# Great Ethicists, Webcomics, Game Theory, and the Iterated Prisoner’s Dilemma

The webcomic Saturday Morning Breakfast Cereal today covers one of my favorite topics: Game Theory and the Social Contract.

The game theory, unfortunately, is not done very carefully. Here’s the Prisoner’s Dilemma setup:

 RAT OUT REMAIN SILENT RAT OUT Both get 1 year in prison 1 goes free, 2 gets 5 years REMAIN SILENT 2 goes free, 1 gets 5 years Both get 6 months

In this table, player 1 chooses the row and player 2 chooses the column. If both REMAIN SILENT, both get 6 months in prison. If each RATS OUT the other, both get 1 year in prison. And if one RATS OUT and the other REMAINS SILENT, the rat goes free and the mute gets 5 years in prison.

Here’s how the comic summarizes things:

So, even though [the bottom right corner] is the best choice, the perfectly rational people pick [the top left corner].

Now, the bottom right corner is emphatically not the “best choice.” For starters, given our setup, we’re not choosing a corner. One person chooses a row, and (completely separately) one person chooses a column. There’s no way to choose a corner, and therefore there’s no “best choice” of corner.

But let’s ignore this nitpick and assume he said “best outcome.” Even this isn’t true. Player 1 would be better off with the top right corner, and player 2 would be better with the bottom left corner. This would seem to disqualify the bottom right corner as being “best.” What he probably meant to say is that it’s better than the top left corner, which happens to be the “rational” (i.e. dominant strategy equilibrium) outcome.

In fact, this is the point of the prisoner’s dilemma: no matter what the other player does, your best choice is to RAT OUT, and so the outcome when “rational” people play is the top left corner. Which is worse for both players than if they’d both REMAINED SILENT. Hence the dilemma. “Everyone act rational” doesn’t always lead to optimal outcomes.

The comic then tries to apply this model to morality. The “great ethicists of history,” it turns out, have been trying to convince people to pick [sic] the bottom right corner. (As mentioned above, we’ll assume that the comic really means that they’re trying to convince people to pick STAY SILENT.) The utilitarian Bentham is pictured trying to convince people that the bottom right corner is utilitarianly awesome. And the damnitarian Jesus is pictured trying to convince people that the top left corner will land you in hell for eternity.

In short, each is trying to artificially change the payoffs of the game. Bentham is trying to convince you to that you should care about the total time spent in prison by both players, not just the time spent by you. It’s easy to see why this is an uphill battle. (Furthermore, all this does is turn the game into a pure coordination game with three different equilibria, one of which is still [RAT OUT, RAT OUT]. Smooth move, Bentham!)

Jesus, on the other hand, is (according to the comic) trying to convince you that the payoff in the top left corner is actually more like infinitely many years in prison. Since game theory doesn’t do well with infinite payoffs, this actually results in a game with no equilibrium, which seems like kind of a dick-ish thing to do. However, it seems weird to condemn someone only if both he and his opponent RAT OUT. A more reasonable Lord-of-the-Universe thing to do would be to give you hugely-negative payoff whenever you RAT OUT, regardless of what your opponent does. And in that case it’s a dominant-strategy equilibrium to keep your mouth shut. (Unfortunately, Jesus ruined his “ethicist” credibility by insisting that the same infinite punishments also apply to people who commit the prisoner’s-dilemma-unrelated “crime” of not accepting him as their personal savior.)

Independent of our ethicists, the game we’ve described is not a particularly compelling model of morality. Life contains plenty of “cooperate or defect” situations, sure, but for the most part these situations occur repeatedly with the same cast of characters. Imagine that you and I play the above-described game day after day after day. (You’ll probably have to change the payoffs to involve money or pain or something, since playing a “go to prison for a year” game every day doesn’t make a whole lot of sense. Just make sure to keep the same strategic structure and relative payoffs.)

It turns out (thanks, Robert Aumann) that when you repeat the prisoner’s dilemma indefinitely, suddenly ratting out isn’t so rational. Imagine that I’m willing to KEEP SILENT for as long as you do, but if you ever RAT OUT then I’ll start RATTING OUT for the rest of time. It’s not hard to see that if you adopt the same strategy, we can land in the bottom right corner over and over and over again, because the one-time payoff from defecting would be vastly outweighed by the ensuing sequence of top-left outcomes. As long as we’re describing repeated interactions, there’s not a lot of a problem.

Another criticism of this line of modeling is that many situations where we’d normally think to apply “morality” are unilateral ones, not strategic ones. “Thou shalt not kill,” “thou shalt not steal,” and similar rules are all decision-theoretic prescriptions, not game-theoretic ones. The Prisoner’s Dilemma (and game theory more generally) describes situations where the outcome to me depends both on my decisions and on yours. But (for example) my decision whether to steal from you is not typically co-mingled with your simultaneous decision whether to steal from me. My decision whether to steal from you probably has more to do with the (implicit or explicit) “social contract” that society has in place.

(Of course you could construct a Prisoner’s-Dilemma-flavored model in which every day you and I decide whether to rob each other, but you’d have a hard time convincing me that your model was in any way a representation of the actual choices and incentives that each of us faces in today’s world.)

In fact, there are some pretty interesting game theoretic considerations that come into play when we think about the (theoretical) adoption of such a social contract, which involves strategy all around. But that’s probably a little much to fit into a webcomic.