# 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.

### Background

Let me remind you of the setting and characters. We consider two anti-correlated public goods, for simplicity let’s call them oxygen and acid. There are three cell types — if you remember from before, the fourth cell type from a factorial over strategies is strictly dominated by the other 3, so we ignore it — DEF, GLY, and VOP. DEF defects in both public goods, so it does not produce oxygen (or, produces less VEGF and thus not calling for oxygen) but uses oxygen, so doesn’t produce acid. GLY does not produce oxygen, but doesn’t use it either thus producing acid. VOP overproduces VEGF, thus calling for more oxygen, and uses oxygen, so doesn’t produce acid. The strategies make up the fractions $x_\text{DEF}, x_\text{GLY}, x_\text{VOP}$ and together sum to one.

GLY looks like the odd one out and we use my factoring trick to take advantage of this. With $p = x_\text{GLY}$ as acid produces, and $1 - p = x_\text{VOP} + x_\text{DEF}$ as the oxygen users. Let q be the fraction of oxygen users that are more active in producing VEGF, so $x_\text{VOP} = (1 - p)q$ and $x_\text{DEF} = (1 - p)(1 - q)$.

In particular, the f(q) that I considered in my sanity-check is linear in q which is the proportion of oxygen producers among the oxygen users, but is not linear in the proportion of oxygen producers among the whole population — i.e. not linear in $x_\text{VOP}$. Let’s consider, instead, a general linear function in $x_\text{VOP}$ given by $f(q) := b_f x_\text{VOP} + f_0$. Of course, we want q to appear in this function, so let’s remember that from our factorization $x_\text{VOP} = (1 - p)q$ and so the equation becomes $f(q) = b_f(1 - p)q + f_0$ and this function is linear both in q and $x_\text{VOP}$. We will keep m(p) the same, since in our factorization $p = x_\text{GLY}$ and so we already have m as linear in both p and $x_\text{GLY}$. The simplest interpretation of the resulting replicator dynamics, especially without the factoring, is to imagine DEF, GLY, and VOP as three different cell types competing under evolutionary forces. But this isn’t the only interpretation. We could also imagine that VOP-DEF are the same cell type, that is optimizing their VEGF production through a learning process like imitation. The resulting intermediate level of VOP-DEF cells are then competing against the GLY cells under more standard evolutionary forces.

If we decide to go from a heuristic to abduction — given his experimental leanings, this is something that Robert is interested — then this factored representation allows us to more easily operationalize the model. The mean acidity of the tumor — or petri dish — can be used as a proxy for p. The concentrations of VEGF can inform q, or maybe $x_\text{VOP}$.

### Linear payoffs

Now it’s time to jump into the math. Define $G(q) := \frac{1}{(1 - p)n}\frac{\partial f}{\partial q} - c$. Note that all the (n + 1) from previous entries have been replaced by just n here. There is a good reason for this, but it is not interesting (or important) for the rest of the derivation. You might even notice the inconsistency in the blackboard pictures above, the left one was early in the week when we didn’t fix n + 1 by n, yet.

Enough with the tangent, recall the equations that govern our dynamics:

\begin{aligned} \dot{p} & = p(1 - p)\Bigg(\frac{1}{n}\frac{\partial m}{\partial p} - (f(q) + qG(q))\Bigg) \\ \dot{q} & = q(1 - q)G(q) \end{aligned}

The nice feature of the new f(q) is that it let’s us cancel the (1 – p) in the denominator of the first summand of G. In particular, we get $G(q) = \frac{b_f}{n} - c$ and thus very simple dynamics for q. If $b_f > cn$ then $q \rightarrow 1$ and if $b_f < cn$ then $q \rightarrow 0$.

The dynamics for p are a little bit more interesting. Suppose that q* = 0: if $b_m > f_0 n$ then $p \rightarrow 1$ and if $b_m < f_0 n$ then $p \rightarrow 0$. But if q* = 1 then:

$\dot{p} = p(1 - p)\Bigg(\frac{b_m}{n} - b_f(1 - p) - f_0 - \frac{b_f}{n} + c\Bigg)$

Now, let p* be the zero of the above gain function. In other words, define:

$p^* = 1 - \frac{c - f_0}{b_f} - \frac{b_m - b_f}{b_fn}$

If p > p* then $p \rightarrow 1$ and if p < p* then $p \rightarrow 0$. In other words, we have an unstable fixed point in p when q = 1.

### Nonlinear payoffs

The key trick to notice, in the above linear analysis, is how much easier life became when we canceled the 1 – p in the denominator of G. To bring this over to the nonlinear case, let us consider oxygen benefit functions of the form $f(q) = (1 - p)F(q) - K(p)$. Here the fitness function is partitioned so that F is the benefit of oxygen to the oxygen users and is completely independent of the proportion p of glycolytic cells; and K is the cost of acidity to the oxygen users and is completely independent of the fraction q of oxygen producers among oxygen users. In that case, our dynamics for q are independent of our dynamics for p and the fixed points are given by q = 0,1 and the q* such that $\frac{\partial F}{\partial q}|_{q^*} = cn$ (equivalently: G(q*) = 0). In graphical terms, we are just plotting $\frac{\partial F}{\partial q}$ as a function of q and seeing where it intersects the horizontal line at height cn. This is just like previous work by Archetti (2013, 2014).

Since the q* equilibria are independent of p that means they hold for any value of p. They divide the phase space into independent sections such that if you start in one section, the dynamics will not take you to another section. In particular, this means that there are no cycles or spirals for these dynamics; although loops from almost all of one strategy back to all of that one strategy with lower levels of that strategy in between are possible.

The interesting case is when we continue onto looking at the fixed points in p. Here, if we look at an equilibrium q* < 1 then we are looking for p* such that $\frac{\partial m}{\partial p}|_{p^*,q^*} = nF(q^*)(1 - p^*) - nK(p^*)$.

If K(p) is a constant k then, graphically, we are plotting $\frac{\partial m}{\partial p}|_{q^*}$ as a function of p and seeing where it intersects a line that starts at height n(F(q*) – k) and decreases with slope nF(q*) until it reaches kn. This is different from Archetti’s previous work (2013,2014); although we recover his work when F(q*) = 0. Consider an oversaturating m; i.e. for a function that increases to a maximum and then decreases. A horizontal line will intersect the derivative of this function at most 2 times, and those are the equilibria that Archetti’s analysis would find. A sloped line, however, can intersect it 3 times, giving us an extra equilibrium point. I don’t know if this extra equilibrium has any biological or clinical significance, but it does let us know that we have a richer set of behaviors. The significance is something that we will figure out in the coming weeks.

Archetti, M. (2013). Evolutionary game theory of growth factor production: implications for tumour heterogeneity and resistance to therapies. British Journal of Cancer, 109 (4), 1056-62 PMID: 23922110

Archetti, M. (2014). Evolutionary dynamics of the Warburg effect: glycolysis as a collective action problem among cancer cells. Journal of Theoretical Biology, 341: 1-8