Adaptive dynamics, game theory and evolutionary population genetics
Hamish G. Spencer, Allan Wilson Centre for Molecular Ecology and Evolution, Department of Zoology, University of Otago, PO Box 56, Dunedin, New Zealand.
Tel.: +64 3 479 7981; fax: +64 3 479 7584;
The most important question that one can ask about a method for scientifically investigating a natural phenomenon is whether that method is of any use. Does it tell us something we did not previously know? Does it lead us to ask novel questions, or better understand the observations under study? It seems to us that, although some of the examples provided by Waxman & Gavrilets (2005) suggest that adaptive dynamics has been a productive tool, many of the studies they cite could have been carried out using existing alternative approaches, which, in some cases, might have led to different – or at least more complete – answers. Thus, we argue that adaptive dynamics fails the utility test on two grounds. Firstly, well-established methods can deal with the problems that the adaptive dynamics approach purports to solve, sometimes by using exactly the same mathematical techniques. And secondly, by avoiding the mathematical simplifications entailed by adaptive dynamics, some of these alternative analyses lead to a better understanding of the underlying biology.
Adaptive dynamics vs. game theory
We first observe that much of adaptive dynamics appears to be indistinguishable from the well-known game-theoretic approach to studying phenotypic evolution (Maynard Smith, 1982). Hence, an evaluation of the utility of adaptive dynamics necessitates an understanding of the role of game theory in evolutionary science. Waxman & Gavrilets (2005) claim that adaptive dynamics ‘grew out of the early work of game theorists’ but is ‘more general than the theory of evolutionarily stable strategies’ (ESSs). The application of game theory to evolution was pioneered by the late Bill Hamilton in his ground-breaking work on the evolution of altruism (Hamilton, 1964) and sex ratios (Hamilton, 1967). These game theoretical approaches, generally brought together under the rubric of ESSs (Maynard Smith, 1974), have been extended, for example, by Eshel (1983) and Christiansen (1991), to include analysis of strategies that are close to what game theorists call Nash equilibria (Houston & McNamara, 1999, pp. 167–169).
Game theory (as applied to evolutionary questions), like adaptive dynamics, emphasizes consideration of the invasibility of populations by rare, newly arising phenotypes (or strategies). Usually, but not always, the population is considered fixed for one type (or strategy) and the stochastic effects of finite population size are ignored. Game theory's ‘pay-off’ matrix contains the crucial fitness parameters that correspond to the invasion fitnesses of adaptive dynamics; both approaches are naturally amenable to incorporating selection schemes that are sensitive to changes in the relative abundance of different types in the population (i.e. frequency-dependent selection). We are left wondering, what is the advantage of adaptive dynamics over traditional game theory?
Secondly, we note, that game theory – and adaptive dynamics – make several significant and problematic mathematical assumptions, admittedly in order to be able to solve otherwise intractable problems. As Waxman & Gavrilets (2005) state, ‘analyzing frequency-dependent selection is notoriously difficult’ and they imply that adaptive dynamics can resolve this difficulty. But they then admit that recombination and segregation ‘make adaptive dynamics inapplicable to sexual populations,’ except where the fitnesses of the corresponding population-genetic framework reduce the problem to asexual haploidy. It really is not much of an advance over standard population-genetic theory if adaptive dynamics works for only such a tiny fraction of diploid models. For example, parent–offspring conflict (Feldman & Eshel, 1982) and other two-locus problems (e.g. Eshel & Feldman, 1984) are out of reach of adaptive dynamics, as are most population-genetic models that involve fitness conflicts between the sexes or between life-history stages.
Adaptive dynamics vs. population genetics
One of the advantages of game-theory models over those of traditional population genetics – sometimes called evolutionary-genetic models – has been their relative mathematical simplicity. By ignoring many of the details of genetic inheritance, game theory has been able to provide significant insight into a number of important biological problems (e.g. the evolution of altruism and sex ratios). Evolutionary-genetic models often become so complicated that useful generalizations cannot be made. Nevertheless, standard population-genetic analysis of the stability of fixation states (to determine what is sometimes called ‘protected polymorphism’) normally gives the same results as adaptive dynamics. These latter methods, however, have the unfortunate limitation that they do not address the further question of the stability of and domains of attraction to possible genetic polymorphisms.
Game theory and (hence) adaptive dynamics are examples of so-called ‘long-term’ evolutionary models evolution (Eshel & Feldman, 1984). This approach considers the phenotypic effects of a range of novel mutations, using these properties to infer their long-term success and hence the ultimate course of evolution. By contrast, ‘short-term’ evolution is concerned with the frequency dynamics of a given complement of alleles (or genotypes). Moreover, for some purposes, the complexities that real genetics causes in the short-term can often be ignored (Hammerstein, 1996; Weissing, 1996; reviewed by Marrow et al., 1996). Over the long term, selection on a succession of novel mutations can drive a population to a phenotypic end-point that can be determined using game-theoretic principles (often an ESS) and it does not matter how the population gets there: the exact dynamics of the mutant-allele frequencies are irrelevant. Unfortunately, however, this extremely equilibrium-eyed view of evolution is not always applicable to questions of evolutionary interest, for instance when cycling of allele frequencies arises. Stable polymorphisms can also cause problems and, indeed, it may be that the predicted ESS cannot be reached via a series of successful mutational invasions.
Advantages of population-genetic theory: two examples
The intricacies of evolutionary genetic modelling can also be worth the algebraic effort because they can illuminate aspects of a problem that may never have been suspected in a game-theoretic approach. We give two examples from our own work, but, of course, there are numerous others, and the possibility of being misled by the different dynamical behaviour of game-theoretic models using inclusive-fitness arguments from that of evolutionary-genetic models using full fitnesses has long been known (e.g. Cavalli-Sforza & Feldman, 1978).
Our first example concerns the well-known but counter-intuitive ability, in many evolutionary-genetic models, of a population's mean fitness to decrease over time. Asmussen et al. (2004) examined an evolutionary-genetic model of frequency-dependent selection at a diallelic locus with complete dominance that closely corresponded to the classical prisoner's dilemma model of game theory. They showed that the reductions in the mean fitness did not arise from either unusual allele-frequency dynamics (such as cycling or oscillations) or strange shapes of the mean fitness function (which behaved qualitatively as in the classical constant-viability model in which decreases in mean fitness are impossible). Rather, the conditions determining the allele-frequency dynamics are decoupled from those governing the shape (and hence the maximum) of the mean-fitness function. Asmussen et al. (2004) went on to show that this sort of behaviour was no mere mathematical curiosity: with randomly generated fitness parameters and initial allele frequencies, mean fitness decreased over 20% of the time, and when doing so, reducing the mean fitness by more than 17% of its original value.
Adaptive dynamics would miss all the richness of this simple model. Although (of course) adaptive dynamics correctly calculates the local stability of the fixation equilibria (i.e. it predicts if a rare mutant will invade or not), its agnosticism about the population's mean fitness precludes any consideration of what might be going on. Any interpretation of the invadibility of a population by a mutant allele would surely imply that adaptation was occurring. What is meant by adaptation when the mean fitness decreases, however, is not altogether clear.
Our second example concerns the disagreements about how to model the genetic-conflict hypothesis for the evolution of genomic imprinting. The evolution of imprinting appears paradoxical because expression at an imprinted locus is effectively haploid and the best-known explanation is Haig's (1992) genetic-conflict hypothesis (although it is by no means the only plausible one; Spencer, 2000; Iwasa & Pomiankowski, 2001). This hypothesis holds that imprinting arose from the conflicting genetic interests of mothers, fathers and their offspring as a consequence of mammalian pregnancy (Haig, 1992). At least three formal models of this verbal hypothesis have been published: Haig (1992 and various other papers) employed standard game theory [i.e. Abrams's (2001)‘ESS method’], Mochizuki et al. (1996) used a hybrid quantitative genetic-game theory approach [i.e. Abrams's (2001)‘quantitative genetic method’] and Spencer et al. (1998) applied traditional evolutionary genetics (reviewed in Spencer et al., 1999).
Haig (1999) was extremely critical of the last of these models, claiming that the evolutionary-genetic approach was inappropriate because it did not apply to long-term evolution. As pointed out by Weisstein et al. (2002), this contention is problematic, because, as we have just shown above, many evolutionary-genetic models result in allele-frequency dynamics that cannot be predicted from simple considerations of invasion conditions (see also Feldman & Eshel, 1982; Eshel & Feldman, 1984; Eshel et al., 1998). In the case of genomic imprinting, only the evolutionary-genetic model of Spencer et al. (1998) was able to predict the potential existence of a stable polymorphism in imprinting status under the genetic-conflict hypothesis; this sort of prediction was outside the domain of the game-theoretic approaches used by Haig (1992) and Mochizuki et al. (1996). This difference may seem an esoteric point, but it turns out to be a pivotal distinguishing prediction between the genetic-conflict hypothesis and the rival ‘ovarian time bomb’ hypothesis (Varmuza & Mann, 1994; Iwasa, 1998): no such a polymorphism was forthcoming in Weisstein et al.'s (2002) evolutionary-genetic model of the ovarian time bomb hypothesis. The apparent polymorphic imprinting status at two human loci – the Wilm's tumour suppressor gene, WT1 (Jinno et al., 1994) and the serotonin-2A (5-HT2A) receptor gene, HTR2A (Bunzel et al., 1998) – thus implies support for genetic conflict over the ovarian time bomb [but only if you admit the applicability of evolutionary genetic models, of course. Haig (1999), however, dismissed any polymorphism as transient and – by implication – of no interest]. In short, evolutionary-genetic modelling of the evolution of genomic imprinting has revealed critical distinctions among competing hypotheses that cannot be derived from game theory or adaptive dynamics.
In summary, we argue that adaptive dynamics is simply a form of game theory, and much of the discussion of the role of game theory in our understanding of evolution applies also to adaptive dynamics. In particular, the relationship between game theory and traditional population genetics, which has become much clearer in the past decade, can usefully shed light on the opportunities and limitations of any novel aspects of adaptive dynamics. Thus we concur with Waxman & Gavrilets (2005) that the practitioners of adaptive dynamics should acknowledge the links between their work and that in the fields of game theory and evolutionary genetics, and not just for reasons of good scholarship: doing so will also save considerable unnecessary repetition and misleading biological interpretations arising from its simplifying mathematical assumptions.