Stag hunt, deadlock, & the sickle cell anemia–malaria game

In last week's post, inspired by William Poundstone's book Prisoner's Dilemma: John von Neumann, Game Theory, and the Puzzle of the Bomb, I owned up to my years-long mistake of calling social dilemmas in general prisoner's dilemmas, and I described a distinctly different social dilemma called chicken. Today I'll describe two more dilemmas from the book.

Stag hunt

The stag hunt is like prisoner's dilemma but with mutual cooperation given the greater good. Here's the payoff table. (Again, like in last week's post, lower numbers are better.)

	Cooperate	Defect
Cooperate	1, 1	4, 2
Defect	2, 4	3, 3

Because in the stag hunt everyone is best off cooperating, there should in theory be no dilemma: the rational choice is to always cooperate. But that only happens on the make-believe planet inhabited in the minds of renown economists, the place where everyone is rational. In the real world the situation is more interesting because cooperating with an irrational player who defects causes you to end up with the worst possible result: a low score of 4. Thus, there's a preventative incentive to defect—just in case your opponent is thinking the same thing.

This makes stag hunt more of a tragedy than prisoner's dilemma and chicken. Whereas in those two games the players are victims of circumstance, the problems born of a stag hunt are self-made owing to a lack of trust.

A good real-world fit for a stag hunt meltdown is nearly any kind of financial bubble, whereby reason is subordinate to greed and fear. I heard more than one person in Phoenix saying after the housing bubble popped that they felt they had to buy a house during the run-up in prices because they feared otherwise becoming forever priced out of the market. This thinking follows the defect before they do destruction of a stag hunt.

Deadlock

The weakest of the four social dilemmas is deadlock.

	Cooperate	Defect
Cooperate	3, 3	4, 1
Defect	1, 4	2, 2

When I read Poundstone's book and first saw the payoff table for deadlock, I tried without success to imagine what this scenario describes. I should have taken a hint from the book's title: Deadlock describes nearly any attempt by two countries to agree to reduce their nuclear arsenal. Cooperation is equivalent to going along with the agreement, and defection is equivalent to breaking the agreement—presumably in secret. In such a scenario, the best outcome for any country is to secretly keep their arsenal while the other country dismantles theirs. Second best is mutual defection, in which case that country at least maintains their nuclear privilege over the have-not countries. The worst outcome is going along with the agreement when the other country defects, in which case there's still a threat of nuclear annihilation and now the country with a dismantled arsenal has no counter-threat.

As its name implies, deadlock leads rational players to always defect, just as in prisoner's dilemma.

Sickle cell anemia and resistance to malaria

So of the four games I described—prisoner's dilemma, chicken, stag hunt, and deadlock—which best describes the conflict inherent in the genetic mutation that leads to increased resistance to malaria but also to having sickle cell anemia? As you may remember, the conflict is symmetrical: any parent having (a single copy of) the mutation benefits from increased malarial resistance, but children of two parents both possessing the mutation may end up with sickle cell anemia.

Imagine the game as being played between the parents, with each parent choosing either to cooperate by not having the mutation or to defect by having the mutation. Here's the payoff table.

	Cooperate	Defect
Cooperate	3, 3	2, 1
Defect	1, 2	4, 4

As I've assigned the values, the best outcome for an individual is to have the mutation but one's mate not to have the mutation. But the second best outcome is switch roles so that one's children still have a chance at getting one copy of the mutation. Mutual cooperation is third best, and mutual defection, which leads to the possibility of children with sickle cell anemia, is worst.

So it turns out the sickle cell anemia–malaria game doesn't match any of the four social dilemmas I described. Indeed, I'm not sure whether it's strictly a social dilemma at all. In an iterated version, the best course would be to take turns cooperating and defecting while the other player does the opposite. In a one-shot version—which is how the game must be played in real life—the dilemma is over choosing who gets to defect, with the loser still getting the second best outcome. Because a sole defection beats mutual cooperation, this game may lack an ingredient necessary for it to be considered social.