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.

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About Artem Kaznatcheev
From the Department of Computer Science at Oxford University and Department of Translational Hematology & Oncology Research at Cleveland Clinic, I marvel at the world through algorithmic lenses. My mind is drawn to evolutionary dynamics, theoretical computer science, mathematical oncology, computational learning theory, and philosophy of science. Previously I was at the Department of Integrated Mathematical Oncology at Moffitt Cancer Center, and the School of Computer Science and Department of Psychology at McGill University. In a past life, I worried about quantum queries at the Institute for Quantum Computing and Department of Combinatorics & Optimization at University of Waterloo and as a visitor to the Centre for Quantum Technologies at National University of Singapore. Meander with me on Google+ and Twitter.

12 Responses to Token vs type fitness and abstraction in evolutionary biology

  1. Rob Noble says:

    A very useful, clear post. Did you deliberately refrain from making an analogy to statistical physics? Perhaps the lack of a universally applicable average can account for failure in the quest for a deeper link between evolution and thermodynamics (cf [1], minutes 22-26).

    [1] https://natureecoevocommunity.nature.com/channels/536-on-the-road/posts/31896-is-there-a-more-fundamental-theorem-of-natural-selection

    • Thank you, Rob!

      I do make the statistical physics vs. thermodynamics analogy in the paper:

      It is helpful to highlight the difference between these two interpretations with an analogy to physics. The setting of statistical mechanics mirrors the fitness for individuals view and defines properties like kinetic energy for individual molecules. Thermodynamics mirrors the effective fitness view and defines properties like temperate for ensembles of molecules. It simply doesn’t make sense to talk of the temperature of an individual molecule. Of course, in simple models like the ideal gas, temperature is just mean kinetic energy; this would correspond to an unstructured (inviscid) population in biology where the effective fitness of a (sub)population is just the average reductive fitness of the individuals that make up that subpopulation. But this ideal case seldom happens in nature. In general, there are many ways like recombination systems, spatial subdivision, and admixture in which structured populations depart from panmixis (mean-field).

      But it doesn’t seem to fit quiet as well in this post, so it didn’t spring to mind as I was writing. It would have made a good footnote. However, I am not sure if I agree with this part of your statement:

      Perhaps the lack of a universally applicable average can account for failure in the quest for a deeper link between evolution and thermodynamics

      My original goal with the ideal gas analogy was to then mention one of the physical systems where temperature is a more complicated function of the constituent parts than just mean kinetic energy. From my vague memories of undergrad, I felt like the 2D Ising Model was an example of this, but I couldn’t quickly dig up and a source to confirm that feeling, so I left only half the analogy. Perhaps, I should do this now that I have some time. Or maybe just a question on the physics stackexchange (where I have been resisting making an account). But long point short: I don’t think that thermo has a universally applicable average, it just has very good toy models like the ideal gas.

      I’ll need to reflect on this more, thank you for bringing it to my attention. I’m also looking forward to watching that video that you linked.

      Philosophers seem to like the analogy between selection and thermodynamic concepts like pressure, temperature, or entropy. As an example from my random recent reading: Desmond (2018) suggests that natural selection is a force like entropic force rather than a Newtonian force. Mostly because “neither its source, nor its effect is spatiotemporally localizable.” I think his argument has some merit, although maybe entropy is not the right thermodynamic ‘force’ to compare to: mostly due to the more central role entropy plays in work like Julian Xue’s entropic drive as seperate ‘forces’ from natural selection and Marc Harper’s information geometry vie of replicator dynamics. To be fair to Desmond, I think entropy was meant as a quick intuitive analogy. Elsewhere in the text, he gives a stronger physical analogy of a population under selection acting like an ideal gas of charged particles in a magnetic field.

      • Rob Noble says:

        Ah yes, I thought I’d seen you make this analogy. As it happens, the video I cited is all about linking entropy and evolution (in a simple toy model).

        • Apparently, the analogy to the kinetic theory of gases is a popular and very old analogy. It was already used by Fisher & others in 1915. Apparently in this paper:

          Fisher, R. A., & Stock, C. S. (1915). Cuénot on preadaptation: a criticism. The Eugenics Review, 7(1): 46.

          I haven’t read the paper, though. And not sure if I am willing to given the venue and Fisher’s history of being a eugenicist. But I wouldn’t be surprised if the connection goes further back and ends up originating in the other direction, with something like Boltzmann having built his theory of gases on inspiration from Darwin. I’ve been meaning to follow this thread for a while now.. I had thought that Boltzmann’s admiration of Darwin was an obscure topic, so it was nice to see it mentioned in that video that you linked.

          • Philip Gerlee says:

            I don’t think you should judge a scientific idea on where it was published. Indeed Fisher was eugenicist (but so was Bertrand Russell in his early days), and his paper on the Fisher equation (which has proven very useful in ecology and oncology) was published in Annals of Eugenics, but the paper itself has nothing to do with eugenics.

  2. Philip Gerlee says:

    To me it seems like a central concept missing in the discussion is importance of probability distributions when calculating population averages. In the ideal gas temperature equals the average kinetic energy, but the distribution of velocities is given by a Maxwellian (normal distribution), which has an impact on other macroscopic properties.

    The same applies to biological systems. If we know how to calculate the token fitness of a particular organism as a function of its relation to other organisms, then the type fitness can be calculated by properly averaging across the probability distribution that describes how the population is structured. The problem is often that this distribution changes over time and one often reduces it mean-field or pair correlations.

    • Do you mean that the importance of probability distribution is missing in typical discussions of token vs type distinction, or in my post in particular? I completely agree with you that the average loses a lot and more information is needed if one wants to translate from tokens to types. In fact, knowing the probability distribution is not enough, one has to know a lot of features of the spatial, reproductive, and genetic structure, too. And that is where the most serious issues seem to crop up for me, because — outside of (increasingly more sophisticated) toy-models — we don’t really have good operationalizations of these features and don’t even know how to focus on them. For example, graph structure is clearly not the right kind of description for most population spatial structure because — as you mention — it can change drastically over time and thus loses most of the advantage one would have gotten from using ‘graphs’. But this doesn’t necessarily mean that diffusion or random walk models are the right approach either. This is further complicated by when there are many ‘reasonable sounding’ reductive models that can all produce similar macroscopic effects.

      That being said, all of these features are captured by the direct measurement of type fitness. This is because you force the physical system itself to figure out how to properly ‘average’ itself. That is why I am trying to advocate for theories built on type-fitness without reference to tokens or particular methods of averaging. Of course, it might end up that for some systems types fitness won’t end up being a scalar value, or all the types might not correspond to ones that feel intuitive and interpretable (and we might need ‘fictitious’ types like ‘free space’ or something else). Where by “won’t end up”, I mean that there is no specification of type fitness as a scalar that can provide sufficiently reliable measurements with sufficiently small error for the physical system and task of interest. But we won’t know that until we measure a bunch of systems, since we don’t have a good grasp on “how” nature abstracts tokens into types.

      • Philip Gerlee says:

        Sorry, but I was a bit unclear. What I meant was that with a theory of how to calculate fitness given certain local conditions and a known probability distribution of local conditions it would be possible to calculate the type fitness.

        More formally: Given a population of N organisms with positions x_i and internal states (e.g. strategy) s_i, a token fitness function F_i(x_1,…,x_N,s_1,….s_N), and a probability distribution P(x_1,…,x_N,s_1,….s_N) it would be possible to calculate the type fitness by taking the expectation of F_i with respect to P.

  3. Andy M says:

    A very clear post, though the letters example is somewhat confusing. As for the substance, I am not aware of meaningful methods to measure token fitness, even in macroscopic organisms. In addition to methodological hurdles, there is almost always a stochastic component in natural or experimental populations, so the token’s fitness is probabilistic – which makes direct measurements not that useful.

    • Thanks!

      On token fitness: I think this is why Abrams distinguishes between two types of token fitness: measured token fitness and tendential token fitness.For the former, he means measures that specifically ignore the stochastic features and are just book-keeping for the number of offspring (or other relevant quantity) that the organism had. In our time-lapse system, this could be things like the inverse of time between a specific cell splitting into two daughters, or more globally: the number off cells that can be traced by splitting to each cell in the initial population. Thus, each cell would have a measured token fitness that is just a summary of what we watched it do. If we wanted to share the measured token fitnesses, we would have to provide a number for each cell that we had seen. Or would you consider this to not be a measure of token fitness?

      After having many measured token fitnesses of many individual cells, through some assumptions of uniformity, we might put together several such individual histories to infer a ‘tendential’ token fitness. This tendential token fitness is suppose to summarize the distribution from which the particular token fitness that we measured was sampled. For agent-based modelers, it is this tendential token fitness that they usually specify when writing their simulations, for example by setting a probability threshold for producing an offspring within a given time-step. Alternatively, we might pick some method for ‘averaging’ the measured token fitnesses and report that average — although that would most likely be an error or a type fitness (depending on what procedure we used to aggregate the individual measured token fitnesses).

      I didn’t go into this too much in the post because I agree with you that token fitness seems inherently less useful in experimental settings. Especially when we have simpler ways to measure type fitness directly. However, it does seem that a lot of people — especially agent-based modelers — think in terms of token fitness. And at least some philosophers — for example Abrams — think that tendential token fitness is casually essential: they think it is what directly causes natural selection. But that is a discussion that I really want to avoid for now.

  4. Andy M says:

    I think measuring token fitness by following individual cells would be very interesting. We have discussed this at some point but did not manage to move ahead as we could not figure out how to do it experimentally (apart from looking at colony size distributions – but these would not be appropriate in the EGT experimental contexts)
    .

  5. Pingback: A month in papers: mostly philosophy of biology | Theory, Evolution, and Games Group

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