Re: “Democracy on the high seas” by Colin Carroll

A slightly modified version of the riddle discussed in detail by Colin Carroll. I recap the key parts of his description below. Rule 4 was added by me in order to eliminate any probabilistic issues from the problem.

You are the captain of a pirate ship with a crew of N people ordered by rank. Your crew just managed to plunder 100 Pieces of Eight. Now you are to propose a division of the 100 PoE, and the crew will vote on the division. The captain doesn’t vote except to break a tie. If the proposal fails, the captain walks the plank, and the first mate becomes captain, the third in command becomes first mate, and so on. Each pirate votes according to the following ordered priorities:

  1. They do not want to die.
  2. They want to maximize their own profit.
  3. They like to kill people, including crewmates.
  4. They prefer to be on good terms with the higher ranked crewmates.

The question is: what division do you, as the captain, suggest?

It is convenient to number the pirates in the reverse order of importance, so that the captain is Nth. This way, if the captain gets killed, the problem neatly reduces to the case N-1. For N\le 4 Rule 4 does not come into effect, so the answers are identical to Colin’s, then they begin to diverge, but for large N the answers are again the same. I summarized the solution in a Google spreadsheet, where the numbers give the distribution of booty, green background indicates who votes for the decision, and black spots are self-explanatory.

The captain survives if and only if one of the following holds:

  • N=1, trivially.
  • 3\le N\le 199. He gets to keep 99-\lfloor N/2\rfloor PoE, except in the cases N=3 and N=4 when he can do better, keeping 100 and 98, respectively.
  • N=199+2^k for k=1,2,3,\dots. Except for N=201, the captain gets nothing, and is only able to survive because of the support of those who are sure to walk the plank after him. Indeed, he can bribe 100 pirates and still needs \lfloor (N-201)/2\rfloor votes from the rest of the crew. These votes come from those who see a black spot hanging over them.

The distribution of looty for N>199 looks a little messy, but eventually it becomes distributed between the 100 pirates that are not directly threatened by a black spot.

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