[Prof. Templeton]

A Classic...

You and your partner committed a crime and are both arrested and interrogated separately. You are offered a chance to confess, in which you agree to testify against you 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 you 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?

[Guest]

1

52.38%....?

EDIT: see Chuck's solution for explanation.

Edited August 8, 2008 by Suicide

[Guest]

11/21

Adding the x->2x equivalence makes this a good problem. The following reasoning is based on the additional assumption that the the marginal utility of staying in prison is constant - in other words, I would dislike spending one year in prison exactly as much as I would dislike spending 20 years over 19 years.

If you choose to confess, your expected displeasure would be p(10+10/2)+(1-p)(0+20/2). When your accomplice confesses also, you each do 10 years which causes a pain of 10 (for you) + 10/2=5 (for him); if you confess but he doesn't, your pain is 0 (for you) + 20/2=10 (for him). Similarly, your expected displeasure if you don't confess is p(20+0/2)+(1-p)(3+3/2). Setting those two equations equal, you are indifferent at a probability of 11/21=.5238 (assuming I did the algebra right).

EDIT: I agree with suicide's post above.

Edited August 8, 2008 by Chuck Rampart

[Prof. Templeton]

Well done both of you, I had thought the indifference part might throw some people.

[unreality]

A Classic...

You and your partner committed a crime and are both arrested and interrogated separately. You are offered a chance to confess, in which you agree to testify against you 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 you 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?

The Problem:

both confess = both serve 10

A confesses, B doesnt = A serves 0, B serves 20

B confesses, A doesnt = B serves 0, A serves 20

both stay silent = both serve 3

probability of partner confessing = p

at what value of p would your partner serve an average of 2 times your own sentence?

my answer...

say that you are A from the above example

Confessing gives you an average of ((10)(p) + (0)(1-p))/2 years and your partner an average of ((10)(p) + (20)(1-p))/2 years

Not confessing gives you an average of ((20)(p) + (3)(1-p))/2 and your partner an average of ((0)(p) + (3)(1-p))/2 years

call your average x... what does p have to be to make your partner's average 2x for both confessing and not confessing?

set it up like this:

((10)(p) + (0)(1-p)) = ((10)(p) + (20)(1-p))/2

((20)(p) + (3)(1-p)) = ((0)(p) + (3)(1-p))/2

simplify to make them easier to look at:

10p = (10p + 20 - 20p)/2

20p + 3 - 3p = (3 - 3p)/2

simplify again:

10p = (20 - 10p)/2

17p + 3 = (3 - 3p)/2

and again:

10p = 10 - 5p

17p + 3 = 1.5 - 1.5p

and again:

15p = 10

18.5p + 3 = 1.5

and again:

p = 10/15 = 2/3 = .6666 if you confess

18.5p = -1.5

and -1.5/18.5 = .081repeating for p if you don't confess

Weird. I think I must've did something wrong, or maybe I didn't understand the problem... you can never have 2x of the other value, so I assumed it had to be averages based on the probability

[Guest]

I did it a different way, but I'm not sure if that's right.

I made a diagram of the situation (this is how I do probabilities)

Y=You

P=other prisoner

C=confess

N=not confess

1 2

YN YC

3 PN NN NC

4 PC CN CC

The probability (p) that your buddy confesses is 1/2 (scenarios 1,4 and 2,4)

The time of sentence to be served based on the above scenarios:

1 2

Y Y

3 P P,Y=0 Y=0, P=20

4 P P=0, Y=20 P,Y=10

so your indifference for serving 10 years, or your partner serving 20 years is 1/2 (scenarios 2,4 and 1,4)

so at what level of p are you indifferent to confessing or not confessing (the scenarios are the same in both cases) so 1/2

[Prof. Templeton]

The Problem:

both confess = both serve 10

A confesses, B doesnt = A serves 0, B serves 20

B confesses, A doesnt = B serves 0, A serves 20

both stay silent = both serve 3

probability of partner confessing = p

at what value of p would your partner serve an average of 2 times your own sentence?

my answer...

say that you are A from the above example

Confessing gives you an average of ((10)(p) + (0)(1-p))/2 years and your partner an average of ((10)(p) + (20)(1-p))/2 years

Not confessing gives you an average of ((20)(p) + (3)(1-p))/2 and your partner an average of ((0)(p) + (3)(1-p))/2 years

call your average x... what does p have to be to make your partner's average 2x for both confessing and not confessing?

set it up like this:

((10)(p) + (0)(1-p)) = ((10)(p) + (20)(1-p))/2

((20)(p) + (3)(1-p)) = ((0)(p) + (3)(1-p))/2

simplify to make them easier to look at:

10p = (10p + 20 - 20p)/2

20p + 3 - 3p = (3 - 3p)/2

simplify again:

10p = (20 - 10p)/2

17p + 3 = (3 - 3p)/2

and again:

10p = 10 - 5p

17p + 3 = 1.5 - 1.5p

and again:

15p = 10

18.5p + 3 = 1.5

and again:

p = 10/15 = 2/3 = .6666 if you confess

18.5p = -1.5

and -1.5/18.5 = .081repeating for p if you don't confess

Weird. I think I must've did something wrong, or maybe I didn't understand the problem... you can never have 2x of the other value, so I assumed it had to be averages based on the probability

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 "pain"(as Chuck put it) as my jail time plus half his jail time. Therefore the pain of both confessing and not confessing cases must be equal in order for me to not care which I pick.

In the confessing case, the total pain is 10p + (20-10p)/2 = 10+5p.

In the not confessing case, the total pain 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

[Prime]

... 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?

I don't agree with your formulation. It seems to imply that it's all the same to you if you serve 50 years while your partner is serving 100 as if you serve 2 years while your partner is serving 4. But even if I'm to interpret it that you feel as bad for each year that you serve as you feel for every two years served by your partner, still something's amiss there. It turns out it's all the same to you if you serve 5 years and 3 months while your partner is serving 14 years and 9 months, as if you serve 11 years and 11 months while your partner is serving only 1 year and 5 months. I say, you should try to be less concerned about your crime partner. It was his idea to rob that liquor store, anyway.

[Prof. Templeton]

I don't agree with your formulation. It seems to imply that it's all the same to you if you serve 50 years while your partner is serving 100 as if you serve 2 years while your partner is serving 4. But even if I'm to interpret it that you feel as bad for each year that you serve as you feel for every two years served by your partner, still something's amiss there. It turns out it's all the same to you if you serve 5 years and 3 months while your partner is serving 14 years and 9 months, as if you serve 11 years and 11 months while your partner is serving only 1 year and 5 months. I say, you should try to be less concerned about your crime partner. It was his idea to rob that liquor store, anyway.

I care about my partner, but more about myself, which is why I become indifferent about confessing when his jail time equals twice mine.