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The five pirates: empate favorable

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In this family of puzzles, pirates vote not for justice, but for survival and self-interest. The decisive detail here is that a distribution is approved if it reaches at least half of the votes.

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 oldest pirate makes a new proposal. They are all 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 cast should the oldest pirate propose?

Hints

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  1. Empieza by the caso of a single pirate and retrocede.
  2. Each pirate compares the current offer with what he would receive if the proposer died.
  3. The proposer only needs to buy the essential votes.

Solution

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Answer: $$98,0,1,0,1.$$ We reason backwards. With a pirate, he keeps the 100 coins: $$100.$$ With two pirates, one vote is enough to reach at least half. The proposer votes for himself, so he can keep everything: $$100,0.$$ With three pirates, at least 2 votes are needed. If the first dies, there would be a case of two pirates: the second would receive 100 and the third 0. The third pirate can be purchased with 1 coin: $$99,0,1.$$ With four pirates, at least 2 votes are needed. If the first one dies, it would be the case of three pirates: $$99,0,1.$$ The third pirate would receive 0 in that scenario, so the proposer offers him 1 coin and keeps the rest: $$99,0,1,0.$$ With five pirates, at least 3 votes are needed. If the first one dies, there would be the case of four pirates: $$99,0,1,0.$$ The third and fifth pirates would receive 0 in that scenario. The proposer already has his own vote, so he buys those two votes with 1 coin each: $$98,0,1,0,1.$$ This distribution gives him enough votes and maximizes what he keeps.

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