Prisoners dilemma (one step further)

When I meant "overall", I envisioned the selection process as two seperate stages, the first being the point where no prisoner has any information, and the second being the point where the prisoner's have all the information they're going to get.

"Overall" would be prisoner A's chances from the first stage, which shouldn't change once the process begins unless it is announced that A will definitely die or will definitely live. The chance A is going to die provided some additional information is given would be the second stage probability. I'm pretty sure these two chances will agree with each other if you consider the chance of that certain piece of information from the second stage being given at all.

Originally posted by Palynka
PBE6 is right although I also don't understand what he means by overall probability.

At each level of info for each individual (after A and C share the info):

A knows he has a 50/50 chance shared with C and B is as good as dead.

C knows exactly the same thing (he has the same info of A.

B doesn't know about it and still puts the odds at 1/3 each ...[text shortened]...

Is there a catch here I'm not seeing? The original problem is much more counter-intuitive.

This matches how I see it, except for the last line. Does it matter what info B has? The question is not about the prisoners 'percieved' chances of survival, but the actual chance of survival?

Once the guard decides to kill B, the chances for the others are 50/50. It does not matter what he tells A or C, these are the actual chances. If the question was about percieved chances, it all changes, but this was not mentioned in the question.

Or am I missing something?

Sorry don't mind me, just passing through. Christ! Nothing but tumbleweeds and old rational choice theory problems around here.

Originally posted by tojo
This matches how I see it, except for the last line. Does it matter what info B has? The question is not about the prisoners 'percieved' chances of survival, but the actual chance of survival?

Once the guard decides to kill B, the chances for the others are 50/50. It does not matter what he tells A or C, these are the actual chances. If the question was abou ...[text shortened]... ces, it all changes, but this was not mentioned in the question.

Or am I missing something?

The actual chance of survival is not a probability since it has been decided already who is going to live and who is going to die (i.e. the guard knows). It's not a random event for the guard, for example.

All probabilities here are each individual's perceived chances.

Originally posted by PBE6


For a thorough discussion of the first part of the question, check out the old "3 Prisoners" thread Thread 11542.

A gentleman named Cribs provided excellent and understandable analysis of this problem in that thread. I recommend reading his posts there. I also made some posts later in that thread, which I also recommend. It is crucial to understand them before attempting to understand this problem.

The problem posed here is identical. A lives with probability 1/3, even after learning what the guard told C. This is because A already knew the information that the guard gave C. C telling A that he knows it as well does not alter A's information, and thus his probability of survival is not altered. A would still prefer to switch with C, and C would still prefer to remain C.

This applies to the standard reading of the problem. However, as Cribs and myself pointed out in the prior thread, another reading of the problem exists, based on how the guard's information is constructed and conveyed to the two parties. In that alternate reading, A's probability is 1/2, and would also be 1/2 in this problem.

Originally posted by tojo


Once the guard decides to kill B, the chances for the others are 50/50. It does not matter what he tells A or C, these are the actual chances.

This is not necessarily correct. Do you understand the solution to the Monty Hall problem?

Originally posted by TheMaster37


Now A finds out that C got the same information from the guard as he did.

Is his chance of survival now 1/2?

It depends on the stochastic processes from which the guard's information derived and which governed how he conveyed that information to the prisoners. In the standard interpretation, A's survival chances remain at 1/3. However, there exist stochastic processes consistent with the problem's formulation that would yield a solution of 1/2 instead. See the above mentioned thread for descriptions of such stochastic processes.

Also, it is misleading to term this problem the Prisoners Dilemma or a variation on the Prisoners Dilemma. That name is taken by a completely different problem in the field of game theory in which prisoners decide to cooperate or defect in order to minimize their sentences, and, unlike this one, actually is a dilemma.

Originally posted by DoctorScribbles
Also, it is misleading to term this problem the Prisoners Dilemma or a variation on the Prisoners Dilemma. That name is taken by a completely different problem in the field of game theory in which prisoners decide to cooperate or defect in order to minimize their sentences, and, unlike this one, actually is a dilemma.

Aye, I found that when I searched on the name. I'm preparing an assignment for one of my classes about these kind of subjects 🙂