The coancestry between two individuals X and Y, ϕ
_{
XY
}, is usually calculated following simple recurrence rules. These rules can be implemented using tabular methods or languages with recursive function support. To calculate the whole set of coancestries for a given pedigree, only two formulae are required [10]:
$$\begin{array}{@{}rcl@{}} \phi_{XX} & = & \left(1+\phi_{FM}\right)/2, \\ \phi_{XY} & = & \left(\phi_{FY}+\phi_{MY}\right)/2, \end{array} $$
((1))
where F and M are the father and the mother of individual X, respectively. These equations operate successively over pairs (X,Y) where X is assumed to be more recent than Y. Let x and x
^{′} be the paternally and maternally inherited copies at a given locus carried by individual X and y and y
^{′} the corresponding copies carried by individual Y. The coancestry between X and Y can then be written as:
$${\fontsize{9.2}{12}\begin{aligned} \phi_{XY} = \frac{1}{4}\left[P\left(x\equiv y\right)+P\left(x\equiv y'\right)+P\left(x'\equiv y\right)+P\left(x'\equiv y'\right)\right], \end{aligned}} $$
where ≡ stands for identical by descent (IBD).
Analogous relationships have been described for gametes [6] as ψ
_{
ab
}=P(a≡b) and recurrence rules have been developed for their pairwise relationships [6,7]:
$$\begin{array}{@{}rcl@{}} \psi_{aa} & = & 1, \\ \psi_{ab} & = & 1/2\left(\psi_{gb}+\psi_{hb}\right), \end{array} $$
((2))
where a and b denote two gametes in the pedigree. Both g and h are the direct ancestral gametes of a, that is, the gametes of the father or the mother if a is a paternal or maternal gamete, respectively. Although Equations (1) and (2) are closely related, they have different interpretations.
We use the threeway (ψ
_{
abc
}), the fourway (ψ
_{
abcd
}) and the twopair (ψ
_{
a
b,c
d
}) gametic relationships as counterparts of the conventional generalized kinship coefficients [4]. These generalized gametic relationships correspond to the probability of three or four gametes to be IBD. Note that these multiple gametic relationships correspond to multiple gametic identities, regardless of the identity by descent with other gametes. For instance, for individuals X and Y, whose paternally and maternally inherited gametes are described above,
$$\begin{array}{@{}rcl@{}} {\fontsize{8.2}{12}\begin{aligned} \psi_{xx'y}=P\left(x \equiv x'\equiv y\right) = P\left(x \equiv x'\equiv y\equiv y'\right)+P\left(x \equiv x'\equiv y\not\equiv y'\right). \end{aligned}} \end{array} $$
ψ
_{
a
b,c
d
} is the probability that gametes a and b are IBD and simultaneously c and d are also IBD. For instance, for individuals X and Y,
$$\begin{array}{@{}rcl@{}} \psi_{xx',yy'} = P\left(x \equiv x'\equiv y\equiv y'\right)+P\left(x \equiv x'\not\equiv y\equiv y'\right). \end{array} $$
The recursive formulae for the whole set of multiple gametic relationships are
$$ {\small\begin{aligned} \psi_{aa} & = 1, \\[2pt] \psi_{ab} & = \frac{1}{2}\left(\psi_{gb}+\psi_{hb}\right), \\[2pt] \psi_{aaa} & = 1, \\[2pt] \psi_{aab} & = \psi_{ab}, \\[2pt] \psi_{abc} & = \frac{1}{2}\left(\psi_{gbc}+\psi_{hbc}\right), \\[2pt] \psi_{aaaa} & = 1, \\[2pt] \psi_{aaab} & = \psi_{ab}, \\[2pt] \psi_{aabc} & = \psi_{abc}, \\[2pt] \psi_{abcd} & = \frac{1}{2}\left(\psi_{gbcd}+\psi_{hbcd}\right), \\[2pt] \psi_{aa,aa} & = 1, \\[2pt] \psi_{aa,ab} & = \psi_{ab}, \\[2pt] \psi_{ab,ac} & = \psi_{abc}, \\[2pt] \psi_{aa,bc} & = \psi_{bc}, \\[2pt] \psi_{ab,cd} & = \frac{1}{2}\left(\psi_{gb,cd}+\psi_{hb,cd}\right). \end{aligned}} $$
((3))
For an easier implementation, Equation (3) can be summarized in a simple set of rules

1.
In any nway relationship, merge groups sharing a given gamete, that is, ψ
_{
a
b,a
c
}=ψ
_{
abac
}

2.
In any nway relationship, discard repeated gametes, i.e. ψ
_{
aabc
}=ψ
_{
abc
} or ψ
_{
a
a,b
c
}=ψ
_{
a,b
c
}.

3.
Given that the probability of a gamete to be IBD to itself is 1, discard groups of identity including a single gamete, i.e. ψ
_{
a,b
c
}=ψ
_{
bc
}.

4.
Identities with a single gamete are 1 and identities with two or more founder gametes at the same group are 0.

5.
Calculate \(\psi _{a\theta }=\frac {1}{2}\left (\psi _{g\theta }+\psi _{h\theta }\right)\), where θ stands for any identity pattern and a for a gamete of the youngest individual. For instance \(\psi _{ab,cd}=\frac {1}{2}\left (\psi _{gb,cd}+\psi _{hb,cd}\right)\) or \(\psi _{abc}=\frac {1}{2}\left (\psi _{gbc}+\psi _{hbc}\right)\).
As long as these rules are correct and regardless of the number of gametes involved, they can be used to calculate identities involving more than two individuals.