George C. Williams
George Williams conceptualized the maintenance of sexual reproduction as a problem of selection within one species or population. He began by considering organisms that include both sexual and asexual modes of reproduction within their complex life-cycles, for example, aphids & rotifers or strawberries & corals. He conceived the cost of sexual reproduction as the cost of meiosis, that is, the cost of reducing the relatedness with the own offspring from r = 1 to r = 0.5, when these organisms meet the time or conditions for switching from asexual to sexual reproduction. With this kin-selection conception he took the maintenance of sexual reproduction as a problem of selection within a population. Herein, he disagreed with Maynard Smith:
“I think that the primary disadvantage of sexual reproduction in relation to asexual is most fruitfully formulated as a paradox of kin selection—an organism devotes resources to the production and care of a more distant (r = 0.5) rather than a close (r = 1) relative. This formulation provides a number of advantages. In its focus on genes identical by descent, kin selection is genetically explicit and relates directly to evolution. Maynard Smith’s economic argument (resources wasted on males) makes it easy to overlook the fundamental distinction between (1) the evolutionary problem of sexual and asexual reproduction as alternative character states in a population, and (2) the purely ecological question of competition between a clone and a Mendelian population.” (Williams 1978, ‘Mysteries of sex and recombination. A review of The Evolution of Sex by John Maynard Smith.’ Quarterly Review of Biology 53: 287–289. Page 298)
“I believe that understanding has been hampered by failure to distinguish the ecological from the evolutionary problem of sexuality. In important ways, insights gained from conceptual or experimental comparisons of sexual populations and competing clones (the ecological problem) may mislead in relation to sexual and clonal reproduction as alternative processes in a population (the evolutionary question with which I am concerned here).” (Williams 1980, ‘Kin selection and the paradox of sexuality.’ In Sociobiology: Beyond nature/nurture? Ed. by G.W. Barlow and J. Silverberg. Boulder, CO: Westview: 371–384. Page 372)The fact that William and Maynard Smith cut the cake differently gets obvious from the way in which Williams treated the maintenance of recombination as not the problem he was at all concerned with:
“I assume that observed chromosome numbers and crossover rates reflect the optimum compromise between maximizing whatever benefits there are in recombination, and minimizing recombinational load. Tighter linkage must reduce recombinational load, but it does nothing to alleviate the cost of meiosis.” (Williams 1975, Sex and Evolution, Princeton Univ. Press, p. 108)That is, reducing replication-rate by fusing gametes is not alleviated by assuming, for example, a species with a genome consisting of one homologous pair of a giant chromosome and no crossing-over between this homologous pair. [Felsenstein and Yokoyama (1976) modelled this problem.] That would exclude recombination through segregating heterologous chromosomes as well as through crossing-over between homologous chromosomes, but it would not pay the cost of reducing r from 1 to 0.5, or the cost of males, or the cost of fusing gametes, or whatever you conceive the cost of sex to be.
John Maynard Smith
John Maynard Smith conceived the maintenance of high recombination rates (not sex) as a problem of within-population selection between alleles that increase and others that decrease recombination rates. [This differs from Williams's within-population problem of a species with a complex life-cycle and both sexual and asexual modes of reproduction within it.] He accepted Williams's criticism of group-selection arguments for this issue (what he called the "balance argument" of Williams). He agreed that this problem requires an immediate individual-level explanation. But he also maintained that the competition between a sexual population and a genetically isolated asexual clone is a case of between-population selection. At this level, he did allow for long-term or group selection to play some (limited) role.
Anyway, his distinction begins in the preface already:
"I am under no illusion that I have solved all the problems which I raise. Indeed, on the most fundamental question - the nature of the forces responsible for the maintenance of sexual reproduction and genetic recombination - my mind is not made up. On sex, the relative importance of group and individual selection is not easy to decide. On recombination, group selection can hardly play a significant role, but it is not clear to me whether the short-term selective forces I discuss are sufficient to account for the facts, or whether models of a qualitatively different kind are needed." (Maynard Smith 1978, The Evolution of Sex, Cambridge Univ. Press, p. ix)
"It may help to classify the various theories; first, according to the time scale on which selection is supposed to act, and then according to the 'unit of selection' - population, individual, or gene." (Maynard Smith 1978, p. 1)
"I do not find it possible to give an unequivocal answer concerning the role of group selection in the maintenance of sexual reproduction. It has played some role, as evidenced by the taxonomic distribution of parthenogens; but it is not the only relevant force, as will be apparent from the review of the balance argument in Chapter 4, section E. But, whatever one may think of the role of group selection in the maintenance of sex, it cannot explain how it started, and it cannot explain the maintenance of high levels of genetic recombination within sexual populations." (Maynard Smith 1978, p. 6)
And so throughout the book. Maynard Smith consistently distinguishes the maintenance of sexual reproduction from that of recombination, the former being an issue of selection between isolated populations and clones, the latter being one of selection between alleles within one population.
Maynard Smith's support for "some role" of long-term or group selection in the maintenance of sex (not recombination) was also defending his earlier publication from 1958 (The Theory of Evolution, Penguin Books, pp. 138-139). It is often forgotten in potted histories about the paradox of sex, that Maynard Smith did already clearly state the cost of males in this early pop-science writing and also embraced the long-term group-selection explanation of the maintenance of sex.
"If the rate of increase of an animal population were limited by the number of eggs which each female could lay, which in turn depended on how much food a female could eat and transform into eggs, then a population consisting entirely of parthenogenetic females would increase twice as fast as would a population of equal numbers of males and females. From the point of view of reproduction, males are a waste of living material. (This argument does not hold for hermaphroditic organisms, or for those animals in which both parents help to feed the young.) The compensating advantage of the sexual process is that it increases the range of potential variation in a population, and therefore, its evolutionary plasticity." (Maynard Smith 1958, p. 138)
"Thus the sexual process is a means of ensuring evolutionary plasticity at the expense of interfering with reproduction. [...] Now if the advantage of sexual reproduction is that it increases the range of potential variation in a population, then the advantage refers to the population as a whole, and not to any particular individual in it. It follows that sexual reproduction has been established as a rule, both in animals and plants, because selection has favoured some populations at the expense of others. This forms a contrast to the the examples discussed in the last chapter, in which the 'unit' selected was the individual and not the population." (Maynard Smith 1958, p. 139)
By the way, Ghiselin (1988, p. 16, in Michod & Levin (eds): The Evolution of Sex), reminisced an instance of Williams reviewing one of his papers and telling him about the twofold cost of sex and that he [Williams] had found it in a book by Maynard Smith (1966), which must have been the second edition of the above quoted Penguin book by Maynard Smith (see also Dagg 2016, On recognising the paradox of sex. Philosophy, Theory, and Practice in Biology. DOI: 10.3998/ptb.6959004.0008.003).
The fact that John Maynard Smith never changed his mind about his hedged support for some role of group (between-population) selection in the maintenance of sex is clear from an interview of Richard Dawkins with John Maynard Smith in 1997 (deposited at the Web of Stories in 2008).