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        riddles >> medium >> Prisoner dilemma redux
        
(Message started by: wowbagger on Sep 27^th, 2002, 4:08am)

Title: Prisoner dilemma redux
Post by wowbagger on Sep 27^th, 2002, 4:08am

Just in case someone wants to compare his/her answer, here's my straigthforwardly calculated result:
You are indifferent to confessing and not confessing at
p = 11 / 21 = 0.5238...

(Couldn't find a thread on this one, sorry if I missed it)

Title: Re: Prisoner dilemma redux
Post by Icarus on Oct 17^th, 2002, 5:17pm

Obviously, I am missing something here. If you do not confess your expected sentence is 3+17p ("expected" in the statistical sense). If you do confess, your expected sentence is 10p. If your first priority is to minimize your jail time, why would you not confess for any value of p?

Can someone point me back in the right direction?

Title: Re: Prisoner dilemma redux
Post by Chronos on Oct 17^th, 2002, 6:17pm

You can't calculate an "expected" sentence without knowing the probabilities for what the other prisoner will say, and you shouldn't just assume that he'll choose randomly.

The idea in game theory (the branch of mathematics which deals with problems such as this) is that you assume that your "opponent" will use the optimal strategy, and based on that assumption, determine what your optimal strategy is.

Title: Re: Prisoner dilemma redux
Post by rugga on Oct 19^th, 2002, 10:12am

You are right in general, Chronos. But the statement of the "redux" problem explicitly says

Quote:

You assume the probability of your partner confessing is p. ... At what value of p are you indifferent to confessing and not confessing?

I agree with Icarus that with that assumption (partner confesses according to some fixed probability) it can't make sense to not confess. I wasn't too sure what the point of this problem was when I first saw it, but I'm guessing it's just to show how a seemingly straightforward analysis can lead to an incomplete (at best) solution. And of course that's what makes the prisoner's delimma so much fun!

Title: Re: Prisoner dilemma redux
Post by Garzahd on Oct 21^st, 2002, 4:38pm

I like game theory. To heck with paper, I want to try solving this in the message composition window.


     C. . .NC
 C 10/10  0/20
NC 20/0   3/3

The C/NC on the left are the options if you confess; the C/NC on the top represents your buddy's confession status. The outcomes are listed as my time / his time.

So the expected jail times if I confess are 10p for me and (10p+20(1-p))=20-10p for him.
If I keep my mouth shut, the expected jail times are (20p+3(1-p))=3+17p for me and 3-3p for him.

Since we're indifferent between x years for me and 2x for him, then let's define "hassle" as my jail time plus half his jail time. Therefore the hassle of both C and NC cases must be equal in order for me to not care which I pick.

In the C case, the total hassle is 10p + (20-10p)/2 = 10+5p.
In the NC case, the total hassle is 3+17p + (3-3p)/2 = 4.5+15.5p.

If 10+5p=4.5+15.5p, then 5.5=10.5p, or p=11/21. Yay.

Title: Re: Prisoner dilemma redux
Post by wowbagger on Oct 22^nd, 2002, 3:56am

Garzahd:
That's exactly how I came up with my answer. Nice description. :)

Rugga:
Maybe we interpret the fragment of the problem statement you quoted differently.

on 10/19/02 at 10:12:49, rugga wrote:

I agree with Icarus that with that assumption (partner confesses according to some fixed probability) it can't make sense to not confess.

I don't think the partner makes a random decision by flipping a coin or rolling a dice or whatever. It is you who assumes a certain probability that your partner will confess. Still, it's his deliberate, wilful decision.

As for your doubt whether it could be sensible to not confess: Look at Garzahd's "hassle" in a case with p < 11/21.
The vital ingredient is of course the quantitative relation that you are

Quote:

indifferent to you serving x years and your partner serving 2x years.

Title: Re: Prisoner dilemma redux
Post by rugga on Oct 22^nd, 2002, 7:43am

I see. I have to admit I ignored the "indifferent" statement because I didn't think it was important. I read "highest priority is to keep yourself out of the pokey" and took that to mean I wanted to minimize my jail time above all else. That's enough to come up with an answer so I just ignored the rest. I'm sure your interpretation is right.

Title: Re: Prisoner dilemma redux
Post by Icarus on Oct 22^nd, 2002, 4:53pm

I didn't ignore the indifference clause, myself. I was just having trouble coming up with a good interpretation of it. The "hassle" interpretation is certainly reasonable, but I have not completely convinced myself yet that it does not assume information not given in the problem. (Three "not"s in one sentence. It's no wonder I'm confused! :P) I'll have to think about it some more. Thanks for the help!