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The Hardy-Weinberg Equilibrium (HWE) can be linked to game theory. This article shows that payoffs, or resources, in a game with alleles as players, determine the frequency of homozygotes. The frequency of **aa** homozygotes in the HWE is an increasing function of the multiplicative difference in own payoffs for each allele. Thus, Mendelian proportions are variable rather than fixed depending on the resources for the alleles. Whereas the concept of evolutionary stable strategy (ESS) is based on non-cooperative competitive settings such as a competition between doves and hawks, this article explores a game theoretic situation where the mating of two alleles is presupposed.

There is an equilibrium concept that is not used by game theorists or by economists but exclusively used by biologists. The Hardy-Weinberg Equilibrium (HWE) is the name of this equilibrium concept [

Maynard Smith proposed the concept of evolutionary stable strategy (ESS) which was based on non-cooperative competitive settings such as competition between doves and hawks [

This article reveals that payoffs for alleles, or resources for players, determine the frequency of homozygotes. The main result of this article suggests that Mendelian proportions are variable rather than fixed depending on the resources for the alleles encountered. Although this result may not influence the academic interests of biologists because the author is not capable of reviewing the existing literature in biology, the introduction of a new equilibrium concept to the field of game theory may well develop yet unknown field of research as applications of the HWE.

An example to apply this approach in economic phenomenon is to investigate diffusion of solar panels on household roofs. We can observe a fraction of cases where a wife and a husband agree to install the solar panel. The wife and the husband have options to choose either to consume fossil energy or to install solar panels. The resources or the payoffs in this game are influenced in various ways such as a subsidy for installing solar panels, the budgets of the husband and wife, the amount of daytime sun in the region and an electric power company scheme to buy surplus electricity from the household. The couple pursues an eco-friendly life when there are large enough payoffs for their installation. Measurement of these variables in actual data is left for empirical research, however.

Section 1 shows the basic logic of the HWE. Section 2 explains how to find a mixed strategy Nash equilibrium given payoffs, or resources to be utilized by each allele. Section 3 shows that the frequency of aa homozygotes in the HWE is an increasing function of the multiplicative difference in own payoffs for each allele. Section 4 discusses how we can apply the HWE notion to the third and fourth generations with different frequencies in alleles. A model in this section shows that mathematical structure of the HWE converges on the structure in the Wiener process. The concluding section sums up the major propositions.

Suppose there are two alleles: the first allele is denoted A and the second is denoted a [

The ratio of the homozygotes to the hetrozygotes given in the above example is p^{2}:2pq:q^{2}. This result is derived only when males and females have the same ratio of two alleles in their frequencies p and q. Now, suppose males and females have different frequencies to create homozygotes and heterozygotes. Suppose further that males have the ratio of p and, and females have the frequency ratio of q and. As shown in

^{a}.

^{a}See for example [

Nash equilibria: two of pure strategy and one of a mixed strategy. The equilibrium in the pure strategy is a combination of (Growth strategy, Eco strategy) and (Ecostrategy, Growth strategy). These equilibriums satisfy the definition of Nash equilibrium where a best response of a player coincides with another player’s best response.

I can calculate the Nash equilibrium in the mixed strategy using the probability given in

We see the following relationships.

If then, or.

If then, or.

If then, or.

The best responses for the male group strategy are shown the above. If the coefficient parameter of p, which is, is positive, it is equivalent to the probability q of the strategic choice of the females which is smaller than. The male group can maximize own expected payoffs by maximizing p in this situation. Therefore, the best response of the male group is to take a pure strategy of. If the coefficient parameter is negative, probability q is larger than. For the male group, the best response is to take strategy of. This means that minimizing p leads to maximization of males expected payoffs. When the coefficient parameter is 0, which indicates, then the expected payoffs for the male group does not depend on p.

Expected payoffs for females are;

We see the following relationships:

If then, or.

If then, or.

If then, or.

From the above the best responses for the female group are as follows: when, it is equivalent to. The female can maximize q to get the highest amount of payoffs. The maximum of q is 1., or when probability, q must be minimized to get the highest payoffs for the female. The minimum is zero. These two cases correspond to pure strategies. If is equal to zero, then. This means that the expected payoff for females is, which does not depend on q.

Thus, two pure strategies are derived. One is that females choose the Eco-strategy when the males choose the Growth strategy. The other is the combination that the females choose the Growth Strategy when the male group chooses the Eco-strategy (p = 0, q = 0). The Nash equilibrium for the mixed strategy is also apparent. The male group assigns the probability of to p and for its {Growth strategy, Eco-strategy}, and the females allocate the combinations of the best response to give to q and for its {Growth strategy, Eco-strategy}. We can show these results as two reaction functions in

Calculating the expected payoffs under this mixed strategy gives

The males and the females obtain payoffs of (50, 50) respectively under the mixed strategy Nash equilibrium. Here we can also get. Consequently, and.

What is more important in relation to the HWE is that the frequency of p and q is simultaneously determined by the existing conditions of the payoffs for the mixed strategy. When biologists calculate the HWE, they observe the aa homozygotes which is supposed to be given by in

is equivalent to in our example of

Proposition 1. The frequency of the aa homozygotes in the HWE is a function of payoffs in a game for two alleles.

I must emphasize that the symmetrical payoffs in

There are eight payoffs in the game, or four payoffs for each allele. We allocate eight parameters as a payoff matrix:. Let us start from the males’ case where we give four parameters, a, b, c and d as payoffs for the males.

If

then, or.

If

then, or.

If

then, or.

We give parameters e, f, g and h for the payoffs for the females’ case;

If

then, or.

If

then, or.

If

then, or.

We can calculate p and q given these parameters. Accordingly we can calculate the expected payoffs of the mixed strategies. The males obtain;

and the females obtain their payoffs;

These results give paradoxical characteristics of mixed strategy.

Proposition 2. Payoffs attained by the mixed strategy for each of the two players do not depend on the other player’s payoffs in the game.

We obtained in the former section that:

and.

Let us denote and.

Then we get;

As shown in Proposition 1, the HWE is dependent on the payoff matrix of the game.

When biologists start off by observing the frequency of the aa homozygotes, which is supposed to be given by, they are looking at or

in our exposition. I see the possibility that payoff e is closer to f and/or payoff a is closer to c. In such cases the multiplication of and approaches to zero. If both results are closer, then converges to zero at an accelerated pace. If, on the other hand, the difference between the two payoffs increases such that e and f and/or a and c have wider gaps in their payoff levels, then the frequency expressed by increases.

Proposition 3. The frequency of the aa homozygotes in the HWE is an increasing function of the multiplicative difference in own payoffs for each allele given and

.

In this article the HWE has been explained through the logic of the game theoretic optimal behavior of alleles. However, questions remain as to how we understand the third and fourth generations that have different frequencies of alleles. One of the expositions of showing the aa homozygotes is shown in

The question is how one can find aa homozygotes in the third generation. I see that there are two ways to obtain the HWE in the third generation from

The sequence is important because frequencies may differ between generations. If we assume that the top two sequences of alleles define the aa homozygotes, the fourth generation with aa homozygotes is directly inherited from the third generation. This frequency is one

hundred percent. If the last two sequences of allele define the aa homozygotes, one half of the parental generation for the fourth generation is aa homozygotes. In the case of the top two sequences, the fourth generation shows aaAA, aaAa, aaaA and aaaa. If we check the third generation, we get aaA and aaa as the ancestors. In the case of the last two sequences, we see those cases at AAaa, Aaaa, aAaa and aaaa. The third generation for them consists of AAa, Aaa, aAa and aaa. The third generation comes from a wider range for the hetrozygotes.

We are now able to allocate the probabilities in any generations. We see that the probability of AA is pq, of Aa is, of aA is and of aa is. We can also calculate from

The probability distribution in

Combining the HWE with game theory produced interesting propositions. The frequency of aa homozygotes in the HWE is a function of payoffs in a game for two alleles. And the frequency of aa homozygotes in the HWE is an increasing function of the multiplicative difference in own payoffs for each allele. Therefore I can conclude that Mendelian proportions are variable depending on the resources or the payoffs for the alleles.

It is interesting to inquire how an evolutionary biologist discerns aa homozygotes from mutation when the frequency is so low that aa homozygotes could not appear over many generations. I can further inquire what is conceived if the resources for the alleles fluctuate, increasing after a rare emergence of the aa homozygotes so that their frequency overwhelms that of the heterozygotes and AA homozygotes. These processes seem to be similar to the emergence of mutations and the selection process of species.

This work was supported by the Japan Society for the Promotion of Science (JSPS), Kakenhi, Grant-in-Aid for Scientific Research (A), 22243032. I am grateful to Professor Marie Anchordoguy at the University of Washington for her academic assistance in my research.