Kevin Aylward B.Sc.

**The Basic Axioms of General Replicator Theory **

The starting point of any good theory are the postulates or axioms of the theory, and the fewer, the better. These postulates can be experimentally tested as justification for the results they predict. The postulates of General Replicator theory are as follows:

1 Traits of Replicators are randomly generated.

2 Traits of Replicators are inherited by "children" from
"parents".

3 Traits of Replicators are selected by the environment.

"Children" and "Parents" are used here as technical terms. They refer to any situation where an entity is produced as the result of another entity. The producer is the parent, the produced is the child. Characteristics or traits can be physical, which are referred to in this document as genes or gene traits, or they can also be idea or concept based, which are referred to in this document as memes or meme traits, but can be also thought of as viruses. An example of a meme is a religion. An instruction of a religion typically says, tells others to believe in this religion. That is, it has instructions contained in it to copy itself.

__General Processes__

The general process is that characteristics are continually being generated in a random manner, usually by varying existing traits, but this is not a necessary assumption. These characteristics, and existing ones, are replicated in children, with the environment acting in a non-random (but usually non-conscious) manner selecting what particular characteristics survive. In principle, it does not matter whether or not traits are randomly generated or deliberately generated. The theory still applies identically. The reason for introducing the concept of random generation, is simply to explain that complicated behavior does not require any conscious or designer type intervention.

__Non Random Selection__

The fundamental reason order arises out of random generation
is that *the basic laws of physics are non random*. For
example, if hydrogen and oxygen are placed in a box at sufficient temperature,
they automatically react to form water. That is, certain physical
structures are more probable than others. Even temperature itself is a
non random selector. However, this "purposeful" selection does
not imply conscious selection.

__Example__

Consider insects being born with random colours, e.g.
brown, pink, green, red...Now suppose that there are birds in the region
that feed on these insects. Further suppose that this particular region
(environment) is a grass field. If the reasonable assumption is made
that any green insects are harder to find and therefore be eaten, it can
be rationalized that the green insects might have a better probability
of survival, and therefore to pass on that green trait to offspring, then the other
colored insects would. Application of the theory, to be
described, *mandates*, that in this particular idealized situation,
after sufficient generations, mostly green insects will be observed.

Comment - variation of traits and generation of new traits are, essentially equivalent as far as these Darwinian axioms are concerned. That is, both methods generate a new trait.

**Replicator Principles**

First we assume that an entity, named a "Replicator" has traits, and satisfies the following postulates/axioms:

1 Traits of Replicators are randomly generated.

2 Traits of Replicators are inherited by "children" from
"parents".

3 Traits of Replicators are selected by the environment.

*Anything* that satisfies these conditions will result in
certain predictable consequences. For example, there is considerable experimental
evidence that all animals satisfy theses conditions. Memes
also satisfy these postulates.

General Replicator Theory is based on statistical analysis, that is, most terms used here refer to means, standard deviations, and general properties of large numbers of many generations. The outline presented here is also greatly simplified in order to illustrate general trends. For example, it is usually assumed in this outline that replication rates are constant, where as in reality, they may vary based on the size of the generated population.

**That which is mostly observed, is that which
replicates the most**

Consider a randomly generated trait that has
the property that it can be, and is replicated. Since, by assumption, in *can*
be
replicated, its copies must also be able to be replicated.
Now, consider various traits continually being replicated,
from parent to child, it its child, to its child.... Suppose that, due
to the constraints of the environment, trait A can replicate at
say, a 1% faster rate than B, that is some environmental factor either impedes
trait B's rate of replication, or aids trait A's rate
of replication. Also assume, for the sake of argument, that the initial
numbers of A and B are equal, although this is not necessary. After 1000
generations the ratio of A/B = 1.01^{1000} = 20,959, by simple mathematics.

This is a very large ratio. So, it can be ascertained that if there is a
*consistent* and *continuous* replication rate advantage of
one trait verses another, the one that is only *slightly* better,
will, given enough time, completely dominate. That is:

1 * "that which is mostly observed, is that which replicates the most".*

It should obviously be noted that, some traits are not consistent over time, for example, getting infinitely bigger then an opponent also has negative consequences in replication rate, so that in practice many competing traits will be observed.

An immediate consequence of 1, is that although traits
of Replicators are inanimate, the net observed effect of well selected
traits is just *as if* the traits take action to maximize themselves.
Using this notion as a convenient fiction usually makes explanations
easier.

The general principle is that traits are usually well replicated at each generation, but that mutations arise such that additional new traits with new behavior are generated. These newer, mutated traits compete with existing traits.

**Replicator Result 1 (RR1) - Fasted Replicating Replicators Observed
the Most**

A continuous, replication rate advantage of one trait over another trait, will result in the advantaged rate Replicator dominating in numbers over the disadvantaged trait Replicator. That is, the best Replicators will be observed the most.

Comment: This almost says nothing new, but simply acknowledging this basic principle, explains much.

**Inherent Selfishness Of Traits**

Consider a random generated trait that aided another competitor trait's
replication rate, to its own *final* replication rate disadvantage.
This would mean that it would not replicate as well as the trait it was
aiding, hence by the above reasoning, that trait would be driven to a
small ratio of its competitor aided trait. On the other hand, if a trait
aided its own replication rate, to the disadvantage of a competing
trait, it would necessarily result in a large ratio to its disadvantaged
competitor trait. This is the principle of "selfishness". A replicating
trait is driven by simple mathematics to be a trait that replicates
itself to the disadvantage of another competitor trait. Any trait that
is not ultimately "selfish", will eventually be overrun by any
other trait that is. Note that "selfishness" is by reference
to its final, outcome. It is not suggested that a trait has any consciousnesses
that makes it "selfish". Its is simply that whatever that is observed,
is just "as if" it is consciously selfish.

Note: It is quite possible to aid another trait, if by doing so, it is attempting to receive a net advantage. A selfish trait must take advantage of any unselfish trait, therefore, that's what we observe.

**Replicator Result 2 (RR2) - Replicated Traits Are
Selfish**

If a trait is observed that is a result of significant
replication and selection, then it * must* be "selfish".

**Replicator Result 3 (RR3) - Pretension of
Replicators ****Wanting to Replicate**

Since the *net* effect of a well generated replicator
is exactly the same as if it consciously wanted to replicate, although
it obviously can't and doesn't, it simplifies explanations to *pretend*
verbally, that they do. That is, in any arguments relating to Replicators,
where phrases such as "the replicator will maximize itself by
taking...", what is actually meant, is that the net effect of random
generation and selection process will result in an effect "just as
if" the Replicator consciously took action. Similar arguments for
the word "designed". That is, traits are not designed, but
exist * as if* they have been.

**Summary**

1 That which is mostly observed, is that which replicates the most.

2 Well replicated and selected traits are selfish.

3 Replicators "want" to increase their numbers and rate of replication, and will do whatever it takes to increase their numbers.

The above are the key results of applying logic to the basic axioms of General Replicator Theory. They are not based on any particular embodiments of a Replicator, and by and large, form a tautology. They stand independently. If a given process can be shown to satisfy, the above axioms, the above results automatically apply to results of that process.

**Combined Replicators**

For their own selfish (as noted above) replication advantage, Replicators can combine with other Replicators, forming a joint Replicator, in an effort to improve the Replication rates of each individual Replicator. In doing so Replicators "select" other Replicators, based on how effective they are in enabling their own traits to be copied, by piggybacking on to the successes of the other Replicator, so to speak.

Some Replicators, e.g. human Replicators, can only replicate by combining with another replicator. These will be referred to as Combined Replicators.

A Replicator that can combine with another Replicator without inhibiting any other immediate Replications that it might be able to perform, is named an M-replicator, from male.

A Replicator that can not combine with another Replicator without inhibiting any other immediate Replications it might have performed, is named an F-Replicator, from female. That is, if a replication, inhibits a further replication, an F-Replicator is not able to increase its replication rate by combining at every opportunity.

**Quality and Quantity**

Combined Replicators can increase their replication rate
by combining with another Replicator based on either *quality* or *quantity*.

A better quality Replicator is simple a Replicator that can replicate at a faster rate than another. This can be an M-Replicator, or an F-Replicator.

A Replicator, replicating by quantity, replicates simple by replicating with another replicator by the increase of numbers such replication ensues. This must be an M-Replicator.

**Quality Versus Quantity**

It is shown here, that under some rather broad conditions, selection based on quantity always results in a faster rate of replication, than that based on selection on quality.

**Observations Of Combined Replicators**

1 A selection based on quantity has the negative attribute that such final Replicator may not have the fastest replication rate, i.e. quality.

2 A selection based on quality has the negative attribute that it reduces the number of replicators that a Replicator can combine with, i.e. quantity.

That is, a quality selection has a negative impact on quantity and a quantity selection has a negative impact on quality. The question then is, which one wins?

__Technical Terms:__

N =* Quantity factor* - average of replications in a
generation period by a replicator. i.e. how many Replicators a
Replicator combines with in a generation period. e.g. 10 mates per day.

Q = *Quality factor *- average number of surviving Replicators
generated per Replicator combination. i.e. how many offspring
Replicators are generated with each combination. e.g. 2 offsprings per
mate (Q is usually < 1). The assumption here is that the offspring
replicators survive for their own generation period.

GRR =* Generation Replication Rate* - GRR=Q.N e.g.
GRR=mates/day x offsprings/mate = offsprings/day = replications/day.

G = *Number of generations*. e.g. after 10
generations.

A = accuracy of the replication - e.g. only 90% of the Replicants are exact copies.

PG = *Population Generated* - The population, or
total number of Replicators after G generations at generation point G
due to generation G. That is prior generations are assumed to have ceased
existence.

PG = (NQA)x(NQa)x(NQa)...= (NQa)^{G}

**Survival Of The Fittest**

The "survival of the fittest" is a popular term that is somewhat vague. Some also claim that it involves a circular argument definition. It is used here as a term to describe the overall statistical effectiveness of a Replicator in replicating a trait, in terms of the above, it is the product NQa

Fittness = NQa

In reality, SOF is better expressed as "that which is observed mostly, is that which replicates the most".

**Example**

The following is an example, that illustrates the basic concept. The intent is not to be too rigorous, but give reasonable justification to the final conclusion. A more detailed analyses can be performed to determine under what exact conditions the results are valid. However, this is out with the intended scope of this document.

We assume here that each Replicator combines with, the *quantity*
factor, N other Replicators and that each combination can replicate with a
*
quality* factor of Q. This is rather simplified, that is the
generation replication
rate, GRR = QN/generation. In words, the replication rate is the product
of how many combinations are generated and how effective that combination
is at replicating. For convenience, we set A=1. Note: a suitable reference value of
Q would be Q=1.

Consider a Replicator that selects Replicators only on quantity.
Further assume that this Replicator combines with another Replicator at
a rate of 10 per day. Each generation will therefore result in a 10x increase
of the existing number of daily Replicators per day. That is, after say,
6 days, the number of Replicators will be
(10x1)x(10x1)x(10x1)x(10x1)x(10x1)x(10x1)x(10x1) = 10^{6 }= 1
million.

Consider a Replicator that selects only on quality. Further assume that the numbers of such better quality are such that the Replicator can only combine with another Replicator at a rate of 5 per day. Further assume, that such a quality Replicator is 1.2 times as effective as an unselected Replicator. After 6 days the number of Replicators will be (5x1.2)x(5x1.2)x(5x1.2)x(5x1.2)x(5x1.2)x(5x1.2)x(5x1.2) = 46,656.

For these particular numbers, it is clear that the larger combination rate wins, that is quantity. Normalizing the QN=QN1/QN2 ratio, we can see that if QN is less than 1, then after a sufficient number of generations, the Replicator that doesn't select will overwhelmingly dominate as the limit of the number of generations G->large number, tends to zero, implying that the number of selective Replicators will also tend to zero. That is for:

PG = (NQ)^{G}

if NQ < 1, PG -> 0

if NQ > 1, PG -> infinite

**Support for the Example**

The key issue is, how does any increase in replication rate described by higher quality be offset by the lower numbers of such available higher quality Replicators:

Assumption by construction - There are unlimited resources such that there are no physical constraints, i.e. mass-energy, to limit Replicator numbers. This is an assumption for the purposes of this example only. General Replicator Theory is easily extendable to include factors such as, for example, N is a function of the populations resource supply.

Comment - N is determined from the number of
M-Replicators found jointly "willing" to combine with
F-Replicators in a generation *time* period. It is not a resource
limitation, only a time limitation.

Key assumption - The number, N, of available quality Replicators falls faster than the quality factor, Q, increases.

Support for the assumption - The gaussian distribution curve.

Gaussian curves for any physical trait show a decline in
a the numbers of a trait that have a value greater than a particular
value away from the mean. Inspection of the gaussian curve shows that
the product of a traits value (replication rate) with the number of
traits at that value or greater, tends to zero as the traits value moves
away from the mean. For example, only 3% are 3 sigma away, and since the
relative sigma is usually much less than 1 (i.e. < sqrt(pi)/2), the
QN product will reduce from unity or below immediately as an increase in
quality occurs. This shows, that for essentially all traits, in practice,
*an increase in selection by quality will necessarily result in a
reduction of available traits at that quality, resulting in a net reduction
of final replication rate compared to selection based on quantity only.*

**Continuous Evolution**

The discreet mathematics identified above is difficult to extend to more complicated problems, so a continuous approximation is usually made such that the difference equations are replaced by differential equations.

The general assumption is that population statistical various in a continuous manner. Consider a small increase in population:

dP = NQ dt

Where N and Q (q) are as defined above, N is the number of mates per generation time, and Q is the number of offspring per mate.

The number of available mates is clearly dependant on the size of the population, so that N=mP, m being a ratio constant. Therefore:

dp = mqP.dt

dP/dt = mqP = fP, f=fitness

The solution to this equation is

P = P_{o} e^{mq}

See also meme/gene Competition Mathematics.

**General Evolution**

In general, there are numerous traits of many independent Replicators, both gene and meme, all competing against each other in a specific environment actually causing its own environment to be a function of the traits themselves. That is, each Replicator will compete with each other, and the memes and genes of each Replicator will compete against themselves. Many traits will co-exist by actuating a prisoners dilemma process, many will flourish, then die as evolution progresses. Exactly what traits will exist an a given generation point requires a detailed mathematical model of the replication rates of each Replicators gene and meme traits, and how the gene and meme traits interact with themselves, and the environment. In general, this is non trivial, and probably the best way to simulate such evolutionary processes, is to use a genetic Algorithm (GA) method on a computer.

**Summary**

If a Replicator can continuously increase its numbers and/or rate of replication by simple combining with another Replicator, i.e. M-Replicator, it will do so, essentially, irrespective of quality of the other Replicator. If a Replicator can not do this, i.e. F-Replicator, it must select replicators based on the quality of the other Replicator to increase its replication rate. That is M-Replicators will replicate either by quality or quantity, F-Replicators will only replicate by quality.

**Appendix**

The relevant formula for the normalized replication rate product RR = Q.N, and normalized mean of 1 is:

GRR = (1 + X).erfc(X)

where erfc() is the complementary gaussian error function.

This product is always less than 1.

These papers may be freely copied only for non commercial use,

provided full credit is given to the author.

© Kevin Aylward 2003 - all rights reserved