## Effective games from spatial structure

December 7, 2018 5 Comments

For the last week, I’ve been at the Institute Mittag-Leffler of the Royal Swedish Academy of Sciences for their program on mathematical biology. The institute is a series of apartments and a grand mathematical library located in the suburbs of Stockholm. And the program is a mostly unstructured atmosphere — with only about 4 hours of seminars over the whole week — aimed to bring like-minded researchers together. It has been a great opportunity to reconnect with old colleagues and meet some new ones.

During my time here, I’ve been thinking a lot about effective games and the effects of spatial structure. Discussions with Philip Gerlee were particularly helpful to reinvigorate my interest in this. As part of my reflection, I revisited the Ohtsuki-Nowak (2006) transform and wanted to use this post to share a cute observation about how space can create an effective game where there is no reductive game.

Suppose you were using our recent game assay to measure an effective game, and you got the above left graph for the fitness functions of your two types. On the x-axis, you have seeding proportion of type C and on the y-axis you have fitness. In cyan you have the measured fitness function for type C and in magenta, you have the fitness function for type D. The particular fitnesses scale of the y-axis is not super important, not even the x-intercept — I’ve chosen them purely for convenience. The only important aspect is that the cyan and magenta lines are parallel, with a positive slope, and the magenta above the cyan.

This is not a crazy result to get, compare it to the fitness functions for the Alectinib + CAF condition measured in Kaznatcheev et al. (2018) which is shown at right. There, cyan is parental and magenta is resistant. The two lines of best fit aren’t parallel, but they aren’t that far off.

How would you interpret this sort of graph? Is there a game-like interaction happening there?

Of course, this is a trick question that I give away by the title and set-up. The answer will depend on if you’re asking about effective or reductive games, and what you know about the population structure. And this is the cute observation that I want to highlight.

## Abstract is not the opposite of empirical: case of the game assay

June 2, 2018 by Artem Kaznatcheev 19 Comments

Last week, Jacob Scott was at a meeting to celebrate the establishment of the Center for Evolutionary Therapy at Moffitt, and he presented our work on measuring the effective games that non-small cell lung cancer plays (see this preprint for the latest draft). From the audience, David Basanta summarized it in a tweet as “trying to make our game theory models less abstract”. But I actually saw our work as doing the opposite (and so quickly disagreed).

However, I could understand the way David was using ‘abstract’. I think I’ve often used it in this colloquial sense as well. And in that sense it is often the opposite of empirical, which is seen as colloquially ‘concrete’. Given my arrogance, I — of course — assume that my current conception of ‘abstract’ is the correct one, and the colloquial sense is wrong. To test myself: in this post, I will attempt to define both what ‘abstract’ means and how it is used colloquially. As a case study, I will use the game assay that David and I disagreed about.

This is a particularly useful exercise for me because it lets me make better sense of how two very different-seeming aspects of my work — the theoretical versus the empirical — are both abstractions. It also lets me think about when simple models are abstract and when they’re ‘just’ toys.

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Filed under Commentary, Meta, Models, Preliminary Tagged with algorithmic philosophy, conference, philosophy of science, replicator dynamics, spatial structure