Pairwise games as a special case of public goods
February 21, 2015 11 Comments
Usually, when we are looking at public goods games, we consider an agent interacting with a group of n other agents. In our minds, we often imagine n to be large, or sometimes even take the limit as n goes to infinity. However, this isn’t the only limit that we should consider when we are grooming our intuition. It is also useful to scale to pairwise games by setting n = 1. In the case of a non-linear public good game with constant cost, this results in a game given by two parameters — the difference in the benefit of the public good from having 1 instead of 0 and 2 instead of 1 contributor in the group, respectively; measured in multiples of the cost c. In that case, if we want to recreate any two-strategy pairwise cooperate-defect game with the canonical payoff matrix then just set and . Alternatively, if you want a free public good (c = 0) then use and . I’ll leave verifying the arithmetic as an exercise for you, dear reader.
In this post, I want to use this sort of n = 1 limit to build a little bit more intuition for the double public good games that I built recently with Robert Vander Velde, David Basanta, and Jacob Scott to think about acid-mediated tumor invasion. In the process, we will get to play with some simplexes to classify the nine qualitatively distinct dynamics of this limit and write another page in my open science notebook.
Let us recall our utility functions from the earlier post, remembering that U(- +) can be eliminated because it is strictly dominated by U(- -):
Where – – are anaerobic (thus acid-secreting) cells that don’t produce VEGF, and the other two are aerobic with + + producing VEGF and + – only using oxygen but not producing VEGF.
Looking at the n = 1 case, our utility functions (or the cells’ fitnesses, if you prefer) simplify to:
Which can be rewritten in payoff matrix form (renormalized by subtracting from all lines) as:
To recreate the qualitative features of our original description of acid, we want . For oxygen as a public good, we want it to be positive and monotonically increasing () and saturating ( > 0).
From looking at the equations, we can notice several things right away:
- U(- -) is strictly dominated by U(+ +) if , otherwise
- U(+ +) is strictly dominated by U(- -) if .
- U(- -) is strictly dominated by U(+ -) if , otherwise
- U(+ -) is strictly dominated by U(- -) if .
- U(+ +) is strictly dominated by U(+ -) if , otherwise
- U(+ -) is strictly dominated by U(+ +) if .
Since we are more interested in cases where no strategy is strictly dominated, we can concentrate on the cases where and and . The first of these conditions results in the simplex edge between ++ and +- having Hawk-Dove dynamics and thus an attracting equilibrium point on that edge with an equilibrium proportion of ++ given by . Combined with all our other inequalities, q* has a useful property: . Finally, if then – – can invade a population made up completely of + + and + – and so the equilibrium is a saddle-point, otherwise it is stable.
Let’s look at the games on the other two edges:
From this we can conclude that:
- If then – – is evolutionary stable against + +.
- If then + + is evolutionary stable against – –.
- If then – – is evolutionary stable against + –.
- If then + – is evolutionary stable against – –, but this contradicts a previous condition that .
This gives us the 6 distinct edge-profiles of the simplex. In fact, 2 of them are equivalent: by relabeling (- -, + +, + -) as (+ -, – -, + +) we can move from the and profile to the and profile. By consulting Bomze (1983,1995) and playing with the inequalities that we’ve derived so far, we can see that there are 9 distinct regimes possible depending on the values of and . The simplexes for the qualitative nature of these dynamics are gives below:
From the above, we can see that:
- if and then no matter where the cancer starts, its dynamics will converge to a polyclonal tumor expressing all three of – –, + +, and + –. During the convergence, we will see cyclic behavior.
- If then it will be possible for the tumour to converge to a homogeneous state made of only – – cells — not producing VEGF or using oxygen — but
- this happens for any initial conditions only if ;
- otherwise if then there are two basins of attraction with the top basin leading to all – – but the bottom one leading to a fully aerobic but still polyclonal tumor with q* of the cells producing VEGF and the rest only benefiting from the delivered oxygen.
- If and then we will have probably the strangest case, with two basins of attraction. One bottom basin will result in a completely aerobic but heterogeneous in VEGF tumour as in the previous point. The top basin, however, will result in a heterogeneous tumour with a mix of anerobic cells and aerobic VEGF producers, but no aerobic VEGF non-producers.
- In all other cases, all initial configurations will lead to an aerobic but heterogeneous in VEGF tumour.
I am not sure which of these possible equilibria are clinically desirable, but I would guess that the completely polyclonal tumour of the first point is the worst possible. On the other hand, a completely aerobic tumour — even if heterogeneous in VEGF production — seems manageable if there is some external way to cut off its oxygen supply. Although this sort of treatment would require a more involved analysis since cutting off the oxygen supply would change and thus the game dynamics. Finally, a completely homogeneous tumour of – – although undergoing the Warburg shift would be tempting to treat with a targeted therapy due to having a single opponent instead of two or three. Of course, I am hoping that Robert, David, Jake, or you, dear reader, will pitch in with more biological considerations.
Bomze, I.M. (1983). Lotka-Volterra equation and replicator dynamics: A two-dimensional classification. Biological Cybernetics, 48 (3), 201-211 DOI: 10.1007/BF00318088
Bomze, I.M. (1995). Lotka-Volterra equation and replicator dynamics: new issues in classification. Biological Cybernetics, 72(5): 447-453.