Individually best, Collectively worse

The prisoner’s dilemma is a game theoretic concept where, in a game of n players (n>1), the individually best strategy for each player, results in a collectively worse outcome for all players.

Consider the following example,

Two people are caught by the police and interrogated separately. They know the following facts:

1. If the first player keeps mum and the other talks, the first person goes to jail, and the second walks and vice-versa.
2. If both players rat each other out, both get a reduced sentence for their cooperation.
3. If neither one talks, both walk free on lack of evidence.

It is important to note that only the police know about the lack of evidence, and hence the actions of each player come down to what they think the other person would do.

We can plot this in a simple 2X2 Grid as follows:

Let’s think about the best strategy for player 1,

If they always keep mum, there are two possible outcomes,

So worst case, longer jail time, best case no jail time.

Now if they always talk, the outcomes are:

So now, worst case, reduced jail time, best case no jail time.

It is clear that in the “Always talk” strategy, the worst case is better than the “Always keep mum” strategy and the best case is the same. Hence we get the optimal strategy, “Always talk”. Following the same logic, player 2 also arrives at the same answer. The net result, the outcome is always that of a reduced sentence and no one ever walks free. So the individually best strategy results in a collectively worse outcome.

Another example is a class test where students have decided to employ “teamwork” on a test. As an individual, if you are the only one who goes to the teacher to explain to them about “teamwork”, you will get a free pass, and the rest of the class is punished. If you don’t go to the teacher but someone else does, you will be in the group that gets in trouble. However, as soon as you go to the teacher, no matter how many others come forward, you will always be safe. So your best strategy is always to tell. When each student arrives at the same answer, the result is the test is rescheduled. Everyone loses by playing the individual best strategy.

So how can we resolve the Prisoner’s Dilemma?

First, there has to be a way to detect cheating, and second a punishment severe enough to alter the incentives in such a way that the individual best strategy no longer remains the best.

Consider the first example, let’s say we have a Godfather outside the who also rewards and punishes loyalty and disloyalty respectively. Now, cheating can be detected if only one player walks, as the only way for that is for him to talk and the other one to keep mum. So the player who talks gets the punishment from the Godfather, which is definitely worse than jail. Now the best strategy for both players could actually be to keep mum as the Godfather might have people inside the jail in case they both talk. In this scenario, because of skewered incentives, the individual best strategy is to keep mum.

In the second scenario, the punishment could be to ostracize the people who came forward. The cheating can be easily detected by finding out the people who were not in the admonished group. In this way, you can weed out people who talk even once and incentivize those who want to stay to comply with the plans.

It is important to realize when we are stuck in a prisoner’s dilemma. The reason why it is so detrimental is that the individually best strategy is “rationally optimal” i.e. we arrive at it by thinking through all possible scenarios and hence there is nothing to correct as such, and in a system where you cannot trust the other people, you are left with no choice but the play your best strategy. Hence it is important to figure out a way to skew the incentives such that everyone plays the strategy that gets the collectively best outcome.

I hope you found this post interesting. Until next time.

Cheers!

Originally published at https://prathmesh6.substack.com on July 14, 2022.