2.1. The basic game

In each individual, the two genes—maternal and paternal—of factor H (where the polymorphism is localized) are viewed as players and the concrete realizations of the SNP (Y or H) as the respective strategy. We are not interested in details in the underlying biological processes but rather in an abstract description of the phenomena. That is why we use the framework of evolutionary games. The pay-off is given by the (average) fitness for the concrete genotype (YY, YH or HH). Moreover, the two heterozygous cases (YH and HY) gain the same fitness. Consequently, the pay-off for both players (genes) is the same and the pay-off matrix becomes symmetric, so that the game has a high degree of symmetry. In game theory, situations where the pay-off matrix is symmetric are called partnership games [52,53].

Let w_YY be the pay-off for the homozygous genotype resulting in YY, w_YH be the pay-off for the heterozygous and symmetric cases (i.e. YH and HY ) and w_HH be the pay-off for the homozygous genotype resulting in HH. We assume that w_YY, w_YH, w_HH are real values and pairwise different. Thus individuals with different specific allele combinations differ in their fitness. From a game-theoretical point of view these three genotypes can be considered as the result of two different strategies realized by the two genes in a concrete individual.

We assume that the given deviating allele frequencies within ethnic groups represent different evolutionarily stable solutions for the trade-off between the false positive and false negative cell detection. Let y*∈(0, 1) denote the evolutionarily stable frequency of the allele realization Y in the population. In particular, populations with homozygous individuals, either pure HH or pure YY, can be invaded by the contrary strategy, respectively (cf. table 1). Thus, there must be a pay-off advantage to a heterozygous individual to have both strategies. In population genetics this case is called overdominance [54]. Considering this, we can derive directly the inequalities

and

The evolutionarily stable frequency y*∈(0, 1) of the allele realization Y in the population can be calculated by using the Bishop–Cannings theorem [55] (see also [56], appendix A):

Although the value of y^* varies in different populations, its values are always higher than [34]; cf. table 1. Together with equation (2.3) we derive the condition

meaning that comparing the two homozygous cases shows that YY gets a higher pay-off than HH. Combining the inequalities (2.1), (2.2) and (2.4), we get

A symmetric game with this pay-off relation is a battle of the sexes game. This is one subtype of the leader game, a classical two-player two-strategy game with mixed Nash equilibria. Here, the follower does not have a disadvantage in pay-off compared to the leader. Thus, it is on the boundary between the battle of the sexes game and another subtype of the leader game (cf. [56]). In the interpretation (cover story) of the battle of the sexes game, that parameter combination means that both partners enjoy the togetherness equally (pay-off w_YH), independent of the event they visit together [56].

The inequalities (2.5) simplify the pay-off matrix. As parameter values can be normalized without loss of generality [57], it can be assumed that

and

The value of y^* can then be reduced with equation (2.3) to

or, expressed for the value of w_YH:

For example, the frequency of the H allele (which equals 1 − y*) is about for Caucasians, about for Hispanics and about for Japanese [34]. The value of w_YH then becomes 2, and , respectively. Therefore, for example, the pay-off matrix for Caucasians becomes

We summarized several studies addressing the factor H polymorphism and calculated the expected pay-offs of the H allele (see equation (2.9)). The examined ethnic groups are organized in cultural areas. The cultural areas coincide with seven of the eight cultural areas identified by Hunter & Whitten [32] (North Asia is missing due to lack of data). For a more detailed view three subareas, i.e. Mesoamerica, Finland and Iceland are included.