Deadlock & Leader as deformations of Prisoner’s dilemma & Hawk-Dove games

January 30, 2018 by Artem Kaznatcheev

Recently, I’ve been working on revisions for our paper on measuring the games that cancer plays. One of the concerns raised by the editor is that we don’t spend enough time introducing game theory and in particular the Deadlock and Leader games that we observed. This is in large part due to the fact that these are not the most exciting games and not much theoretic efforts have been spent on them in the past. In fact, none that I know of in mathematical oncology.

With that said, I think it is possible to relate the Deadlock and Leader games to more famous games like Prisoner’s dilemma and the Hawk-Dove games; both that I’ve discussed at length on TheEGG. Given that I am currently at the Lorentz Center in Leiden for a workshop on Understanding Cancer Through Evolutionary Game Theory (follow along on twitter via #cancerEGT), I thought it’d be a good time to give this description here. Maybe it’ll inspire some mathematical oncologists to play with these games.



As context, I want explicitly look at the games we measured in Kaznatcheev et al. (2017). The below graph is based on figure 4b from that paper:

Let’s take a moment to decipher this figure. I won’t go into the details of the experimental system, or how measurements are done (for background, see the paper and two early posts on using time-lapse microscopy to count cancer cells and measure population dynamics). What matters is that there are two types of cells, which we’ll call parental and resistant, that play a game in the petri dish. We looked at their interaction in four different conditions: no drug (DMSO), no drug + fibroblasts (DMSO + CAF), drug (Alectinib), and drug + fibroblasts (Alectinib + CAFs); each corresponds to a different game. The resulting matrix game can be plotted in a two dimensional game space spanned by an x-axis of the relative fitness of a resistant focal in a parental monotypic culture (value of gain function for resistant at parental proportion p = 1) and y-axis of relative fitness of a parental focal in a resistant monotypic culture (value of gain function for parental at parental proportion p = 0). The four games we measured are also given as matrices (with entries multiplied by a factor of 100) by their label.

The top left game here is Leader, and the other three are Deadlock. And I’ll use the specific matrix entries of the DMSO + CAF game (Leader) versus the DMSO game (Deadlock) as an illustration for defining the two games.

The Deadlock game is a social dilemma that is in some ways the opposite of the popular Prisoner’s Dilemma (PD) game (in fact, Robinson & Goforth (2005) — a book I wasn’t familiar with, and I’m thankfull to a review for introducing me — call it the anti-PD). If we interpret parental as cooperate and resistant as defect then similar to PD, each player wants to defect regardless of what the other player does (because 4.0 > 2.5 & 2.7 > 2.4) but hopes that the other player will cooperate (because 4.0 > 2.7). However, unlike PD, mutual cooperation does not Pareto dominate mutual defection (because 2.5 < 2.7) but is instead strictly dominated by it. Thus, the players are locked into defection — hence the name. And the boring nature.

But maybe the game should be called PD with an overly-eager prosecutor. In the typical story of the PD, you imagine two prisoners being interrogated separately. If one of the prisoners defects by ratting out the other then the snitch will walk away free, while the other one will serve a very long sentence. But if they both snitch on each other then both can be convicted of the crime and both will serve a medium sentence. In the typical PD, however, if both prisoners cooperate by staying silence then the prosecutor doesn’t have enough evidence and has to convict them of a less charge, resulting in short sentences for both prisoners. Deadlock would correspond to a variant with an overly-eager and crooked prosecutor. Without evidence to convict the prisoners of the crime (if both cooperate and stay silent), the prosecutor instead makes up outrages new charges and convicts the prisoners of this, resulting in a long sentence for each.

The Leader game is one of Rapoport’s (1967) four archetype 2 x 2 games and a social dilemma related to the Hawk-Dove (HD) game. HD has many other names, like Chicken or Snowdrift and Robinson & Goforth (2005) call Leader as Benevolent Chicken.

As you might expect from the many names, there are many stories that people tell about HD. I want to adapt an analogy for HD that is used by Joshua Greene in Moral Tribes. Consider two people in a canoe. Each person can either steer (lead) or paddle-hard (work). The goal is to get down the river, which provides some reward. If both people paddle-hard then they’ll reach the reward but both will be tired from paddling, resulting in an overall medium payoff for each. If a person steers then that person gets less tired and the one paddling gets more tired; so the steerer gets a very big payoff, but the single paddler gets a small payoff. However, if both steer and thus no one paddles then the two people never reach their goal and so get no payoff. This is the classic HD game, and this story is closest to the Snowdrift story with paddle-hard = shovel and steer = wait.

Leader can be interpreted as this canoe version of HD on a turbulant river. If both paddle then they still get a medium payoff from being tired and banged up by the current. However, if a person steers then not only do they reach the goal but they also avoid getting banged up since the steerer can navigate the cannoe around obstacle: thus the steerer (leader) get a very good reward and the paddle-hard (worker) gets a good reward. As before, if both lead then nobody works and then they get nowhere and thus no reward.

Let me repeat in the context of the Leader game measured in DMSO. If we interpret parental as ‘lead’ and resistant as ‘work’ then similar to HD, mutual work is better than both leading (because 3.0 > 2.6) and thus no work being done but each player would want to lead while the other works (because 3.5 > 3.0). However, unlike HD, mutual work is not better than the “sucker’s payoff” of working while the other player leads (because 3.1 > 3.0). Rapoport (1967) sees this as a tension with a player switching from a “natural” point of mutual work to lead and thus benefit both players (3.5 > 3.0, 3.1 > 3.0) but if the second player also does the same and becomes a leader then all benefit disappears (because 2.6 is the smallest payoff — i.e. no reward in the story).

Thus, we might not have measured exciting games like PD or HD, but we measured to deformations of this game: Deadlock and Leader. More importantly, Kaznatcheev et al. (2017) shows that adding drug or eliminating fibroblasts can qualitatively change the game from Leader (DMSO + CAF) to Deadlock (Alectinib + CAF or DMSO). Thus, we can think about treating the game. It’d be interesting to see if we could explain this shift between deformed PD and HD in terms similar to how Chow & Elena (2017) shift from PD to HD.

Note that these intuitive stories are meant as heuristics; what Rubinstein (2012) calls a fable. In the context of our empirical abduction to the simplest effective game, we should not take these fables too seriously since an effective game is a summary of population-level properties (Kaznatcheev, 2017). This means that the matrix entries should be interpreted as direct interactions between cells, but as general couplings between subpopulations corresponding to different strategies. The coupling term includes not only direct interactions but also indirect effects due to spatial structure, diffusible goods, contract inhibition, etc.

References

Chao, L., & Elena, S. F. (2017). Nonlinear trade-offs allow the cooperation game to evolve from Prisoner’s Dilemma to Snowdrift. Proceedings of the Royal Society: B: 284(1854): . 20170228.

Greene, J. (2014). Moral tribes: Emotion, reason, and the gap between us and them. Penguin.

Kaznatcheev, A., Peacock, J., Basanta, D., Marusyk, A., & Scott, J. G. (2017). Fibroblasts and alectinib switch the evolutionary games that non-small cell lung cancer plays. bioRxiv, 179259.

Kaznatcheev, A. (2017). Two conceptions of evolutionary games: reductive vs effective. bioRxiv, 231993.

Rapoport, A. (1967). Exploiter, leader, hero, and martyr: The four archetypes of the 2 × 2 game. Systems Research and Behavioral Science, 12(2): 81-84.

Robinson, D., & Goforth, D. (2005). The topology of the 2×2 games: a new periodic table. Routledge: London & New York.