Statistics are mathematically simple, but applications often suffer from 'the Monty Hall problem problem': the exact assumptions from which the mathematics derive their validity often remain unstated; in such a case the whole statistical number juggling becomes a dubious affair.
Even when the assumptions are clear, it's very tempting to draw false inferences - in which the reader unknowingly misreads or changes the assumptions on which the statistics are based.
This is why I love the writeup above: not only does it spell out the mathematics in great detail, it is also very careful in pointing out the exact assumptions, the exact situation in which these numbers have practical meaning.
The logic, or should I say calculation, of Maynard Smith's paradox is indisputable. The problem I see with it - note: I am not a biologist - is that it presents a situation that is utterly unrealistic.
Asexual reproduction happens by means of cell division - the development of this complex process is in itself a major feat of evolution. Sexual reproduction can be seen as the opposite: it requires that genetic material from two different cells somehow combines into a new cell.
But the sexual process is really just an extension of the asexual one. In cell division, the genetic material splits in two, then each of the halves are duplicated to form a full copy, around which the rest of the cell regroups to form two cells. In sexual reproduction, the split halves then leave the cell to combine with other halves, and a new cell has to form around it.
So at the DNA level, the process isn't very different; in a 'primordial DNA soup' without cells, both mechanisms could develop. But when cells are formed, the processes become very different: with genetic recombination, the DNA halves must leave the cell in order to combine with another matching half. (Sexual reproduction is a particular form of genetic recombination.)
It's clear how both possibilities, that is, DNA duplication and DNA recombination, can preexist as mechanisms within the primordial DNA soup, and these processes can be retained as the process of cell formation develops, leading to cell division (asexual reproduction) and a mechanism to send out 'DNA halves' to other cells (sexual reproduction).
What I fail to see is how one mode of reproduction can easily develop from the other, and it seems utterly preposterous to suggest that one of the two could be created as the result of a genetic mutation, which is what Smith apparently presupposes.
Rather: when an organism - i.e. something with one or more body cells - has both modes of reproduction, they exist as completely different mechanisms, separate or even operating in parallel, without any competition at all.
Therefore, assumption 1 is totally unwarranted. I won't even discuss assumption 2.
By way of illustration, consider some examples.
- single-cell organisms with sexual reproduction (do they exist?)
- the paradox predicts that the very large majority of reproductions occurs asexually, i.e. by cell division; does actual evidence contradict this in any way?
- the strawberry
- offshoots from strawberry plants can form roots and completely separate plants - but there is no direct competition with sexual reproduction, which allows new strawberry plants to form in completely different locations - assumption 1 cannot even be said to apply here
- human beings
- soon, we will be able to clone humans; even if we could only clone ourselves, we can't expect cloning would spread as rapidly as in The Boys from Brazil - assumption 1 is demonstrably false
- human body cells
- this organism actually uses both modes of reproduction very successfully - asexual reproduction to form human bodies from a single cell, and sexual reproduction to spawn off new bodies before the present ones collapse; these mechanisms don't compete at all, but complement each other - both are essential to our continued existence, and to that of all sexually reproducing multicellular organisms; there is no paradox at all.