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Matthew Wampler-Doty wrote:
> There is a well known problem in game theory regarding 10 "Voting
> Pirates" and a split of 100 gold pieces of treasure. The problem may be
> presented a variety of ways; I will give one for the purposes of
> demonstration.
>
> Each pirate has a unique rank from 1 to 10. Each round the highest
> ranking (say with rank R) pirate offers a split to the other pirates,
> and they all vote to approve or not (all R of them). The head pirate
> needs at least a 1/2 majority for his split to go through, and if it
> fails to go through, he is killed, and then game is replayed with one
> fewer pirates.
>
> A unit of Gold may not be split. A pirate prefers N+1 units of gold
> over N units, always. A pirate prefers to keep his life over any amount
> of gold.
>
> For a presentation of the "classical" solution to this problem, I refer
> to Wikipedia: http://en.wikipedia.org/wiki/Pirate_game
>
> But another, less palatable solution to this problem goes as follows:
> All 10 pirates survive, the pirate with rank 10 gives himself all of the
> gold and the others nothing, and they all vote to approve of this.
>
> Why is this a solution? Because for any one pirate, if they are voting
> to approve, they cannot change their vote unilaterally and get change
> the outcome that the head pirate wins the majority. This makes this
> horrible solution a Nash Equilibrium, technically.
>
> Another solution is the "naive" everyone gets 10 gold pieces solution,
> and no pirate dies. This would happen if the pirates all held the
> contingent strategy that if the pirate with rank 10 didn't split the
> gold evenly, they'd all vote to kill him. Again, no pirate has any
> unilateral power, and the first pirate certainly wants to keep his life,
> so he provides and they all vote conform to the voting strategy.
>
> One can place more restrictions on the pirates. One could demand that
> they all have complete contingent strategies, for instance. I will
> contend (and prove, if necessary) that having complete contingent
> strategies, and subgame perfect equilibria don't "cure" the pirates of
> their apathy.
>
> In fact, I have yet to find a truly acceptable, formal criteria to pick
> out the classical solution to this problem. Any ideas?
>
> On a side note, the riddle and the problem of apathy came up in a class
> I had under an Economics professor. His solution was to try to ban
> apathy, and demand that agents "vote their preferences." I was later
> surprised to discover that he had written this article back in
> 2004: www.slate.com/id/2107240/">http://www.slate.com/id/2107240/
>
> Go figure.
>
> Matthew P. Wampler-Doty
Suppose a pirate never accepts an outcome when by following a certain
strategy he can enforce a better outcome for himself. Call such a
pirate rational of order 1.
Suppose a pirate never accepts an outcome when by following a certain
strategy he can enforce a better outcome for himself, given that all
the pirates are rational of order 1. Call such a pirate rational of
order 2.
And so on.
Assume the pirates are rational of order n for every positive integer
n.
Then the outcome is the one presented in the Wikipedia article.
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