# The Prisoner’s Dilemma

## Problem

Three prisoners, $$A$$, $$B$$ and $$C$$, with apparently equally good records have applied for parole. The parole board has decided to release two of the three, and the prisoners know this but not which two. A warder friend of prisoner $$A$$ knows who are to be released. Prisoner A realizes that it would be unethical to ask the warder if he, $$A$$, is to be released, but thinks of asking for the name of one prisoner other than himself who is to be released. He thinks that before he asks, his chances of release are $$\frac{2}{3}$$. He thinks that if the warder says "$$B$$ will be released", his own chances have now gone down to $$1/2$$, because either $$A$$ and $$B$$ or $$B$$ and $$C$$ are to be released. And so $$A$$ decides not to reduce his chances by asking. However, $$A$$ is mistaken in his calculations. Explain.

## Solution

Prisoner $$A$$’s mistake is not considering the entire sample space of pairs of prisoners that could be released, which is $$\Omega = \{AB, AC, BC\}$$. In the case of $$AB$$, the warden will say that $$B$$ will be released, and $$A$$ has a $$100\%$$ chance of release. Similarly, in the case of $$AC$$, the warden will say that $$C$$ will be released, and $$A$$ has a $$100\%$$ chance of release. In the case of $$BC$$, the warden will say that either $$B$$ or $$C$$ will be released, and $$A$$ has a $$0\%$$ chance of release. Each outcome has a $$\frac{1}{3}$$ chance of occurring. When combined, this gives the overall chance of release with the warden’s statement as:

$P(A\;Released) = \left(\frac{1}{3}\right)1 + \left(\frac{1}{3}\right)1 + \left(\frac{1}{3}\right)0 = \frac{2}{3}$

Thus, asking the warden has no effect on prisoner $$A$$’s chances of being released.