Six pirates and one dollar

This is taken from Skiena’s “The Algorithm and Design Manual” book. It’s a bit more interesting that the standard version.

Description:

Reconsider the pirate problem above, where only one indivisible dollar is to be divided. Who gets the dollar and how many are killed?

For context see previous post.

Solution Link to heading

Keeping in mind the priorities are the same:

1. staying alive
2. getting the $1 (can be fulfilled only if 1. is still valid)

From the perspective of the Captain he knows that other pirates won’t be happy with him claiming the $1, so will vote against him. He’s potentially facing a 5:1 vote kill, but he also knows that the other 5 pirates primarily want to save their skin, and can recognise that when they end up being the senior-most pirate, they’ll also get killed, since every other pirate can turn on them.

Captain can give the dollar to Senior 1, but others have no reason to accept. In particular the guy to convince is Junior 1, because he can see that if every other senior gets killed and he’s left with Junior 2, he’ll claim the price. Junior 1 will simply reject any proposal that doesn’t involve him getting the dollar.

Similarly, Junior 2 will reject any proposal that doesn’t get him the dollar, he can afford to do this and not get killed.

So this is how I see the game being played out:

1. in the first round, the Captain gets killed because regardless of his proposal the other seniors have no interest in keeping him alive, so they’ll vote against him, so long their vote doesn’t affect their own lives. Ditto for every other pirate for that matter.

2. we’re now in a 5-man game, and Senior 1 also gets killed for the same reason as the Captain.

3. in round 3 we’re now in a 4-man game with Senior 2, Mid rank, Junior 1, and Junior 2 left. Senior recognises that if he keeps the $1, every other pirate will turn on him and he’ll get killed. So the best he can do is give the money to someone else. He can choose any allocation because his vote and whoever receives the dollar will be 50% of the remaining. If for example he proposes (0, 1, 0, 0), the proposal will pass even if Junior 1 and Junior 2 reject it, because Mid rank, the beneficiary, will certainly accept it.

In summary: