Thursday, October 11, 2007

DT RG: Carcel Confusion

This pertains to the reading group on decision theory, but any comments are welcome.
It's my version of the Three Prisoners Paradox. I read about the original paradox years ago in a book about the Monty Hall Problem, and I always assumed that it was created as a variant of this problem. However, Wikipedia recently taught me that it's much older than the MHP, and that it's due to Martin Gardner.
Here goes the story (I assume you know about the Monty Hall Problem; otherwise, read the wikipedia entry first):
Three prisoners, A, B and C, are awaiting their execution. It's known to them that one of them will be pardoned, but part of their punishment is that they may not know who prior to the day of the execution. They are kept in separate cells in different buildings.
One day, as the prison guard comes to check on prisoner A, A begs him to give him a hint concerning his fate. Of course the guard declines, but A keeps begging. At least, A suggests, the guard could tell him the name of only one of the others who will be executed for sure. That way A would still not know whether he will die or live, and the guard wouldn't have disobeyed his orders.
The guard thinks it through and mercifully agrees to give the required information: B will die. A thanks the guard and thinks to himself: "Well, at least I know that my chances to get out alive are 50% now."
So far, so good. Anyone familiar with Monty Hall will see that A is wrong. His chances are still 33%, the guard's revelation has gained him nothing. To draw the analogy to Monty Hall, he should switch fates with C if only he could. If we, the audience, were the type of people who bet on people's lives and deaths, we should put our money on C's staying alive.
That's the Three Prisoners Paradox as I remember reading it in the book. Now my appendix:
The guard passes by the cell of poor B, who is sound asleep, and finally comes to C. Here, a similar scene as before unfolds. C implores the guard to tell him something about his situation. The guard recalls his talk with A, goes through the reasoning once again to make sure he's not disobeying his orders, and tells C that B is going to die. "Whoopy!", thinks C, "So my chances to survive are 50%!!" But of course we know that this is false, his chances are 33%.
To sum up, we have A at a survival-chance of 33%, doomed B at 0% and C at 33% as well. But that seems a bit odd...
I've been puzzled by this for a long time, and I've asked a bunch of people and received a bunch of interesting and interestingly different answers. I think I know what's wrong, but I'm never quite sure (about 66% most of the time), so I await clarification(s)!


Sebas said...

I do not agree with you.
The survival probability of A does not increase (considering the same analysis as the ones explained in the Monty Hall case:the guard will always reveal the name of a prisoner that is going to die). The subjective survival probability of A is 1/3 and 2/3 for C (he should change his position if he could).

The analysis is identical for C, his survival possibility remains by 1/3 while, for him, the survival possibility of A's has increased (just follow the Monty Hall analysis).

I think that you miss one point: No one knows what the guard tells to the others. The survival chance for B is also 33%, because the answer that he would to his corresponding demand would be A (or C).
For an external observer like you in the situation that you are presenting the survival-chance for each position is 1/2 for A, 1/2 for C and 0 for B.

What do you think?

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