Return to equilibrium


Jason Collins


July 30, 2011

A post on Cheap Talk reminded me about an old paper of Bill Hamilton’s on the potential for extraordinary sex ratios. Apart from its importance for the particular topic (Hamilton considered it to be one of his best papers), it is one of the more interesting expositions that what is good for the individual (or more specifically, the gene) may not be good for the species. It also raises the implicit question of how quickly something can return to equilibrium.

First, take Fisher’s argument for the equality of sex ratios, as stated by Hamilton:

  1. Suppose male births are less common than female.
  2. A newborn male then has better mating prospects than a newborn female, and therefore can expect to have more offspring.
  3. Therefore parents genetically disposed to produce males tend to have more than average numbers of grandchildren born to them.
  4. Therefore the genes for male-producing tendencies spread, and male births become commoner.
  5. As the 1:1 sex ratio is approached, the advantage associated with producing males dies away.
  6. The same reasoning holds if females are substituted for males through out. Therefore 1:1 is the equilibrium ratio.

All well and good. Then Hamilton frames the following situation (remembering that for humans, men have an X and a Y chromosome, while women have two X chromosomes):

Suppose the Y chromosome has mutated in a way which causes it always to win in the race to fertilize. A male with the Y mutant then produces nothing but sons. Provided these sons, who also carry the mutant, cannot be in any way discriminated against in the unrestricted competition for mates (a situation which is implied if mating is random for the whole population), the Y mutant will have a constant selective advantage. As the mutant spreads, the population sex ratio will become more and more male-biased and the population itself will become smaller and smaller; finally the population will be extinguished, after the last female has chanced to mate with a male carrying the mutant.

Starting from a population of 1,000 with one mutant male, it takes only 15 generations to drive the expected number of females below one. A similar situation can arise with an X-linked mutation, although the path to all females and species extinction is slower.

While this is a theoretical example and rests on the assumption that there is no discrimination against the mutant males or females, there are a few real-world cases of the X-linked drive to all females. At the time of Hamilton’s paper (I’m not sure if this has changed) only one case of the Y-linked drive was known. In that case, a mosquito has a sex-determining gene (not a whole chromosome) but the path to all males has been restrained by several other genes.

That brings me to the economics. Fisher’s basic principle, which is the best starting point for discussions of sex ratios, sounds much like a neo-classical economic description of the world. If things tend away from equilibrium, there is a clear strategy that can be exploited - and we would expect it to exploited - that will return the system to equilibrium. However, when someone plays a novel, fast acting strategy, things can move quickly. The question is whether the responding strategy is played immediately or may take some time. In the above case of the mosquito, restraining mutations have occurred, preventing extinction.

Some evolutionary or neo-Schumpeterian economics seeks to deal with this, particularly in the form put forward by Nelson and Winter. Firms search for technological and organisational solutions based on habitual methods and those which succeed in finding them replicate and spread. The process is not immediate and can be lumpy and crude. Arnold Kling, with his patterns of sustainable specialisation and trade (PSST), also notes that time is an important consideration. Take the following:

It is the task of entrepreneurs to organize the economy so that people produce stuff that has value. Sometimes, entrepreneurs are not quite up to that task. Then you get unemployment. There is an incentive for entrepreneurs to try to figure out ways to create patterns of sustainable specialization and trade that utilize workers who are currently unemployed.

The system gets a bump. How long does it then take for a strategy to be developed that deals with it?