The prisoner’s dilemma in R

The prisoner’s dilemma in R: why defection dominates a single encounter and mutual cooperation collapses, the starting point for the evolution of cooperation.

Cooperation is a puzzle because cheating pays. Two animals that both help each other do better than two that both cheat, yet whatever the other one does, an individual gains by cheating. That tension has a canonical model: the prisoner’s dilemma. It is the foundation of everything in the evolution of cooperation, and this post builds it in a few lines of base R to show exactly why cooperation is hard.

The payoffs

Two players each choose to cooperate or defect. The four outcomes have standard names: mutual cooperation pays the reward R, mutual defection the punishment P, and a defector against a cooperator gets the temptation T while the cooperator gets the sucker’s payoff S. A prisoner’s dilemma is any game with T > R > P > S.

Tt <- 5; R <- 3; P <- 1; S <- 0     # temptation, reward, punishment, sucker
# focal player's payoff given (focal move, opponent move), C = cooperate, D = defect
payoff <- function(me, opp) {
  if (me == "C" && opp == "C") R else if (me == "C" && opp == "D") S else
  if (me == "D" && opp == "C") Tt else P
}
cat(sprintf("T=%g > R=%g > P=%g > S=%g\n", Tt, R, P, S))

T=5 > R=3 > P=1 > S=0

Defection dominates

Ask the only question that matters to a selfish player: whatever my opponent does, am I better off cooperating or defecting?

cat(sprintf("opponent cooperates:  D pays %g vs C pays %g  ->  defect\n", payoff("D", "C"), payoff("C", "C")))

opponent cooperates:  D pays 5 vs C pays 3  ->  defect

cat(sprintf("opponent defects:     D pays %g vs C pays %g  ->  defect\n", payoff("D", "D"), payoff("C", "D")))

opponent defects:     D pays 1 vs C pays 0  ->  defect

Against a cooperator, defecting pays 5 instead of 3. Against a defector, defecting pays 1 instead of 0. Defection wins in both columns, so it is a dominant strategy: a rational player defects no matter what, and a population of defectors cannot be invaded by a lone cooperator. Defection is the only evolutionarily stable strategy of the one-shot game, the same logic that makes all-defect an equilibrium in any single social encounter.

The dilemma

Here is the catch that gives the game its name.

cat(sprintf("both defect (the ESS): %g each\n", payoff("D", "D")))

both defect (the ESS): 1 each

cat(sprintf("both cooperate:        %g each\n", payoff("C", "C")))

both cooperate:        3 each

Two defectors earn 1 each; two cooperators would earn 3 each. Individual rationality drives everyone to the outcome that is worse for everyone. Nothing is wrong with the reasoning; the trap is real. This is why cooperation in nature demands an explanation, and why so much of social evolution is the search for what changes the payoffs or the structure enough to escape it.

library(ggplot2)
d <- expand.grid(me = c("C", "D"), opp = c("C", "D"))
d$pay <- mapply(payoff, d$me, d$opp)
ggplot(d, aes(opp, me, fill = pay)) +
  geom_tile(colour = "white", linewidth = 2) +
  geom_text(aes(label = pay), colour = "white", size = 7) +
  scale_fill_gradient(low = "#93a87f", high = "#275139", guide = "none") +
  labs(x = "Opponent move", y = "Focal move") +
  theme_minimal(base_size = 13)

Figure 1: The focal player’s payoff for each move against each opponent move. Defect (bottom row) beats cooperate (top row) in both columns, so defection dominates, even though mutual cooperation (3) beats mutual defection (1).

An honest limit

The one-shot dilemma is a starting point, not a description of most animal life. Its stark conclusion, always defect, rests on a specific and often false assumption: that the two players will never meet again and that nothing links this encounter to the next. Real interactions repeat, happen among relatives, are watched by others, and play out in space. Each of those changes the game, and each is a route out of the trap. The next posts follow the most studied one, repeated interaction, where the shadow of future encounters can make cooperation pay after all. The dilemma is worth understanding precisely because it sets the bar every explanation of cooperation has to clear.

Related tutorials

• Direct reciprocity and tit-for-tat
• Network reciprocity on a lattice
• Checking a reciprocity model
• The hawk-dove game and the ESS

References

Axelrod R, Hamilton WD 1981. Science 211(4489):1390-1396 (10.1126/science.7466396)

Trivers RL 1971. Quarterly Review of Biology 46(1):35-57 (10.1086/406755)