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prisoner's dilemma

Two drug-traffickers are arrested for drug running and
are placed in separate cells. The district attorney
tells them of the following:


        If both do not confess, then the district
attorney will drop the drug-charges for lack of
evidence, and will instead charge them with
        a minor offence. Both of them will get one
year in jail.
        If both confess, each will get five years in
jail.
        If either one confesses and the other remains
silent, the one who confesses will be released (for
turning in evidence) and the other
        will be jailed for ten years.


    Furthermore, each of them is only concerned with
getting the smallest sentence for himself. Also, since
they are separated, neither of
    them knows the other's decision, and we assume
that they are acting rationally.


    If you were one of them, what would you do?

shailesh kumar
Monday, May 06, 2002

I think a confession is more "profitable" here...

If the other guys is silent, confessing will let you go free, silence will buy you a year in jail.

If the other guy confesses, doing so as well will put you in jail for five, but staying silent will double that term.

Additionally, because the other guy is in the same position as you are, you can assume that he *WILL* confess (being a rational person), and therefore you should too. This way you will get 5 years and not 10.

levik
Tuesday, May 07, 2002

I think this one is a paradox. Confessing seems to be the logical move, but this doesn't give the best result. The best result (one year each), can't be gained by reasoning about what the other will do.

Paul Viney

Paul Viney
Tuesday, May 07, 2002

I think it would be a paradox if the condition was that the prisoners were "friends" who care about what happens to one another.

As it was stated - all you care about is minimizing your own time - I think my logic proves that the optimal way out is spilling the beans.

levik
Wednesday, May 08, 2002

i don't know the answer to this question either. My reasoning was like :
let probab of my confessing be p and his be q.

my sentence = pq(5) + p(1-q)0 + (1-p)q10 + (1-p)(1-q)1
=q( 9 - 4p) + 1 - p
if p  = 1 = > 5q.
if p = 0 => 9q + 1.
so p = 1 to minimize the weightage on q.
i will confess.

shailesh kumar
Thursday, May 09, 2002

This is the classic problem of game theory. As it turns out, it is not the mathematics of the problem that is difficult. What you have (correctly) calculated is the 'dominant' strategy for each prisoner, i.e., the strategy that will yield the best result regardless of the other guy's strategy.  There is another equilibrium, called the 'Nash equilibrium' (yeah the beautiful mind guy), which is defined as:
The common strategy at which neither player has the incentive to improve his position by adopting a better strategy. Obviously, the situation in which both prisoners refuse to confess is a Nash equilibrium (but not the only one ).
Which strategy will be adopted in real life is not a matter of mathematical calculation. It is  a matter of psychology. It depends on social concepts of fair and unfair behaviour, whether the players see an opportunity to repeat the game (so that by punishing others, they can get people to behave), etc.

Ravikiran
Thursday, May 09, 2002

I would let the imaginary Ed Harris in my mind tell me what to do.

Chris Shieh
Thursday, May 09, 2002

Since each of the prisoners is also aware that the other one is as intelligent as he is, therefore both of them know that whatever decision they take, the other one is going to take the same decision.
So they will opt for not confessing.

vivek gupta
Friday, May 10, 2002

by that logic, each of the prisoners now knows that the other will not confess.  since their decisions are indepedent, there is nothing to stop one of them from confessing at the last moment in the interest of the smallest sentence.  it's important to remember that the prisoners are not friends, they are selfish.  they want the smallest sentence for themselves.

since a prisoner has control *only* on his own decision, the governing rule would be to minimize risk *even* if the other prisoner did the opposite of what you expected him to do.  in other words, create a win-win situation for you.  the *only* decision you can make where you "win" (get the least possible sentence) irrespective of the other prisoner's decision is to confess.

Vin
Friday, May 10, 2002

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