The altruistic prisoner's dilemma

I had a little thought the other day about the prisoner's dilemma. Readers familiar with the basics of the concept might skip ahead to the third paragraph, but in case they are not, here is the dilemma in brief. Imagine two prisoners, both accused of two crimes awaiting trial. One carries a sentence of 5 years, another carries a sentence of 15. The police have sufficient evidence to produce a compelling case for both prisoners of the lesser crime, but insufficient evidence to produce a compelling case against either prisoner for the greater crime. A testimony from either prisoner, however, will be provide enough evidence for the police to have the other convicted. The police therefore makes an offer to each prisoner: if you betray the other prisoner we will not prosecute you for the lesser crime. There are four possible outcomes to the scenario. If prisoner A testifies and the prisoner B does not, prisoner A is convicted of neither crime (the police only has sufficient evidence against the other prisoner for the greater crime, and has honoured their agreement about the lesser one) and prisoner B is convicted of both. If B testifies and A does not, the inverse occurs. If they both testify, A and B are both convicted only of the greater crime (the police honours the agreement about the lesser crime, but now also has evidence against both prisoners for the greater crime). Finally, if neither testifies, A and B are both convicted for the lesser crime but not the greater. This can be summarised by the following table, with numbers representing the years in sentence received by each prisoner in each eventuality:

		Prisoner A
		Don't testify	Testify
Prisoner B	Don't Testify	5,5	0,20
	Testify	20,0	15,15

As the table makes clear, for any given move prisoner A makes, prisoner B will always receive a lower sentence by testifying than not. The same is true for prisoner A. But this means that if both prisoners are only interested in minimising their own sentence, the outcome will be that both are sentenced for 15 years, i.e a worse outcome than if neither had testified. The beauty of the thought experiment is that it doesn't matter if the prisoners realise this, or even if they agree somehow to coordinate their efforts. Provided neither prisoner has a means of enforcing what is agreed, rational self interest will always lead to the same bad outcome*.

In popular culture, and to some lesser extent in political theory, this parable is used as a demonstration of potential problems of rational self interest, and the circumstances in which it can lead to bad outcomes. But what if we changed the parameters of the game? Imagine, now, that both prisoners were purely selfless in their behaviour. Imagine their only priority was minimising the sentence of the other player. If we wanted a narrative to this, perhaps we could imagine some love story between the two. With the game currently set up, that produces a good outcome. But what if we now imagined the police had enough evidence against both parties for the crime carrying the greater sentence, but not the lesser? If we suspended disbelief for a moment and imagined the police were willing to let either party off prosecution for the greater crime provided they testified against the other for the lesser crime (perhaps the lesser crime was politically significant, or helped the police in some other case etc.) we would then have the following payoff matrix: 

		Prisoner A
		Don't testify	Testify
Prisoner B	Don't Testify	15,15	0,20
	Testify	20,0	5,5

It's true that prisoner A could always reduce their sentence by testifying agains the other prisoner. But in this example, prisoner A doesn't care about that. Prisoner A only cares about reducing the sentence of prisoner B. And they will always reduce the sentence of prisoner B by not testifying. The same is true for prisoner B. But now we have a truly unfortunate result. By acting in a rationally altruistic way, they have now both ended up giving the other player a longer sentence than if they had both betrayed the other! 

What does this illustrate? Possibly nothing much at all. If we were looking for morality tales, or things which can usefully be applied to real life situations, the first, more conventional dilemma probably has a wider range of plausible applications. But in the abstract, it does demonstrate one thing: the prisoners dilemma does not have to be about failures of self interest, it can just as easily be about the circumstances in which preference coordination fails more generally. 

* Provided the game is only played a finite, known number of times.