# Price

### From Evolution and Games

## Contents |

## Introduction

Evolution is all about transitions. One generation gives birth to the next, which gives birth to yet another generation, in an endless chain of transitions. These transitions don't just go anywhere; some transitions are more likely than others. First of all we beget offspring that is likely to bear some similarity with ourselves. And how likely we are to get offspring at all, and how many, tends to depend on certain heritable characteristics. The relative likelihood of some transitions and the relative improbability of others imply that we can make informed guesses as to where this endless chain of transitions came from and where it is going.

The Price equation zooms in on transitions from one generation to the next. This suggests that it could be a fundamental tool for understanding this process of evolution. It was formulated by George Richard Price in 1970 in an article in Nature, and acclaimed by some, maybe even quite a few, to be a fundamental ingredient for models of evolution (Gardner, Frank, Grafen). We think not. The reason why it is hard to identify it as such is that it is all about probability theory and statistics, and those two topics are just not that easy. The following tutorial will help by looking at what proper probability theory or statistics would do and compare that to what the Price equation does to numbers that represent those transitions. There is an article with the same message, but this tutorial makes it simpler, by allowing you to play around with models, data and samples yourself. At some points you may find that the steps the tutorial takes are evident and all too easy, but it is possible that after a few steps, as simple as they are, you may nonetheless feel confused enough to make you want to retrace your steps and be happy, the second time around, that they are that simple. If this is not the case you will just have an easy journey to a deep understanding of why the Price equation is not what some say it is.

## Price 1.0: A very simple model

We will begin with the simplest real model that we can think of. No sex, no shrinking or expanding populations, no frequency dependence, no complications whatsoever. Just two individuals that are ready to reproduce. Asexually. This gives us a version of the Price equation that is even simpler than the first version by Price himself. And it will give us probability theory and statistics that is relatively nice and easy too, and therefore an excellent start.

## Price 2.0: A model with sexual reproduction

If we add sexual reproduction, everything gets slightly more complicated, the proper statistics as well as the Price equation. This makes the statistics much more fun, and it makes the failure of the Price equation to be of any help even more salient. We like to keep it relatively simple though, so for the moment we do not assume actual different sexes, but we do assume that every individual is diploid.

## Price 3.0: A model with frequency dependence

We go back to the asexual model, but now we will take a model with frequency dependence. This will be a bit more complicated, but we can still use a basic, well known model. Individuals are randomly paired to play a game. The game gives them payoffs according to a payoff matrix

## Extra links

- Please check the paper
**Group selection and inclusive fitness are not equivalent; the Price equation vs. models and statistics**. van Veelen, Garcia, Egas, Sabellis. In press. It is about the use of the price equation and the theory of inclusive fitness. Link

- John Maynard Smith talks about the Price equation and how this approach always felt unnatural to him. Since some of us remember a feeling of doubt when first confronted with the Price equation, it may be a relief to hear him say 'I'm not going to tell you what Price's theorem is, because I don't actually understand it'.