In this family of puzzles, pirates vote not for justice, but for survival and self-interest. The approval rule has a peculiarity that changes the entire reasoning.
The five pirates: favorable tie
Riddle statement
Five pirates, ordered from oldest to youngest, must divide 100 coins.
The oldest proposes a distribution and everyone votes, including him. If the distribution receives at least half of the votes, it is approved; If not, the proposer dies and the next most senior pirate makes a new proposal.
Everyone is perfectly rational. Their priority is, in this order: survive, get as many coins as they can and, if everything else doesn't matter, throw another pirate into the sea.
What distribution should the most veteran pirate propose?
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Solution
Answer:
We reason backwards from the simplest case.
A pirate. He keeps the 100 coins: his vote is enough.
Two pirates. To reach at least half, one vote is enough. The proposer votes for himself and can keep everything:
Three pirates. At least 2 votes are required. If the first dies, the resulting case is two pirates: the second would receive 100 and the third, 0. The third pirate accepts any offer greater than zero, so the proposer gives him 1 coin:
Four pirates. At least 2 votes are required. If the first dies, the resulting case is three pirates: the second would receive 99, the third would receive 0, and the fourth would receive 1. The third pirate would receive 0 in that scenario, so he accepts any positive offer. The proposer buys the vote with 1 coin:
Five pirates. At least 3 votes are needed. If the first dies, the resulting case is four pirates: the second would receive 99, the third 0, the fourth 1, and the fifth 0. The third and fifth pirates would receive 0 in that scenario. The proposer already has his own vote and buys those two with 1 coin each:
That distribution obtains exactly the three votes necessary and maximizes what the veteran keeps.