Subsistence Strategies, Uncertainty and Cycles of Cultural Diversity

Our (Mark Lake's and mine) paper on "The Cultural Evolution of Adaptive-Trait Diversity when Resources are Uncertain and Finite" have been accepted for a special issue of Advances in Complex Systems!
We basically extended the work we've done for the conference on Cultural Evolution in Spatially Structured Populations (see blog entry), focusing more on the dynamics of cultural evolution for traits which are: 1) adaptive (instead of being neutral) and hence determining changes in the reproductive rate; 2) characterised by negative frequency dependence (we've actually explored initially both positive and negative frequency dependence and a combination of the two, but that's another story/paper); and 3) produces stochastic yields. 
In practice, we developed an ABM (written in R) where agents forage based on a specific trait they possess. The yield of the foraging activity is associated to some degree of uncertainty and is restricted by two types of frequency dependence. In the S-mode model we've explored scenarios where different traits represents different technology or behaviour which are adopted for harvesting a shared resource, while in the I-mode model we've explored scenarios where each trait harvests a separate and independent resource (e.g. different preys). We then allowed agents to reproduce, die, innovate and learn (with frequency z)  using a model (payoff) -biased transmission following the model proposed by Shennan (2001), and measured the diversity of traits using  Simpson's diversity index. The model showed many interesting properties, here are some which I thought were particularly notable:

• High values of z (frequency of social learning) have negative impacts in both I-mode and S-mode models if some degree of stochasticity in the payoff. 
• When traits share the same resource, the highest rate of cultural evolution occurs with values of z which determines a limit cycle between moments of low and high diversity. 
• When traits are harvesting independent resources, the highest rate of cultural evolution occurs with values of z which determines the adoption of largest number of different traits (highest richness) with patterns similar to the Ideal Free Distribution.  When the frequency of social learning is too high, novel traits are lost by the innovators before this is transmitted to the rest of the population.

The negative impact of high reliance on social learning is perhaps the most interesting outcome and relates to what is known as the survivorship bias. Suppose a population of n individuals adopting the same trait A, which determines a normally distributed payoff (with mean μA and standard deviation σA).  At a given point in time an individual innovates and adopt a novel trait B, with a payoff which is on average higher than A (thus μA > μB). With a frequency z some individuals will copy the most successful (thus the individual with the highest payoff) individual among k randomly sampled individuals (with k being our sample window of observation for each agent). If the  σA=σB=0 the payoff will be always the same, and thus trait B will always produce a higher yield than A. This means that the novel trait will be adopted by approximately zn agents, and that the innovator will stuck to B (which will be always higher than A).   However if σA > 0 < σB (or in other words if the payoff have some degree of uncertainty) something different will happen. Since the number of individuals adopting trait A is by definition higher than the number of individuals adopting B, and since the payoff is stochastic, some lucky individuals with trait A are likely to have a payoff higher than μB. If these individuals are among the k sampled individuals of the innovator, the innovator will switch back, erroneously underestimating his own new trait. If z is low, the innovator is unlikely going to do this (it won't rely on social learning) while a proportion zn will have some chance to adopt trait B. If the number of individuals adopting this trait exceeds a certain number this will unlikely got lost, and hence can spread and invade trait A.  The survivorship bias tells a similar story. Suppose you are a businessman and decide to adopt a specific market strategy because you've read on forbes that some guy was successful on this. You have a model (the guy on forbes) which is successful and you explain this based on the strategy he used. However forbes won't mention you that maybe there are 10,000 other businessman who adopted the same market strategy but actually failed. The same applies for music industry. You see people making a lot of money, and so you decide to learn and give a try. And you ignore that hundreds of thousands of people did the same, and failed. The pattern is probably stronger here, because the success rate is not normally distributed, but much more skewed (in fact its likely to be a power law, see here). The small tail of very successful individuals are much more visible than the other (majority) of people. In our model the  shape of the distribution is different, but nonetheless the few successful people are regarded as a representative of a trait in a model biassed transmission. 

So what's the moral in all of this? If there is any, although it's an obvious, a bit cheesy, over-mentioned advice, is to "believe in yourself and not rely to much on copying successful individuals". They're in most case just lucky, and you might have something bigger in your hands.