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The pirate's treasure

The pirate's treasure



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5 pirates with a highly developed logic decide to distribute the 100 gold coins of their last booty.

The method used to distribute the treasure is as follows: the fiercest pirate of the five will make a proposal to distribute the treasure that will be put to the vote. If half or more of the votes (including yours) is favorable to your proposal, this is how the spoils are distributed. Otherwise the pirate will be thrown overboard to the sharks and the next one at the level of ferocity will make a new proposal following the same procedure as before until the less fierce pirate is reached.

These are pirates so bloodthirsty that they prefer to throw a partner overboard if they have the opportunity to get another proposal in which they get the same number of coins.

What is the proposal that the fiercest pirate must make to get as many coins as possible avoiding being swallowed by sharks?

Solution

You might think that the solution to this problem is to distribute as many coins as possible to avoid being shark grass and sacrifice our benefit even if nothing is further from reality.

To solve this problem we will assume the simplest possible case and we will go back in time. We number the pirates in order of ferocity being 5 the fiercest pirate and 1 the least fierce. If we reached the situation where only the two least ferocious pirates remained because the proposals of the rest of the pirates had caused their companions to have thrown them overboard, the pirate with a higher level of ferocity would propose to keep the 100 coins since his own vote is enough to get 50% of the votes.

The next fiercest pirate, 3, knows this situation and it is enough that he gives a coin to pirate 1 to give him his vote since he knows that if 3 is thrown overboard he will not have loot based on the above. This would ensure the pirate 3 two votes, his and 1 vote, so he could keep 99 coins without risk of being grazed by sharks.

Pirate 4, comes to the same conclusion so that in order to get much of the loot he knows he has to bribe another pirate. In this case and since we have commented that these are pirates so bloodthirsty that they prefer to throw a partner overboard if they know they will receive an equal proposal later, we cannot bribe pirate 1 since he knows that if 4 is thrown by the Overboard will get the same benefit based on the conclusion of the previous paragraph, so Pirate 4 must bribe Pirate 2 with a coin since in case they threw 4 overboard, they would not get any benefits as we have seen.

Finally, pirate 5 knowing all of the above will be enough to bribe pirates 1 and 3 with a coin to each since they both know that if they throw pirate overboard 5 they will not get any benefit from pirate 4. We might think that pirate 1 could vote in favor of throwing pirate 5 overboard since he has another chance of getting a coin when they are left only 3 pirates but since it is logical pirates knows that pirate 4 would propose a strategy that would surely be accepted and that would leave him out of the cast.