I have some beginning thoughts on the game theory of polyamory that I'd like to share.
First, let us suppose that there are four people in a small rural area, say Wyoming, in the middle of nowhere. These are the only four people in a thousand square miles.
Let's say that it is a makeup of two men and two women, aged 27. They are of similar body of appearance, so that there is no advantage between any of them or difference save for their differences in sex and sexual orientation.
Now, let's suppose that they are all bisexual, and derive equal benefits personally to any type of pairing. Let's give them names. For the women, we will name them Alice and Beth. For the men, we will name them Chris and David.
And let's say that all relationships have an accumulative benefit value peak of 1. So that each person derives equal benefit of 0.5 from the relationship.
If Alice and David pair, or if Alice and Beth pair, or if Chris and David pair, or if Chris and Alice pair, or if Chris and Beth pair, or if David and Beth pair, they will all derive the same value of 0.5 each.
If we go with two pairings, Alice and Chris, and Beth and David, each person has a 0.5 benefit, making a total community benefit of 2.
But let's add something else to the equation. Let's assume that there is a 10% benefit of the happiness of the other person. So that so long as there is a benefit to one person, this gives extra benefit to the other person. In this way, we could say that so long as Alice wants to be paired to Chris, and is not forced, her happiness will be of 10% greater benefit to Chris. If Alice didn't want to be paired with Chris, or at least preferred David, her happiness and satisfaction would be lower. Thus, while Chris still gets a 10% benefit, he will not get the same amount of benefit.
In other words, Chris's own happiness will suffer if both parties hapiness isn't maxized.
So, in this scenario, because all happiness is equal, all parties receive the full 10% benefit, and the community has a total benefit of 2.2. The extra 10% mutual happiness benefit to each person is 0.05.
In normative society of monogamy, the focus is on maximizing the benefit of the other party. This not only benefits the other partner directly, but maximizes the 10% mutual benefit to each other. It is therefore in society's benefit to maximize bliss between partners, to promote mutual benefit for all.
But the question is, does monogamy itself maximize benefits for everyone, even in perfect pair coupling?
Let's assume that we introduce other pair couplings in a polyamorous way, but that we put a price on this. Let's say that each additional partner only brings half of the happiness and pleasure of the previous. Therefore, the first partner gives us 0.5 happiness, the second brings 0.25 happiness, and the third brings us 0.125 happiness. The mutual 10% benefit of the first partner is 0.05% hapiness, the second is 0.025 happiness, and the third is 0.0125 happiness.
Let's introduce a fully maximized situation for the four people.
Alice, Beth, Chris, and David are all in relationship with each other. Each has a primary benefit of 0.5+0.25+0.125. In this scenario, we see a diminishing benefit, so that the number is approaching, but never reaching 1. Those of you who are math lovers will recognize this pattern.
In this society, each person has a base happiness of 0.875. This brings a total happiness of 0.875*4, or 3.5.
The mutual 10% happiness for each pairing is 0.05, 0.025, and 0.0125. For each person, this is a benefit of 0.0875.
When added to all, we see a total mutual 10% benefit of 0.0875*4, or 0.35. This brings the total community benefit total to 3.85.
Remember, the original community benefit for monogamy was 2.2. We've now increased this to 3.85, or an additional 1.65. This is a 75% improvement over monogamy.
But we don't live in a perfect society, and no all pairings may match. So let's take one person and place them into monogamy with one of the three left in complete polyamory. Would this one person benefit from polyamory, even if they only they participate in monogamy?
In the original monogamy, the full benefit to any one person is 0.55. Under this system, their partner would have a total of three partners. Let's say that Chris is monogamous ande his only partner is Alice. But Alice is partnered with Beth and David as well.
Alice has a benefit of 0.5, 0.25, and 0.125, or 0.875
Beth has a benefit of 0.5 and 0.25, or 0.75
Chris has a benefit of 0.5.
David has a benefit of 0.5 and 0.25, or 0.75.
Alice has a mutual 10% benefit of 0.05, 0.075, and 0.075, or 0.2
Beth has a mutual 10% benefit of 0.0875 and 0.075, or .1625.
Chris has a mutual 10% benefit of 0.0875.
David has a mutual 10% benefit of 0.0875 and 0.075, or .1625.
Alice has a total benefit of 1.075.
Beth has a total benefit of .9125.
Chris has a total benefit of 0.5875.
David has a total benefit of .9125.
So, as we can see even in this scenario, everyone exceeds the previous maximum of 0.55 benefit, even Chris who has a 0.5875 benefit. What Chris is benefitting from is Alice's extra relationship happiness and satisfaction, which is improving his own relationship.
We can take this to the other extreme, say a single person who has three relationships, but where those relationships have no other relationships. Let's assume that Alice has a relationship with Beth, Chris, and David, but that Beth, Chris, and David only have relationships with Alice.
In this scenario, Alice has a total base benefit of 0.5+0.25+0.125, or 0.875. Everyone else has a total base benefit of 0.5. This gives Alice a total mutual 10% benefit of 0.05*3, or 0.15. This brings her total benefit to 1.025. Everyone else has a total mutual 10% benefit of 0.0875. This brings each of their totals to 0.5875.
Even in this scenario where only one person is polyamorous, the entire community still enjoys a higher level of benefit, even if they are monogamous.
The percentage of the mutual benefit can be adjusted to any rate, the arbitrary number doesn't matter. The benefit percentage can even lower for subsequent relationships. But so long as this is a positive percentage, one truth holds:
Even if one person is polyamorous in society, all of their partners benefit over the partners of strictly monogamous relationships.
How is cost handled? It is handled by the law of diminishing returns for each subsequent relationship. We are not simply adding up full benefits. Each person requires a higher and higher cost, thus the lowering of benefits.
One might revise this to say that after a certain number, the cost outweighs the benefits. But, so long as we can go to even the minimum required to establish polyamory, two partners, with a positive return, this model holds up until we reach the point where cost exceeds benefit. We can even introduce a negative mutal benefit percentage after a certain point. But again, this model holds up so long as the minimum of two partners has a positive mutual benefit.
While I will be working on more advanced modeling, including a triad minimal benefit model to prove this particular game theory outcome under the most strict conditions.
But, so long as positive returns can be shown for two relationships, this model proves the increased benefit of polyamory over monogamy.
So... you're welcome. ;)