Arjun Kakkar
The new version of this article can be found here. – http://wp.me/P1XIHd1w
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INTRODUCTION
The purpose of this model is to provide a more interesting, new approach to homosexuality than is usually seen. This is an effort to apply game theory in a way that has never been done before. Before I begin, let it be clear that these are my thoughts on the subject and there is a high probability that I made some massive error during the process of writing this text. In that scenario, please forgive me. With that out of the way, let’s begin.
Setting up the game
The strategic form of any game has three components: players, strategies and payoffs. If we are to attempt to model sexual preference in a game then that is where to start from. The players in this model are going to be the people participating in reproduction. Two players are going to be paired with each other to see how they perform against each other. The different strategies are going to be the choice between homosexuality and heterosexuality. The payoff matrix for the game is going to be decided in the subsequent paragraphs.
Simplifying assumptions
The choice of strategies within the strategy set is free, i.e. the choice between homosexuality and heterosexualtiy is on the person and not pre decided by genetic or environmental factors, and thus sexual reproduction has no role to play as an influencing factor in the choice of strategies.
All players are rational thinkers and act in order to maximize their payoffs only.
The payoffs of the players are decided only on the basis of reproductive success and personal standing of the individual.
Assessing payoffs
Let us take the payoff of a heterosexual pairing to be 10 for both players for the sake of simplification. Obviously a homosexual and heterosexual pairing and vice versa will yield a payoff 0 for both players as reproductive success is nil. Figuring out homosexual pairing payoffs is a little tricky. In order to do that we need to calculate the comparative reproductive success of homosexuals to heterosexuals through assisted reproductive techniques like surrogacy or sperm donation. We also need to add the negative payoff of being homosexual because of social taboos. This could vary from person to person.
According to the 2008 American census, 20% of all homosexual couples have children below the age 18. This would then give homosexual pairing a payoff of 2. Now from this we subtract a variable value (let this value be x, where x is a positive rational number). Thus the final payoff of both players in a homosexual pairing is (2x). We will deal with the different values of x and the resulting payoffs because of those values later. The resulting payoff matrix is shown below.

Player 1/Player 2 He Ho
He 10,10 0,0
Ho 0,0 (2x),(2x)
DOWN TO BUSINESS
Now that we have our game set up and running, it’s time to start playing and getting some results. In order to do this we have to analyze the play of the game for different values x.
The different scenarios are as follows:
1. 0x2, i.e. (2x) is negative
Each different scenario has remarkably different outcomes.
Scenario 1 [(2x) is positive]
For this scenario let us assume that (2x) is some positive number a less than or equal to 2. The payoff matrix now becomes as follows.

Player 1/Player 2 He Ho
He 10,10 0,0
Ho 0,0 a,a
The above game is a classic example of a coordination game. The Nash equilibria in the following game can be shown in the following manner.
b1(He)=He, Where b1(He) stands for the best response of player 1 to the
strategy He of player 2.
b1(Ho)=Ho, as a is greater than 0
b2(He)=He, as 10 is greater than 0
b2(Ho)=Ho, as a is greater than 0
Hence the two Nash equilibria in this game are (He,He) and (Ho,Ho).
This shows that for all positive values of (2x), Ho is an evolutionarily stable strategy along with He, which was anticipated.
This game is called a coordination game because the entire population will be coordinated at one of these two equilibria and not at both at the same time.
Along with these two visible pure strategy equilibria, there also exists a mixed strategy equilibrium. This can be calculated by taking the following basic assumption:
Let there be a probability of encountering the strategy Ho. Thus the probability of encountering the strategy He is (1). A mixed strategy equilibrium exists when player 1 is indifferent towards the strategy choice of player 2. This will occur only when the expected payoff with strategy He of player 1 is the same as the expected payoff with strategy Ho.
i.e. [(1)10]+[0]=[(1)0]+[a]
1010=a
=1010+a, (1)=a10+a
This mixed strategy equilibrium can be thought of as a threshold limit beyond which the game coordinates to one of the equilibria.
Thus, when given a choice, in a population where 1010+a of the people are playing Ho, a player would be indifferent towards choosing He or Ho. But, if even a slight deviation occurs due to any reason the mixed equilibrium would be disturbed and with successive repetitions of gameplay the game would become coordinated at one equilibrium.
What this means is that if most of the population is homosexual (specifically more than 1010+a of it) it would be a better strategy to be homosexual, as it would have a better expected payoff. Whereas if most of the population is heterosexual, it is more beneficial to be heterosexual as is in our society.
This relationship can be described mathematically by the following inequality derived from the mixed strategy equilibrium definition.
10(1)>a, if 1(Ho) if <1010+a, (where 1(He) is the expected payoff of player 1 when player 2 plays He)
and, 1(He)1010+a
Thus, when Ho is small, the payoff of being heterosexual is strictly greater than the payoff of being homosexual. In that scenario there are absolutely no incentives for choosing the strategy Ho (being homosexual). But this is not the case, in reality around 7% of the population identify themselves as homosexual. According to our deductions so far, this is somewhat surprising. The only possible explanation for this is that sexual preference is not a personal choice but driven by genetic and environmental factors (this effect was completely ignored for simplification). This is the first important result I want to emphasize.
The second, perhaps more subtle deduction is that since both strategies are Nash equilibria and are evolutionarily stable, they only dominate when the other strategy is small in number.
There is also another very important thing that can be seen through this model. The maximum value of a that can be obtained is 2 and thus the minimum possible value of is ⅚. This means that for the equilibrium to shift to (Ho,Ho), more than ⅚ of the population has to choose Ho. This is not possible even on randomization of strategy choices. Thus even if strategy choices are made by the flip of a coin, the equilibrium will be (He,He).
If you take a close look at how the payoff of homosexual pairing was assigned you will see that the range of a can change over time. This can happen through advancement of technology and increase in social acceptability of homosexuality. This will be especially helpful in relating the different scenarios of values of (2x).
Scenario 2 [(2x)=0]
For this scenario the payoff matrix is going to be as follows.

Player 1/Player 2 He Ho
He 10,10 0,0
Ho 0,0 0,0
For the calculation of Nash equilibria as above, the best responses are shown below.
b1(He)=He, as 10 is greater than 0
b1(Ho)=He or Ho, as both are 0
b2(He)=He, as 10 is greater than 0
b2(Ho)=He or Ho, as both are equal to 0
It is clear that (He,He) is a strict Nash equilibrium and that (Ho,Ho) is a weak Nash equilibrium.
Also He is evolutionarily stable whereas Ho is not as (Ho,He)>(Ho,Ho) is not true.
There also exists a mixed strategy Nash equilibrium which is equivalent to the pure strategy Nash equilibrium (Ho,Ho). This means that if everyone is playing Ho, the game is at equilibrium, but as soon as one person deviates and plays some other strategy, everybody loses their incentive of playing Ho and gain a large incentive of playing He.
The equilibrium (Ho,Ho) can be thought of as a highly unstable state which is the crossover point between scenario 1 and scenario 3.
Thus we can see an increasing amount of plays of strategy He as payoffs for Ho decrease.
Scenario 3 [(2x) is negative]
Let us assume that (2x) is a negative rational number given by z. Now the payoff matrix looks like this.

Player 1/Player 2 He Ho
He 10,10 0,0
Ho 0,0 z,z
For the calculation of the Nash equilibrium as above, the best responses are shown below.
b1(He)=He, as 10 is greater than 0
b1(Ho)=He, as 0 is greater than a negative number
b2(He)=He, as 10 is greater than 0
b2(Ho)=He, as 0 is greater than a negative number
It is clear that there is only one Nash equilibrium in this game, i.e. (He,He) and Ho is not evolutionarily stable.
There is no mixed strategy nash equilibrium, and there is only one strategy set in this game that is a solution.
In a scenario where the social pressures are so immense that x has become much greater than 2 and driven the payoff of being homosexual far below 0, even genetic influences cannot cause Ho to be played.
If the other extreme scenario is considered, (Ho,Ho)>(He,He). Genetic influences impacting He to be played will become too redundant to be considered as people will have incentives to choose Ho more often.
DEDUCTIONS AND PREDICTIONS
Payoff function for Ho pairing
Taking a look at the above model, it can be seen that (Ho,Ho) has been written as (2x). I argue that this function can be written as (ux) where both u and x are functions of time. This is because, with time, as technology develops homosexual reproductive success will increase and u will increase. Similarly x will also keep decreasing with time as society starts accepting homosexuals just as they do heterosexuals.
Using data from the (http://www.gallup.com/poll/154529/halfamericanssupportlegalgaymarriage.aspx and http://williamsinstitute.law.ucla.edu/wpcontent/uploads/GatesBadgettNCFRLGBTFamiliesDecember2011.pdf) and plotting a regression from the given data x=0.113t+232.98 and u=0.025t48.46.
y=x(t) y=u(t)
(Both equations are linear regressions and do not give a perfect estimate but are the closest approximation I could do. They are just being used to provide a sense of the theme.)
Now the payoff function is given by (t)={0.138t281.437, t2061.77}
{0.025t48.46, t>2061.77 }
y=(t)
We can therefore map the point at which each of the scenarios will play out.
It can easily be seen from the graph that we are currently in scenario 3 in which the payoff of choosing Ho is negative and it’s still a strictly dominated strategy. The only reason that we observe homosexuality at a fixed rate is because of external factors and not because of strategic decisions or personal preference.
Scenario predictions
As soon as the switch occurs and scenario 2 begins (depending on where the payoff hits 0) we are going to see a sudden change. All those who were being prompted to choose Ho because of genetic or environmental reasons but were stopping themselves from doing so are going to be free to make the choice of being homosexual and it will not be a dominated strategy anymore, but an equilibrium. The first scenario will come into play only when t>2039.4.
We can also use the equation of (t) as a in the first scenario for all the positive values of (t).
Thus, a=(t) , t>2039.4
=1010+a , a={0.138t281.437, t2061.77}{0.025t48.46, t>2061.77}
This equation can be used to calculate the minimum fraction of the population that has to chose strategy Ho so that the equilibrium is shifted to (Ho,Ho), with respect to time.
Of course there are limitations to the equation as it is based on a regression line. As the value of a increases the rate of change of a will decrease till it reaches 0 at some point. That will be the limiting point beyond which no matter what the value of t, (t) will remain constant.
There is one more idea that can be drawn from this model. For the value of (t) to be 10, the value of t has to be 2338.4. This value of t will give a payoff of 10 only when the function remains linear and doesn’t become constant after some time. This is virtually impossible.
This means that only for a very rare case there may arise a situation in which (Ho,Ho)>(He,He).
Conclusions
Here are the few points that I have established through the creation and use of this model:
• It is impossible for the strategy choice Ho (or homosexuality in general) to be a completely personal choice, especially in the scenario in which we are playing the game right now (Scenario 3).
• Ho is evolutionarily stable and a strict Nash equilibrium only when (t)>0.
• When Ho is being played more frequently, one has more incentive to play Ho than He and vice versa.
• If randomization occurs at any time, i.e. players choose Ho and He on the basis of a coin toss, equilibrium will always shift towards (He,He).
• Upto t=2039.4, Ho is a dominated strategy, but after that point it becomes a strict nash equilibrium.
• The payoff of playing Ho (being homosexual) can almost never be greater than payoff of playing He (being heterosexual). But the expected payoffs can sway in any direction.
http://arjunkakkar.wordpress.com/gametheoryandhomosexuality/