[Paleopsych] Thorbjorn Knudsen: The Evolution of Cooperation in Structured Populations
checker at panix.com
Mon Jan 30 00:16:47 UTC 2006
This may be of interest Howard. References to Eschel.
Thorbjorn Knudsen: The Evolution of Cooperation in Structured Populations
Constitutional Political Economy, 13, 129-148, 2002.
tok at sam.sdu.dk
Department of Marketing, School of Business and Economics, University of
Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
[This paper, the second in an issue of _Constitutional Political Economy_
devoted to using tools of evolutionary biology to help understand
political behavior, is excessively mathematical. Since none of the values
of variables are actually measured (and I'm not sure how often they do get
measured in biology itself!), purely verbal arguments will be just as
[I can send the PDF to anyone who wants it. The PDF --> TXT conversion
does not handle formulas very well, so I got rid of most of them. Alt-172,
¼, which shows up on ISO-8859-1 as one-quarter, is the equals sign (=) in
the PDF, for example.
[When you find the going tough, speed to the fourth and fifth sections.
Viscosity means stickiness, but here the author takes it to mean genetic
or spatial proximity.]
Abstract. The present paper analyses the evolution of costly cooperation
in a multi-group population. Building on insights first developed in
modern biology the idea of viscous population equilibria is introduced [a
population is said to be viscous when a (sub)population of players is
spatially or genetically clustered). A simple model then analyses how the
combined effect of viscosity within multiple subgroups and different
levels of between-group segregation influences the evolution of
cooperation. The results suggest that a key issue in the evolution of
cooperation is the shifting balance between the need to protect
cooperators and propagation of the tendency to cooperate.
JEL classification: C72, D71.
Keywords: viscosity, multi-group selection, structured populations,
1. The Evolution of Cooperation in Structured Populations The interaction
between cooperators and defectors is often modelled in terms of the
Prisoners' Dilemma (PD henceforth) where two players each have two
possible strategies: defect or cooperate. In face of the negative fate of
cooperators implied by non-cooperative game theory and epitomised in the
famous PD-game, there is an obvious difficulty in sustaining the argument
of persistent costly cooperative behaviour. 3 Nevertheless, examples of
costly cooperative behaviour are replete both in biological (Hamilton
1975; Sella and Lachmann 2000) and social populations (Axelrod 1997;
Becker 1996; Elster 1989; Hechter 1988; Hodgson 1994).
Hamilton (1964) demonstrated that cooperation could be sustained in
viscous populations, i.e., when close relatives are more likely to
interact than they are with distant ones, e.g. due to the formation of
local herds and colonies. This analysis was extended in Hamilton (1975)
and recently supplemented by research on geographical dispersal (Nakamura,
Nogami, and Iwasa 1998). Since most of these studies consider single
populations, it would be interesting to see how the interaction of
multiple clustered populations influences the evolution of team play.
In the ensuing, I address this problem in a model where each of a number
of finite interacting subgroups is populated by members of which a
proportion is cooperators and the rest are defectors. A member of a
subgroup meets another subgroup member and receives the PD pay-off in
terms of an expected number of offspring.4 The PD pay-off is thus a
measure of fitness in a biological model. In a model relevant for the
social realm, the PD pay-off may denote the number of students educated by
a specific university (department) or the number of new team members
trained by a specific team (in a private company as well as in a state
institution such as the U.S. Navy Seals).
The purpose of the present paper is to analyse a population of multiple
interacting subgroups in which each subgroup is more or less clustered.
Thus, in addition to the described within-group interaction and death,
there is migration between groups. Migration is modelled as a tendency
that the difference in population mixture between segregated subgroups
will vanish. This type of dynamics seems to have a number of close
parallels both in the biological and the social realm. One example of such
structural dynamics is associated with the interaction within and between
departments in private corporations, universities and military
organisations. Another relates to the interaction within and between
strategic groups of private corporations and yet another to the
interaction within and between nation states.
Extending the analysis of viscosity (clustering) from single-to
multi-group populations, the present work is closely related to Hamilton
(1964, 1975) and Myerson et al. (1991). Also, the multilevel problem
mentioned in Sober and Wilson (1998) is given consideration. Other related
work is Becker (1976), Bergstrom and Stark (1993), DeCanio and Watkins
(1998) and Eshel et al. (1998). The paper is organised as follows. First,
the concept of viscosity that defines clustering in a population is
related to the evolution of cooperation in structured biological and
social populations. Then, a simple model illustrates how the combined
effect of viscosity within multiple subgroups and different levels of
segregation between the subgroups influence the evolution of cooperation.
Some implications for theories of innovation are discussed followed by the
2. Structured Populations and Viscous Population Equilibria
In the following, I consider the dynamics of population structure on two
levels: within and between multiple subgroups. That is, in addition to
within-group dynamics, I consider different levels of segregation between
At this point, it is useful to provide the definition of viscosity used to
model clustering within subgroups of a population. Thus, in accordance
with Hamilton (1964, 1975), a viscous population is one where there is
spatial or genetic clustering. More precisely, the degree of clustering
can be defined in terms of a viscosity parameter d referring to the
probability of interacting with a neighbour who plays a similar strategy.
For a cooperator this implies that the probability of interaction is the
sum of the probabilities of two forms of interaction: (1) the probability
of interacting with random strangers (cooperators or defectors), and (2)
the probability of interacting with players that are definitely
cooperators. An equivalent definition applies to defectors. When an
interaction is of the first type, I shall refer to it as "random." The
second type of interaction is referred to as, "non-random." When the
interaction is random, the individual meets cooperators and defectors with
a probability in proportion to their population mixture. When the
interaction is non-random, the individual definitely meets its own
strategy (including the interaction of the individual with itself]. Thus
for cooperators, the probability of meeting cooperators (including self )
within subgroup i is defined as:
P(Csubi,Csubj) = deltai + (1 - deltai)Csubi*
[That's how I can best type it. When I got is
PðCi; CiÞ¼di þð1 diÞC* ð1Þ
where C* i ¼ Ci /
( Di + Ci) is the proportion of cooperators in subgroup i, di is the
viscosity parameter specific to subgroup i and P(Ci, Di) ¼ 1 . P(Ci, Ci).5
As the viscosity parameter di increases, the subgroup is increasingly
clustered or viscous, and, the probability that a cooperator meets a
cooperator increases for a given mixture of cooperators and defectors.
A different way to look at this is that N samples were obtained from
subgroup i by a random procedure and then correlated. If N is arbitrarily
large, the viscosity parameter di is an expression of the correlation as
measured by the correlation coefficient. In biological models, a positive
correlation coefficient is often referred to as positive assortment. The
viscosity parameter di, therefore, provides a convenient compact
representation of a statistical relationship between members of any
population whatever its source (spatial structure or genetic relatedness).
Therefore, whatever its source, I shall refer to a subgroup as being
viscous if di > 0, meaning that there is a statistical relationship
between subgroup members that deviates from the expectation under a random
distribution of subgroup members. In addition to the viscosity parameter
that regulates the interaction of cooperators and defectors within each
subgroup, I define an equivalent population parameter, c, that regulates
between-group segregation. The frequency Fic of cooperators in subgroup i
is defined as:
i ¼Ci =ðCi þDiÞ; ð2Þ
[My typing gives Fsubi-superc = Csubi divided by Csubi + Dsubi.]
The frequency of cooperators in each of the j neighbouring subgroups after
migration is defined as:
[I am going to expunge the math. Ask me for the PDF.]
where c is the segregation parameter bounded between 0 and 1. All L
subgroups live on a one-dimensional lattice folded into a ring and the
summation is taken over j < L adjacent subgroups. Starting from subgroup 1
(randomly assigned), there is migration between the subgroups 1, 2, ..., j
in time step 1, between 2, 3, ..., j + 1 in time step 2, and so on (since
the lattice is folded into a ring, there is no boundary problem). C* i is
the proportion of cooperators in each of the subgroups i ¼1, 2, ... L.
Note that c ¼ 1 [typing in, this is gamma = 1] is an expression of
complete segregation between subgroups and c ¼ 0 is the situation with no
segregation between subgroups. As can be seen by comparison, c is the
population level equivalent of the within-group viscosity parameter di.
2.1. Viscosity and the Evolution of Cooperation in Biological Populations
In biology, the evolution of cooperation is associated with two distinct
ideas: (1) a gene- centred view which dissolves a conflict between the
gene and its carrier, and (2) viscous populations in terms of genetic
relatedness (referred to as positive assortment by the biologist) or
spatial clustering (referred to as viscosity by the biologist). Much has
been said about the first point (e.g. Dawkins 1976). Apparent altruism on
part of a gene- carrying individual is sometimes a clever strategy
favouring the gene's multiplication. What perhaps needs more emphasis in
this story is that the potential immortality of genes opens the
possibility of infinitely repeated games and thus altruism on part of the
gene. Therefore, the universal selfishness of genes promoted by Dawkins
(1976) may be an overstatement. We shall leave this matter here and in the
ensuing focus on the second point, viscous populations where the
expression of genetic relatedness or spatial clustering explains the
persistence of cooperation. The evolution of cooperation in biological
populations has been associated with the clustering of organisms in
geographical space. The original formulation was due to Hamilton (1964):
With many natural populations it must happen that an individual forms the
centre of an actual local concentration of his relatives, which is due to
a general inability, or disinclination of the organisms to move far from
their places of birth. (Hamilton 1964: 10).
Hamilton (1964) further developed a model, which distinguished between two
effects: a diluting effect and an inclusive fitness effect. Inclusive
fitness was defined as a measure of the total effect of an organism's
behaviour upon all genes identical by descent whereas the diluting effect
measured the total effect on other organisms. The sum of the two, then,
accounts for the total effects on reproduction due to a particular
organism. Whereas the diluting effect was shown to influence the level of
altruism, the inclusive fitness effect was shown to be more fundamental,
determining the direction and progress in frequencies of altruistic genes.
Crucially, the possibility that altruism may evolve by natural selection
was shown to depend upon inclusive fitness. A vivid illustration of the
idea, provided by Hamilton (and before him Haldane) is that we should
expect that no one is prepared to sacrifice his life for any single person
but that everyone will sacrifice it when he can thereby save more than two
sisters, four half-sisters, or eight first cousins etc.8 Moreover, in
comparison to the classical model, where advantages are conferred directly
by a gene to its carrier, another gene, which conferred similar advantages
to its sibs, would progress at exactly half the rate, i.e., the more
indirect the transfer of reproductive potential, the slower the rate of
evolutionary progress. This point indicates a general trade-off between
probabilistic dilution and the speed of progress, or, in different terms,
the need to balance protection and propagation of cooperation depending on
the rate at which cooperation is "produced" within each subgroup.
Considering the case of multiple subgroups, Hamilton (1975) used the Price
(1970, 1972) equations to further clarify how between-group selection
favouring altruistic traits, to some extent, but never completely, may
slow down within-group selection favouring selfishness. Briefly, the
higher the between-group differences, the more important is between-group
selection compared to within-group selection, however, within-group
selection is always stronger. Unless a mechanism can be devised that
continuously supplies new between-group variance, altruism cannot prosper.
Two extreme models of multiple subgroups are analysed by Hamilton (1975).
The first assumes persistent groups with no extinction and no
between-group variation. Since, in this model, between-group variance is
reduced as the best group increasingly dominates, altruism cannot evolve.
The analysis of a model at the opposite extreme in which groups break up
completely and reform in each generation leads to the same conclusion,
altruism cannot evolve. However, slightly changing the assortment
procedure so the correlation in two samples of randomly selected members
of a group is F can make the model work. Hamilton (1975) assumes this is
achieved by having a fraction F of groups made pure for each type and the
remainder formed randomly. Then, if the altruist gives up k units of
fitness to add K units to joint fitness, it is relatively straightforward
to show that the criterion for positive selection of altruism is (Hamilton
[I'll type in this one: K/k > 1/F.]
k > 1=
As Hamilton (1975) further notes, this criterion is completely general for
asexual models with non-overlapping generations, and also holds for
diploid biological models. In other words, it is general to both social
and biological populations. Moreover, Hamilton (1975) provides the
following useful general definition of F later adopted by Myerson et al.
(1991) in terms of the viscosity parameter (d): ...the existence of the
positive correlation F could be interpreted as implying in this case that
there is a chance F that the K units of fitness are definitely given to a
fellow altruist, while with chance [1 . F ) they are given (as they always
were) to a random member of the population.
Three general conclusions flow from Hamilton's (1975) model. The first is
that between-group segregation is important in determining the level which
altruism can reach, a point also emphasised by Eshel (1972). Second,
altruism can evolve if the population is genetically related or spatially
clustered so the viscosity parameter F [¼ d) > 0. Third, the population
dynamics introduced by migration may be a source of spatial clustering but
mere segregation between multiple subgroups is not enough to substitute
for viscosity within the individual subgroups.9 Segregation between
subgroups clearly helps sustain cooperation but without the presence of
viscosity at the subgroup level, cooperators will eventually wither away.
Thus viscosity within subgroups (and not different levels of segregation
between subgroups) may lead to the co-existence of cooperative and
defective strategies in a biological population. The mechanism that
accounts for this result is the spatial clustering or genetic relatedness,
which gives rise to uneven dispersal of strategies within a subgroup. When
strategies are unevenly distributed, the probability of meeting a relative
programmed with a similar strategy may be sufficiently high to ensure that
the cooperative strategy will be fixed in the population. This would be
the case when local herds or colonies build up as in the examples of the
social insects provided by Hamilton (1964, 1975).
2.2. Viscosity and the Evolution of Cooperation in Social Populations
Since deterministic biological models are not obviously relevant in the
social realm, it is important to identify conditions that justify their
application. As aforementioned, we often encounter structured social
populations, such as evolutionary cells and strategic groups within
interacting industries. Although the underlying source of the prevailing
population structure in terms of persistent subgroup heterogeneity may
differ from case to case, it is important to emphasise that the population
structure needs to be persistent. Therefore, a general source of
between-group segregation in social populations is the opportunity costs
involved in changing membership from one subgroup to the next. As the
opportunity costs of moving between subgroups approach zero, nothing
prevents the differences between subgroups to vanish. A number of sources
for opportunity costs associated with membership of social populations
such as sentiment, club membership and industrial entry barriers can be
envisioned. But what are the sources of within-group viscosity in social
Viscosity refers to the tendency for agents with a stable predisposition
towards a certain type of behaviour to cluster in geographical space or in
gene-space. In a previous paper (Knudsen and Foss 1999) it was argued that
behavioural programmes, specifically the tendency to cooperate or defect,
could be viewed as a semi-stable trait acquired in a particular subgroup,
i.e., a production team in a business organisation. As pointed out by,
among others, March and Simon (1958), Nelson and Winter (1982) and
Langlois (1995), the role of behavioural programmes and routines is not
just the storing of production knowledge, but also the storing of
behavioural knowledge. If it is further accepted that behavioural
programmes and routines involve a tacit component acquired in time-
consuming face-to-face interaction, particular behavioural traits will not
only tend to be relatively stable, they will also spread slowly in a
larger subgroup. Therefore, in larger subgroups close neighbours should be
more likely to share behavioural dispositions than distant neighbours.
Thus we have arrived at a social equivalent of viscosity, i.e., clustering
in social space.
Professionalization is obviously an example of a possible source of stable
social stereotypes acquired by face-to-face interaction during long
periods of training (Van Maanen 1973). Surely, such stereotypes spread
relatively slowly and unevenly in the social landscape. As described by
Nelson and Winter (1982), training in business organisations plays a
similar role. According to Nelson and Winter (1982), new organisation
members acquire the routines that carry the firm's productive knowledge
through face-to-face interactions. The reason that face-to-face
interaction is necessary is that knowledge is sticky, routines contain a
tacit component that can only be acquired through an emulation procedure
involving learning by doing with other team members (Knudsen 2001).
A further source of viscosity in social populations, originally pointed
out by Hamilton (1964), is provided by discrimination, i.e., when norms
dictate that frequent social interaction with some is more appropriate
than with others.
Having provided equivalents to within-group viscosity and between-group
segregation in social populations, I next turn to a model that illustrates
how their combined effect influences the evolution of cooperation. In
order to strip the analysis down to essentials, I shall not consider
reciprocity, i.e., strategies relying on discrimination and memory are
ignored. If cooperation can evolve in a model without reciprocity, surely,
the introduction of reciprocity will allow cooperation to prosper in yet a
number of additional situations.
3. The Model
3.1. The Evolution of Cooperation in Isolated or Single Groups
Before presenting the case of multiple subgroups, I first turn to Myerson
et al.'s (1991) useful extension of Hamilton's (1964, 1975) insights.
Myerson et al. (1991) define a viscosity parameter (d), a viscous
population equilibrium and, taking the limit as viscosity goes to zero,
they define a fluid population equilibrium. In the present article the
analyses are limited to non-cooperative games and use the PD pay-offs
shown below in Table 1a. Furthermore, I refer to Myerson et al. (1991) for
the generalisation that at least one viscous equilibrium exists in a
finite strategy set.
For a moment, consider a one-shot PD-game where pay-offs are utilities (or
profits). Let strategy C meet its kin with probability P(Ci, Ci) ¼di +[1
di) C* i and its opponent D with probability P(Ci, Di) ¼
1 P(Ci, Ci) as previously defined (equation 1 and note 5). Likewise, D
meets D with probability P(Di, Di) ¼di +[1 di) D* i and its opponent C
with probability P(Di, Ci) ¼
1 P(Di, Di).10 In this case, the expected value of the pay-off associated
with a cooperator's (defector's) strategy profile s is:
Table 1b below shows the relevant adjusted PD pay-offs where C is the
strategy cooperate and D is the strategy defect.11 Note that the expected
value of the pay-offs uC(s) and uD(s) can be read off directly from Table
1b if we, for a moment, interpret the population states C* i and Di * as
the components of a mixed strategy.12 Note that the
normalised adjusted pay-offs, which are used in the analysis of the
replicator dynamics in later sections of the paper, are derived from the
pay-offs in Table 1b and are for reasons of convenient comparison shown
here in Table 1c.
From the adjusted pay-off matrix shown in Table 1b, it is straightforward
to see that cooperation can only evolve when the probability of meeting a
fellow cooperator is at least 1/2, i.e., when 1/2 . di, the unique viscous
equilibrium is C and everybody receives pay-off 3. When di . 1/3, the
unique viscous equilibrium is D, and when 1/ 3 < di < 1/2, the unique
equilibrium is mixed. The thrust of the example, is that cooperation can
evolve in a single subgroup if the viscosity parameter di is sufficiently
high, i.e., if spatial or genetic clustering is sufficiently high. One
limit of the example, however, is that fitness is suppressed. What would
happen if we interpreted the pay-offs as an expression of the individual's
fitness in terms of the number of offspring produced per time unit?
In the remainder of the paper, the game is thought of as a meeting between
two individuals programmed to play the pure strategy C or D. The
individuals live in subgroup i and the state of the subgroup is defined as
the vector whose components C* i and D* i are the shares of subgroup i
programmed to play C and D, respectively. Within each subgroup, the
players meet in pairs according to the probabilities P(Ci, Ci), and P(Di,
Di)as previously defined.
Assuming a population with non-overlapping generations, we can use
Hamilton's (1975) general criterion for altruism to evolve, i.e., K/ k >
1/F as explained in the above. The question is, what values of the
viscosity parameter di ( F in Hamilton's model), can sustain cooperation
when the opportunity cost to the altruist is k and the benefit added to
the group is K?
With the above pay-offs, and assuming a population where all defect, k ¼ 1
and K ¼ 3 so cooperation can evolve when F > k/K ¼ 1/3.13 As indicated by
this calculation, some level of cooperation will prosper for 1/3 < di <
1/2 in models of non-overlapping generations where fitness is expressed,
i.e., in growth models where pay-offs denote differential increases (or
decreases) in the actual number of particular types of agents.14 Even if
this result can be obtained directly from an inspection of the eigenvalues
associated with the replicator dynamics, as shown in the following
section, it provides a useful connection to Hamilton's original model.
The result that cooperation can evolve when F [¼ di) > 1/3 demonstrates
that Hamilton's (1975: 140) assortment procedure can be respecified to
work in a population that is not divided into subgroups or demes.
Consequently, Sober and Wilson's (1998: 76-7) claim that the significance
of relatedness for the evolution of altruism is to increase genetic
variation among subgroups and thereby increasing the importance of group
selection needs further qualification. The general cause of altruism in
Hamilton's (1975) model is viscosity and although viscosity may be
produced by an assortment procedure involving multiple subgroups, as
specified by Hamilton (1975), this is not really necessary.
However, this clarification points to a further limit of the above
example. What would happen if we allowed for differences in viscosity in a
population of multiple subgroups? In particular, it would be interesting
to see how the combined effect of viscosity within subgroups, and, the
level of segregation between subgroups, influences the range of feasible
equilibria, an issue not addressed by Myerson et al. (1991). To shed light
on this problem, the ensuing sections study how viscosity within and
segregation between multiple subgroups influence the range of stable
states where a proportion of cooperators is sustained.
3.2. Evolution with Viscosity within and Segregation between Multiple
This section describes a simple model of evolution in a multi-group
population. The model illustrates how different levels of within-group
viscosity and between-group segregation influences survival of
cooperation. To simplify matters, we use the above PD pay-offs and note
that conclusions will not change qualitatively as long as this pay-off
structure is unaltered.
3.2.1. Interactions Between Individuals Within Subgroups
Assume a finite subgroup i of Ni players programmed to play C and D in the
proportions C* i , D* i . In each time unit t, all agents within subgroup
i meet in pairs according to the probabilities P(Ci, Ci), and P(Di, Di) as
previously defined. As a result of the meeting, a C and D player will
receive the number offspring specified by the pay-offs in Table 1.
3.2.2. Analysis of One Subgroup
Before the multi-group model is presented, it is useful to analyse the
replicator dynamics in the case of a single subgroup (identical to the
case of complete segregation). The replicator dynamics provides an
expression for the growth rate of the population share C* i (D* i ) in
terms of the difference between the number of offspring (pay-off)
currently obtained by C-players and the current average pay-off (Weibull
1997). The replicator dynamics, is therefore a convenient basis for
computing the steady states of the within-
group dynamics in the case of a single subgroup. Since the growth rate is
independent of the background birth-and death rates in the case of a
single subgroup (Weibull 1997), I first introduce death when the
multi-group model is analysed. It should also be noted that although there
exists a multi-group version of the replicator dynamics ( Hofbauer and
Sigmund 1998; Weibull 1997), it is not readily tailored to the purpose of
the present multi-group study. Using the normalised adjusted pay-offs
shown in Table 1c, the replicator dynamics for a single completely
segregated subgroup can be defined as (Weibull 1997):
3.2.3. Analysis of Multiple Subgroups
I now proceed to illustrate how between-group dynamics [0 c < 1) alter
the values of Ci * and Di * when di is allowed to vary across subgroups. I
use a simulation model since it is not straightforward to obtain an
analytic solution for n subgroups when there are differences across more
than two subgroups in di and c < 1.15 A model of 20 subgroups i of size Ni
with NiC cooperators and NiD defectors is studied. The results are
independent of the value of Ni, which is set to 50. The expected value of
the pay offs to the Ni agents in each subgroup are according to the
adjusted pay offs shown in Table 1b:
3.2.4. Between-Group Segregation
As previously defined, c regulates between-group segregation. In the
present model there are L ¼20 subgroups and there is migration between j ¼
4 neighbouring subgroups at each time step.16 The probability that
cooperators will migrate between the 4 neighbouring subgroups i and j was
previously defined in equation (2).
After each period, the Ci and Di strategists are reduced with a factor w.
w is 2 with certainty plus a random component drawn from a uniform
distribution [0,2], so E(w) ¼3.
3.3. The Combined Influence of Within-Group Viscosity and Between-Group
This section presents the results of the analysis of the model described
in the above. Two configurations with initial maximum differences in
subgroup composition were analysed.
In the first configuration, all 20 subgroups were identical and completely
mixed in period one, i.e., 50% cooperators and 50% defectors. In the
second configuration, 10 subgroups were populated entirely by cooperators
and the remaining 10 subgroups were populated entirely by defectors. Apart
from the extreme situation where all subgroups were completely segregated
(c ¼1), the equilibrium proportions of cooperators converged in the two
configurations. Averaging 100 runs of the model, 50 for each configuration
obtained the results reported in the ensuing. Each run encompassed 150
periods since stability was usually obtained in less than 100 periods. The
analyses cover the following topics: (1) the evolution of cooperation in
completely segregated subgroups (c ¼1), (2) the evolution of cooperation
when there is no segregation between subgroups (c ¼0), (3) the evolution
of cooperation when there is some segregation between subgroups [0 < c <
1),(4) the evolution of cooperation for low and high average subgroup
viscosity d ¼1/20 Sdi when there is some segregation between subgroups [0
< c < 1), and (5) the sensitivity to initial conditions.
3.3.1. The Evolution of Cooperation in Completely Segregated Subgroups
The subgroups are completely segregated when there is no migration between
neighbouring subgroups. The evolution of cooperation in completely
segregated subgroups (c ¼ 1) is identical to the evolution of cooperation
in a single subgroup covered in the above analysis. Thus, stable
proportions of both Ci * and Di * are sustained for 1/3 < di < 1/2. For di
¼ 3/7, 50% of the subgroup are cooperators. Unless, subgroups are
initially pure (Ci * ¼ 1), it is impossible to sustain cooperation for any
di < 1/3. That is, the population viscosity average d for the 20 subgroups
needs to be larger than 1/ 3 for cooperation to survive. How does the
level of segregation between groups change this fact?
3.3.2. The Evolution of Cooperation when there is Complete Migration
As c ! 0, migration between neighbouring subgroups increases and it
becomes increasingly difficult to sustain cooperation. When c ¼ 0, the
cooperators in subgroup i can only survive when all 1/ 3 < di implying an
average 1/3 < d. Some degree of segregation between subgroups [0 < c < 1),
however, introduces a delay that allows cooperation to be sustained for
very low average d's (in the present study d > 0.05).
3.3.3. The Evolution of Cooperation when there is some Segregation Between
In the case where di has the same value in all subgroups (di ¼ d for all
i), increasing the segregation between subgroups (c ! 1) does not help to
sustain cooperation. For (di ¼] d < 1/3, cooperation cannot be sustained
even when c ¼ 0.99. For 1/3< (di ¼] d < 1/2, the stable proportion of
cooperators is identical to the case of completely isolated subgroups but
as the level of segregation increases, the number of periods to reach the
steady state also increases. For (di ¼] d ¼ 1/2, (and slightly above 1/2),
however, segregation does make a difference. For very high levels of c,
cooperation will drive out defection. As segregation decreases, however, a
small proportion of defectors can survive. Since the reduction of group
size (death) after each period includes a stochastic component, the
defectors' survival point, defined by d, increases slightly (from 0.50 to
Figure 1 shows the case where d ¼ 1/2.
As shown in Figure 1, if c ¼ 0.99, the population consists only of
cooperators (C* ¼ 1) when di ¼ d ¼ 1/2. In the ensuing, the fitness
landscape in Figure 1 is explained in more detail.17
3.3.4. The Evolution of Cooperation when Subgroup Viscosity Differs
Introducing differences in viscosity (di) across subgroups for a given
average d changes matters. For a given level of the population average d,
maximum differences in subgroup di's are analysed. Maximum differences are
obtained when some subgroups
are set to a given value of di and di ¼ 0 for the remaining subgroups.
When di ¼ 0 for the remaining subgroups, a population average d ¼ 0.50
can, for example, be obtained by the following values of di, 0.50 in all
20 subgroups, 0.60 in 16 subgroups and 0.40 in one subgroup, 0.70 in 14
subgroups and 0.20 in one subgroup, and so on, till 10 out of 20 subgroups
have di ¼ 1.00. Note that maximum differences include the case of maximum
variance, i.e., when the average d is obtained by setting subgroup di's to
one and zero.
Figure 1 above shows the results for population average d ¼ 0.50. For very
high segregation (c ¼ 0.99), the highest proportion of cooperators (C*] is
sustained when di's are set as low as possible. By contrast, for very low
segregation (c ¼ 0.00), the highest proportion of cooperators is sustained
when di's are set as high as possible. For extreme values of segregation
(c ¼ 0.00 or c ¼ 0.99) only cooperators survive (C* ¼ 1). Some level of
segregation (c) between the extreme values allows some defectors to
survive (C* < 1) for the same value of average d [¼ 0.50). For example,
when c ¼ 0.20, about 85% cooperators survive independent of the subgroup
values of di (see Figure 1).
As shown in Figure 1, for average d ¼ 0.50, increasing segregation (c)
benefits cooperation when there is minimum differences in viscosity across
subgroups (all di ¼ 0.50). Decreasing segregation (c) benefits cooperation
when there are maximum differences in viscosity across subgroups (di's set
to 1.00 and 0.00). These conclusions do not hold for values of the
population average di different from 0.50. In the following, I describe
how changes in levels of the population average d alter the fitness
Figure 2. Stable proportions of C* for alternative values of c and average
d ¼ 1/20 Sdi ¼ 0.05 for 0.30 . di . 1.00.
3.3.5. The Evolution of Cooperation for Low Average Subgroup Viscosity (d
It is possible to sustain an evolutionarily stable proportion of
cooperators even for very low values of the population average d when the
level of segregation is sufficiently high. Figure 2 shows the fitness
landscape for average d ¼ 0.05, i.e., one subgroup with di ¼ 1.00 and the
nineteen remaining di's ¼ 0.00, one subgroup with di ¼ 0.90 another with
di ¼ 0.10 and the remaining eighteen di's ¼ 0.00 etc.
The first thing to notice is that cooperation cannot be sustained for d <
1/ 3 when segregation is zero. It is necessary with 0.40 . c to sustain
cooperation. Generally, the evolutionarily stable proportion of
cooperators increases in c. Moreover, for very high levels of c [¼ 0.99),
the sustainable proportion of cooperators increases as di approach 0.50.
Indeed, for all d < 0.50, the maximum evolutionarily stable proportion of
cooperators is sustained for c ¼ 0.99 and di ¼ 0.50 for the relevant
number of subgroups (two in the present model when di ¼ 0.05). For lower
levels of c (. 0.80 when d ¼ 0.05), the sustainable proportion of
cooperators increases as di approach 1.00 for the relevant number of
subgroups (see Figure 2). For low values of average d, the general issue
is the trade off between the protection of cooperators provided by high
levels of segregation (high c) and the need to reproduce new cooperators
fast enough (high di). For very low levels of average d (0.05 in the
present model), the necessary speed in replication can only be maintained
when the subgroup is protected against invasion by very high levels of c.
In this case, the evolutionarily stable proportion of cooperators shrinks
as c decreases. As average d increases from very low levels towards 0.50,
the pressure against cooperators eases. Gradually, for every level of c,
the value of di that sustains the maximum level of cooperators approaches
0.50. In consequence, raising average d from 0 towards 0.50 increases the
smoothness of the fitness landscape but the peak at (c ¼ 0.99 and di ¼
3.3.6. The Evolution of Cooperation for High Average Subgroup Viscosity
(0.50 . d)
For high levels of d, the general issue is the trade off between the speed
in replication of new cooperators and the protection of cooperators caused
by high levels of segregation (high c). Viewed from the perspective of
defectors', the issue is identical to the cooperator's problem when
average d is low. For high levels of average d, the defector's general
problem is the trade off between the protection provided by high levels of
segregation (high c) and the need to reproduce new defectors fast enough
(increases in di). Therefore, increasing c for high levels of d, slightly
decreases the evolutionarily stable proportion of cooperators. Even when d
is quite high, i.e., average population d ¼ 0.80 as shown in Figure 3,
some proportion of defectors is sustained.
The explanation is that d ¼ 0.80 implies di ¼ 0.00 in four subgroups and
di ¼ 1.00 in sixteen subgroups. When the diffusion of cooperators is
slowed down (by increasing c), some proportion of defectors is sustained.
Even if this is the case, when d increases, the cooperators' fitness
surface becomes increasingly smooth and approaches 1.00.
3.3.7. Sensitivity to Initial Conditions
Apart from the extreme case where c ¼ 1.00, initial conditions will not
influence the evolutionarily stable proportion of cooperators. In other
words, initial differences in the mix of strategies and the number of
agents in each subgroup will not change the stable proportions of
surviving cooperators, however, the subgroup and population growth rates
will change. Furthermore, the implications of the above conclusions hold
for any pay-offs that respect the PD pay-off structure. The initial
subgroup composition matters for growth, however. When average d is low,
populations with maximum initial differences among subgroups outgrow
populations with completely mixed subgroups (for identical population d
and independent of c). By contrast, high average d in combination with low
c favour populations where subgroups initially are completely mixed,
however, uncertainty about the outcome increases in c.
In sum, the above analysis showed that within-group viscosity is necessary
for cooperation to evolve whereas between-group segregation has an
important role in influencing the level of cooperation, i.e., the stable
fraction of cooperating agents. Clearly, within-group viscosity is the
more important source of cooperation. Having said that, it was striking to
notice that some level of cooperation could prosper even for very low
average values of the viscosity parameter. Put differently, only one
viscous subgroup was needed for high levels of segregation. The analyses
implied that a key issue in the evolution of cooperation is the balance
between protection and propagation of the tendency to cooperate. As we
have seen, high levels of segregation helps protect cooperation when
viscosity is low, however, reduces the propagation of cooperation when
viscosity is high. In the following, we turn to this issue and consider
the role of viscosity in different stages of revolution, broadly
understood as overthrowing old ideas through cooperative efforts.
4. Viscosity and the Diffusion of Innovations
When new innovations threaten to overthrow received wisdom they are likely
to be met by negative reactions and to be opposed by the potential losers.
Therefore, development in the progress of technology and science includes
moments of revolution. As indicated by Mokyr (1990), resistance to
innovation is a common phenomenon including historical examples such as
the prolonged resistance to new technology in the textile trade, the edict
which in 1299 banned the use of Arabic numbers by Florentine bankers and
the delayed introduction of printing into Paris caused by resistance from
the scribes guild.
Comparable resistance to scientific change has been observed in economics,
as beautifully illustrated by Shackle (1967), in biology (Mayr 1982) and
undoubtedly most other disciplines. Historical examples include the
declaration of Mandeville's 1723 edition of the Fable of the Bees as a
public nuisance by the Grand Jury of Middlesex and the subsequent assail
on his work by eminent contemporary writers (Torrance 1998). More recent
examples include the rejection of Darwinian theory as unscientific and its
long period of eclipse (Cronin 1991), the reluctance in mainstream
economics to renounce the assumption of perfect competition and,
subsequently, to consider process and learning (Shackle 1967).
Since the distribution of benefits flowing from new ideas probably can be
characterised by a low average and a high variance, some resistance is
clearly apposite. In numerous situations new ideas are clear improvements,
however; at least viewed from the perspective of society. Since an
invention in its early phases threaten the private interests of the
potential losers, its propagation often depends on the protection of the
inventor from those who stand to benefit from the suppression of the
invention (Mokyr 1990). In other words, there is a conflict between those
who oppose the invention and the cooperating revolutionaries who aim to
As the above analyses have shown, the general issue in such situations is
the trade off between the protection of cooperators provided by high
levels of segregation (high c) and propagation in terms of the need to
reproduce new cooperators fast enough (high di].
The trade-off is well known to planners of political revolutions.
Cooperating comrades are organised in cells and the segregation between
cells needs to be high enough to protect against invasion of subversive
elements. Moreover, the rate of conversion of newcomers to the cause
(pay-offs in our model) depends on the probability of interacting with
other genuine proponents of the cause. Seasoned members should meet each
other more frequently than they meet newcomers and the segregation between
cells should be high. Early phases of successful revolutions seem to have
As the above model illustrates, the problem in early phases of political,
technological or scientific revolutions is to strike a proper balance
between interacting with like-minded people and running the risk of
betrayal or losing faith by meeting new prospects. In later phases of
revolution, when the faithful are plenty and their presence clustered
within a local population, the cause will prosper by decreasing
segregation. Similar considerations can be associated with the evolution
of cooperation in other structured populations such as research teams
within interacting universities, strategic groups within interacting
industries and rock musicians playing in alternative "interacting" bands.
In all these cases, changes in structural dynamics will influence the
protection and propagation of cooperation.
Successfully overthrowing existing wisdom is facilitated by particular
structural conditions that need adjustment as support is gained. The above
model suggests how the general trade-off between protection and
propagation may be balanced. It also complements existing research on
viscous equilibria by introducing a multi-level selection environment,
however, to maintain focus, the model ignores the issue of how structural
dynamics interacts with differences in information processing power across
It should be mentioned that prior research on multi-level selection
reported in Sober and Wilson (1998), to some extent, is challenged but
also supported by the above results. Sober and Wilson (1998) claim that
the significance of relatedness for the evolution of altruism is to
increase genetic variation among subgroups and thereby increasing the
importance of group selection. As indicated in the above, this claim
relies on a specific interpretation of Hamilton's (1975) argument. It is
viscosity, which allows cooperation to evolve, both for Hamilton (1975)
and in the model analysed in the present work. Although between-group
segregation can influence the level of cooperation, it can never sustain
cooperation in the absence of viscosity. Here it should be noted that the
above analyses excluded the possibility of empty subgroups and thus a
dynamics that includes the birth and death of subgroups. If this
possibility is included, the conclusion may or may not be reversed
depending on the definition of the transition probabilities, which
determine in which order and at what rate cooperators and defectors arrive
at empty subgroups. It should further be noted that since segregation
between subgroups may actually compensate for clustering within subgroups,
the simulation lends support to Sober and Wilson's (1998) claim that group
selection and individual selection are best seen as distinct causes that
may interact to sustain costly cooperation.
Concerning the general issue of altruism versus egoism, we reach similar
conclusions as prior research. In a local interaction model where players
choose strategies by imitating successful players, Eshel et al. (1998)
showed that altruists need protection to survive. Protection is provided
by local interaction patterns that allow groups of altruists to share the
net cost of altruism, however, mutant egoists or defectors can always
invade altruists or cooperators. In Eshel et al.' s (1998) model, the
limit to expansion of egoism is the fall in pay-off to egoism, which at
some point leads imitators to become altruists. Even if Eshel et al.'s
(1998) model is different from the one studied above, it shares its
central feature; cooperation cannot survive in a completely unstructured
(fluid) population. By contrast, when factors such as spatial or genetic
clustering introduce constrained non-random interaction patterns, it is
possible for some proportion of cooperators or altruists to survive. As in
Eshel et al. (1998) this result does not rely on reciprocity.
In sum, the simple model analysed in the present work, showed how the
combined effect of viscosity within multiple subgroups and segregation,
defined as limits in the migration of members between these, influence the
evolution of cooperation. In particular, the model showed that segregation
(slow migration between subgroups) is important because it helps to
sustain the weaker strategy. The present work has assumed that
within-group viscosity existed in biological and social populations. This
is obviously an empirical question. Nevertheless, it would be interesting
to identify the dynamics, which could "produce" viscosity as an outcome of
an evolutionary process. The above results suggest that we should look for
a dynamics, which combine slow migration with the possibility of
persistent subgroup heterogeneity. This suggests that a sufficiently large
delay in replication caused by slow migration [a transient fitness
reduction) may be enough to produce the positive correlation between
subgroups, which can be interpreted as an expression of viscosity.
1. The author appreciates helpful comments on earlier drafts from Rabah
Amir, Mie Augier, Markus Becker, Ken Binmore, Nicolai J. Foss, Geoffrey M.
Hodgson, Josef Hof bauer, Oddvar M. Kaarbøe, Michael Lachmann, Daniel
Levinthal, Brian Loasby, James G. March, Trond Olsen, Ludo Pagie, John
Pepper, Walter W. Powell, Christian Sartorius, Markus Schmidt, Sara J.
Solnick, Robert Sugden, Viktor Vanberg, Jack Vromen, Ulrich Witt and
2. Phone: +45 6550 3148, Fax: +45 6615 5129.
3. The terms "team play," "altruism" and "cooperation" are used
synonymously to denote cooperative behaviour, which at a real cost to its
provider adds a greater benefit to the group.
4. The model, therefore, is different from the multi-population models
described by Weibull (1997) where members from one subpopulation can
interact directly with members from other subpopulations.
5. The probability of a (Di, Di) meeting is P(Di, Di) ¼ di +[1 . di) Di *,
where Di * is the proportion of defectors in subgroup i and the
probability of a (Di, Ci) meeting is P(Di, Ci) ¼ 1 . P(Di, Di).
6. The frequency of migrating cooperators from subgroup i is ( Fic . Z] /
Fic: if the sign is positive, cooperators leave subgroup i, and if the
sign is negative cooperators arrive to subgroup i.
7. Since P(Ci) ¼ Z . c [Z . Ci *] ¼ c Ci * +[1 . c)Z.
8. Since, on average, a human being shares 50% of its genes with its
sisters, 25% with half-sisters etc.
9. An exception includes the situation where the pay-off to a defector
meeting another defector < 1 provided there is sufficient initial
differences in the size of sub-populations (see Sella and Lachman 2000).
10. The probabilities P(Ci, Ci) and P(Di, Di) provide a complete
description of the interaction between players.
11. The pay-offs in 1b can be obtained directly from the matrix A ¼
(3,0;5,1) shown in 1a, i.e., by adding [1 . d) A and dA' where A' ¼
12. We can do this since a population state is formally identical with a
mixed strategy (Weibull 1995).
13. The joint benefit K to the group is calculated by inclusion of the
14. An economic model where fitness is expressed would imply that the
appropriate criterion of success is actual asset accumulation. Profits in
such models, in contrast to most models analysed in evolutionary
economics, must be seen as a medium which translates into asset
15. Weibull (1997) considers alternative approaches to multipopulation
replicator dynamics (e.g. Hofbauer and Sigmund 1998) without viscosity. In
the present work, we study different levels in viscosity among a number of
16. If the condition j < L is fullfilled, the obtained results are not
sensitive to the particular number L of subgroups analysed or the
particular number j < L of subgroups among which there is migration in
each time step.
17. Figure 2 shows a fitness landscape for cooperators in terms of the
actual evolutionarily stable proportions C* for alternative levels of
between-group segregation (c) and within-group (di) viscosity. In biology,
fitness is commonly defined in terms of the expected number of offspring.
In economics matters are less settled and fitness can be defined in terms
of profits or growth (actual of expected). Using the latter definition, I
refer to Figure 1 as a fitness landscape.
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