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Confess or Not

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.


Puzzle ID:#8823
Fun:*** (2.35)
Difficulty:*** (2.64)
Submitted By:lesternoronha1****
Corrected By:Winner4600




You and your partner in crime are both arrested and questioned separately. You are offered a chance to confess, in which you agree to testify against your partner, in exchange for all charges being dropped against you, unless he testifies against you also. Your lawyer, whom you trust, says that the evidence against both of you, if neither confesses, is scant and you could expect to take a plea and each serve 3 years. If one implicates the other, the other can expect to serve 20 years. If both implicate each other, you could each expect to serve 10 years. You assume the probability of your partner confessing is p. Your highest priority is to keep yourself out of the pokey, and your secondary motive is to keep your partner out. Specifically, you are indifferent to you serving x years and your partner serving 2x years. At what value of p are you indifferent to confessing and not confessing?

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Jan 17, 2003

are you mr. nash?
Jan 29, 2003

Ah, but of course it could also be said that, whatever happens, you are better off confessing.
If your partner confesses, and you don't you get 30 years. If you do you get ten years. So you are better off confessing.
If your partner doesn't confess and you don't, you get three years. If you do, you get no years, so again, you are better of confessing.
Apr 01, 2003

This seems like an unlikely answer given that the conditions for both of you are identical and yet we are told that the point of indifference is when your partner serves double your time. An answer of about 50% should see you both doing equal time on the average. I'm still working on it!
Apr 18, 2003

i would keep my mouth shut cus im not a rat
Jul 02, 2004

MY BRAIN HURTS!!!!!!!!!!
Jan 26, 2005

It's a nice teaser. I agree with the given answer - I don't see why it troubles some people.
Jan 04, 2006

AAAhhhrrgghh ! don't even want to try to solve this one.
Feb 07, 2009

Nash had less to do with the Prisoner's Dilemma than von Neumann. In any case, the answer is wrong. There is no point in calculating a Nash equilibrium point for a single game that is not zero-sum.

It is the classic Prisoner's Dilemma -- Given that my partner has chosen one choice or the other, and his choice is not influenced by my choice, there is no reason for me not to confess. Of course, I'd LIKE my partner to think that I might not confess, thereby making a chance that he doesn't. But whether he has confessed or not, I always get a lesser sentence by confessing, so it is foolish for me ever not to.

When the game is repeated, as in von Neumann's experiments, it makes sense not to confess (or "betray" in his terms). But in a single game (or the last round of a multi-game) there never is a reason, assuming I am acting purely selfishly.
May 14, 2013

Question wording is too complicated. Should be made simpler.

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