Natural selection is often regarded as a result of severe competition. Defect seems beneficial for a single individual in many cases.However, cooperation is observed in many levels of biological systems ranging from single cells to animals, including human society. We have yet known that in unstructured populations, evolution favors defectors over cooperators. On the other hand, there have been much interest on evolutionary games^1,2^ on structured population and on graphs^3-16^. Structures of biological systems and societies of animals can be taken as networks. They discover that network structures determine results of the games. Together with the recent interest of complex networks^17,18^, many researchers investigate real network structures. Recently even economists study firms&#x27; transactions structure^19^. Seminal work^11^ derives the condition of favoring cooperation for evolutionary games on networks, that is, benefit divided by cost, _b/c_, exceeds average degree, (_k_). Although this condition has been believed so far^20^, we find the condition is _b/c_ (_k~nm~_) instead. _k~nm~_ is the mean nearest neighbor degree. Our condition enables us to compare how network structure enhances cooperation across different kinds of networks. Regular network favors most, scale free network least. On ideal scale free networks, cooperation is unfeasible. We could say that (_k_) is the degree of itself, while _k~nm~_ is that of others. One of the most interesting points in network theory is that results depend not only on itself but also on others. In evolutionary games on network, we find the same characteristic

Tomohiko Konno

English

Nature Precedings

A condition of cooperation. Games on network
Tomohiko Konno1,2
1Graduate School of Economics, University of Tokyo
2Japan Society for the Promotion of Science
Natural selection is often regarded as a result of severe competition. Defect seems beneficial
for a single individual in many cases. However, cooperation is observed in many levels of
biological systems ranging from single cells to animals, including human society. We have yet
known that in unstructured populations, evolution favors defectors over cooperators. On the
other hand, there have been much interest on evolutionary games1,2 on structured population
and on graphs3–16. Structures of biological systems and societies of animals can be taken as
networks. They discover that network structures determine results of the games. Together
with the recent interest of complex networks17,18, many researchers investigate real network
structures. Recently even economists study firms’ transactions structure19. Seminal work11
derives the condition of favoring cooperation for evolutionary games on networks, that is,
benefit divided by cost, b/c, exceeds average degree, 〈k〉. Although this condition has been
believed so far20, we find the condition is b/c > 〈knn〉 instead. 〈knn〉 is the mean nearest
neighbor degree. Our condition enables us to compare how network structure enhances
cooperation across different kinds of networks. Regular network favors most, scale free
network least. On ideal scale free networks, cooperation is unfeasible. We could say that
〈k〉 is the degree of itself, while 〈knn〉 is that of others. One of the most interesting points in
1
network theory is that results depend not only on itself but also on others’. In evolutionary
games on network, we find the same characteristic.
There are only two kinds of agent: cooperator and defector. A cooperator pays a cost, c, for
each neighbor to receive a benefit, b. On the other hand, a defector pays no cost and its neighbor
does not receive any benefit. We also assume b > c. The payoff of the game can be described by
the following payoff matrix
⎛
⎜⎜⎜⎝
C D
C b− c −c
D b 0
⎞
⎟⎟⎟⎠ (1)
In deterministic game, D-D is the unique Nash equilibrium and will be taken. Although cooper-
ation by all the agents raises payoff of them, it is difficult to maintain cooperation because for a
single agent defect is always beneficial than cooperation. Thus cooperation will collapse. In an
unstructured population, where all the agents interact each other, defectors tend to have higher av-
erage gain than cooperators, so that cooperators would extinct as a result of natural selection. This
is true for deterministic replicator equation21,22 and stochastic game of finite population23. In the
model of present paper, a network structure is introduced. Agent occupies each vertex of the net-
work. They play games only with the agents on adjacent vertices. If there are l neighbors around
the cooperator regardless of cooperators or defectors, he pays cost, lc. In addition, if j of them are
cooperators he is benefited jb from the cooperators. His fitness from the game is jb − lc. If the
defector has i cooperators around him, his fitness is ib. The game proceeds as follows. For each
time step, randomly chosen one agent on the network dies. Then adjacent agents compete for the
2
empty vertex proportional to their fitness. The fitness of each agent is given by the constant term,
baseline fitness, plus the payoff of the game. When the payoff of the game is relatively smaller than
the baseline fitness, we call this as weak selection. The idea of the weak selection is that there are
many factors determining the whole fitness, the game in consideration is only one of the factors.
Reproduction of strategy can be taken as generic and cultural. The former is biological selection,
the latter is social phenomena.
The interest of the present paper comes from evolutionary biology and complex networks.
We need to explain a few things about complex networks. Degree, k, is the number of edges the
vertex has. If all the vertices have the same degree, the network is called regular network. If not,
it is called non-regular network. 〈k〉 is the mean degree. 〈knn〉 = 〈k2〉/〈k〉 is the nearest neighbor
mean degree. For instance, if you choose one vertex randomly, the mean degree of which is 〈k〉.
If you look at vertices linked to the randomly chosen one, the mean degree of which is not 〈k〉
but 〈knn〉. The fact that 〈k〉 = 〈knn〉 plays an important role in complex network theory. In our
point of view many interesting results24,25 comes from 〈k〉 = 〈knn〉. The intuitive reason why they
differ is that vertices are likely to be linked to the vertices with larger degree. We need to explain
in what respect we can say a network favors cooperation. First we prepare a network where all the
vertices are occupied by only defectors, then we replace one of them by a single cooperator, then
we start the evolutionary games until either when all the vertices are occupied by only defectors or
only cooperators. We iterate the same games and have the probability that only cooperators occupy
the vertices. This probability is called fixation probability. If selection neither favors nor opposes
cooperation the probability is 1/N , N is the size of networks. If the fixation probability is larger
3
than 1/N , we say that the network structure favors cooperation. Thus the condition b/c > 〈knn〉 is
like a threshold between network favors cooperation or opposes.
We derive the condition by mean field approximation. Mean field approximation is an useful
method which replaces something by mean value to obtain analytical solution, which has been
used especially in Physics. In mean field picture, a vertex is surrounded by the vertices with degree
〈knn〉 as illustrated in Fig.2. First we need to examine what happens on the vertex with 〈knn〉. In
mean field picture, as a vertex is surrounded by vertices 〈knn〉, so the vertex 〈knn〉 is surrounded by
the vertices 〈knn〉, which is illustrated in Fig.2 and called 〈knn〉 network hereafter. Next, we study
the fixation problem on the 〈knn〉 network. The point is that this network is regular network with
degree 〈knn〉. We are able to use the results of previous studies11,12, that is, on regular network with
degree k the condition of favoring cooperation is b/c > k. Thus, on 〈knn〉 network the condition is
b/c > 〈knn〉. It is necessary for the whole network to be fixed that at least 〈knn〉 network is fixed.
Unless 〈knn〉 is fixed, the whole network is not fixed either. Thus that is the necessary condition.
However, we will see that it is not only necessary condition but also sufficient condition in mean
field picture. What happens if 〈knn〉 network is fixed? Remind the rule of the game, for each
time step one vertex is randomly chosen to die and the adjacent vertices compete for the empty
vertex proportional to their fitness. In mean field picture, any chosen vertex is surrounded by the
vertices with degree 〈knn〉. Because we assume that 〈knn〉 is already fixed, any vertex surrounding
randomly chosen vertex is a cooperator. Since the empty site will be taken by the adjacent players
and all of them are cooperators yet, the agent on the site has no choice but to be a cooperator. As
we have seen, we only have to study the fixation problem on the 〈knn〉 network in our mean field
4
picture. The condition of favoring cooperation on the 〈knn〉 network is b/c > 〈knn〉. As we have
discussed, this is also the condition for the whole network. On regular networks 〈knn〉 is always
equal to k, thus the rule becomes b/c > k.
Because we have the condition b/c > 〈knn〉, we can compare how three kinds of represen-
tative networks: regular networks, random networks26, and scale free networks favor cooperation
most. Scale free network is the network that has degree distribution P (k) ∼ k−gamma. We need
to fix 〈k〉 across the networks. Even though 〈k〉 are the same, 〈knn〉 can be different. This fact
enables us to compare across different kinds of network. Among three networks, regular network
favors cooperation most and scale free network opposes most. On ideal scale free network which
has infinite number of vertices with 2 < γ ≤ 3, many real scale free network falls this parameter
range, cooperation is unfeasible because 〈knn〉 → ∞.
We check two points whether the condition b/c > 〈knn〉 holds well and which condition
b/c > 〈k〉 or b/c > 〈knn〉 is better by numerical simulation. Seen from Fig.4 the condition
b/c > 〈knn〉 holds well and is better than b/c > 〈k〉 which has been believed before. However,
〈knn〉 does not become approximate threshold for networks with too large deviation of degree
distribution like scale free network Fig.4 (e). Of course the rule, b/c > 〈k〉, does not hold well for
such a network either. Since it is derived by mean field approximation, it is not hard to understand
that in such a case it does not hold well. We need to make another step forward to understanding the
condition when the deviation of degree distribution is too large. Nevertheless, for all the networks
in Fig.4 if b/c > 〈knn〉 the networks favor cooperation and this condition is sufficient. On the
5
other hand, b/c > 〈k〉 is not a sufficient condition. There are regions where b/c > 〈k〉 satisfies but
networks oppose cooperation.
Here is the conclusion. We derive the condition of favoring cooperation on networks. Al-
though it has been believed that b/c > 〈k〉 is the condition, we show that b/c > 〈knn〉 is the one.
We also show that among three kinds of network: regular network, random network and scale
free network, regular network favors cooperation most and scale free network least. On scale free
network with infinite number of vertices and 2 < γ ≤ 3 cooperation is unfeasible.
Acknowledgements We acknowledge Zbigniew Struzik for his interest and Naomichi Hatano for
helpful discussion.
Competing Interests The authors declare that they have no competing financial interests.
Correspondence Correspondence and requests for materials should be addressed to T.K.
(email: tomo.konno@gmail.com).
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Figure 1 The rule of the game. All the vertices are occupied by individuals. An individ-
ual derives payoffs, P, from the game. The fitness is 1 − w + wP , where w 	 1 means
weak selection. w = 1 means strong selection. For each time step, randomly chosen
individual dies, then the empty vertex will be occupied by the neighbors in proportional to
their fitness. In the figure, the fitness of cooperators is FC = 2 − 2w + w(5b − 6c). The
fitness of defectors is FD = 2 − 2w + 3wb. The vertex will be occupied by a cooperator
with probability FC/(FC + FD).
Figure 2 Mean Field. (a) In mean field approximation, every vertex adjacent to any
vertex with arbitrary degree, k, is replaced by the vertex with degree 〈knn〉. (b) The vertex
with 〈knn〉 is also surrounded by the vertices with 〈knn〉.
Figure 3 Intuition for the rule. Suppose that all the vertices with 〈knn〉 are occupied
by cooperators as illustrated in (b). Because every vertex is surrounded by the vertices
with degree 〈knn〉 in mean field picture and they are already occupied by cooperators as
supposed, any vertex with arbitrary degree, k, is surrounded by cooperators. Remind
the rule of the game that an empty site will be occupied by the neighbors. Now, all
the neighbors of any vertex are cooperators in mean field picture, subsequently overall
network will be occupied by cooperators.
Figure 4 Numerical Simulations. N stands for the size of networks. y-axis means ratio
defined by the fixation probability of a neutral mutant, 1/N , divided by the fixation prob-
ability of cooperator. Dashed lines are ratio=1. If ratio is less than 1, the network favors
10
cooperation. For each simulation, the value of b/c is (〈knn〉 − 〈k〉)t + 〈k〉. t is the control
parameter. t = 0 corresponds to b/c = 〈k〉, while t = 1 corresponds to 〈knn〉. x-axis
means t. For every simulation, after each 2N time steps we build new networks because
the fixation probability may be dependent on specific realization. The games are under
weak selection, w = 0.01. The values of 〈k〉 and 〈knn〉 are merely examples, they vary
across realizations. (a) This network is the mixture of vertices with degree 5, 6, 7, 8, 9,
and 10. The number of five kinds of vertices are the same. We iterated 400000 steps
for each. 〈k〉 = 7.5, 〈knn〉 = 7.9 (b)-(d) Random networks with 〈k〉 = 10, 12, 14. N=600,
600, 700. Iteration is 72000, 72000, and 84000. 〈knn〉 = 〈k〉+ 1 (e) Scale free network with
γ = 2.2 and N = 500. 〈k〉 = 3.7, 〈knn〉 = 6.6 From t = 1 to t = 0, iterations are 25000,
25000, 25000, 50000, 100000, and 150000.
11
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