Marginal fitnesses and equilbria

After a little algebra it's possible to write down how allele frequencies change in response to viability selection:²

$$p_i' = \frac{p_iw_i}{\bar w} \quad ,$$

where $p_i = \sum p_i w_{ij}$ is the marginal fitness of allele $i$ and $\bar w = \sum p_i^2 w_{ii} + \sum_i\sum_{j>i} 2p_ip_jw_{ij}$ is the mean fitness in the population.

It's easy to see³ that if the marginal fitness of an allele is less than the mean fitness of the population it will decrease in frequency. If its marginal fitness is greater than the mean fitness, it will increase in frequency. If its marginal fitness is equal to the mean fitness it won't change in frequency. So if there's a stable polymorphism, all alleles present at that equilibrium will have marginal fitnesses equal to the population mean fitness. And, since they're all equal to the same thing, they're also all equal to one another.

That's the only thing easy to say about selection with multiple alleles. To say anything more complete would require a lot of linear algebra. The only general conclusion I can mention, and I'll have to leave it pretty vague, is that for a complete polymorphism⁴ to be stable, none of the fitnesses can be too different from one another. Let's play with an example to illustrate what I mean.