Malecot's method of coancestry

Malecot's coancestry coefficient, $$f$$, refers to an indirect measure of genetic similarity of two individuals which was initially devised by the French mathematician Gustave Malécot.

$$f$$ is defined as the probability that any two alleles, sampled at random (one from each individual), are identical copies of an ancestral allele. In species with well-known lineages (such as domesticated crops), $$f$$ can be calculated by examining detailed pedigree records. Modernly, $$f$$ can be estimated using genetic marker data.

Evolution of inbreeding coefficient in finite size populations
In a finite size population, after some generations, all individuals will have a common ancestor : $$f \rightarrow 1 $$. Consider a non-sexual population of fixed size $$N$$, and call $$f_i$$ the inbreeding coefficient of generation $$i$$. Here, $$f$$ means the probability that two individuals picked at random will have a common ancestor. At each generation, each individual produces a large number $$k \gg 1$$ of descendants, from the pool of which $$N$$ individual will be chosen at random to form the new generation.

At generation $$n$$, the probability that two individuals have a common ancestor is "they have a common parent" OR "they descend from two distinct individuals which have a common ancestor" :


 * $$f_n = \frac{k-1}{kN} + \frac{k(N-1)}{kN}f_{n-1}$$

''What is the source of the above formula? Is it in a later paper than the 1948 Reference.''
 * $$ \approx  \frac{1}{N}+ (1-\frac{1}{N})f_{n-1}. $$

This is a recurrence relation easily solved. Considering the worst case where at generation zero, no two individuals have a common ancestor,
 * $$f_0=0$$, we get


 * $$f_n = 1 - (1- \frac{1}{N})^n.$$

The scale of the fixation time (average number of generation it takes to homogenize the population) is therefore


 * $$ \bar{n}= -1/\log(1-1/N) \approx N. $$

This computation trivially extends to the inbreeding coefficients of alleles in a sexual population by changing $$N$$ to $$2N$$ (the number of gametes).