Why has evolution favored sexual reproduction using two mates over sexual reproduction using three mates?

Why has evolution favored sexual reproduction using two mates over sexual reproduction using three mates?

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Many animals species require two members - a male and a female - to sexually reproduce. Why has nature (or the process of evolution) chosen to favor the form of sexual reproduction which requires two members? Why didn't nature encourage the form of sexual reproduction which involved three members?

The reason for sexual reproduction being a better choice over asexual reproduction is justified by saying that sexual reproduction involves mixing of genetic information from two living beings and hence if one set of them had an error, the other would compensate for it. With the same kind of reasoning, we can argue that sexual reproduction involving more than two organisms coming together would cause better genetic variation, shuffling of mutations, etc. and thereby produce a better offspring.

However, most of the organisms as we see today have evolved to reproduce sexually using two beings of the same species.

Consider an animal species which has 24 pairs of chromosomes.

Sexual Reproduction Type 1:

There are two sexes: male and female. During the event of sexual reproduction, the male transfers 24 chromosomes to the female where the chromosomes from both the sexes are mixed to produce an offspring.

Sexual Reproduction Type 2:

There are three sexes: male, female and [insert word]. During the event of sexual reproduction, each member transfers 16 chromosomes to the female where it gets mixed to produce an offspring.

We don't observe sexual reproduction of type 2 (or it is rare). Doesn't the type-2 form of sexual reproduction cause better mixing of genetic information than type-1 form of sexual reproduction? Despite the aforementioned advantage, nature has chosen to encourage the type-1 form of sexual reproduction. Why is the type-1 form of sexual reproduction more common than the type-2 form of sexual reproduction?

This question can be extended to "Why has evolution favored sexual reproduction involving two mates than sexual reproduction involving $n space (n > 2)$ mates?

Let's break it down: The chance that two individuals of distinct gender meet at one location in space time is tremendously higher than for three. We could do some math here (assuming biased random walks of points in three classes on a 2d surface etc.) but that is not necessary I guess. As far as we know sexual reproduction evolved long before complex sensory systems and/or brains (namely 1250 million years versus 500 million years) that allow for active search behavior beyond chemotaxis. Therefore a random walk would be a reasonable first approximation for organism behavior to simulate the situation back then. Having two genders seemed to have been a good trade-off between the chance to meet and advantages in terms of mixing the genetic information. Note that this answer might not be valid for organisms being able to reproduce in both ways, sexual as well as asexual.

Quite simply two mates is simpler both functionally and logistically. It is basic statistics you are far more likely to find 1 individual from 50% of the population once, than 2 individuals each representing 33% of the population at the same time. That's three factors making two sexes statistically easier. Also most sexual organisms do not have chromosome counts divisible by 3, some don't even have numbers evenly divisible by 2.

There is still debate about why sex is favored at all, so it is difficult to go into greater detail for other possibilities, but it is far more likely for two individuals to cross paths than three or more, that alone would be enough for a selective pressure favoring two over three most of the time. And of course not now most sexual organisms are built to handle that 2 individual mating they inherited and could not mechanically handle a third.

Though some would disagree, sexual reproduction has distinct advantages over clonal reproduction specifically because it provides a way to shuffle homologous blocks of genes within a population. But increasing the number of parents required beyond two would not provide enough greater "shuffling" power to compensate for the logistical impediments to n-parent matings.