Surprise Execution meets Variation on Prisoners Dilemma


The puzzle, again, was this:

Two nasty, distrustful thieves get locked up in adjacent jail cells in
a jailhouse in some small town in Kansas. The cells are each large
steel-barred enclosures a few yards apart.

The next morning, news hits that a monster tornado has touched down
that is now demolishing everything in its path. Furthermore, the
newscast states that the local jailhouse lies right in the path of the
storm, and its subsequent obliteration is imminent. All within the
jailhouse hear the newscast, and the jailer rushes to grab his
belongings and run for safety in a nearby storm cellar. Before he runs
out, however, the locked up thieves plead to be released so they to
can take shelter.

The jailer grabs two key rings [We'll assume for this that each key
ring has exactly 12 keys] - one for each prisoner - and tosses each
set through the bars to each of the two thieves. After he leaves, the
prisoners frantically try to open their respective cells when they
suddenly come to learn that the jailer tossed the wrong key ring to
each prisoner. That is, each now has the keys to the other jail cell.

As they glance at one another - each now holding the keys the other
needs - they also notice that the jailer has left the safe open, with
piles of cash available for the taking.

Staring at each other they quickly realize - and verbalize - the
screwed up situation they find themselves in, specifically:
In the spirit of cooperation they should quickly toss the correct keys
through the bars into each other cells, thereby allowing their mutual
escape and splitting of the loot in the safe. But, each being entirely
distrustful of the other, fear that if they toss the set of keys they
now hold, the other will not reciprocate and instead free themselves
and leave for safety taking all of the loot in the safe.

[In fact, for the sake of this puzzle, we'll assume that if either guy
is freed before the other one, he WILL double-cross the other guy,
take the money and leave the other guy to die.]

They both realize all of this and are staring at each other terrified
as they now begin to hear the roar of the approaching tornado. If the
two thieves are, and know each other to be, rational - if not trustful
- can they reason their way out of this situation?


One solution went like this:

Maybe if there are multiple keys on the ring, and they can't tell
which is which, they can agree to toss each other the keys one at a
time, simultaneously...

... If they
both want to live, they have to agree not to try their individual keys
until all keys have been thrown.

I think when the puzzle says that they're both "rational," we have to
assume that B won't just try each key as it's thrown to him.

If they both have exactly 12 keys, and they both agree to throw one
key at the same time, then I don't see where B benefits from
immediately trying A's key. Sure, there's a 1-in-12 chance that B
will get free and then keep the money, but if the key doesn't work,
then there's about a 1-in-12 chance that he just signed his own death
sentence when A tries his key and discovers that it does work.


I was perfectly satisfied with this answer until a friend pointed out
to me that, when it came to key #12 (the final key), what incentive
does either guy have to throw?

Sure, they can count "1! 2! 3!" But why would either guy throw that
final key? Can we, in fact, assume that neither guy will throw key
#12?

Oh well, I thought, maybe they each only end up with an 11-out-of-12
chance of living...

But wait a second. Suddenly the ELEVENTH key just became the final
key! Now, why would they throw that?

And then, working backwards, I started to find myself backing into the
"You will be executed on a day you don't expect it" situation!


I can't picture what these guys will now do, assuming they're both
following the rules as stated. They won't let the other guy live
unless they have to, and they know he'll do the same.

Maybe they have to throw some number of keys -- say, between 9 and 11
somewhere, just to have a reasonable chance of getting out... Or does
the Surprise Execution dilemma really kick in and reduce their chances
of living to about 50-50 each?
.

Relevant Pages

  • Re: Surprise Execution meets Variation on Prisoners Dilemma
    ... let's surprise him by not posting on day 1. ... The cells are each large ... The jailer grabs two key rings [We'll assume for this that each key ... there's a 1-in-12 chance that B ...
    (rec.puzzles)
  • Re: What do you open with this 85 : - /AQJ104/-/KQJ98653
    ... The chance that it will make is actually closer to 5/9 not 4/9. ... Finally copy the row of five cells and paste them on the next 999 ... probability that partner holds one or both of them. ... and there are is the offsetting case where a heart ruff kills you at trick 1 ...
    (rec.games.bridge)
  • Re: Variation on prisoners dilemma
    ... jail cells in a jailhouse in some small town in Kansas. ... jailer grabs two sets of keys - one for each prisoner - and tosses each ... escape and splitting of the loot in the safe. ...
    (rec.puzzles)
  • Re: HP and Functions
    ... What else is in those cells? ... doesn't make sense to excel? ... Any chance that you're getting a "formula too long" message? ... Mike S. wrote: ...
    (microsoft.public.excel.worksheet.functions)
  • Re: Finding Duplicated Text within Columns
    ... While there is a chance this formula will not always be ... FALSE indicates that the cells of the same ... peach banana melon grape apple orange ... >I need to compare the columns and find which rows are ...
    (microsoft.public.excel.misc)
Loading