Twenty Bank Robbers -- Game theory:)

Here's a puzzle for our game theorists.

Twenty cypherpunks robbed a bank. They took 20 million bucks. Here's
how they plan to split the money: they stay in line, and the first guy
suggests how to split the money. Then they vote on his suggestion. If
50% or more vote for his proposal, his suggestion is adopted.

Otherwise they kill the  first robber and now it is the turn of guy #2
to make another splitting proposal. Same voting rules apply.

The question is, what will be the outcome? How will they split the
money, how many robbers will be dead, and so on?

igor

>Here's a puzzle for our game theorists.
>
>Twenty cypherpunks robbed a bank. They took 20 million bucks. Here's
>how they plan to split the money: they stay in line, and the first guy
>suggests how to split the money. Then they vote on his suggestion. If
>50% or more vote for his proposal, his suggestion is adopted.
>
>Otherwise they kill the  first robber and now it is the turn of guy #2
>to make another splitting proposal. Same voting rules apply.
>
>The question is, what will be the outcome? How will they split the
>money, how many robbers will be dead, and so on?
>
It seems to me that the last two guys in line will _almost always_ vote for killing the suggestor.

the exceptions being for extreme suggestions like "let's split the money between #19 and #20", which I figure will get voted down by #s 2 thru 18.

-- Marshall

Marshall Clow     Aladdin Systems   <mailto:mclow@mailhost2.csusm.edu>

"We're not gonna take it/Never did and never will
We're not gonna take it/Gonna break it, gonna shake it,
let's forget it better still" -- The Who, "Tommy"

On Thu, 25 Jul 1996, Marshall Clow wrote:

> >Here's a puzzle for our game theorists.
> >
> >Twenty cypherpunks robbed a bank. They took 20 million bucks. Here's
> >how they plan to split the money: they stay in line, and the first guy
> >suggests how to split the money. Then they vote on his suggestion. If
> >50% or more vote for his proposal, his suggestion is adopted.
> >
> >Otherwise they kill the  first robber and now it is the turn of guy #2
> >to make another splitting proposal. Same voting rules apply.
> >
> >The question is, what will be the outcome? How will they split the
> >money, how many robbers will be dead, and so on?
> >
> It seems to me that the last two guys in line will _almost always_ vote for killing the suggestor.
> 
> the exceptions being for extreme suggestions like "let's split the money between #19 and #20", which I figure will get voted down by #s 2 thru 18.

Starting at the end, and working to the beginning:

If only 19 and 20 are left, 19 has 50% of the vote, and can take all. #20 
loses out.

Therefore, with 18,19, and 20 left, 20 will take whatever 18 offers, as 
it is better than getting nothing.

With 17,18,19,20 left, 17 should propose a split with 20, who will likely 
get a smaller cut from 18, because of the above.

With 16, 17, 18, 19, 20, three votes are needed, reducing the take for 
the majority, so no one other than #16 is acting in their best interest 
to approve a split, except for #19 trying to avoid losing out to 17/20 in 
the next round. Not enough for a majority.

Follow this forward, and find that any even numbered cypherpunk on 
Ritalin with UZI bankrobber is useless, as an additional person is needed 
to form the 50%-or-better as compared to the next round.

All even-numbered cypherpunks should then expect a short life expectancy. 
(You are number 6; who is number 1?)

Therefore, punk #2 should propose that the money be split equally between 
all even-numbered / disadvantaged punks, as they will otherwise all get 
bumped off by odd punks. The odd punks, of course, get nothing in #2's 
proposal.

Taking this one step further, and assuming that all clever punks have 
realized this, punk #1 proposes that the evens will perform the above 
split, if #2 is allowed to advancve this proposal, and therefore the only 
profitable option open to odd punks is to spilt the money between 
themselves, giving the even punks nothing.

Of course, if one of the punks is our recently arrived ^h^h^h^h
(just remembered I'd taken the pledge ...)  ;)

Rabid Wombat
Nocturnal Diseased Marsupial

The moral of this story is that it is good to be an odd punk ...

> 
> -- Marshall
> 
> Marshall Clow     Aladdin Systems   <mailto:mclow@mailhost2.csusm.edu>
> 
> "We're not gonna take it/Never did and never will
> We're not gonna take it/Gonna break it, gonna shake it,
> let's forget it better still" -- The Who, "Tommy"
> 
> 
>

At 9:09 -0500 7/25/96, Igor Chudov wrote:


>Here's a puzzle for our game theorists.
>
>Twenty cypherpunks robbed a bank. They took 20 million bucks. Here's
>how they plan to split the money: they stay in line, and the first guy
>suggests how to split the money. Then they vote on his suggestion. If
>50% or more vote for his proposal, his suggestion is adopted.
>
>Otherwise they kill the  first robber and now it is the turn of guy #2
>to make another splitting proposal. Same voting rules apply.
>
>The question is, what will be the outcome? How will they split the
>money, how many robbers will be dead, and so on?
>
>igor

I've read the differing scenarios and they seem to fall into two groups -
Either #1 makes an offer good enough to get 9 others to vote for it (and
thus save his life) or there is a blood bath ending with #19 getting all
the money and #20 with nothing but his life.

I think that if the blood bath occurs it will not get to the 2 survivor
stage. I think it will end at the 4 (or possibly 3) survivor stage. I base
this analysis on #20's best outcomes and interests. He will survive no
matter what (assuming that we ignore the cases where those who do not get
their fair share wack those who got money and take it) so this is not an
issue for him (he has no way of being killed for being too greedy). The
amount of money that he will get is totally dependent on what the current
"split proposer" offers him as an incentive to vote for the split (as
noted, once #19 gets to the top of the queue, he can [will?] grab
everything and cut out #20 so it is in #20's financial interest to vote for
a prior robber who will offer him some of the money). Since the vote must
be 50% or better for the proposed split, once it is #17's chance (ie: When
we are down to only 4 robbers and 2 yes votes will "win"), he can get #20's
vote by offering him at least 50% of the money (more than 50% will be an
incentive to #20 to take the deal since if he goes thumbs down to a 50/50
split with #17, #18 will only need to offer #20 the same 50/50 deal ["its
then 50% or nothing"] after #17 gets killed). #17 can hedge his bet by
offering #18 some of the rest (assuming a secret vote or all voting at the
same time in ignorance of how the other voted) since it might gain his vote
(a split between 17&20 leaves #18 out in the cold so he might go for some
money as opposed to none [in the case where #20 goes for 17&20 split]).