The drug dealers' dilemma

Probably the most famous example in game theory is the prisoners' dilemma. The general story goes something like this (with lots of variants; this is the version I use in ECON100):

Bonnie and Clyde are two criminals who have been captured by police. The police have enough evidence to convict both Bonnie and Clyde of the minor offence of carrying an unregistered gun. This would result in a sentence of one year in jail for each of them.

However, the police suspect Bonnie and Clyde of committing a bank robbery (but they have no evidence). The police question Bonnie and Clyde separately and offer them a deal: if they confess to the bank robbery they will get immunity (and be set free) but the other criminal would get a sentence of 20 years. However, if both criminals confess they would both receive a sentence of 8 years (since their testimonies would not be needed).

The outcome of the game is that both criminals have a dominant strategy to confess. Confessing results in a payoff that is always better than the alternative, no matter what the other criminal decides to do.

The New York Times reports on a new real-world take on this game:

When the sheriff in Franklin County, Ky., posted a flier on Facebook asking local drug dealers to snitch on their competition, the response was more than a little incredulous.

That is, until a tip sent to a phone number on the flier led to an investigation that helped the sheriff arrest a local drug dealer...

“Asking drug dealers to turn in other drug dealers,” Sheriff Melton said. “It’s comical, and it’s working.”

Here's the flier they used:

And here's why it works, for drug dealers who have short time horizons. For simplicity, let's assume that there are two drug dealers only (A and B). In this short-run non-repeated game each drug dealer has a simple choice: Snitch on their competition, or stay silent. If both drug dealers stay silent, then make low profits (because they have to compete with each other). If one stays silent and the other snitches, then the snitch makes high profits while the other goes to jail. If both snitch, then both go to jail. The game in normal (payoff table) form is as follows:

What happens in this game? Consider Drug Dealer A first. They have a weakly dominant strategy to snitch. This is because the payoff is never worse than snitching (and sometimes it is better). If Drug Dealer B stays silent, Drug Dealer A is better off snitching (because high profits are better than low profits). If Drug Dealer B snitches, either strategy is equally good for Drug Dealer A (since they go to jail either way). So Drug Dealer A shouldn't choose to stay silent, because the better option (at least some of the time) is to choose to snitch.

Now consider Drug Dealer B. They also have a weakly dominant strategy to snitch. This is because the payoff is never worse than snitching (and sometimes it is better). If Drug Dealer A stays silent, Drug Dealer B is better off snitching (because high profits are better than low profits). If Drug Dealer A snitches, either strategy is equally good for Drug Dealer B (since they go to jail either way). So Drug Dealer A shouldn't choose to stay silent, because the better option (at least some of the time) is to choose to snitch.

So, both drug dealers acting in their own best interests leads both to snitch. If you're working out the Nash equilibriums, there are three - the only outcome that isn't a Nash equilibrium is the outcome where both drug dealers remain silent (to see why, take any of those outcomes and consider whether either player would be willing to change strategy - since they wouldn't (or would be indifferent to the change in strategy) those three outcomes are Nash equilibriums).

Now, the game isn't quite that simple of course, because the game isn't played just once (a non-repeated game, which is what the payoff table shows), but is played many times. You could think of the two drug dealers as making the choice to snitch or stay silent every morning when they wake up. The payoffs to the repeated game are the sums of the profits (low or high) or jail time from every play of the game.

So what happens in a repeated game? The 'best' choice for each drug dealer may be to cooperate (i.e. to remain silent). If either drug dealer snitches, then that is likely to cause the other drug dealer to snitch the next time the game is played. Snitching only has a short-term payoff, but ultimately leads to a worse outcome for everyone. So, the threat of reciprocal snitching may be enough to ensure an uneasy alliance results for drug dealers who have long time horizons (those who recognise this as a repeated game). Which suggests that there is probably such a thing as 'honour among drug dealers', and this police strategy will not work for long.

[HT: Marginal Revolution]