[Paleopsych] Thorbjorn Knudsen: The Evolution of Cooperation in Structured Populations

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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, 
population dynamics.

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

i ;

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

[omit tables]

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).

3.2.5. Death

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 
Between Subgroups

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 
< 0.50)

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 
0.50 remains).

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 
propagate it.

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 
these characteristics.

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.

5. Conclusion

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 
altruists pay-offs.

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 
non-fixed populations.

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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