Effective games from spatial structure

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.

In the above figure, there is clearly an effective game happening. As the proportion of strategy B increases, both strategy A and strategy B benefit. In fact, if we look at just the fitness functions then the story of a public good or a benefit-cost prisoner’s dilemma game would be consistent with this graph. We can think of strategy B as paying a constant cost — the cost is the distance between the parallel lines — to produce a public good that benefits both type A and B cells equally — the benefit is the slope of the parallel lines.

If you were to write down the corresponding game matrix, you would get . Where type D corresponds to the first row and column, and type C corresponds to the second row and column. From this, you’d see it isn’t a traditional benefit-cost Prisoner’s dilemma since you’d need benefit b = 2 and cost c = 3. While we usually think of traditional PDs as having b > c .

But this would still certainly feels like a textbook example of a non-cell-autonomous process; even if not the most standard one.

In fact, for the similar Alectinib + CAF condition shown at right, Kaznatcheev et al. (2018) make this conclusion. They say that the fitness functions in this condition (as well as several other ones that aren’t close to conforming to parallel lines) as non-cell-autonomous.

Unfrotuntaly, this is a conclusion about the effective games. And it doesn’t need to hold for the reductive game.

As I’ve discussed before (see Kaznatcheev, 2017; 2018), it can be difficult to infer reductive games from the measurement of effective games. Many different reductive games in different population structures can map to the same effective game. But if we have a particularly nice population structure then we can invert the reductive to effective abstraction, and uniquely infer the reductive game.

So let’s do that.

Let’s suppose that we know that the effective game we measured is actually implemented by a reductive game played on a random 3-regular graph with birth-death updating. I’m picking this population structure not because it is a common structure in nature, but just for the purpose of this illustration. I’m picking this population structure because the Ohtsuki-Nowak transform provides us with a way to map from reductive to effective games.

As I’ve derived before, in the case of two strategy games, the ON transform maps reductive games of the form to effective games of the form where the game coordinates change according to the following affine transformation:

where k is the degree of the graph, and for Birth-Death updating or for Death-Birth updating.

In particular, if we take the reductive (i.e. token) fitness of type D cells to be 1 and the reductive fitness of type C cells to be 0. In other words, each type has constant token fitness independent of who they interact with — something that most people wouldn’t describe as a game-like dynamic. But if we insist writing it dowb this cell-autonomous fitness as a payoff matrix then we get the reductive payoff matrix: .

And at this point, it should be no surprise that on a 3-regular random graph with Birth-Death updating, this reductive ‘game’ becomes the effective game given by .

In words: two kinds of cells that each have cell-autonomous token fitness at the reductive level can have non-cell-autonomous type fitness at the effective level if their population in spatially structured. Spatial structure can create an effective game where there is no reductive game.

When the spatial structure is simple — as it is for the first-order approximation of the ON-transform — then the resulting effective game is also relatively simple. But I would also expect this general ‘effective game creation’ result to hold for more complicated and realistic population structures. Except instead of creating a simple linear effective game of two parallel fitness functions, it could create a much more complicated non-linear game that depends on the details of the spatial structure itself.

To make things even more difficult: if space can create an effective game from a constant fitness reductive ‘game’ then space can also hide a reductive game in a constant fitness effective ‘game’.

This can be most easily seen if we invert the affine transform from reductive to effective games into one from effective to reductive games:

Now, we can consider effective observations where the cell-autonomous type-fitness of type C is 0 and of type D is 1. This would correspond to the effective ‘game’ of or X = 1, Y = 0. When you plug this in above, you find that for a 3-regular graph with Birth-Death updating, the resulting reductive game is non-cell-autonomous: .

If we were judging just from the effective fitness function, we might conclude that the system has a cell-autonomous system. But at the reductive level, it would actually have frequency dependent fitness that is masked by the spatial structure.

Thus, in space, we have another complication to consider if we’re trying to make reductive statements from effective measurements. More importantly, this complication is important even if we’re only interested in broad qualitative conclusions like distinguishing cell-autonomous vs. non-cell-autonomous processes.

References

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: 115(8): E1709.

Kaznatcheev, A., Peacock, J., Basanta, D., Marusyk, A., & Scott, J. G. (2018). Fibroblasts and Alectinib switch the evolutionary games played by non-small cell lung cancer. bioRxiv, 179259.

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