Re: Twenty Bank Robbers -- Game theory:)

[jim bell <jimbell@pacifier.com>]

At 03:03 PM 7/25/96 -0400, Simon Spero wrote:
>On Thu, 25 Jul 1996, jim bell wrote:
>> My guess?  They all agree to kill whoever made that suicidal rule.  
>> Otherwise, all but two would end up dead.
>
>But the people at the start of the line know that if they don't 
>hang together, they will end up dead, and if that they act purely 
>selfishly only the last two will benefit. Because they want to stay 
>alive, a better solution for the first person to propose equal shares, 
>which would be opposed by the last two players, but supported by the rest.
>He could also split the money only amongst the first half of 
>the gang, since he only needs half the votes.

Yes, my answer was quick, flip, and partly wrong.  It turns out the answer is probably indeterminate, because the amount people want to live is indeterminate.  Consider:


If two were left, #2 would get everything by the rules.  (he would propose, "I get everything!"  The vote would be 1-1, or 50%, which would win.)

If three were left, #2 knows that if #3 is eliminated, he would win as above.  #1 knows this as well, and is motivated to make a deal with #3 to prevent this.  #3 is also motivated to deal, because if he can't get an agreement he's not only out of the money, he's dead.  How they choose to split up the money is unknowable, I suspect, because of the "death" aspect.  #3 could also deal with #2 if #1's terms were onerous.  This problem would be simpler to analyze (and probably determinate) if anyone whose proposal was rejected was simply out of the game, rather than dead.


There's another complicating aspect. Voting order is important.  According to the rules, #3 must make a proposal, which needs to be voted on.  Obviously, #3 will vote for it.  But even if he's come to some agreement with #2, will #2 vote yes?  If #2 votes no, 3's gone and #2 wins everything.   So #3 couldn't trust #2 to vote yes. particularly if #2 voted last.  If #1 voted last, and #2 defected, #1 might vote for it, _IF_ it was more desireable than "zero" for him.

Could #3 make a proposal like this:  "I propose that the money be split up among all who vote for this proposal."   #1 would have to vote for it, else he'd get nothing.

Jim Bell
jimbell@pacifier.com