Five pirates (Arrrthur, Barrry, Charrrles, Darrrian and Edwarrrd) have stolen 100 doubloons and have agreed on a scheme to decide how to divide them:
Arrrthur must propose a division of the booty, and all the pirates then vote on this proposal.
If it fails to get a strict majority, Arrrthur is executed, Barrry proposes a division, and so on.
Each pirate follows a strategy whose descending priorities are: 1) remaining alive 2) getting maximum gold 3) killing others.
What division does Arrrthur propose?
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9 comments:
This is what I have so far:
Edwarrrd will never die, so he will always vote no to cause more carnage, unless he concludes that some upcoming division may pass and give him less gold than the presently proposed division.
This is impossible if it becomes Darrrian's turn, because the only upcoming division is Edwarrrd's own.
At Charrrles' turn, Edwarrrd would not conclude that any upcoming division will pass because he has veto power over Darrrian's division, and he will veto any proposal by Darrrian. However, if it gets to Charrrles, Charrrles and Darrrian know that if it gets to Darrrian, Darrrian is a dead man. So Charrrles can propose to keep all the doubloons for himself, and Darrrian will support him to stay alive and the division will pass.
Therefore, if it gets to Barrry, Edwarrrd will support any division that gives him at least one doubloon.
This is correct, as far as it goes.
Carrying on, then.
All pirates know that if Charrrles gets his turn, he can ask for all the doubloons. Darrrian has no choice but to support Charrrles, because if he doesn't, even an offer to give all the doubloons to Edwarrrd will be refused and he will walk the plank.
Barrry, knowing this, can buy the vote of Edwarrrd by promising a single doubloon, and with his own vote get a 50% share of votes. To get a majority, he can buy the vote of Darrrian with one doubloon too, since the best Darrrian can hope for from Charrrles is his life.
Arrrthur can use the same logic. If his division does not pass, Barrry will give himself 98 doubloons and one to Darrrian and one to Edwarrrd. To buy a majority, he just gives two doubloons each to Darrrian and Edwarrrd and keeps 96 for himself.
Edwarrrd would be wise to conspire with Darrrian and promise to accept a 50-50 division, thereby getting them both better results for Priorities 2 and 3, but Darrrian knows that Edwarrrd's piratical instincts and unfailing logic will overrule his promise when the time comes.
Got it in two.
I think I might have a sligtly more optimal solution for Arrrthur.
Much like Steve, let me work backwards:
Round 5:
Edwarrrd takes 100 gold, everyone else is dead.
Round 4:
Edwarrrd will vote no, since round 5 is optimal. Darrrian can't achieve 51% of votes and dies.
Round 3:
Darrian will vote yes for 0 gold since at least he lives, Charrrles can keep 100 gold since he will have 2/3 of the yes votes and wins. The voting will never get past round 3.
Round 2:
Darriaan, and Edwarrrd will vote yes for 1 gold since if it gets to round 3 they will get nothing. Barrry can win with 98 gold, it never gets to round 3, and Charrrles gets nothing.
Round 1:
Arrthur can buy up Darrrian and/or Edwarrrd with 2 gold each, (1 more than the next round to make up for the lack of murdering). This is Steve's answer. The thing is that Charrrles will vote yes for 1 gold since if it gets to round 2 he will get nothing. Thus Arrrthur can obtain two other votes for a total of 3 gold, (1 to Charrrles, and 2 to Darrian), and can keep 97 gold for himself.
Matt's distribution seems flawless to me. Can the original poster share the source of this nut? Is there an official solution?
I don't recall the source any more. Matt's improvement certainly seems valid and I don't see any further improvements.
Wikipedia has an article on the Pirate Game. The result is slightly different as a tie vote is sufficient to pass a proposed distribution.
There is a link to a Scientific American article on the problem by Ian Stewart.
Unsurprisingly, Arrrthur can do a little better in the case where a tie passes a proposition: 98 to Arrrthur, 1 to Charrrles and 1 to Edwarrrd.
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