**Polymorphism of Recombination Strategy in
Evolutionary Games**

**a Theoretical Study**

Recombination is the process of the formation of a new combination of genes on a
chromosome as a result of crossing over. Although the evolutionary function of
recombination is not fully understood many research works support its long-term
effect on the population level. The majority of these works investigate the
relationship between the recombination and fitness of an organism, defined as
the organism's ability to survive and reproduce in a particular environment.
Others, go farther, and propose that recombination is negatively
associated with fitness, breaking down unfit combinations of genes with a higher
probability than fitter ones.

For an association between recombination and fitness to occur, it is sufficient
that some correlating factor s exists, which is associated with fitness f and
affects the recombination rate r. Hadany and Beker studied the performance of
Fitness-Associated Recombination (FAR) using a series of simple haploid models
and showed that the allele for FAR tends to increase from scarcity in any
population with a uniform recombination rate. However, their model restricts FAR
to some kind of an on-off behavior where a fitness-associated recombination
either exists or not. In this project we show that FAR is not a Boolean but a
multi-value feature defined on a continuum, which determines the recombination
strategy. We study various classes of strategies within the boundaries of
association between fitness and recombination, their behavior and their
evolutionary effects. In addition, we simulate evolutionary games to show the
ability of the a strategies to spread

in a population originally dominated by another strategy. We also extend these
simulations to account for more complex models of

varying linkage values, nonrandom mating and mutations in regions controlling
the strategy.

**Methods**

We suggest a more general evolutionary model of fitness associated
recombination. This model is based on the non-facultative approach and a new
continuous formulation of equilibrium equations.

**Continuous strategies -** The new model extends the
strategy notion formulated by Hadany and Beker and instead of 2 trivial
options (z(x)
= const and a logical
function) enables arbitrary continuous strategies. This extension is of utmost
importance since the natural evolution is obviously not restricted to the
trivial cases of simple discrete functions as in the original model. Even though
we do not claim that our model resembles the true evolutionary process, still
this generalization opens up a wide range of potentially effective evolutionary
approaches.

**Linkage - **We use a non-facultative
approach of reproduction. Thus, the linkage between the fitness loci and the
strategy locus is independent of the fitness of the organism, its strategy, or
the recombination within the fitness loci. We test a variety of linkage factors
and examine the impact on the individual and population levels.

**Mutations in strategy locus** - We propose another
extension which allows mutations not only in the fitness loci but also in the
strategy locus. Such a possibility serves as a balancing factor and causes
organisms with high fitness (which usually possess the winning strategy) to
switch their strategy and vice versa.

**Non-random mating - **In real world the mating process in
not random and the individuals choose their pair mate according to some
criteria, for example fitness.

Random mating is a factor assumed in the Hardy-Weinberg equilibrium, and thus,
usually applicable to populations without migration. However, when evolution or
natural selection occurs, random mating is not the logical choice as in
evolution traits favorable to survival are preferred to others.

We consider two mating scenarios which mimic the natural mating process more
precisely:

1. Uniform fitness-dependent mating where organisms tend to chose their pairs
mate with fitness similar to theirs. More formally, the probability of an
organism with relative fitness f1 to mate with another organism with relative
fitness f2 is drawn from a normal distribution with mean f1 and standard
deviation equals 1. As reference, we also consider the opposite paradigm where
the probability is drawn from a normal distribution with mean 1-f1.

2. Free-will fitness dependent mating where organisms with
higher fitness tend to mate more than organisms with lower fitness. We also
consider the opposite theory that organisms with low relative fitness tend to
reproduce more in order to increase the probability of producing a viable
offspring with high relative fitness.

We note that mating paradigm is population dependent and therefore it is logical
to examine both paradigms suggested here.

We use computer simulations to investigate the effect of the different aspects
of the enhanced evolutionary model.

Our simulations mimic an evolutionary process in closed populations without
migration, and include selection, mating, recombination, mutations and survival.

There, an organism has 10,000 loci that determine its fitness through a given
formula that takes into consideration the number of mutated loci and the
relative damage inflicted by a single mutation. In addition, a locus that
controls the recombination strategy of the organism is linked to the fitness
loci via a pre-defined parameter.

In the general framework of these simulations which is sketched in the figure
below, a random pair of organisms is selected for mating, where recombination
within the fitness loci occurs with probability proportional to the
recombination rates of the two parents.

The result of this mating is a single offspring on which we inflict mutations,
the number of which is drawn from a Poisson distribution with a pre-defined
parameter.

The offspring survives with probability proportional to its absolute fitness.
Additional recombination between the fitness loci and strategy locus may occur
with a fixed probability determined by the linkage parameter. This recombination
occurs independently of decision regarding recombination within the fitness
loci.

In this setting all mutations are reversible. In each of the runs additional
features which are explicated are added and distinct it from the others.

We examined the ability of a variety of continuous strategies to spread in
a population originally dominated by another strategy. For each two strategies,

zi, zj, we measured the winning percentage of strategy zi in a population
dominated by strategy zj and vice versa in a set of 1000 games. If the winning
percentage of zi is significantly higher than the initial fraction (called
domination barrier) of organisms with strategy zi in a population of the
opposite strategy zj and the opposite is not true, we say that strategy zi
dominates strategy zj. Our domination games consist of 3 rounds:

1. All strategies vs. UR - all strategies (with fixed parameters) compete
against UR.

2. Sigmoid vs. UR - sigmoid strategies with different parameters
parameters compete against UR. This round is of utmost interest since sigmoid
functions are the generalization of the simple FAR strategy presented by Hadany
and Beker.

3. Sigmoid duel - sigmoid strategies with different parameters compete
with each other.

In the first two setups changes in the models (such as linkage, non-random
mating, etc.) were also considered. For the sake of fairness in all simulations
the average recombination rate of the competing strategies is equal.