The Assumption of Weak Selection?

Question

I was reading this question and I failed to fully understand the introductory part of it.

The OP (@Artem Kaznatcheev) says:

Most analytic models like to assume weak selection because it allows the authors to Taylor expand the selection function and linearize it by dropping terms that are higher order in the stength of selection.

I don't fully understand it. Can you help me making sense of why assuming weak selection allows one to Taylor expand the selection function. I am hoping someone would answer by presenting a mathematical model that at first does not assume weak selection and show why assuming weak selection allows the use of a Taylor series to linearize the function. I'd like to understand which terms fall down and which terms left with this assumption.

Answer 1

This is my take in this, without experience of using the Taylor series to analyse evolutionary game theory problems.

As you know the Taylor series expansion of $f(x)$ at point $a$ can be written as:

$f(x)= f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f^{(3)}(a)}{3!}(x-a)^3 + ... + \frac{f^{(n)}(a)}{n!}(x-a)^n + ...$

Often factors above second order are dropped as a simplification. However, to use the Taylor expansion, the function must be differentiable at point $a$, and to be differentiable it must be continuous.

As @Artem Kaznatcheev writes in his question, "...with weak selection meaning the game modifies overall fitness only slightly..." and "... it is typical to model organisms as having a base fitness that is modified slightly by the game interaction...". These statements imply that the overall fitness function/surface can be assumed to be relatively smooth and continuous (i.e. the payoff is negligible), which means that the Taylor series expansion can be used. If the game payoffs for a single game would determine a large proportion of the overall fitness, the fitness surface would be discontinuous.

An example where a Taylor series expansion is used to simplify the fitness function under weak selection is given in Andre & Godelle (2006) (see equation 20 ff).

You might also find this supplement to Nowak et al (2010) interesting (a rather controversial paper), especially page 8.

Answer 2

I think at @fileunderwater provides a good explanation of the basics math behind it, and some good references. I would like to go more into detail about the modeling decisions and benefits of assuming weak selection and why it is done in the literature.

When you are making evolutionary models, especially one in evolutionary game theory, the first place is to start with a differential equations model of inviscid populations (this usually means that you assume very large population size). These models often result in dynamics of the form $\dot{x}_i = x_if_i(\vec{x})$ where $f_i$ is a non-linear function corresponding to the relative fitness but where the non-linearity depends on the strength of selection. Making a weak-selection argument, often allows you to come close to linearizing $f_i$ using Taylor expansion. This reducese your equations to a form like $\frac{dx_i}{dt} = x_i(\vec{c}_i^T\vec{x} - \phi)$ where $\phi$ is an average fitness. Although the transients of this dynamic are still complicated, the equilibrium is usually relatively easy to analyze and reduces to looking at the matrix $C = [\vec{c}_i^T]_{1 \leq i \leq n}$ with its columns given by your linearization of the $f_i$s.

Of course, this is only a first step, and the current trend in the literature is to move to a setting where you have spatial structure. Here, weak selection gives you a further bonus because you can use it to separate the timescales of local (dominated by the constant term) and global (dominated by first-order in $w$ -- the selection strength) network dynamics. This allows you to use the powerful pair-approximation technique. A great example of its use is Ohtsuki & Nowak's (2006) solution to the evolutionary dynamics of games on $k$-regular random graphs (you can find a brief overview here).

The final settings where this is useful is for studying the interaction of mutation and selection in finite populations. In this case, weak selection allows you to look at the selective force as a small perturbation of the mutator-balanced steady state. In the setting of EGT this rquires you to define some new solution concepts for what it means for one strategy to be "better" than another but then some interesting connections (see Antal, Nowak, & Traulsen (2009) for more) and very general analysis of the mutation-selection equilibrium can be done (Antal et al., 2009).