In this family of puzzles, each pirate weighs not only what he gets now, but what he would get if the proposer were eliminated. The approval threshold changes everything.
The five pirates: strict majority
Riddle statement
Five pirates, ordered from oldest to youngest, must divide 100 coins.
The oldest proposes a distribution and everyone votes, including him. To be approved, the distribution must receive more than half of the votes; 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:
Two pirates. With a strict majority of 2 votes, 2 are needed votes: the proposer needs the other to vote in favor. If he dies, the other keeps only the 100 coins; He will not accept anything less. The proposer must offer you exactly 100 and is left with nothing:
Three pirates. 2 votes are needed. If the proposer dies, the resulting scenario deals:
The second pirate would receive 0; It is enough to offer him 1 coin so that he prefers to vote in favor. The third party would receive 100 and will never accept less. The optimal distribution is:
Four pirates. 3 votes are needed. If the proposer dies, the resulting scenario distributes:
The proposer already has his own vote and needs two more. Buying the second one would cost 100 coins; On the other hand, the fourth pirate would receive 0 and accept for 1, and the third would receive 1 and accept for 2. The optimal distribution is:
Five pirates. 3 votes are needed. If the proposer dies, the resulting scenario distributes:
The proposer has his own vote and needs two more. The second pirate would receive 97 coins: impossible to buy. The fourth would receive 0 and accepts for 1; the fifth would receive 1 and accepts for 2. The proposer keeps 97 coins: