Mathematical Modelling 101 – The Price Equation

So in this post I’m going to assume you know absolutely nothing about anything. If you know something about something this probably isn’t what you’re looking for. If you’re looking for something which will go into depth on how the price equation is derived this probably isn’t what you’re looking for either. If you simply want to know what the price equation does and how to use it at face value then welcome! You’ve found the right place.

The price equation is used to calculate how the average value of any variant can change within a population from generation to generation.

Here I will cover everything you need to know to understand the equation and slot in the right values:

Covariance

Covariance is the measure of how much 2 things change together and can generally be calculated by the following equation:

cov(x, y) = E(xy) – E(x)E(y)

cov(x, y) is the value of how much two random variables (x and y) change together.

E(xy) is the expected value of the product of x and y and E(x)E(y) the product of their expected values.

The price equation

Imagine a population of individuals who had varying degrees of variant S. (this is usually a characteristic expressed by genes in biology, but for the purposes of linguistics this could just as well be the use of a word, phrase or other varying item which can have different degrees of communicative success or ‘fitness’.)

If we are to keep track of variant S, where the population is divided up into subpopulations (i = 1, 2, 3, …) dependant on how much of variant S they have. ni is the number of individuals in subpopulation i and the value of S within each subpopulation is zi.

w = fitness of whole population
wi = fitness of subpopulation i
w ‘ = fitness in next generation of whole population
wi ‘ = fitness of next generation of subpopulation i
z = average amount of variant S found in whole population
zi = average amount of variant S found in subpopulation i
z ‘ = average amount of variant S found the next generation of population
zi ‘ = average amount of variant S found the next generation of subpopulation i
n = number of individuals in the population
ni = number of individuals in subpopulation i
n ‘ = number of individuals in next generation of population
wi ni = number of offspring in next generation of subpopulation i
ni ‘ = number of offspring in next generation of subpopulation i

so:

wi ni = ni

therefore:

wi = ni ‘/ni

So, amount of change from zi to zi ‘ can be summed up as follows:

The amount of change from z to z’ can also be summed up in this way.

And now you should know everything needed to know in order to understand the price equation!

George Price

Further Reading

To read about George Price’s extraordinary life go here: http://en.wikipedia.org/wiki/George_R._Price

http://www.biology.ed.ac.uk/research/groups/gardner/publications/Gardner_2008.pdf

For a clear, in depth derivation and expansion on the price equation:

McElreath & Boyd (2007). Mathematical Models of Social Evolution: A guide for the perplexed. University of Chicago Press. Amazon link.

  • Lea Luke

    You write:
    z = average amount of variant S found in subpopulation i
    zi = average amount of variant S found in whole population

    Isn’t this a misprint?

  • http://www.replicatedtypo.com/authors/hannah-little/ Hannah

    Oh yeah. Thanks. I’ll fix it.

  • http://www.replicatedtypo.com/authors/hannah-little/ Hannah

    Done. Sorry about that.