or more strategies becomes extinct (some compone nts of n are zero) will be
called non-FDCEP.
In this paper we are primarily interested in studying the circumstanc es in which FDCEP
are the preferred attractors of the dynamics, since in that case individual specialization and
global diversity will arise dynamically in the community.
Diﬀerences between PRD and GRD
If the initial point of a trajectory in GRD is homogeneous, the traje ctory remains homoge-
neous for all time, and evolves according to (1.2) except that the t ime is speeded up by a
factor of N − 1. The sum ¯xi ≡ (1/N ) ∑ N
α =1 pα
i equals the probability that strategy i is being
played in the entire community, and is therefore the analogue of xi in PRD. We can ask how
¯xi evolves in GRD. It is easy to see that

˙¯xi = N[¯xi
∑
j
aij ¯xj −
∑
k,j
xikakj ¯xj − 1
N
∑
j
xijaij + 1
N
∑
k,j
xikjakj ], (2.2)
where xik ≡ (1/N ) ∑
α pα
i pα
k and xikj ≡ (1/N ) ∑
α pα
i pα
k pα
j . The r.h.s. of (2.2) is not propor-
tional to the r.h.s. of (1.2), except for homogeneous trajectorie s in which case xik = ¯xi ¯xk,
xikj = ¯xi ¯xk ¯xj. Thus in general ¯ xi does not follow the PRD. One might have hoped that
when the number of agents N is large ¯xi follows PRD, but even that is not the case due to
variation among the agents. For example, at the corner n, the diﬀerence between xik and
¯xi ¯xk is ni(N δij − nj)/N 2, which is comparable to the former two even for large N (except
for homogeneous corners).
One of our results in this paper is that even though variation among a gents, which
is generic in GRD, causes the evolution of ¯ xi to be diﬀerent from PRD, under suitable
conditions ¯xi nevertheless converges to the interior equilibrium point of PRD.
B. The interior equilibria of GRD
Consider an interior equilibrium point (IEP) p of (1.1). By deﬁnition no pα
i is zero in the
interior of J N . Therefore the bracket [ ] on the r.h.s. of (1.1) must vanish for all α, i . Deﬁne
xα
0 ≡
∑
β ̸=α
∑
i,j pα
i aijpβ
j , and vα
i ≡
∑
β ̸=α pβ
i . Then
∑ s
i=1 vα
i = N − 1 ∀ α , and the interior
equilibrium condition can be written as
BXα = (N − 1)E0 ∀ α, (2.3)
where Xα ≡ (xα
0 , v α
1 , v α
2 , . . . , v α
s )T , B is the s + 1-dimensional matrix
B =









0 1 1 · · · 1
−1
−1 A
.
.
.
−1









, (2.4)
and E0 is the s + 1-dimensional unit vector (1 , 0, 0, · · ·, 0)T .
It is not diﬃcult to see (details in [8]) that (2.3) has an isolated solution if and only if
the following condition holds:
A1: ui ̸= 0 ∀ i, and all ui have the same sign, where ui denotes the co-factor of B0i.
For the sake of notational simplicity, we have denoted the cofacto r of B0i by ui instead
of u0i thereby suppressing the ﬁxed ﬁrst index. Under the above condit ion ( A1) det B =∑ s
i=1 ui ̸= 0, and the solution is unique and given by vα
i = (N − 1)xi ∀ α . Here,
xi = ui/ detB (2.5)
is nothing but the ith coordinate of the unique isolated interior equilibrium point of PRD.
(Note that A1 is also the necessary and suﬃcient condition for PRD to have an isolat ed IEP,
which, if it exists, is unique.) Since pα
i − pβ
i = vβ
i − vα
i = 0, it follows that the equilibrium
point is homogeneous and given by pα
i = xi. Thus we have proved
Theorem 2.1 There exists at most one isolated equilibrium in the interior of J N . It exists
if and only if A1 is satisﬁed and then it is homogeneous (all agents pursue the same m ixed
strategy), and coincides with the isolated interior equilibrium point of PRD, pα
i = xi ∀ α, i .

III. CORNER EQUILIBRIA OF GRD: DIVERSIFICA TION WITH
SPECIALIZA TION
A. Stability of corner equilibria
The IEP of GRD is always unstable to small perturbations. This is a con sequence of the
following theorem proved in [1]:
Theorem 3.1 An equilibrium point of (1.1) is asymptotically stable if and only if it is
a strict Nash equilibrium. Further, any compact set in the relative int erior of a face cannot
be asymptotically stable.
Note that strict Nash equilibria are perforce pure strategy Nash e quilibria and therefore
correspond to CEP. As a consequence of this theorem, a traject ory either eternally moves
around in the relative interior of some face or the interior of J N coming arbitrarily close to
its boundaries and corners (the case of non-compact attractor ), or it converges to a corner of
J N . It is possible to construct payoﬀ matrices for which there are no a symptotically stable
corners in J N , whereupon the former situation obtains. However, our numerica l work with
3 × 3 payoﬀ matrices suggests that this happens rarely (i.e., in a relative ly small region of
R3×3); for most payoﬀ matrices asymptotically stable corners do exist f or most values of
N. Further, we randomly generated ten 3 × 3 payoﬀ matrices and numerically integrated
the GRD equations for long times for each payoﬀ matrix with ten rand omly chosen initial
conditions. When this was done with N = 5, in 90 out of the 100 cases the dynamics
converged to a corner. With N = 10, all 100 cases converged to a corner. This suggests
that typically, at large N, not only do asymptotically stable corners exist, but also that
their basins of attraction cover most of J N . Thus corners seem to be the most common
attractors in GRD. These are numerical indications and need to be m ade more precise. In
our interpretation of the model, a corner corresponds to a fully sp ecialized community. The
above theorem and numerical evidence therefore suggest that s pecialization of all the agents
is the most common outcome in GRD.
At the CEP n, the payoﬀ to an agent playing the jth pure strategy from the other N − 1
agents is
Pj =
s∑
k̸=j
ajknk + (nj − 1)ajj =
s∑
k=1
ajknk − ajj = Pj − ajj , (3.1)
where Pj ≡
∑ s
k=1 ajknk. If this agent were to suddenly switch to the ith pure strategy ( i ̸= j),
all other agents remaining at their respective pure strategies, th en for this agent the payoﬀ
would change to ∑ s
k̸=j aiknk − aij(nj − 1) = Pi − aij. Thus the increase in payoﬀ for an agent
playing the jth pure strategy at the FDCEP n (and this assumes nj ̸= 0) in switching to
the ith pure strategy is
λ ij = Pi − Pj − hij, h ij ≡ aij − ajj . (3.2)
Therefore, n is a strict Nash equilibrium if for every j such that nj ̸= 0, the conditions
λ ij < 0 (3.3)

are satisﬁed for all i ̸= j. At a FDCEP, all nj are nonzero and this is a set of s(s − 1)
conditions. At a non-FDCEP where only s′ < s components of n are nonzero, the number
of conditions is smaller, s′(s − 1).
From Theorem 3.1, these are identical to the conditions for the asy mptotic stability of
the FDCEP’s associated with n. In fact, one can show that λ ij given by (3.2) are precisely
the eigenvalues of the Jacobian matrix of (1.1) linearized around a co rner of J N characterized
by n.
B. Stability of fully diversiﬁed corners
Theorem 3.2 : Let PRD admit an isolated IEP x. That is, condition A1 holds [cf.
Section II]. Let n, n′ be any pair of asymptotically stable FDCEP’s of GRD with N ≥ s.
Then
(i) all components of the diﬀerence n′ − n are bounded by a function of A alone, not of N,
and
(ii) lim N →∞
ni
N = xi.
The proof is given in Appendix A.
The signiﬁcance of this theorem is that it characterizes the FDCEP t hat are attractors
of the dynamics. If the community is going to end up in a fully specialized and diversiﬁed
conﬁguration, the theorem quantiﬁes the relative weights of all st rategies that will obtain in
that conﬁguration: these relative weights are forced to be ‘close’ to the IEP conﬁguration
given by (2.5). The theorem does not guarantee the existence of a stable FDCEP. One can
prove the existence of an inﬁnite set of values of N at which stable FDCEP are guaranteed
to exist under the conditions of the theorem. One can also identify s uﬃcient conditions for
the existence of stable FDCEP for any N ≥ s. These are presented in [8].
C. Instability of non-fully diversiﬁed corners
We would like to deﬁne GRD as possessing diversity if all trajectories in the faces Fk
become unstable at some time or the other with respect to perturb ations that take them away
from these faces. With this in mind we now study corners at which one or more strategies
become extinct and determine the conditions under which all such co rners become unstable.
Then under small perturbations the population will dynamically ﬂow ou t of such corners,
eliminating specialized conﬁgurations that do not carry the full diver sity of strategies. As
remarked earlier, the number of conditions to be satisﬁed by a non- FDCEP to be stable is
less than the number to be satisﬁed by a FDCEP. Thus a priori, things seem to be loaded
against diversiﬁcation. As we shall see, some further structure w ill need to be imposed on
A in order to make the non-FDCEP unstable. At this point we do not hav e the general
conditions for arbitrary s, but some insight gleaned from special cases s = 2, 3.
s = 2, N arbitrary
In this case conditions (3.3) can be studied exhaustively. There are generically four cases.

Case 1: a11 > a 21 and a22 > a 12: Both ( N, 0) and (0 , N ) are asymptotically stable, other
corners are not.
Case 2: a11 > a 21 and a22 < a 12: ( N, 0) is the only asymptotically stable corner.
Case 3: a11 < a 21 and a22 > a 12: (0 , N ) is the only asymptotically stable corner.
Case 4: a11 < a 21 and a22 < a 12: The only asymptotically stable corners ( n1, n 2) (with
n1 + n2 = N) are those for which n1 ̸= 0, n2 ̸= 0, and furthermore, (n2−1)
n1
h12 < h 21 <
n2
(n1−1)h12) if n2 < N − 1, and (n2−1)
n1
h12 < h 21 if n2 = N − 1.
Cases 2 and 3 correspond to dominated strategies. (The cases wit h one or more equalities
instead of inequalities have been disregarded as nongeneric. In any case, they are not diﬃcult
to handle.) The case of interest to us is the last one, which shows dive rsiﬁcation. It is
convenient to introduce the
Deﬁnition : A is diagonally subdominant if aii < a ji ∀ j ̸= i, {i, j } ⊂ S.
That is, hij > 0 ∀ i ̸= j. From the above exhaustive list it follows that the condition
A2: A is diagonally subdominant,
is the necessary and suﬃcient condition for non-FDCEP to be unsta ble (for generic A). If
A2 is satisﬁed, the only asymptotically stable CEP are the FDCEP, for wh ich Theorem 3.2
applies. (Note that for s = 2, A2 implies A1, the IEP is given by pα = 1
h12+h21
(h12, h 21) for
all α , and the inequalities involving n in Case 4 above are equivalent to the statement that
(1/N )(n1, n 2) must be close to this IEP for arbitrary N and converge to it as N → ∞.)
Note that GRD remains invariant under addition of an arbitrary cons tant to any column
of the payoﬀ matrix. Thus we may replace aij by hij, thus obtaining a matrix which under
A2 has zero diagonal elements and nonnegative oﬀ-diagonal elements . It is interesting that
these conditions also arise in PRD in the context of population genetic s and ecological
models [9] as well as in models of catalytic networks of chemically react ing molecules [10].
s = 3, N arbitrary
For s = 3, A2 no longer implies A1; the latter is an independent condition. We now state
Theorem 3.3 For s = 3, if both A1 and A2 hold, then there exists a positive number N0
depending on A such that for all N > N 0, all non-FDCEP are unstable.
The proof of this theorem can be found in Appendix B.
We remark that while N0 is ﬁnite, it may, depending upon A, be much larger than three.
The above result can be further generalized (with the imposition of a n additional condition)
to prove that for s = 3 all points in Fk are unstable for suﬃciently large N [8]. Note that
our notion of diversity for GRD is related to the notions of ‘permanen ce’, ‘persistence’, etc.
introduced for PRD (see [2] and references therein). PRD is said to exhibit permanence if
every interior solution has components that remain bounded away f rom zero by a common
constant δ > 0. Strong persistence, in turn, is the weaker requirement where δ is trajectory
dependent and persistence the even weaker requirement that ea ch component of an interior
trajectory not converge to zero. The biological implications are ob vious: the concept is
clearly related to survival of species. The corresponding phenome non here is the survival of
policies. The conditions for, e.g., permanence in PRD (see [2]) may quite generally play a
role in discussions of diversity in GRD.

IV. DISCUSSION AND CONCLUSIONS
To summarize:
(i) We have considered the equation (1.1) as a model of evolution of a community of
N agents, each agent being capable of performing any mix of a set of s strategies, and
modifying her mix depending upon the payoﬀ received from other age nts. We have studied
some properties of the attractors of this system to gain insight on how the community is
expected to evolve.
(ii) This model can exhibit specialization of the agents into pure strat egies. Evidence
for this comes from the previously known Theorem 3.1, supported w ith our numerical ob-
servations. While individual specialization seems to be the most commo n outcome in this
model, it would be interesting to characterize more precisely the circ umstances in which
specialization is guaranteed, i.e., when corners are the only attract ors, and when not.
(iii) We have shown that under suitable conditions, while each agent sp ecializes to a single
pure strategy, it is guaranteed that the community as a whole pres erves the full diversity
of strategies. These are that the community be suﬃciently large ( N should be larger than
a number N0 that depends upon the payoﬀ matrix), and the payoﬀ matrix itself s hould
satisfy A1 (existence of an isolated interior equilibrium point) and A2 (diagonal entries
of A be smaller than other entries in the same column). These guarantee (for upto three
strategies) that all corners where one or more strategy become s extinct are unstable to small
perturbations (Theorem 3.3). To identify suﬃcient conditions for la rger s (and necessary
and suﬃcient conditions for s ≥ 3) is a task for the future. The appearance of a lower limit
on the size of the community in this context (which could be much large r than the number
of strategies) is interesting.
(iv) Within the set of conﬁgurations where the community would exhib it full specializa-
tion and diversity (the FDCEP), we have given a quantitative criterio n as to which ones
will be the attractors (Theorem 3.2). ni/N (where ni is the number of agents pursuing the
pure strategy i at the attractor) is forced to be close to xi and equal to it in the large N
limit, where xi is given by (2.5) and is the relative weight of the ith strategy at the interior
equilibrium point of PRD. This constraint is a consequence of the ﬁne b alance that exists
for every agent at a strict Nash equilibrium; any strategy switch fo r any agent reduces her
payoﬀ. This ﬁne tuning, caused by the interaction of the agent with other agents, is a kind
of organization exhibited by the system.
The conditions for the instability of non-FDCEP (Theorem 3.3) may als o be relevant to
the question: when does a society accept an innovation? For consid er a community of a
large number of agents but with only two strategies, 1 and 2, at a st able corner where n1
agents pursue the pure strategy 1 and n2 = N − n1 agents the pure strategy 2 (neither n1
nor n2 is zero). Since this corner is assumed stable, the 2 × 2 matrix A satisﬁes condition
A2 (diagonal subdominance). Now imagine that a new strategy 3 arises thereby enlarging
the payoﬀ matrix to a 3 × 3 matrix A′ containing A as a 2 × 2 block. In the new context the
earlier state of the community will be described by a three vector n = (n1, n 2, 0), which is in
the face F3. Now if the new payoﬀ matrix satisﬁes A1,A2, and N is suﬃciently large, then
from Theorem 3.3, this conﬁguration is unstable with respect to per turbations in which one
of the agents begins to explore the new strategy. Thus if this agen t were to explore the new
strategy ever so slightly, her payoﬀ would increase and a small pert urbation of the community

would grow until it settles down in another attractor. The new attr actor if described by
Theorem 3.2 would have the property that a ﬁnite fraction of the po pulation pursues the
new strategy: the innovation has been accepted by the society. T hus the conditions A1,
A2 of Theorem 3.3 indicate what the payoﬀs of a new strategy (innovat ion) should be with
respect to the existing ones, if the innovation is to be guaranteed a cceptance. (Conditions
that are both necessary and suﬃcient for diversity would consider ably strengthen the above
remarks.)
It is worth mentioning that conditions A1, A2 are not equalities but inequalities. Thus
there is no ﬁne tuning of parameters needed; the behaviour discus sed above emerges when-
ever parameters cross certain thresholds.
It may be interesting to consider the ‘economic signiﬁcance’ of cond itions which play an
important role in preserving the full diversity of strategies. For ex ample, diagonal subdomi-
nance, when translated as ‘each pure strategy gives more payoﬀ t o other pure strategies than
to itself’, carries a shade of an ‘altruism’ of sorts (at the level of st rategies, not individuals).
Note that PRD with a payoﬀ matrix in which diagonal entries are zero a nd oﬀdiagonal ones
greater than or equal to zero is called a ‘catalytic network’ [2]. The g eneral message might
be that if the initial set of allowed strategies is chosen with the ‘right v ision’ (read ‘right
payoﬀs’), then, even a community of identical and selﬁsh individuals, if large enough, will
exhibit diversity and accept only the ‘right’ innovations.
APPENDIX A
The proof of Theorem 3.2 in Section IIIB follows:
Proof: Note that Pi − Pj ﬁgures in both λ ij and λ ji. Therefore the s(s − 1) conditions
(3.3) can be written in terms of s(s − 1)/ 2 “double-sided” inequalities
− hji < P i − Pj < h ij. (4.1)
Deﬁne zi ≡ Pi − Pi+1 for i = 1 , . . . , s , with Ps+1 ≡ P1. Then zi = ∑ s
j=1 cijnj with cij ≡
aij − ai+1,j , where it is again understood that as+1,j ≡ a1j. Now, since all the nj are not
independent, let us express zi in terms of only n1, . . . , n s−1 by eliminating ns = N −(n1+· · ·+
ns−1). This gives zi = yi + cisN where yi ≡ ∑ s−1
j=1 dijnj and dij ≡ cij − cis, i, j = 1, . . . , s − 1.
With this notation, consider the subset of s − 1 inequalities obtained by setting j = i + 1 in
(4.1), with i = 1, . . . , s − 1. These involve zi and take the form
− hi+1,i − cisN < y i < h i,i +1 − cisN, i = 1, . . . , s − 1. (4.2)
These inequalities mean that for any stable FDCEP n, the yi, which are linear combinations
of n1, . . . , n s−1, are constrained to be in an open interval of the real line. While the lo cation
of this interval is N dependent, it follows from (4.2) that the size of this interval is ﬁnite ,
independent of N, and depends only on the payoﬀ matrix (for yi the size of the interval is
hi+1,i + hi,i +1).
If the s − 1 dimensional matrix D = (dij) has an inverse, we can invert yi ≡ ∑ s−1
j=1 dijnj to
express the nj in terms of yi. Then, (4.2) will get converted into inequalities for n1, . . . , n s−1.
Since the vector ˜y = (y1, . . . , y s−1) in the s−1 dimensional cartesian space whose axes are the
yi is constrained by (4.2) to lie in a (rectangular) parallelepiped, the vec tor ˜n = (n1, . . . , n s−1)

in the s − 1 dimensional cartesian space whose axes are the ni will also lie in a (in general
oblique) parallelepiped which is the image, under D−1, of the rectangular parallelepiped
in y-space deﬁned by (4.2). Again, while the location of the parallelepiped in n1, . . . , n s−1
space will depend upon N, its size, i.e., its extent along any of the coordinate axes, will be
independent of N. This is because the matrix D−1, if it exists, depends only on the payoﬀ
matrix and not on N. Therefore, if D−1 exists, the diﬀerences in ni, i = 1 , . . . , s − 1 for
all FDCEP are bounded by some function of A alone, not of N. The same is true for ns
also since
∑ s
i=1 ni = N. One can show [8] that det D = det B. The existence of D−1 is thus
guaranteed by condition A1, completing the proof of the ﬁrst part of Theorem 3.2.
To prove the second part of Theorem 3.2, divide all sides of (4.1) by N and take the
limit N → ∞ . This yields lim N →∞[ Pi
N − Pj
N ] = 0. Deﬁning x′
i ≡ limN →∞(ni/N ) along an
appropriate subsequence independent of i, this is equivalent to the statement that ∑ s
k=1 aikx′
k
is independent of i, which implies that x′ is the same as x, the IEP of PRD. ✷
APPENDIX B
The proof of Theorem 3.3 in Section IIIC follows:
Proof: Any non-FDCEP n must belong to some Fk, in this case to F1, F2 or F3. For
every i such that the component ni of n ∈ Fk is nonzero, consider the eigenvalue
λ ki = Pk − Pi − hki. (4.3)
From the discussion of Eq. (3.2) it follows that if any one (or the large st) of the λ ki at n
is greater than zero, then the CEP n is unstable against perturbations in which an agent
pursuing the pure strategy i moves towards strategy k (i.e., the perturbations which restore
the extinct strategy k will then grow).
For concreteness consider F3. Corners of F3 are of two types.
Case 1: Only one strategy survives at the corner. Then n = ( N, 0, 0) or (0 , N, 0). In the
former case (4.3) implies λ 31 = (N − 1)h31 and in the latter case λ 32 = (N − 1)h32. By A2
both corners are unstable.
Case 2: Both strategies 1 and 2 survive at the corner of F3. Then n = (n1, n 2, 0) with both
n1 and n2 positive integers and n1 + n2 = N. There are then two eigenvalues from (4.3),
λ 31 = h31n1 + h32n2 − h12n2 − h31, and λ 32 = h31n1 + h32n2 − h21n1 − h32. Let us assume
that this corner is stable, hence both λ 31, λ 32 are negative. The condition λ 31 < 0 (upon
eliminating n1 = N − n2) reduces to ( h12 + h31 − h32)n2 > (N − 1)h31. Since n2, N − 1, h 31
are all positive this means that the combination h12 + h31 − h32 is also positive, and
(N − 1)h31
h12 + h31 − h32
< n 2. (4.4)
Similarly λ 32 < 0 implies that h21 + h32 − h31 is positive (as can be seen by eliminating n2)
and further,
n2 < (N − 1)(h21 − h31)
h21 + h32 − h31
+ 1. (4.5)
Combining the two, we get

(N − 1)h31
h12 + h31 − h32
< (N − 1)(h21 − h31)
h21 + h32 − h31
+ 1, (4.6)
which can be rearranged into the form
(N − 1)[−h21h12 + h21h32 + h31h12] < (h12 + h31 − h32)(h21 + h32 − h31). (4.7)
But the quantity in [ ] on the l.h.s. of this inequality is just u3 (as evaluated from the
deﬁnition given in A1), which is positive. (The positivity of det B and hence u1, u 2, u 3 also
follows from A1 and A2.) Thus we have
N < (h12 + h31 − h32)(h21 + h32 − h31)
u3
+ 1. (4.8)
Note that the r.h.s. is a function of A alone and is ﬁnite, say N0(A). If N is chosen larger
than N0(A), this inequality is violated. That is, for N > N 0(A), the corner of F3 under
consideration cannot be stable. We have thus proved that under A1,A2, all corners of F3
are unstable for N > N 0(A). Similarly one may consider F1, F 2, which will yield the same
result but with diﬀerent ﬁnite bounds in place of N0(A). We can henceforth use N0 for the
largest of the three. The claim follows. ✷

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