## From linear to nonlinear payoffs in the double public goods game

If you recall, dear reader, around this time last year, Robert Vander Velde, David Basanta, Jacob Scott and I got excited about the Archetti (2013,2014) approach to modeling non-linear public goods in cancer. We’ve been working on this intermittently for the last year, but aim to focus now that I have settled in here at Moffitt. This means there will be a lot more cancer posts as I resume thinking careful about mathematical oncology. Although I didn’t update the blog in the summer, it doesn’t mean that nothing was written. The work below is mostly from when I visited Tampa in late July. As are these two blackboards:

In this project, we are combining growth factor production (Archetti, 2013) and acidity (2014) as a pair of anti-correlated public goods. The resulting dynamics cannot be understood by studying just one or the other good. The goal is to explore the richer behaviors that are possible with coupled social dilemmas. At the start of the year — in my first analysis of the double public goods game — as a sanity check I considered the linear public goods $f(q) = b_f q$ and $m(p) = b_m p$. After a long meeting with Robert a few month ago, I think that these were misleading payoffs to consider. I jotted these notes after the meeting, but forgot to release them on the blog. Instead, you get to enjoy them now while I refresh my memory.