The Prisoner’s dilemma is a ‘game’ where we imagine 2 people are in cahoots for a crime, the police drag each of them off to interrogate. If you rat on the other person (defect), you get a reduced sentence if the other guy keeps quiet (the sucker), the other guy then gets a heavier sentence (the sucker payoff). If both of you remain silent (cooperate), you get a moderate sentence. If both of you rat on the other you get the worst punishment each. So the payoff looks something like 1–1 if both tell, 5–1 if one tells, 3–3 if both remains silent. We can display this as a payoff matrix:
Our cells are constantly dividing, and with each division tiny errors get introduced into the DNA much like a game of broken telephone. Luckily, the mechanism for copying is remarkably robust and the error rates are very small. Once, in awhile though the DNA get mutated at a site which regulate the growth or death or reproduction of the cell. The result is a cell that starts behaving as a free agent. Accumulate enough oncogenic (cancer causing) mutations and you get a cell that grows and reproduce without regulation. In other words cancer.
If we consider cancer cells as defectors and normal cells as cooperators, intuitively we might think that the cooperators are doomed against an invasion of defectors. While this is often the case, the picture becomes more nuanced if we consider their structure.
If we imagine cells arranged in a line (such as in colonic crypts where stem cells replace existing cells in a linear fashion):
We see that no matter what type it is the first cell in the chain would be the only type in the whole chain after sufficient time. This structure effectively suppresses propagation of mutants since the mutation would have to happen in the first cell for it to propagate.
If we consider a directed cycle, with a payoff matrix:
whereby the cell A can be replaced by cell B if the b>d. However,there are a few ways that the payoff can be decided, 2 basic ways are: a) cell n interacts with cell n+1 to decide the payoff, b) cell n interacts with cell n-1
In case a, we can intuit that red invaders will successfully take over if b>c. While in case b, the condition for red to invade is that a>d. This is surprising in that it implies that the unconditional cooperator mentioned in the prisoner ‘s dilemma can actually take over a system resident by defectors.
With the introduction of structure, the winning party is no longer determined by only the payoff matrix but also the graph and the orientation. There are also certain unique structures such as the superstar that strongly amplifies such that even a dominated strategy (a<c and b<d) can conquer the whole system if b>c (think of it as the skinny kid with mild social skills doing the impossible by virtue of being the protagonist).
While certain structures are definitely more biologically relevant than others . I still found it interesting to see how graphs and games can inform us about how cancer might spread and how structures of other types of networks govern their dynamics. Personally, this gave me more questions than answers, but isn’t that the case for all the best games?
Reference:
Lieberman, E., Hauert, C. & Nowak, M. Evolutionary dynamics on graphs. Nature 433, 312–316 (2005). https://doi.org/10.1038/nature03204
