Double public goods games and acid-mediated tumor invasion

Although I’ve spent more time thinking about pairwise games, I’ve recently expanded my horizons to more serious considerations of public-goods games. They crop up frequently when we are modeling agents at the cellular level, since interacts are often indirect through production of some sort of common extra-cellular signal. Unlike the trivial to characterize two strategy pairwise games, two strategy public-goods have a more sophisticated range of possible dynamics. However, through a nice trick using the properties of Bernstein polynomials, Archetti (2013,2014) and Peña et al. (2014a) have greatly increased our understanding of the public good, and I will be borrowing heavily from the toolbag and extending it slightly in this post. I will discuss the obvious continuation of this work by considering more than two strategies and several public goods together. Unfortunately, the use of public goods games here — and of evolutionary game theory (EGT) more generally — is not without controversy. This extension is not meant to address the controversy of spatial structure (although for progress on this, see Peña et al., 2014b), but the rigorous qualitative analysis that I’ll use (mostly in a the next post on this project) will allow me to side-step much of the parameter-fitting issues.

Of course, having two public goods games is only interesting if we couple them to each other. In this case, we will have one public good from which everyone benefits, but the second good is anti-correlated in the sense that only those that don’t contribute to the first can benefit from the second. A more general analysis of all possible ways to correlate two public-goods game might be a fun future direction, but at this point it is not clear what other correlations would be useful for modeling; at least in mathematical oncology.

By the way, if you are curious what mathematical oncology research looks like, it is often just scribbles like this emailed back and forth:

equations

I’ll use the rest of this post to guide you through the ideas behind the above sketch, and thus introduce you to the joint project that I am working on with Robert Vander Velde, David Basanta, and Jacob Scott. Treat this as a page from my open research notebook.

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