Supplements to Procedural Analysis
More Advanced Matters
Evolutionarily Stable Strategies
No-one is infallible and it is practically inevitable that mistakes and omissions in the PA theory will appear in the publications and on this website. The only way not to make mistakes is not to do anything. Certainly, the danger of error is greater in a situation like the one in which I find myself, without the continuous peer review which would be provided in an orthodox academic environment. (In a rigidly enforced orthodoxy of course, the danger of greater and far more dangerous errancy exists.)
Fortunately the mistakes which have been found so far – although nonetheless embarassing – have been minor. Most times a fix involves changing a single word, or adding one in qualification.
One overdue matter is the treatment of PA in respect of John Maynard Smith’s concept of the ESS (evolutionarily stable strategy). In fact JMS’s influence set a formative stamp around 1986 when I attended one of his last lectures at Sussex University. I dearly wish I had recorded it, held in the large circular BIOLS lecture theatre. It was JMS’s exposition of the basic notion of evolutionary stability, and analysis in terms of costs and benefits, as recalled from that lecture, that influenced my later investigations into human behaviour. Somewhat belatedly therefore, I would like to discuss PA is relation to the ESS.
Procedural Analysis is a conflation of evolutionary psychology, sex differences and game theory. PA itself consists of three major components:
PA is concerned with the analysis of distinct moves in the “evolutionary game,” especially the competition between male and female, what is sometimes called the “evolutionary arms race” between men and women. In contrast, the ESS model is applied to populations. Sometimes that population, for simplicity, is assumed to be infinite, and it is generally implicit that the population is large.
Here, a trait is regarded as viable if the behaviour is both advantageous and heritable. That is, it has a reason to exist (advantageous) and a means to exist (heritable). Thus the trait is viable evolutionarily; that single behaviour expressed by a single individual can persist. This may be compared with JMS’s concept of evolutionary stability.
Individual traits, and sequential actions performed under the influence of those traits, is the chief focus of PA, while the ESS mathematically models the optimal behaviour of populations.
The female signals, the male responds. The propensity of the female to signal a prospective mate is so pronounced that it must surely be an ESS. The strategy confers numerous advantages to the female (detailed elsewhere) and hence increases her fitness (often, as here, defined as the number of grandchildren which ensue). If the strategy were not advantageous, it would neither be so prevalent nor could it be an ESS. However, signalling is not the only strategy the female can adopt. Let us try then, to put this aspect of PA in the context of the ESS.
|Adopting the notation employed by JMS (p. 24), strategy I is an ESS if||(2.9)|
|W (J, I ) < W (I, I )||(2.9a)|
|for all J ≠ I, where W(J, I ) is the fitness of a single J strategist in a population of I strategists|
|or||W (J, I ) = W (I, I )||(2.9b)|
|and W (J, Pq, J, I ) < W (I, Pq, J, I )||(2.9c)|
|where q is small and W (J, Pq, J, I ) is the fitness of a J strategist in a population P consisting of qJ + (1 − q) I.|
Condition 2.9c stipulates that the fitness of a J strategist in a population consisting of I strategists with a small proportion of J strategists is less than the fitness of an I strategist in that same population.
Suppose the dominant strategy I is to signal, and J is any one of a number, possibly infinite, of alternative strategies. Let the alternative strategy be that the female approaches the male i.e. goes up to him and makes an Approach Statement. Other alternatives exist, but the female approaching the male will certainly serve for the purpose of example.
In a population consisting otherwise of I strategists, a single J strategist approaches her chosen mate. In Dawkins ‘Battle of the Sexes’ (quoted by JMS on p. 130) a cyclical male/female game is defined with
|Male:||Faithful or Philanderer|
|Female:||Coy or Fast|
The female approaching the male is equivalent to her being “Fast.” It will, in general, pay the male in this circumstance to philander.2 ‘Easy Come, Easy Go’ applies; the male has his wicked way and then deserts. Then the female has been exploited and her fitness is reduced: she must raise the progeny single-handedly and that progeny will be less likely to form successful and productive unions of their own. Then we can apply expression 2.9a. However, we cannot be certain that this holds in all cases and for all possible alternative strategies, as required.
Suppose, exceptionally, that the female approached a faithful male, and secured as successful and productive a union as if she had signalled and waited for an eager male to approach her. Then 2.9b is satisfied and this would give rise to a sub-population qJ. That is, the female progeny of the exceptional (mutant) female adopting this strategy J themselves adopt strategy J. We then have P = qJ + (1 − q) I and apply the alternative set of conditions 2.9b, 2.9c for an ESS defined by JMS. The variable q can be thought of as a percentage, which must be small.
It seems reasonable to suppose that 2.9c will be satisfied for the sub-population qJ who approach a male, since females adopting this strategy are more likely to be deserted, and have less productive unions. It should be remembered that our model is phylogenetic, i.e. our assumptions are based on our long evolutionary past, during which time the State did not fulfil the role of male provider for the “single mother.” Even so, these assumptions are likely to still possess validity today.
The J strategy in this model is any action besides signalling. It is easy to imagine alternatives to signalling which would be less advantageous, such as retreating or doing nothing (assuming that it was not the “Going Quiet” signal), but less easy to imagine alternatives that confer greater benefit.3 This is not to say that none exist, but we are here concerned with a population and its aggregate.
I hope to have demonstrated some confluence between PA and the ESS. Notwithstanding, I would not claim that the above is proof that female signalling is an ESS, especially with the likelihood of mixed strategies being employed and the difficulty of completely specifying what a human individual will do “in any situation in which it may find itself.”
Page references are to John Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, Cambridge, 1982