From the perspective of an organism trying to propagate its genes, sex is like a trade: I’ll put half of your DNA in my offspring if you put half of my DNA in yours. I still pass one copy of my genes onto the next generation per unit of investment in children, so it’s a fair deal. And it doesn’t impact the average fitness of my kids very much, since on average my partner’s genes will be about as good as mine.
But the trade has transaction costs, so I’m only going to do it if I get some benefit. In this post I’ll tell a particularly simple story about the benefit of sex. I think this is basically equivalent to the standard story, but I find it much clearer. It also makes it more obvious that we don’t require group selection, and that the benefit is very large.
Why doesn’t sex change the average fitness of my kids? The possibility of a “lucky” kid who gets the better genes from both of us is offset by the possibility of an unlucky kid who gets the worse genes from both of us. If the effects of genes are linear, the average fitness will be exactly the same as the parents. In practice I expect it to be slightly lower because of convexity and linkage disequilibrium.
But sex increases the average fitness of my grandchildren, because my fittest children will be responsible for a disproportionate fraction of my grandkids. More precisely, if an organism with fitness dX has (1+dX) kids per generation, then the total fitness of my grandkids is E[dx * (1 + dX)] = E[dX] + E[dX^2]. So increasing variance by 1 unit is as good as increasing average fitness by 1 unit.
Reproductive decisions are naturally a tradeoff between average fitness and variance. Sex slightly lowers the average but increases the variance. If you try to get the same amount of variance with random mutations, you’ll have to totally tank your kid’s expected fitness, because your current genome is well-optimized, and you’ll also pass on fewer genes to the next generation (since some of yours got destroyed). In fact, it’s hard to think of any way to get similar benefits without exchanging genes.
Variance becomes linearly more important over time. For the fitness of my grandkids, 1 unit of variance is worth 1 unit of fitness; for great-grandkids, variance is twice as valuable; for great-great-grandkids it’s three times as valuable. I don’t think we have to look too many generations ahead before the variance bonus from sex outweighs the costs. For example, if genetic fitness differs by 5% between my offspring, and sex reduces fitness by 1%, then sex breaks even within about 6 generations.
The sideways view 
Great Post!
I see how sexual reproduction with different sexual partners increases variance in fitness among your children. Did you also mean mean to claim that sexual reproduction with just one sexual partner increases the variance in fitness among your children? If so, can you explain how that works?
Isn’t redundancy a major first-generation advantage? Say an organism accidentally mutates a protein important in late-stage development: then having a codominant non-mutated allele can provide redundancy for offspring to survive. When a genome gets big enough that you expect a couple mutations per generation, this seems like a very good trade-off. Even if you were to try to recover this redundancy by replacing sex by diploid parthogenesis, sexual reproduction has the advantage of being more stable to environmental change (so if your lineage gradually lost a gene only beneficial in certain environments, e.g. a component of the immune system that defends against a particular pathogen, your children might still be resilient to this change because of sexual reproduction). I might be wrong here, as this is very much not my field.
Having thought for 10 minutes, of course this is not an improvement over self-cloning unless you mate with a partner from an independent evolutionary branch, which your model precisely captures at the “first adopter” level. Sorry and thanks for the great explanation.
Is the variance still as beneficial if you take into account the gambler’s ruin risk (ie there’s no coming back from 0)?
I thought of sex quite differently: not in terms of an abstract concept like “variance”, but in terms of concrete ideas like “improvements” and “suckiness”.
I used to be a Christian, and as a software developer I had a novel intuition against evolution that I never heard anyone else express. I observed that the DNA code was like a computer program, and that if you start randomly changing bits in a (human-built) program, the chance of “improving” the program any way is infitessimal, while the chance of breaking it, or making it function less well, is probably thousands, tens of thousands, or hundreds of thousands of times higher. Despite a natural selection step, it didn’t seem plausible to overcome such an immense force of entropy.
(I also figured that thought that “intelligent design” was necessarily a perfectly reasonable concept, because if natural evolution of intelligent beings was possible, surely those intelligent beings could design more intelligent beings, and thus the concept of evolution implied the possibility of intelligent design, and thus I thought myself clever. I digress, but I like digressing. Against this, the standard argument against intelligent design was unpersuasive. This argument said that if it could be shown that it was possible for a thing to have evolved, this was proof that it had indeed evolved and intelligent design didn’t happen. It wasn’t until much later that I heard a better argument from Yudkowsky, namely that “a human would be out of place in a community of /actually/ intelligently designed life forms” – the whole argument is less than a paragraph, as Yudkowsky seems to feel that this is so obvious that there is no need to write a whole essay on it. My church has an counterargument, mind you, that we’re poorly designed so that life will suck more, because cancer, arthritis, diseases and cognitive biases are all part of God’s mysterious plan, which He will explain to us after we die and you just need to be patient. Then I found out that my church was false. So now I say, damn, God is an incredibly bizarre guy, and isn’t it weird how the “7 days” in Genesis was code for “4 billion years”, just the sort of time frame that makes evolution plausible.)
Anyway, So! Sex? What is it good for? You see, it takes better advantage of the ultimate motor of evolution: death.
If you have 20 mutations in your genes, maybe 9 are harmful, 1 is beneficial, and 10 have a negligible effect we can ignore. I’m pulling these numbers out of my ass, but bear with me: the argument should work across a range of different values.
In asexual reproduction, you’re stuck with those 9 harmful mutations, and if you’re allowed to reproduce freely, you tend to suffer from accumulation: 18 bad + 2 good, then 27 bad + 3 good, and so on until your genome sucks enough to kill you outright or at least let the other local microbes beat you to the food source so that you starve. In bacteria-land, this still somehow works out, apparently because there are billions of bacteria. If the ratio of good:bad mutations is 1:9, then 1 bacteria in every billion is extremely lucky and receives 9 good mutations and 0 bad mutations over a certain period of time, allowing it to compete really well for food, so it makes zillions of copies of itself and spreads throughout the local microbiome. (Most of these copies will get harmful mutations, and so the cycle begins again; even so, the surviving bacteria can be more fit than before.)
Sexual reproduction offers something far less dystopic. If a bacteria gets 9 harmful mutations, it is shit out of luck unless the bad mutations spontaneously un-mutate, and the chance of that is just about negligible. If a human gets 9 harmful mutations, it’s a different story. Each offspring – which, by the way, we get by having sex – isn’t life grand? – each offspring has a 50% chance of not having one of the bad mutations, so if you have 8 kids, you can expect each one to have 4.5 bad mutations from you, plus, who knows, maybe 4.5 from your partner for a total of 9. But wait, you can have multiple kids! So you have 8 children, because that was once very popular, and probably one of those children has only 6 bad mutations instead of 9. Also, what about the 1 good mutation? About 2 of your kids were lucky enough to have both good mutations, and 25% of the time, the kid with only 6 mutations also gets the 2 good mutations! So each generation of humans is likely to produce a few exceptional kids with unusually few bad mutations plus a few good mutations, and if the other humans like what they see (in a sexual way, wink wink), those exceptional kids will spread their genes around copulously.
So, assuming the mutation rate is slow enough, and assuming that the ratio of bad to good mutations is not extremely high, sexual reproduction should eventually allow bad mutations to be eliminated, by killing off the people who hold them and destroying their souls – you see why people stay religious? Evolution is fukking horrible! Meanwhile good mutations not only keep the species alive, but improve it. This allows sexually-reproducing species to improve in fewer generations, with much fewer malfunctioning incarnations, than bacteria suffer.
Now, what about my original argument, that the ratio of bad-to-good mutations should be much, much greater than 9:1? Meh. Exercise for the reader. I can’t write all day, you know.