Token vs type fitness and abstraction in evolutionary biology

There are only twenty-six letters in the English alphabet, and yet there are more than twenty-six letters in this sentence. How do we make sense of this?

Ever since I first started collaborating with David Basanta and Jacob Scott back in 2012/13, a certain tension about evolutionary games has been gnawing at me. A feeling that a couple of different concepts are being swept up under the rug of a single name.^[1] This feeling became stronger during my time at Moffitt, especially as I pushed for operationalizing evolutionary games. The measured games that I was imagining were simply not the same sort of thing as the games implemented in agent-based models. Finally this past November, as we were actually measuring the games that cancer plays, a way to make the tension clear finally crystallized for me: the difference between reductive and effective games could be linked to two different conceptions of fitness.

This showed a new door for me: philosophers of biology have already done extensive conceptual analysis of different versions of fitness. Unfortunately, due to various time pressures, I could only peak through the keyhole before rushing out my first draft on the two conceptions of evolutionary games. In particular, I didn’t connect directly to the philosophy literature and just named the underlying views of fitness after the names I’ve been giving to the games: reductive fitness and effective fitness.

Now, after a third of a year busy teaching and revising other work, I finally had a chance to open that door and read some of the philosophy literature. This has provided me with a better vocabulary and clearer categorization of fitness concepts. Instead of defining reductive vs effective fitness, the distinction I was looking for is between token fitness and type fitness. And in this post, I want to discuss that distinction. I will synthesize some of the existing work in a way that is relevant to separating reductive vs. effective games. In the process, I will highlight some missing points in the current debates. I suspect this points have been overlooked because most of the philosophers of biology are focused more on macroscopic organisms instead of the microscopic systems that motivated me.^[2]

Say what you will of birds and ornithology, but I am finding reading philosophy of biology to be extremely useful for doing ‘actual’ biology. I hope that you will, too.

Let’s return to the start, but then take a different route.

There are only twenty-six letters in the English alphabet, and yet there are more than twenty-six letters in this sentence. The type-token distinction was introduced by Peirce (1906) to make sense of sentences like this. Types are abstract descriptive concepts while tokens are objects that instantiate concepts.^[3] An example might be more useful: the are twenty-six letter types in English but more than twenty-six letter tokens occurred^[4] on your screen or paper. To give a less linguistic example: I — the writer of this post — am a token of the writer type. More relevant to biology: I am also a token of the human type; and I am a token of the brown-eyed type; and — at the time of writing — I am a token of the bearded type.

Given this imprecise definition of token and type, let us go deeper into evolutionary biology and turn to something else that, in the words of Stearns (1976), “everyone understands but no one can define” — fitness. Here, I want to turn to Abrams’ (2012) taxonomy of fitness concepts. He nests two distinctions — one based on attribution and one based on role^[5] — for a total of four fitness concepts.^[6] In this post I was to focus on just his top-level distinction of token fitness vs type fitness:

• Token fitness concepts attribute fitness as a property of a particular individual organism: “token fitnesses reflect an individual’s complete set of genes, heritable and non-heritable phenotypic properties, and any details of surrounding environmental variations that can affect eventual reproductive success or success of descendants” (Abrams, 2012). It is the way an agent-based modeler might conceptualize fitness: an attribute of each individual agent, something that might be shaped by interactions with other agents and the environment but that ‘resides’ in the individual agent. These fitnessess are sometimes also called individual fitnesses (e.g. Sober, 2013). or organismic fitnesses (e.g. Pence & Ramsey, 2015).
• Type fitness concepts attribute fitness as a property of a type. This is closer to how a population geneticist conceptualizes fitness: an attribute of a genotype or phenotype which might be instantiated in many individual organisms. These fitnesses are more commonly known as trait fitnesses (eg. Sober, 2013; Pence & Ramsey, 2015).

My biggest misstep in the first draft of Kaznatcheev (2017) was saying that what I called ‘effective fitness’ is attributed to a population instead of a type. This was read by some as if I was treating populations as a token and thus as if my distinction was between individual vs group selection.^[7] With the new type-token terminology, I hope that I can avoid this misreading. I realize now that the distinction I was trying to make was that ‘reductive fitness’ is token fitness and ‘effective fitness’ is type fitness.

Hopefully, my examples in the definitions of the two kinds of fitness have you thinking that they aren’t independent of each other. In particular, what if a population geneticist uses agent-based models: is she using token or type fitness? She could be using both, or more specifically: using token fitness to get type fitness. The most popular view of type fitness holds that a type’s type-fitness is the average token-fitness of the tokens of that type. This is equivalent to Pence & Ramsey’s (2015) first concept of trait fitness, except restated in the language of tokens and types instead of organisms and traits.^[8]

For any mathematician, alarm bells should be rininging when they see the word ‘average’: what kind of average? Arithmetic mean, geometric mean, median, mode, or some sort of weighted mean? Most philosophers I’ve read — including Sober (2013) and Pence & Ramsey (2015) — assume it means arithmetic mean. This is especially bizarre — although convenient to his argument against trait fitness — in cases like Sober (2013) where Gillespie’s (1977) work on the impact of variance on evolution is explicitly discussed, since in that case switching to geometric mean eliminates the need for looking at arithmetic mean and variance together (Orr, 2007).^[9]

Abrams (2012) avoids the pitfall of over-committing to arithmetic mean when connecting his two metrological kinds of fitness. For Abrams (2012) statistical type fitness is defined in terms of statistics on sets of measurable token fitnesses. He recognizes that the the statistics in question might not be just the arithmetic mean and could easily involve geometric means, higher moments or other mathematical transforms. But this still puts token fitness as methodologically prior to type fitness. It imagines us measuring many token fitnesses and them somehow combining those measurements into a statistical type fitness. Although this approach might by typical of many studies on macroscopic organisms, I don’t think it is typical of the methodology on microscopic systems. Instead, biologists often directly measure the (statistical) type fitness in microscopic experiments and use it as a primitive from which other quantities of interest are derived.

We can make sense of this by first looking at Pence & Ramsey’s (2015) second conception of trait fitness in the language of tokens and types: type fitness is a quantity that is, given a model of population dynamics, predictive of the future dynamics of the type in the population. In the case of this post, I’ve conveniently fixed the model of population dynamics to the replicator equation. Now, we can invert this conception to give us an experimental definition suitable for microscopic systems: type fitness is the quantity that describes the measured changes of the type in the population. This allows us to directly measure type fitness from the population fold change, logit of change in frequency, or population growth rate. These procedures do not make direct reference to token fitness of individual cells but they still provide a measurement of the type fitness.

Of course, type fitness is still a consequence of the interactions of the various tokens.^[10] As such, we can think of a microscopic experiment and our subsequent operationalization of type-fitness as a physical implementation of ‘some statistic’ on tokens. However, this statistic might not necessarily be on token-fitness but on tokens and their interactions more generally. More importantly, this statistic might be difficult to reverse engineer and replace by a simple formula. This is why our hypothetical population geneticist might be using agent-based models. In a typical situation, she will specify the token-fitnesses — or the procedure for generating them — for each token and then combine it with complicated spatial structure, genetic structure, or demographic structure. She then simulates these tokens in an agent based model. She does this simulation because there is not a simpler way to get the trait fitnesses from this structured population.^[11] In other words, the simulation itself — like the experiment — becomes her statistic. In particular, this type fitness allows her to abstract over the details of the simulation. Or, for the experimenter, type fitness abstracts over the complexities of spatial (or other) structures that we don’t know how to model or measure. And if our questions can be expressed at the level of types and the error generated by this abstraction is sufficiently low, then this approach never needs to explicitly reference tokens.

Notes and References

1. This gnawing tension has a pre-history from my time as an undergraduate at Tom Shultz’ Laboratory for Natural and Simulated Cognition. We would present our models during lab meetings and I would often argue that a certain model doesn’t implement the game or the fitness that the modeler thinks it does due to various conditions that might not seem directly related to the payoffs that agents received. The most memorable case was in a model with sexual reproduction where certain strategies were allowed to reproduce only with certain other strategies: thus effectively nullifying any payoffs they might have received. In hindsight, these debates were probably manifestations of the plural notions of fitness that I present here.
2. Over on twitter, Philippe Huneman and Lucie Laplane offered some comments on philosophy of biology’s macroscopic bias. Their consensus is that the bias it’s there but things are improving: “there are now some philosophers working on all kinds of organisms”. Huneman attributes the established bias to philosophers following the evolutionary biologists. At its foundation, the modern synthesis did not focus much of microbes and defined many philosophically interesting topics — like species — exclusively for macroscopic sexual organisms. Kokko (2017) argues that this focus on system that are more similar to us can distort evolutionary biology through the questions we ask and what we consider appropriate answers.

   Thankfully, philosophers have recently spread out to microscopic systems, usually with different questions in focus. Huneman suggests the most crucial domains with microscopic focus as: biological individuality; tree of life and horizontal gene transfer; and issues related to modeling of and experiments on microbes. All of these are fascinating to me, and although I can’t read everything, I am hoping to read some of the philosophy of modeling and experiments, since I expect that will fit well with my recent focus on operationalization.

3. It is not clear to me that this sentence is a good ‘definition’ of type and token, since it seems to replace them by equally vague terms like ‘abstract descriptive concept’ and ‘instance object’ which take up more text but don’t necessarily add clarity. But hopefully giving enough description of types and token will convey the idea.
4. There is an extra distinction here between tokens and occurrences, which I currently don’t see as relevant to this post nor as uncontroversial. In particular, all discussions of tokens vs occurrences that I’ve read seem to commit me to viewing tokens as physical spatio-temporal objects (in the case of letters: the pixels on your screen or ink on your page). I am happy with using token to mean both token and occurrence in this post. Hopefully without producing extra confusion.
5. Here I am borrowing from Pence & Ramsey (2013). They independently — or at least without citing Abrams (2012) — discuss a classification of fitness based on its attribute (organismic vs trait, for them) and role (conceptual vs. metrological). I find the roles analogous to Abrams (2009) nested level. In particular Abrams’ (200) parametric/tendential fitness serves the conceptual role and his statistical/measured fitnesses serves the metrological role.
6. Abrams (2012) actually discusses five conceptions of fitness, the first is purely mathematical fitness and the other four are his nested categorization. However, his latter four conceptions are much better developed, and purely mathematical fitness becomes more of a vague catch all. In my view, I want to take Abrams’ (2012) latter four conceptions as interpretations of fitness and the purely mathematical fitnesses as the things to be interpreted. In particular, due to the focus of Kaznatcheev (2017) on just replicator dynamics, I have a single purely mathematical fitness in need of different interpretations.
7. When I first circulated Kaznatcheev (2017), the mention of populations was a particular issue for Andriy Marusyk. For him, it didn’t make sense to focus on the fitness of a whole population since that just leaves us with the growth rate of everything in a petri dish treated as one big whole. At the time, I didn’t fully understand this objections, and tried to sidestep it by referring to ‘(sub)populations’. This created confusions in my view of evolutionary dynamics as the population’s decision making process. Wells & Richmond (1995) convinced me that my conceptions of ‘(sub)populations’ was wrong. Thankfully, types save me from this. Now, I can think of the population as composed of tokens of different types.
8. My constant rephrasing in terms of type-token is in part motivated by Pence & Ramsey’s (2015) third conception of trait fitness: “Trait fitness is the reproductive advantage of the individual conferred by possessing the trait.” For me, this creates a mixed metaphor since the first conception treats the trait more like a ‘container’ of tokens, and the third conception treats tokens as a ‘container’ of traits. I think this ambiguity is considered a feature, not a bug of traits. With types such ambiguity is absent. Although a token can be of many types, it is not natural to think of a token as a ‘container’ of types. As such, it is less natural to pose this third conception, which helps reduce the conceptual tensions noted by Pence & Ramsey (2015) to just the tensions between the first and second conceptions. And it is this tension that is central to my point about the two types of evolutionary games.
9. Ironically, I think that Sober (2013) and the many philosophers and biologists that make similar mistakes are actually making a mathematical type error in confusing additive and multiplicative fitness. Of course, we can change multiplicative fitness into additive fitness with the log-transform, and thus let us take arithmetic means but sometimes it might be more complicated but still doable like a logit. In the case of replicator dynamics, we know that the final mathematical type of fitness is additive, saving us some headache. This is part of the reason why I focus on discussing different interpretations of this one single purely mathematical fitness.

   However, if we want to bounce back and forth between replicator dynamics on graphs and inviscid replicator dynamics then we will again need a transform between the fitnesses, or between the games. In this way, I think that things like the Ohtsuki & Nowak (2006) transform serve as ways to implicitly translate between token and type fitness while abstracting over a spatial structure. Similarly, the derivation of inclusive fitness from individual fitness is a way to get a type fitness from a token fitness while while abstracting over a relatedness structure. However, both of these are just wild guesses that I need to develop more.

10. From the little bits of the philosophy literature that I’ve read, it seems that philosophers are primarily concerned with the causal role of the various conceptions of fitness. At this time, this is not of interest to me. And as such, I am not committed to either the statisticalist or causal view of type-fitness. However, I hope that a statement like ‘type-fitness is a consequence of the tokens and their interactions’ is not a controversial statement.
11. As I mentioned in the second half of footnote [9], in some cases there are non-obvious but still analytic ways to translate between the tokens and their complex interactions and the resultant type fitness. However, as I stress in Kaznatcheev (2018), for this to be useful in practice, we want to learn how to invert the statistic. We want to be able to start with type-fitnesses as some information about the structure and then recover information about token fitness (or Abrams’ (2012) tendential token fitness). Thus, this is the next step for me — operationalizing local interactions.

Abrams, M. (2012). Measured, modeled, and causal conceptions of fitness. Frontiers in Genetics, 3: 196.

Gillespie, J.H. (1977). Natural selection for variance in offspring number: a new evolutionary principle. Am. Nat. 111:1010-1014.

Kaznatcheev, A. (2017). Two conceptions of evolutionary games: reductive vs effective. bioRxiv: 231993.

Kaznatcheev, A. (2018). Effective games and the confusion over spatial structure. Proceedings of the National Academy of Sciences: 201719031.

Kokko, H. (2017). Give one species the task to come up with a theory that spans them all: what good can come out of that?. Proc. R. Soc. B, 284(1867): 20171652.

Ohtsuki H, & Nowak MA (2006). The replicator equation on graphs. Journal of Theoretical Biology, 243(1): 86-97.

Orr, H.A. (2007). Absolute fitness, relative fitness, and utility. Evolution, 61(12): 2997-3000.

Peirce, C.S.S. (1906). Prolegomena to an apology for pragmaticism. The Monist, 16(4): 492-546.

Pence, C. H., & Ramsey, G. (2015). Is Organismic Fitness at the Basis of Evolutionary Theory? Philosophy of Science, 82(5): 1081-1091.

Sober, E. (2013). Trait fitness is not a propensity, but fitness variation is. Studies in History and Philosophy of Biological and Biomedical Sciences, 44(3): 336-341.

Stearns, S.C. (1976). Life-history tactics: a review of the ideas. The Quarterly Review of Biology, 51(1): 3-47.

Wells, J. V., & Richmond, M. E. (1995). Populations, metapopulations, and species populations: what are they and who should care? Wildlife Society Bulletin, 23(3): 458-462.