where dwi/dx is the effect of a change in x on the fitness of individual i, and pi and mi are coefficients of patrilineal and matrilineal relatedness of the category to which individual i belongs. Inequality (1A) specifies that extra x reduces patrilineal inclusive fitness. This condition maintains silence of paternally derived alleles. Eq. (1B) specifies that x is a local maximum of matrilineal inclusive fitness (Haig, 1997). These conditions are equivalent to
where s indexes symmetric kin (individuals for whom ms = ps) and a indexes asymmetric kin (individuals for whom ma ≠ pa). An individual’s symmetric kin include herself, her offspring, and her grandoffspring, but most other categories of kin are asymmetric, including “fullsibs,” because of uncertainty of paternity. Thus, the right-hand sides of 1A and Eq. (1B) can be considered to represent the marginal effect of x on the individual’s own survival and reproduction (individual fitness).
Eq. (2B) describes a tradeoff in the maximization of matrilineal inclusive fitness. At the ESS, the marginal effect of x on individual fitness is balanced by a marginal effect of opposite sign on indirect fitness obtained via asymmetric kin. If the value of Eq. (2B) is negative, then extra x increases individual fitness at a cost to matrilineal asymmetric kin. If the value of Eq. (2B) is zero, then x simultaneously maximizes both components of inclusive fitness (most plausible if x has no effects on matrilineal asymmetric kin). If the value of Eq. (2B) is positive, then extra x increases the fitness of matrilineal asymmetric kin at a cost to individual fitness.
Substitution of the right-hand side of Eq. (2B) for the left-hand side of 2A yields
which can be rearranged to give
where j indexes matrikin (individuals for whom mj > pj) and k indexes patrikin (individuals for whom mk < pk). This partition allows kin to be assigned to three mutually exclusive classes: symmetric kin (mi = pi),