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5. Group selection. In this case, the system is assumed to be split in groups,
each of which can become extinct, or give origin to another one. A group can
be invaded by defectors, but in this case it will succumb in competition with
all-cooperator groups, that have larger payos.
15.5.5. Eective treatment of strategies
The analysis of strategies with memory is quite complex, due to the large number
of variables. As shown by Nowak [70], it is often possible to study the evolutionary
competition among strategies by using an eective payo matrix W , that species
the average gain of a strategy against the others, given the parameters of the prob-
lem. In this way the evolution of the system is just given by the interaction weights
of Eq. (15.30), and the analysis is quite simpler.
For the competition between two strategies A and B with a parameter p, three
conditions can be found [70], according with the position of the saddle point p
c
that
separates the basins of all-A (p = 0) and all-B (p = 1)distributions:
1. If p
c
= 0, only the distribution all-B is present. This generally corresponds to
all-defectors.
2. If p
c
is near zero, the strategy all-A is evolutionary stable: in an innite popu-
lation, a single (or small but dispersed) mutant B cannot invade, while a bunch
of B (in neighboring sites) can.
3. If p
c
> 1=2, the A startegy is risk-dominant: by mixing at random the popula-
tion, in a larger number of sites A is dominant.
4. If p
c
> 2=3 then the A strategy is advantageous. The analysis of nite popula-
tion of size N shows that the xation probability of a mutant by random drift
for at or weak selection is 1=N. For p
c
> 2=3, the xation probability of a
cooperator mutant in a nite population of defectors is greater than that given
by random drift.
15.5.6. Self-organization of ecosystems
The nal goal of a theoretical evolutionary theory is to explain the macro-patterns
of evolution (speciation, extinctions, emergence of new characteristics like coopera-
tion, formation of complex ecosystems) as the result of self-organization of individ-
uals [71]. The main idea is that at the molecular level evolution is either neutral,
or lethal. The latter corresponds to unviable phenotypes, for instances, mutations
that lead to proteins that do not fold correctly. Within this assumption, the tness
space for a single \replicator" (an ancient bio-molecule) is like a slice of a Swiss
cheese. We assume that there is one big connected component. By considering only
the paths on this component, the origin of life should correspond to one largely
connected \hub", where entropy tends to concentrate random walks (induced to
mutations).
