Game theory and developmental robusticity

17 minute read

The introduction of game theory into evolutionary biology is often credited to George Price and John Maynard Smith. This is for good reason; together they were able to generalize Hamilton's (1967) work on parental investment strategies. By doing so, they provided an account of the evolution of variant strategies of many kinds from a gene-centered perspective.

Before their landmark contribution, there had been earlier forays attempting to integrate a game theoretic perspective into evolutionary terms. Hamilton's own work was immensely important, since his "unbeatable strategy" concept was a clear precursor of the ESS concept. Additionally, we should include the series of papers by Richard Levins (e.g., 1963), which considered the optimum adaptive solutions for various problems relating to spatial and temporal changes in the environment. These didn't explicitly involve the terminology of game theory, but dealt with the mathematical conditions under which variant strategies would pay off.

But the earliest major paper was Richard Lewontin's "Evolution and the Theory of Games," published in 1961 (and presaging Maynard Smith's own 1982 book of the same title). This paper introduced the concepts of game theory to many biologists, and some of them started playing with the ideas in interesting ways that weren't ultimately integrated into the later development of evolutionary game theory.

Waddington on game theory

One of these interesting early efforts is a short paper by C. H. Waddington (1965), which served as an introduction to a symposium on the evolution of colonizing species. Waddington discussed Lewontin's 1961 paper in some detail, and used it to frame the problem of colonizing versus noncolonizing species. He took these two alternatives as possible strategies that a species might adopt in its contest against Nature.

This is an unusual formulation from the perspective of later evolutionary game theory, and reminiscent of "good of the species" Wynne-Edwards-type thinking, but Waddington lays it out very clearly:

In the "evolution game" which it is playing, a species has to contend with unforeseen eventualities which the future may bring -- a new parasite, a new predator, possibly an Ice Age. Another element of uncertainty arises from the fact that there may be several different ways in which the species makes a place for itself within the whole ecological network available for its exploitation -- it could change its food habits or length of life cycle, or it could migrate to another locality, and so on. The game it is playing is perhaps best formulated as a zero-sum three-person game, the players being (1) the species or population under consideration, (2) the whole environment, organic as well as inorganic, that impinges on that species, and (3) the bio-system that would occupy the living sace of first player, if it were eliminated. This third player may include species which are also involved in the second player, but by formulating the game in this way, the third player is reduced to a dummy whose only function is to absorb the gains and losses of the first player; in this way we retain the advantages of dealing with a zero-sum game and only have to consider the moves of two players, the species and Nature (Waddington 1965:2).

In the next sentence, Waddington defines the "score" of the game as the number of offspring produced by the first player in the succeeding generation, which of course is easily scaled to the population mean fitness. In my mind, this shows the "game theoretic" description here to be a conceit, since he is not really describing a situation different from the ordinary assumptions of population genetics. Indeed, his description of the "third player" is entirely superfluous from this point of view.

But I find the conceit illuminating, because it reminds us that there are other species there to absorb Nature's gains. A species must compete in reproductivity at a high level due to these interspecific interactions, or it will not last. Later in the book, Lewontin describes some models where the typical viability fitness of the mean genotype is far less than unity, which of course means that the fertility of these genotypes must be very high, indeed, for them to manage to survive. These are only modeling questions, but the possibility of losses against the field are important to the models.

He goes on to describe the game as an interdemic process, in which different populations within a species may adopt different strategies:

A population has, of course, no intelligence of its own which would make it possible for it to choose which move to make, i.e., to adopt a strategy. But no large population is fully panmictic; it is always broken up, if only by distance, into a number of smaller subpopulations which are partially independent in genotype. Each subpopulation will make a somewhat different move, some of which will be more successful and others less; a global or "Monte Carlo" strategy will emerge as that sequence of moves that has proved most successful up till the stage the game has reached. As we shall see, there are really many different games going on simultaneously, affecting different levels of individual and population organization, and each game elicits a corresponding strategy (Waddington 1965:2-3).

The point of his paper is to suggest that colonizing may be a strategy that is adaptive under some circumstances and not in others. The theme is developed later in the volume by Edward O. Wilson, Ernst Mayr, and Lewontin, and I may post a bit on those contributions later.

What I found provoking in Waddington's paper was this passage:

At a fundamental biochemical level, there are alternative strategies possible in the organization of the genetic control of enzyme seqeunces. Consider a metabolic pathway in which successive steps are catalyzed by enzymes P, Q, R, S, T, .... As Kaeser (1963) has pointed out, in some cases it is found that one of these enzymes, say R, is rate-determining for the whole sequence of steps, the throughput being highly dependent on the quantity or activity of R, but little affected even by considerable changes in the activities of the other enzymes; in other metabolic pathways, all of the enzymes may have more or less equal importance in controlling the over-all flow through the system. If the first strategy is adopted, the system is little affected by mutations or environmental effects controlling the nonrate-determining enzymes, but is very sensitive to effects on R; with the second strategy, the system is affected somewhat by mutations or other influences on any of the enzyme proteins, but is not affected drastically by any of them. The second would therefore seem to be the Minimax strategy, but a species might often be able to get away with the first gambit, in which it would only rarely suffer any loss of efficiency, at the expense of failing completely in a few individuals (Waddington 1965:4).

With this "first strategy", Waddington is describing a strategy for developmental robusticity: resistance to alteration in the developmental program due to alterations in the genetic background. For example, some developmental programs continue to generate adaptive outcomes even if there is a knockout mutation in one of their essential genes. We could say that these systems "degrade gracefully," so that many kinds of mutations lead to phenotypes that are not markedly reduced in fitness. Yet a few of the key genes in most systems cannot tolerate such changes. These developmental programs have evolved in such a way that reduces the impact of most mutational variants in their essential genes, but has emphasized the impact of others.

Now, this kind of structure might be a necessary consequence of genetic and developmental networks. Maybe it just isn't possible to build a genetic system like Waddington's hypothetical every-gene-equally-crucial example.

But the current trend in evo-devo is to propose that such network structures (so called hub-and-spoke networks) are themselves selected based on optimizing some biological property, such as modularity or reliability. Optimality theory and game theory are closely conceptually related to each other -- Maynard Smith was a central figure in the introduction of both to evolutionary biology -- but few studies of developmental processes seem to have explicitly focused on the idea of alternate developmental strategies.

Switches, canalization, and genetic variation

Waddington is best known for his concept of developmental canalization (I posted a quick review of the topic early last year). In this paper, he suggests canalization as one of a set of four developmental strategies that might be adaptive in different environmental contexts:

In order to meet the demands of differeing environmental effects on development, and on selective pressures, a species has, in general, to preserve considerable genetic variation within its populations. But this variation can be deployed in a number of ways: (a) The species can become very good at producing one particular phenotype under almost any circumstances, relying upon the environment always offering a possibility for this phenotype to get by. This leads to the evolution of systems of developmental canalization of the phenotype...

In other words, genetic evolution tends to reduce the effect of environmental variance on the phenotype. That insensitivity to environmental (and background genetic) variability is canalization.

...(b) The species can become good at doing one or another of a few alternative things. This leads to switch mechanisms between canalized phenotypes, e.g., in species which have hot and cold weather or aquatic and terrestrial forms. (c) The species can allow the environment to have a strong influence on individual ontogeny, provided it is ensured that the environmental modifications are toward the selection optimum for that particular environment. This leads to the evolution of developmental systems which are highly adaptable. (d) The species can have a development which is relatively unaffected by normal environmental variations, but in which most genetic changes come to phenotypic expression, and can rely on its wealth of genetic variation always to throw up some phenotypes near the selection optimum. This leads to systems in which there are considerable random or periodic changes in the gene pool from time to time, but little genearl long-term movement in any particular direction (e.g., fluctuations in inversion frequencies according to season, as in some species of Drosophila (Waddington 1965:4-5).

This list is worth remembering: (a) canalization, (b) developmental switches, (c) environmental variance, (d) genetic variance.

I did a little noodling around and found that a few people have followed up on this idea that developmental robusticity versus plasticity may be treated in a game theoretic perspective. For many purposes, the benefits and drawbacks of a given developmental program may be examined without reference to the idea of strategies. Still, plasticity itself is presumed to be adaptive to changing environments, so that the system of benefits and drawbacks in particular environmental contexts (and their frequency) may be usefully considered in ESS terms. Some have picked up on this analogy and mentioned the relation between a plastic developmental program and a "mixed strategy" ESS solution.

Lively (1986) considered the case of "developmental switches" in a game theoretic context. Picking up on the work of Levins (1963) and Levene (1953), Lively examined the circumstances under which organisms may adapt a stress-tolerant phenotype. Such phenotypes may be adaptive to low-resource environments, or cold, or high-predator environments. The possibility of such stress-tolerant phenotypes presents some complexities for interpreting

If the inducing cue [in the environment] is highly correlated with the harsh patch and rare in the benign patch, a conditional strategy can be stable over a wide range of patch frequencies, and this range increases with increasing cost to the stress-tolerant morphology. Hence, a change from one patch type to the other over geological time could result in a correspondingly "rapid" change in morphology without speciation or even any genetic evolution. This would happen without a trace of intermediate forms. Care must be taken, therefore in interpretation of the fossil record when developmental conversion is suspected to be an alternative to strict genetic determination of morphology (e.g., Reyment 1982) (Lively 1986:569).

The relevance of such stress-tolerant phenotypes might seem to be clearer for short-lived, high-predation animals than for hominids.

But there are obvious applications of the idea of developmental strategies in human evolution. For one thing, the maturation time is a prime target of research into fossil humans. A long series of papers has been devoted to uncovering whether Neandertals developed on the schedule of modern humans or some other (presumably more ape-like) schedule. Only recently has this literature brought in the substantial variation in dental development time among recent human populations.

As yet, the subject of nutrition-induced variation in development time has not been a major topic in papers examining skeletal development in Neandertals or other early humans. The normal phenotypic response to developmental stresses in humans is to elongate the developmental span, delaying maturation and/or truncating body size. The relevance of stress-tolerant phenotypes is reasonably clear in this context.

Lively also elaborates on the game-theoretic properties of such a system, including a surprising observation about variant strategy morphs:

The present study also shows that interactions among morphs are required for a mixture of unconditional strateies (i.e., genetically determined polymorphism) to ba an evolutionaily stable state of the population. Hence, we have the apparent paradox that, in the absence of mutualistic effects, genetically determined morphs must compete in order to coexist. This result is analogous to results gained from genotypic models, which show that density dependence is required for the maintenance of allelic polymorphisms (Maynard Smith 1962, 1970; Anderson and Arnold 1983). The competitive interactions within and between morphs may be completely symmetrical (all eij = e), provided there is some cost associated with the stress-tolerant morphology [if the stress-tolerant morphology had no costs, it would be the only ESS]. Asymmetrical competition that favors the nontolerant morph in the benign patch further increases the range of patch frequencies over which genetically determined morphs may coexist, but this region is narrow even under hte best of conditions. This narrowness may explain why so few genetically determined polymorphisms are observed amon randomly dispersing organisms (Lively 1986:569, emphasis mine).

I also found some work that places development into the perspective of another of my current interests, information theory. Thomas Getty (1996) suggested that developmental plasticity may be interpreted as a signal reception problem. An organism will maximize its fitness if it can adopt the pattern of development that is most adaptive to the environment in which it lives. If we consider a population in which individuals may find themselves in two environments with different requirements, then a plastic developmental program might allow individuals to develop in the way appropriate to one of these environments. This is the classic problem also treated by Levins with relation to environmental heterogeneity. Under certain patterns of different environments, a population may optimize its fitness by retaining or evolving plasticity.

Getty points out that a plastic developmental program with conditional expression of different phenotypes requires some way to detect which environment the organism is in:

The genotype, in effect, discriminates among possible environments on the basis of cues. Although it has long been reconized that the evolution of phenotypic plasticity hinges on the availability of reliable cues (Levins 1963; Lively 1986), most recent analyses do not explicitly consider the role of cue discrimination (e.g., Houston and McNamara 1992; Kawecki and Stearns 1993; de Jong 1995; Via et al. 1995). In contrast, Moran (1992) focuses on the role of cues and concludes that they are a dominant factor limiting conditional developmental switching. At the heart of her analysis is a probabilistic relationship between (proximate) environmental cues and (ultimate) environmental quality within generations. This probabilistic relationship suggests that "reliable cues" are like "noisy signals" in signal detection theory (SDT). In both cases, there are risks that a response will not correctly match the ultimate conditions. I want to show that it is useful to think of this aspect of phenotypic plasticity as a signal detection process (Getty 1996:378).

The term, "bet-hedging" comes up frequently in these kinds of considerations. Sometimes organisms simply don't have access to high-quality (i.e., minimal noise) signals of their environments, at least not at the important stages of early development when significant phenotypic choices must be made. But the higher the fitness payoff from betting correctly on a phenotype-environment match, the higher the risk an organism should be willing to take on the basis of its necessarily limited information.

Predicting the future

Of course, the value of better information-processing mechanisms is clear: if an organism can predict its environment with more certainty, then it can adapt with greater plasticity because the risk of a poor match between phenotype and environment will be greatly reduced.

This relation comes up again and again from the consideration of phenotypic plasticity and environments. Generally, phenotypic variation is adaptive to more predictable environments.

Sound counter-intuitive? Surely, it seems that the way to adapt to a more variable environment is with a variable phenotype?

Indeed, not so. Phenotypic variation is just as likely to make an organism less adapted to a varying environment as more adapted. This is worse than a wash if the environment is unpredictable. The best that selection can do in an unpredictable environment is to minimize heritable (i.e., the genetic component of) phenotypic variation. This is purely a loss-cutting measure, since in generations where the environment is very different from the maximally adapted value, the population fitness will tank. Still, a variable population would be worse, because much of a phenotypically variable population will do poorly even under the mean environment!

Levins (1963) shows that a more predictably changing environment allows another response. In such an environment, the presence of phenotypic variation can allow selection to track environmental changes. So predictability is the key to adaptability by selection.

Levins considered the case where the environment was predictable because it was temporally autocorrelated -- in other words, one generation's environment predicts the next. We may instead consider predictability that emerges from signals available in the environment. An organism that can detect such signals has a way to maximize its adaptation -- choosing the phenotype that is most adaptive to the current environment.

In terms of Waddington's four developmental possibilities; Levins is testing two of them: the canalized phenotype versus the phenotype with substantial genetic variance.

As Getty describes, the signals that would permit an organism to choose between these strategies are typically noisy: they entail substantial errors of reception. You might bet on a cold year because the groundhog sees its shadow, but how often is the groundhog right, really?

It is an open question whether human intelligence once enhanced fitness by better reading the signs that predicted future environments. I think this is unlikely because there is a scaling problem -- take an organism with a lifespan as long as a human, and try to predict the course of the environment in a given area over that timespan. It's certainly beyond me.

But one argument is that human culture really constitutes Waddington's option (c): human behaviors are extensively induced by the environment, allowing a "highly adaptable" response to changing environments.

Information about the environment may smooth the fitness function, so that a given amount of environmental fluctuation presents a smaller fitness cost. This could occur either because information reduces mortality (allowing individuals to survive immediate shortfalls), or because it permits higher fertility. Both are intimately related to energy (how much food is available), nutritional ecology (how much protein is available), and group structure (how many mates are available).

This implies a certain kind of ecology for ancient humans, one with some surprising correlates. More on that later.


Getty T. 1996. The maintenance of phenotypic plasticity as a signal detection problem. Am Naturalist 148:378-385.

Hamilton WD. 1967. Extraordinary sex ratios. Science 156:477-488. </a>

Levene H. 1953. Genetic equilibrium when more than one ecological niche is available. Am Naturalist 87:331-333.

Levins R. 1963. Theory of fitness in a heterogenous environment. II. Developmental flexibility and niche selection. Am Naturalist 97:75-90.

Lewontin RC. 1961. Evolution and the theory of games. J Theor Biol 1:382-403.

Lewontin RC. 1965. Selection for colonizing ability. Pp. 77-94 in The Genetics of Colonizing Species, Baker HG, Stebbins GL, eds. Academic Press, London.

Lively CM. 1986. Canalization versus developmental conversion in a spatially variable environment. Am Naturalist 128:561-572.

Maynard Smith J, Price GR. 1973. The logic of animal conflict. Nature 246:15-18.

Waddington CH. 1965. Introduction to the symposium. Pp. 1-7 in The Genetics of Colonizing Species, Baker HG, Stebbins GL, eds. Academic Press, London.