General Replicator Theory 

Fundamentals

Kevin Aylward B.Sc.


Index

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

Definitions

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 = 1.011000 = 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:

"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) = 106 = 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 = Po emq

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