An allele is one of several possible forms of a gene. Most of multi-cellular organisms are diploid, i.e., their chromosomes form homologous pairs. Each pair of chromosomes has a pair of alleles at each genetic locus. Thus, n different alleles may form n square different allele pairs, as two alleles in each pair may not be the same. Different allele pairs are considered to be different genotypes, which may result in different phenotypes or in other words, different genetic traits.
The fitness of all different allele pairs or in other words, all different genotypes at a given genetic locus can then be given in a matrix with the rows corresponding to the choices for the first allele and the columns to the choices for the second allele in the allele pair. Again, n different alleles will give n different choices for both the first and second alleles in the allele pair, and hence an n by n fitness matrix.
With such a fitness matrix, a genetic selection problem can then be formulated as an evolutionary game with the choices of the first and second alleles in the allele pair at a given genetic locus as the strategies for player I and II. Here, player I can be considered as an individual with a specific choice of allele at the given locus. The choice could be one of the possible alleles or a combination of them with each selected with some chance. The former corresponds to a pure strategy, while the latter to a mixed one.
On the other hand, player II can be considered as a typical individual in the given population. This individual could have only one of possible alleles at the given locus or a combination of them with each selected with some chance. The strategy of this individual really is the same as the composition of alleles at the given locus in the whole population. Therefore, it is also the allele profile of the population for this particular genetic locus.
The fitness matrix for the genetic selection game is symmetric. Such a game corresponds to a special type of quadratic program: The equilibrium states of the game correspond to the KKT points of the quadratic program, and the evolutionarily stable states of the game correspond to the strict local maximizers of the quadratic program. Therefore, the equilibrium states and their stabilities of a genetic selection game can be analyzed via the solution of the corresponding quadratic program (Huang et al. 2015).