Abstract
The ‘phenotypic gambit,’ the assumption that we can ignore genetics and look at the fitness of phenotypes to determine the expected evolutionary dynamics of a population, is often used in evolutionary game theory. However, as this paper will show, an overlooked genotype to phenotype map can qualitatively affect evolution in ways the phenotypic approach cannot predict or explain. This gives us reason to believe that, even in the long-term, correspondences between phenotypic predictions and dynamical outcomes are not robust for all plausible assumptions regarding the underlying genetics of traits. This paper shows important ways in which the phenotypic gambit can fail and how to proceed with evolutionary game theoretic modeling when it does.
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Notes
In contrast, when selection is not frequency dependent, selection will eventually lead to a population composed of the most fit homozygote (although the heterozygote phenotype may affect the speed of this evolutionary process).
In real life situations these maps are often one-to-many or many-to-many because environmental factors affect phenotypes as well as genetic factors. However, environmental factors are often ignored in models for simplicity, so it is assumed that the maps are one-to-one or many-to-one.
One classic example of heterozygotes exhibiting traits that differ from either homozygote is in primroses where homozygotes have either red or white flowers while the heterozygote's flowers are pink.
Hines and Bishop (1984a) explore cases where the heterozygote strategy is not a convex combination of mixed strategies, but their model is different from the ones presented here in that both the homozygotes and the heterozygotes employ mixed strategies, and mix from the same set of possible pure strategies. This is discussed in further detail in the “ESS methodology” section.
I assume for simplicity that homozygotes play pure strategies, but they could also play mixed strategies and similar analysis would apply.
Genotype to phenotype maps have been investigated with a variety of different aims. Within evolutionary game theory, for instance, Binmore and Samuelson (2011) evaluate the effect of second order forces (e.g. drift or mutation) in models with different underlying genetics.
This assumes there are only two alleles in the population, but of course the dynamics can be extended to account for more than two alleles. The same is true of the Hardy–Weinberg dynamics.
This is calculated from the fitness of the A1A1 homozygote times its frequency, plus the fitness of the heterozygote times pq. The frequency of these heterozygotes is 2pq, but since they only pass on the A1 allele half of the time, in calculating the fitness of A1, the heterozygote fitness is multiplied by half their frequency.
This is just the weighted average of the fitness of each genotype.
This case has also been considered by Tennant (1999), but the sorts of dynamic and stability concerns addressed here were not at issue.
It is, however, still in some sense 'intermediate' between the two homozygote strategies because an organism with this type of strategy will sometimes play hawk and sometimes play dove, depending on what type of organism it interacts with. If we are thinking in terms of alleles encoding for a ‘dose’ of some gene product, we can think of the retaliator as having one ‘dose’ of whatever causes the disposition for hawkish behavior and one ‘dose’ of whatever causes dovish behavior—the retaliator has some disposition toward peaceful behavior, as seen when it interacts with a peaceful organism, but also has the disposition to act aggressively if faced with another aggressive organism. The same sort of explanation can be given in example 3 for why tit-for-tat can be considered an intermediate phenotype.
Selective pressures can also lead toward a steady state of all retaliators or of some combination of retaliators and doves. This set of states can be evolutionarily important in that it can attract a large portion of the possible initial populations. However, states within this set are not stable equilibria in the sense defined above, but are instead what is called Lyapunov stable. That is, when there is some mutation, evolutionary pressures will not drive the population back to the state in which it started. Evolutionary pressures will instead drive the population to a state near where it started. A somewhat similar situation occurs with tit-for-tat in example 3. This sort of phenomenon is also taken to show the limitations of ESS methodology, but will not be addressed in detail here (see Huttegger and Zollman 2013, for a discussion).
An altruistic act in this sense is just one that decreases the fitness of the actor and increases the fitness of another.
There are also steady states composed of mixtures of altruists and organisms playing tit-for-tat, but these are not stable equilibria in the sense described above.
I should make it clear that I will use the term heterozygote advantage in the loose sense just to mean that the heterozygote tends to have the highest fitness. Cases of heterozygote advantage, while seemingly rare (Bubb et al. 2006), are of interest in evolutionary biology. For one thing, they are a way of maintaining genetic variability in diploid populations. However, the significance of genetics in the pure strategy case is not dependent on the heterozygote being the most fit. As can be seen in Fig. 1Ba, the heterozygote may simply shift the location of the rest point away from the population ESS.
This might not be the case if, for example, in the Hawk–Dove game each phenotype played hawk more than half the time.
Linkage describes the fact that genes located close together on the same chromosome tend to be inherited together.
Epistasis occurs when a gene or genes at one locus affect or ‘modify’ the effects of a genes or genes at another locus.
For instance, one might argue that a translocation could allow for a copy of each allele to be on one chromosome. Then, an organism could be homozygous with respect to each locus (one locus is homozygous for the allele encoding sickle cells, the other locus homozygous for the allele encoding normal blood cells). However, these mutations are generally not considered in a long-term evolutionary models, which assume a specific number of loci. One way of thinking about this point is that these sort of mutations change the genetic system— a trait with one-locus inheritance, for example, would become a trait with two-locus inheritance (Grafen 1984, 65).
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Acknowledgments
I would like to thank Justin Bruner, Simon Huttegger, Cailin O’Connor, Brian Skyrms, Kyle Stanford, two anonymous referees, and audiences at the Philosophy of Science 2014 meeting for helpful comments.
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Appendix 1: Hawk–Dove–Goose
Appendix 1: Hawk–Dove–Goose
One might be concerned that the analysis in this paper is somehow affected by the fact that the heterozygote strategies described in the pure strategy case for the Hawk–Dove game and the iterated Prisoner’s Dilemma are more cognitively complex than in the mixed strategy case. To ease this concern, I consider an alternative approach to modeling the heterozygote pure strategy in Hawk–Dove. A natural suggestion is that the heterozygote does not condition their behavior on the type of their counterpart, but the level of aggression displayed is somewhere between hawkish and dovish. We might call such an individual a ‘goose’.
When a goose interacts with a hawk, the hawk gets the resource because it is more aggressive. Likewise, when a dove interacts with a goose, the goose gets the resource because it is more aggressive. What happens when two geese meet? There are two plausible options. First, the two geese may get enter a conflict and end up splitting the resource and the cost of conflict. In this case we would reasonably assume that the cost of conflict for two geese is less than the cost for two hawks, as the level of aggression displayed is less. The payoffs for the interaction among these three types is shown in Table 4.
For option 1, the replicator dynamics leads to a polymorphic equilibrium where a third of the population are geese and two-thirds of the population are hawks. By contrast, a genetic model with heterozygous geese leads to one stable polymorphic equilibrium at p ≈ 0.20, where the population is roughly 3.9 % doves, 31.7 % geese, and 64.4 % hawks.
The second option is that the two geese end up peacefully splitting the resource. The idea behind this is that geese are aggressive enough to signal aggression but not aggressive enough to actually engage in conflict. So both will signal aggression, but back down once they see the aggressive signal. The payoffs for the second option are shown in Table 5.
In option 2, for the replicator dynamics there is one stable equilibrium, composed of 50 % geese and 50 % hawks. For the genetic model there is one polymorphic equilibrium at p ≈ 0.26, where about 6.8 % are doves, 38.6 % are geese, and about 54.6 % are hawks. In either case, we can see that the genetic models differ from the phenotypic models and, more importantly, differ from the genetic mixed strategy case discussed in the main text.
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Rubin, H. The phenotypic gambit: selective pressures and ESS methodology in evolutionary game theory. Biol Philos 31, 551–569 (2016). https://doi.org/10.1007/s10539-016-9524-4
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DOI: https://doi.org/10.1007/s10539-016-9524-4