fixation is $\prod_{i=1}^n[1-P(s_i,u)]$, regardless of the order of their appearance. The probability that at least one mutant goes to fixation is therefore $$\label{overall-fix-prob-multi}
  \pi = 1-\prod_{i=1}^n[1-P(s_i,u)]\,,$$ which reduces to Eq. (\[overall-fix-prob\]) in the case of $n=2$. Using an extension of Kimura’s well-known result for $P(s_i,u)$ derived in the Appendix, Eq. (\[kimura-approx\]), we can simplify this expression even further. We find in the limit $N\rightarrow \infty$: $$\label{pieq}
\pi = 1- \exp\left[-2\left(\gamma_{1}+\gamma_{2}+ \cdots +\gamma_{n}\right)\right]\,,$$ with $\gamma_i=(1+s_i)(1-u)-1$ for $u<s_i/(1+s_i)$, and $\gamma_i=0$ otherwise. Moreover, in the absence of deleterious mutations, for $u=0$, $\pi$ becomes $$\pi = 1 - \exp\left[-2\left(s_{1}+s_{2}+ \cdots +s_{n}\right)\right]\,.$$ In this limit, the overall probability of fixation is the same as the probability of fixation of a single mutant with selective advantage $s$ equal to the sum of the selective advantages of all invading mutants, $s=\sum_i s_{i}$. Such a hypothetical mutant is extremely unlikely in clonal populations, because all beneficial mutations would have to hit the lineage sequentially, but could occur when beneficial mutations are shared via recombination. In the limit that all $\gamma_i$ vanish (that is, for very large mutation rates or for vanishing $s_i$), Eq. (\[overall-fix-prob-multi\]) becomes $$\pi = \frac{n}{N}\,. \label{neutral}$$

Expected fitness increase and expected number of fixed mutants
