| Synopsis: Here, we will show that the only times where rule #1 for determining a pure ESS might not be true are when the invading strategy arrives at high frequencies. This is simply not possible when a new strategy is created by mutation and so when we consider the ultimate causes of variation in evolution, rule #1 holds nicely. On the other hand, if we are concerned with the evolution of a small population, it is possible that rule #1 might not hold if a large proportion of migrants arrive in some population. |
This page deals with two separate questions. The broad question is one of the useful range prediction made by models -- i.e, the range of possibilities a model presents versus what is biologically likely and meaningful. The narrow question deals with understanding the circumstances where the two rules we learned for determining whether or not a strategy is a pure ESS are correct. Most of this page deals with this narrow question.
The Problem
Recall that for a pairwise game involving strategy A vs. B where A is common and B exceeding rare, A is a pure ESS if:
1. E(A,A) > E(B,A) (press here to review the logic of this)
or if :
2. E(A,A) = E(B,A) -- AND -- E(A,B) > E(B,B)
If you think about these two rules, you might be bothered a bit. Mathematically, they seem overly simplistic. The reason is that the fitness of a strategy is actually determined by the all possible payoffs to a strategy times the frequency of each (press here to review). One could imagine cases where E(A,A)>E(B,A) is true but where A was not be a pure ESS. This would be the case if :
Let's find out how large these factors would need to be to upset rule 1.
Either continue reading or skip the math and go to the graphs and discussion
The Problem
Recall that for strategy A to be a pure ESS then:
eq. 1. W(strat. A) > W(strat. B)
expanding this using the expressions we learned earlier (re-expressed with A and B as the strategies):
eq. 2. E(A,A)*a + E(A,B)*b > E(B,A)*a + E(B,B)*b
where freq. of Strategy A = a
and freq. Strategy B = b = 1 - a
| Important Note: all of the discussion from here on will assume that strategy B, as the invader, has a frequency less that 0.5. Perhaps it might be better to think of B simply as the minority strategy, a situation that an invader would generally find itself in! |
Now, when would A not be a pure ESS? Either:
eq. 3a. {E(A,A) - E(B,A)} * a = {E(B,B) - E(A,B)} * b
OR
eq. 3b. {E(A,A) - E(B,A)} * a < {E(B,B) - E(A,B)} * b
Of these two expressions, eq. 3a is the most useful since it describes the minimum condition for to prevent strategy A from being a pure ESS -- for strategy A to be a pure ESS the left side of eq. 3a must be larger than the right.
Next, to reduce the number of variables and hopefully make things a bit easier to comprehend, let's lump all the payoffs into a ratio R:
eq. 4. R = {E(B,B) - E(A,B)} / {E(A,A) - E(B,A)}
Let's see what this ratio means:
Be cautious here -- remember that both the numerator and denominator simply give the difference in payoffs per contest of rare or common contests. By themselves they do not predict whether or not a strategy is an ESS. One other note -- we will not consider what happens when either the numerator, the denominator, or both are negative numbers -- feel free to think about it, but let's try to keep things simple here. |
If we now substitute eq. #4 (definition of R) into eq. 3a:
and then solve for b:
Now we can answer our original question: "under what circumstances is rule#1 invalid?" A graph and some calculations based on eq. 5b will illustrate the answer. For both exhibits, we select a series of different frequencies of strategy B, starting from rare (as was assumed when me made up rule #1) to very common and then find the value of R for each. Here's the graph:
What does this graph tell us? Basically, it says that as strategy B becomes rarer, the difference in the payoffs of rare encounters (E(B,B) - E(A,B)) in favor of B over A required to offset an advantage favoring A (E(A,A) - E(B,A) increases exponentially! Here is the same information in tabular format:
Freq. B |
{ E(B,B) - E(A,B)} / {E(A,A) - E(B, A)} |
0.001 |
999 |
0.002 |
499 |
0.005 |
199 |
0.010 |
99 |
0.020 |
49 |
0.1 |
9 |
0.2 |
4 |
0.5 |
1 |
To take one set of values from this table: if B is at a freq. of 0.001 then for A not to be an ESS when E(A,A) - E(B,A) > 0:
eq. 6 E(B,B) - E(A,B) >= 999 * {E(A,A) - E(B,A)}
It would seem totally improbable that E(B,B) would be so large and so rule 1 represents a good and useful approximation.
The only circumstances where we we should start to worry about rule #1 is when a large proportion of invaders enter a population of A strategists. This may not be completely far-fetched. For instance, it is not unreasonable to assume that a small local population of 10 individuals could be invaded by nearly as many migrants (but it would be very unreasonable to assume that such an invasion was through mutation (see note)). Furthermore, even if a large numbers of migrants arrive, it would be very unusual for them to possess radically different allele (strategy) frequencies -- usually migrants come into populations from nearby populations that are similar genetically.
Realistic models: Thus, rule #1 is a good one for most situations, it represents an appropriate simplification of the complete rule for determining an ESS. Rule #1 works because the magnitude of payoffs favoring the invader in rare contests required to offset those that favor the common strategy in common interactions become unbelievably large the rarer B becomes. And, since we are usually interested in how a totally new strategy would enter a population by mutation or learning, not by migration from another population, the normal course of invasion will involve very, very rare invaders.
In biology, others sciences, and life in general it is very useful to find appropriate simplifications of complicated models or rules. The justification for these simplifications should always be understood so that they are not misapplied, but when properly used, they are great time savers.
Endnote
About Mutation, Migration, and Evolution: Any time new alleles are added to a population, variation increases and the raw material for evolution of that population has been increased. Thus, migrations are of great potential importance in evolution of populations as, of course, are mutations. We don't know enough about the evolution of species to discount the idea that a new mutation could increase in one population, for instance by genetic drift, and then migrate to another population with a different environment and now be amplified by selection and become an important adaptation. Thus, migration is always mentioned as a possible immediate source of genetic variation or means to change allele frequencies.
However, since mutation is the the ultimate means for a new genetically based strategy to enter a population and since mutation rates are low (about 1 in 10^5 or 10^6), it is probably best to analyze invasions by assuming that the invader is a very rare mutant.