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Matthew Wampler-Doty wrote:
> Rupert wrote:
> > Rupert wrote:
> >> Rupert wrote:
> >>> 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.
> >> I should be a bit clearer about what I mean. When I say a pirate never
> >> accepts an outcome when he can enforce a better one, I mean he never
> >> accepts the *possibility* of an outcome when he can enforce a better
> >> one. Thus the pirates are maximiners.
> >>
> >> The pirates are maximiners of order 1. That is, of all the available
> >> strategies, they pick the one such that the worst possible outcome is
> >> as good as possible.
> >>
> >> And the pirates are maximiners of order 2. That is, of all the
> >> available strategies which are consistent with being a maximiner of
> >> order 1, they pick the one such that the worst possible outcome is as
> >> good as possible, given that all the other pirates are maximiners of
> >> order 1.
> >>
> >> And so on.
> >>
> >> This forces the solution in the Wikipedia article.
> >>
> >> I don't think we need to assume that the pirates are full maximiners. I
> >> think we just need to assume that death has infinite disvalue (the
> >> pirates will never allow the possibility of death when it can be
> >> avoided) and the pirate will never select a strategy when there is
> >> another strategy that strongly dominates it (is guaranteed to produce a
> >> better outcome).
> >
> > I shouldn't have said "strongly dominates". I should have said "weakly
> > dominates". A strategy weakly dominates another if it is guaranteed to
> > produce an outcome at least as good, and there is a possibility that it
> > will produce a better outcome.
> >
>
> After posting I thought about this problem a little bit, and I too
> concluded that including the assumption that all the pirates must all
> have weakly dominant strategies was the proper criterion needed to
> select the classical solution.
>
> I'll take our convergence on the same answer as me as a strong
> indication of it's correctness :)
>
> Thank you greatly!
> Matthew P. Wampler-Doty
I proposed the problem to a friend of mine. He misinterpreted it,
thinking that the pirate who proposed a distribution did not have the
right to vote on it (though he had the right to make a casting vote if
there was a tie). This is an interesting variant of the problem. The
distributions go like this:
2 pirates: (0,100)
3 pirates: (99,1,0)
4 pirates: (0,99,1,0)
5 pirates: (98,1,0,1,0)
6 pirates: (0,98,1,0,1,0)
etc.
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