Multiplicative versus additive fitness and the limit of weak selection

Previously, I have discussed the importance of understanding how fitness is defined in a given model. So far, I’ve focused on how mathematically equivalent formulations can have different ontological commitments. In this post, I want to touch briefly on another concern: two different types of mathematical definitions of fitness. In particular, I will discuss additive fitness versus multiplicative fitness.[1] You often see the former in continuous time replicator dynamics and the latter in discrete time models.

In some ways, these versions are equivalent: there is a natural bijection between them through the exponential map or by taking the limit of infinitesimally small time-steps. A special case of more general Lie theory. But in practice, they are used differently in models. Implicitly changing which definition one uses throughout a model — without running back and forth through the isomorphism — can lead to silly mistakes. Thankfully, there is usually a quick fix for this in the limit of weak selection.

I suspect that this post is common knowledge. However, I didn’t have a quick reference to give to Pranav Warman, so I am writing this.
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Argument is the midwife of ideas (and other metaphors)

In their classic book Metaphors We Live By, George Lakoff and Mark Johnson argue — very convincingly, and as I’ve reviewed before — that “[m]etaphor is one of our most important tools for trying to comprehend partially what cannot be comprehended totally” and that these conceptual metaphors are central to shaping our understanding of and interaction with the world we are embedded in. Based on the authors’ grounding in linguistics, part of their case proceeds by offering examples of, by my count, over 58 different metaphors and metonymies in our everyday language; and given their book’s intentions, they chose a particularly pertinent first case: ARGUMENT is WAR.[1]

They show this metaphor in action through some example of common usage (pg. 4):

What do you want me to do? LEAVE? Then they'll keep being wrong!Your claims are indefensible.
He attacked every weak point in my argument.
His criticisms were right on target.
I demolished his argument.
I’ve never won an argument with him.
You disagree? Okay, shoot!
If you use that strategy, he’ll wipe you out.
He shot down all my arguments.

Notice that the even the xkcd I borrowed for visual reinforcement is titled ‘Duty Calls’, an expression usually associated with a departure for war. With our awareness drawn to this militaristic structure, Lakoff and Johnson encourage the reader to ask themselves: how would discussions look if instead of structuring arguments adversarially, we structured them after a cooperative activity like dance?[2]

Argument is Therapy

Unfortunately, I don’t know how to think about argument as dance. So instead, I will start by considering ARGUMENT is THERAPY. Whereas a classical conception of medical therapy is to eliminate a disease — eliminate an impurity in the body. A philosophical therapy eliminates false beliefs. This imagery is widely used, especially by movements like the New Atheists. It is a central part of memetics and we can see it in action with claims that religion is a “virus of the mind” (Dawkins, 1993). It is a great rhetorical tool to suggest that your opponent’s belief is tainted and unclean. It also turns the tables on who benefits from an argument. Here, if you lost the argument then you have been cured of your false belief and thus benefitted. I have not gained anything from convincing you, except maybe some moral smugness at a job well done.

Argument is therapy — imperfect like all metaphors — can draw our attention to important aspects of how arguments and beliefs develop. In particular, we can look at the accompanying metaphor that IDEAS are GERMS. This puts the idea first and human minds as the environment that they inhabit. We can then reason about the dynamics of the resultant knowledge ecology.

By being repeated again and again, ideas and lines of argument mutate, with some becoming more effective and more repeated. Creativity in argument becomes mere mutations in the mind. In analogy to evolution,[3] the (apparent) design we see in well-constructed ideas moves from the individual thinker to the process of selective propagation through minds of the community. Ideas develop through sharing, discussion, and argument — not through individual deliberation. Even if the individual does not intend it, their participation in argument allows the community to pursue cooperative discovery and through variation learn something new.

Unfortunately, it is important to remember that reproductive success is not goodness. And this is easy to see in pop-culture and the meme factories of the internet, where some of the most wide-spread ideas are far from the best. Instead, they are the most comforting, the most enraging, the simplest, or the easiest to share. This is made worse when two ideas that differ in content can set up totems-of-the-other around which minds can gather, rage, and continue the idea propagation. This draws in bystanders, who instead of synthesis join one side or the other, and increase the pool of hosts.

Even when proponents of one idea do interact with the other, the argument-is-therapy metaphor (just like the ARGUMENT is WAR metaphor) prevents them from finding a synthesis. Instead, it is too easy to take a position of moral smugness, and see yourself as the doctor that will cure your interlocutor of their mind disease. Of course, they too view themselves as doctors, not patients. And nothing is achieved.

Argument is Gift Exchange

The medical metaphor’s paternalism of elevating myself above my interlocutor does not sit well with Catarina Dutilh Novaes. Instead, she suggests that ARGUMENT is GIFT-EXCHANGE. Viewing an idea as a gift to be given highlights that the idea is something valuable that I want to share. This is in sharp contrast to a germ that is to be avoided or cured. In the medical metaphor for argument, it doesn’t matter how we cure a bad thought-germ — revelation, propaganda, and brainwashing are just as good as a philosophical argument. But with a gift, there is an emphasis on honesty and the desires of the recipient. An argument is a better gift because it is sharing the reasons you have for an idea — things you find valuable — instead of propaganda that might proceed by lies and misdirection. Thus, Dutilh Novaes’ metaphor draws our attention to what sets apart (philosophical) argument from other ways of instilling beliefs or sharing ideas.

The endorsement of an argument by the informant as something valuable also provides extra information for the interlocutor. Just like receiving a gift from someone close to you changes your perception of the value of that gift. With this, Dutilh Novaes aims to draw attention to a current topic of interest in social epistemology: the reliability of the informant.

The argument-is-gift-exchange metaphor is more egalitarian in its conception, and probably more conducive to learning from arguments. Although, it is still easy to forget the ‘exchange’ aspect, and just view yourself as donating ‘gifts’ to the destitute that is your interlocutor. In fact, packaging beliefs, ideas, and the outcomes of arguments as discrete states instead of an ongoing process might distance this view even further than therapy from my understanding of argument. It suggests that the purpose of argument is to convey wholly-formed quanta of thought, not to generate them. Generation is left to the separate apparatus of reason.

Argument is Midwifery

But it is not clear to me that reasoning can be separated from argument. To understand this, let’s start with The Pragmatic Programmer, and the a classic tale of a software engineer and his rubber ducky. When the engineer had to debug his code, he would turn to his rubber ducky and explain each line of the code to it. In the process, he would stumble on his own mistake, something that he missed during his initial conception of the program.

Why was this effective? By explaining to his ducky, the programmer didn’t just share his premeditated reasons for why he wrote certain lines of code. In some cases, he actually had to generate those reasons as he spoke. Eventually discovering ones that clash with his intent for the program. And although in common usage, arguing is not the same as explaining, I would say that a philosophical argument — especially as we often present them now — shares enough in common with an explanation. An explanation of something the interlocutor does not already believe.

This view puts argument as a central player in reasoning. It suggests that we reason through argument or (at least) in anticipation of an argument. A view that would be at home in an ancient Athenian assembly.

In Plato’s dialogues, Socrates constantly engages in discussion, argument, and dialectic with his interlocutors. He uses two metaphors to describe this activity and his role. In the Apology, Socrates describes himself as a gadfly that through his questioning bites at the noble steed that is the citizenry of Athens to encourage them out of intellectual and moral sluggishness. This is a metaphor I’ve borrowed before for theorists.

But my concern in this post is Socrates’ second metaphor. Phaenarete — Socrates mother — was a midwife and, in the Theaetetus, Socrates views himself as continuing his mother’s trade. Except instead of helping women birth children, he helps his interlocutors develop ideas. Socrates views himself as barren: he is not gifting an idea that he developed, nor curing his interlocutor of a mistaken belief. For he knows only one thing: that he knows nothing.

Prior to a discussion, Socrates does not know the idea that will be born, nor if the idea currently held by his interlocutor is wrong (and thus in need of curing). Socrates views himself as a servant to his interlocutor, just as a midwife to a mother. He does not push his own ideas on them, just like a midwife does not give her own child to the expectant mother. Instead, through argument, Socrates helps his interlocutors develop and explore their own; just like a midwife helps a mother deliver her own baby. ARGUMENT is MIDWIFERY.

I prefer this metaphor over both the medical and the gift-exchange metaphors. I think it is best at humbling the informant, without making them irrelevant. It places the final idea — or baby — as the focal point, with the interlocutor — or mother — as the hardest working agent. This metaphor associated good argumentation with a certain skill — midwifery — that can be developed over time. And it presents an argument as a fluid process that responds in different ways to different ideas and interlocutors — not as an exchange of prepackaged thoughts.

As with all metaphors, the argument-is-midwifery metaphor is a tool that only lets us comprehend partially the concept of philosophical argument. In the contemporary setting, most arguments are presented as single written monologues that aim to convince us of a conclusion. We can view this as a retelling of an argument that the writer had with herself, but this seems to be a stretch. Although for me, the question becomes: is monologue the right way to present philosophical arguments?


  1. Although Lakoff & Johnson were thinking about any sort of argument, whether it be convincing your roommate to take out the trash, or that the two-aspects is superior to the two-objects as interpretations of Kant’s transcendental idealism, for the rest of this article I will only concentrate on the latter kind of cognitive arguments — mostly because TheEGG seldom tries to convince you to do more house chores.
  2. When comparing to dance, Lakoff & Johnson obviously expect us to ignore the bboy battles that were emerging in New York while the authors were releasing their book at the start of the 80s. But the particular alternative doesn’t matter as much as the question itself.

    In his TEDx talk, Daniel H. Cohen asks the same sort of question, although calling the structuring of arguments ‘modes’ instead of ‘metaphors’. Along with an introduction, he stresses the difficulty of changing these entrenched metaphors but suggests some ways forward for moving toward more cooperative modes. I recommend watching the talk if you haven’t read Metaphors We Live By, or if you just like TED-style talks:

  3. Here, I think it is important to go beyond simple caricatures of adaptationism and consider carefully Julian Xue’s supply driver evolution. The individual mind’s treatment of ideas is the mechanism for generation variation, and it is far from an unbiasedly random process. This creates a certain structure to the mutations and that structure can drive the direction of evolution of ideas, even when natural selection is the motor. I suspect that this can be pushed further to strike a nice balance between the role of the individual and the role of the community in the concepts of knowledge (and truth).

Evolutionary dynamics of acid and VEGF production in tumours

Today was my presentation day at ECMTB/SMB 2016. I spoke in David Basanta’s mini-symposium on the games that cancer cells play and postered during the poster session. The mini-symposium started with a brief intro from David, and had 25 minute talks from Jacob Scott, myself, Alexander Anderson, and John Nagy. David, Jake, Sandy, and John are some of the top mathematical oncologists and really drew a crowd, so I felt privileged at the opportunity to address that crowd. It was also just fun to see lots of familiar faces in the same place.

A crowded room by the end of Sandy's presentation.

A crowded room by the end of Sandy’s presentation.

My talk was focused on two projects. The first part was the advertised “Evolutionary dynamics of acid and VEGF production in tumours” that I’ve been working on with Robert Vander Velde, Jake, and David. The second part — and my poster later in the day — was the additional “(+ measuring games in non-small cell lung cancer)” based on work with Jeffrey Peacock, Andriy Marusyk, and Jake. You can download my slides here (also the poster), but they are probably hard to make sense of without a presentation. I had intended to have a preprint out on this prior to today, but it will follow next week instead. Since there are already many blog posts about the double goods project on TheEGG, in this post I will organize them into a single annotated linkdex.

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Modeling influenza at ECMTB/SMB 2016

This week, I am at the University of Nottingham for the joint meeting of the Society of Mathematical Biology and the European Conference on Mathematical and Theoretical Biology — ECMTB/SMB 2016. It is a huge meeting, with over 800 delegates in attendance, 308 half-hour mini-symposium talks, 264 twenty-minute contributed talks, 190 posters, 7 prize talks, 7 plenary talks, and 1 public lecture. With seventeen to eighteen sessions running in parallel, it is impossible to see more than a tiny fraction of the content. And impossible for me to give you a comprehensive account of the event. However, I did want to share some moments from this week. If you are at ECMTB and want to share some of your highlights for TheEGG then let me know, and we can have you guest post.

I did not come to Nottingham alone. Above is a photo of all the current/recent Moffitteers that made their way to the meeting.

I did not come to Nottingham alone. Above is a photo of current/recent Moffitteers that made their way to the meeting this year.

On the train ride to Nottingham, I needed to hear some success stories of mathematical biology. One of the ones that Dan Nichol volunteered was the SIR-model for controlling the spread of infectious disease. This is a simple system of ODEs with three compartments corresponding to the infection status of individuals in the population: susceptible (S), infectious (I), recovered (R). It is given by the following equations

\begin{aligned}  \dot{S} & = - \beta I S \\  \dot{I} & = \beta I S - \gamma I \\  \dot{R} & = \gamma I,  \end{aligned}

where \beta and \gamma are usually taken to be constants dependent on the pathogen, and the total number of individuals N = S + I + R is an invariant of the dynamics.

As the replicator dynamics are to evolutionary game theory, the SIR-model is to epidemiology. And it was where Julia Gog opened the conference with her plenary on the challenges of modeling infectious disease. In this post, I will briefly touch on her extensions of the SIR-model and how she used it to look at the 2009 swine flu outbreak in the US.
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Computational kindness and the revelation principle

In EWD1300, Edsger W. Dijkstra wrote:

even if you have only 60 readers, it pays to spend an hour if by doing so you can save your average reader a minute.

He wrote this as the justification for the mathematical notations that he introduced and as an ode to the art of definition. But any writer should heed this aphorism.[1] Recently, I finished reading Algorithms to Live By by Brian Christian and Tom Griffiths.[2] In the conclusion of their book, they gave a unifying name to the sentiment that Dijkstra expresses above: computational kindness.

As computer scientists, we recognise that computation is costly. Processing time is a limited resource. Whenever we interact with others, we are sharing in a joint computational process, and we need to be mindful of when we are not carrying our part of the processing burden. Or worse yet, when we are needlessly increasing that burden and imposing it on our interlocutor. If you are computationally kind then you will be respectful of the cognitive problems that you force others to solve.

I think this is a great observation by Christian and Griffiths. In this post, I want to share with you some examples of how certain systems — at the level of the individual, small group, and society — are computationally kind. And how some are cruel. I will draw on examples from their book, and some of my own. They will include, language, bus stops, and the revelation principle in algorithmic game theory.
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Hamiltonian systems and closed orbits in replicator dynamics of cancer

Last month, I classified the possible dynamic regimes of our model of acidity and vasculature as linear goods in cancer. In one of those dynamic regimes, there is an internal fixed point and I claimed closed orbits around that point. However, I did not justify or illustrate this claim. In this post, I will sketch how to prove that those orbits are indeed closed, and show some examples. In the process, we’ll see how to transform our replicator dynamics into a Hamiltonian system and use standard tricks from classical mechanics to our advantage. As before, my tricks will draw heavily from Hauert et al. (2002) analysis of the optional public good game. Studying this classic paper closely is useful for us because of an analogy that Robert Vander Velde found between the linear version of our double goods model for the Warburg effect and the optional public good game.

The post will mostly be about the mathematics. However, at the end, I will consider an example of how these sort of cyclic dynamics can matter for treatment. In particular, I will consider what happens if we target aerobic glycolysis with a drug like lonidamine but stop the treatment too early.

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Multiple realizability of replicator dynamics

Abstraction is my favorite part of mathematics. I find a certain beauty in seeing structures without their implementations, or structures that are preserved across various implementations. And although it seems possible to reason through analogy without (explicit) abstraction, I would not enjoy being restricted in such a way. In biology and medicine, however, I often find that one can get caught up in the concrete and particular. This makes it harder to remember that certain macro-dynamical properties can be abstracted and made independent of particular micro-dynamical implementations. In this post, I want to focus on a particular pet-peeve of mine: accounts of the replicator equation.

I will start with a brief philosophical detour through multiple realizability, and discuss the popular analogy of temperature. Then I will move on to the phenomenological definition of the replicator equation, and a few realizations. A particular target will be the statement I’ve been hearing too often recently: replicator dynamics are only true for a very large but fixed-size well-mixed population.

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Systemic change, effective altruism and philanthropy

Keep your coins. I want change.The topics of effective altruism and social (in)justice have weighed heavy on my mind for several years. I’ve even touched on the latter occasionally on TheEGG, but usually in specific domains closer to my expertise, such as in my post on the ethics of big data. Recently, I started reading more thoroughly about effective altruism. I had known about the movement[1] for some time, but had conflicting feelings towards it. My mind is still in disarray on the topic, but I thought I would share an analytic linkdex of some texts that have caught my attention. This is motivated by a hope to get some guidance from you, dear reader. Below are three videos, two articles, two book reviews and one paper alongside my summaries and comments. The methods range from philosophy to comedy and from critical theory to social psychology. I reach no conclusions.

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EGT Reading Group 56 – 60

Since my last update in February, the evolutionary game theory reading group has passed another milestone with 5 more meetings over the last 4 months. We looked at a broad range of topics, from life histories in cancer to the effects of heterogeneity and biodiversity. From the definitions of fitness to analyzing digital pathology. Part of this variety came from suggested papers by the group members. The paper for EGT 57 was suggested by Jill Gallaher, EGT 58 by Robert Vander Velde, and the second paper for EGT 60 came from a tip by Jacob Scott. We haven’t yet recovered our goal of regular weekly meetings, but we’ve more than halved the time it took for these five meetings compared to the previous ones.

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Acidity and vascularization as linear goods in cancer

Last month, Robert Vander Velde discussed a striking similarity between the linear version of our model of two anti-correlated goods and the Hauert et al. (2002) optional public good game. Robert didn’t get a chance to go into the detailed math behind the scenes, so I wanted to do that today. The derivations here will be in the context of mathematical oncology, but will follow the earlier ecological work closely. There is only a small (and generally inconsequential) difference in the mathematics of the double anti-correlated goods and the optional public goods games. Keep your eye out for it, dear reader, and mention it in the comments if you catch it.[1]

In this post, I will remind you of the double goods game for acidity and vascularization, show you how to simplify the resulting fitness functions in the linear case — without using the approximations of the general case — and then classify the possible dynamics. From the classification of dynamics, I will speculate on how to treat the game to take us from one regime to another. In particular, we will see the importance of treating anemia, that buffer therapy can be effective, and not so much for bevacizumab.

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