# 5 logical pirates find a 100 coin chest

• Bob-omb
In summary, a group of pirates with different ranks are trying to figure out how to split a chest of coins. After deciding to let the most senior pirate make a decision, the rest of the pirates will vote and if at least half agree, the decision stands. If not, the senior pirate is thrown overboard and the next senior pirate decides. To maximize their gold without being thrown overboard, the pirates will split the coins among all the odd-numbered pirates, with the exception of the last two. This means that 203 pirates will get 100 coins, while 204 and above will vote yes and receive their share. For a larger group of 524488 pirates, the coins can still be split in the same manner.
Bob-omb
they are trying to figure out how to split the coins.

However all of the pirates are different rank in that Pirate A is great than Pirate B and so on, such that Pirate E is the lowest rank.

they have decided that let the most senior pirate decide how to split the coins, then all the pirates will take a vote. if at least half the pirates agree with the decision then that is how they split, if more than half disagrees, the most senior pirate is thrown overboard and the next senior pirate decides how to split the gold.

assuming all of the pirates wants to maximize the gold they get without being thrown overboard, how would the gold be split and why?

A - 60%; b - 0%; C - 20%; d - 0%; E - 20%

Reasoning: If A, B, and C are tossed overboard, the game reduces to a dictator game between D & E with D getting 100%.

But before getting to that point, you have to have a game between C, D, & E. D will vote to reject no matter what, so the C, D, & E game is really an ultimatum game between C & E. Being human pirates, E will reject any offer below 20% (the tipping point is actually about 16%, but acceptance at exactly the tipping point would be far from certain). Below that percentage, the psychological rewards (and corresponding physiological rewards) of punishing C's unfairness outweigh the monetary rewards, even if C getting 0 money were the only punishment (personally, I think adding in the reward of tossing C overboard would increase the threshold for rejection, but no one has done that type of experiment).

Before getting to the CDE game, you have to have a 4 player B, C, D, & E game. C will surely reject and D will surely receive 0 for rejecting. E gets no advantage either way. The fact that D would suffer for rejecting while E would not essentially reduces this to a 2-player ultimatum game between B and D with 20% once again being the "unfairness" threshhold.

But, before getting to the BCDE game, a five player game has to be played between A, B, C, D, & E. Finally, you have a different game - a 3 player ultimatum game between A, C, and E (since C and E would suffer if the game went to 2-player game between B & D). You would think this would change the dynamics but, surprisingly, a 3 player ultimatum game requiring unanimous consent runs very consistent with the 2-player version. A keeping 6 times as much as C and E is the tipping point, but A would want to keep less than 4 times as much as C and E to be certain he wouldn't get tossed overboard - especially since the act of tossing a pirate overboard is pleasing in itself and would add to the natural tendency to reject unfair offers.

http://ethesis.helsinki.fi/julkaisut/eri/hecer/disc/140/ultimatu.pdf

(Not quite the answer that "pure logic" dictates, but surprisingly similar once allowances for human psychology are made.)

There is a major exception to the "unfairness" threshhold. If the responder can leave comments on an index card that the proposer may or may not read, the responder will accept a more unfair offer. Of course, that experiment didn't allow the responder to use profanity in their responses and that would never work for pirates. (But it does strengthen my feeling that the opportunity to throw a pirate overboard if they reject the offer would increase how much the responder would have to receive. There's a definite psychological reward for punishing unfairness and that psychological reward is worth some money.)

http://www.andrew.cmu.edu/user/exiao/ef.pdf

There's also another exception. If the responder has to state the minimum offer he'll accept before any offer is made, the threshold for rejection is higher - possibly because it's hard to imagine how unfair people can be until it actually happens? And when confronted by the reality of unfairness, people become more accepting of it?

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This is not a whole puzzle. There are at least next tree questions:

1. How many pirates is not able split 100 coin chest ? Found smallest number !

2. How many pirate is able split 100 coin chest, when this number is bigger than in question 1 ?

3. 524488 pirates and 100 coin chest. Is able to split coins ?

[color="#black"]1. 203 pirates
2. 204 pirates (100 coin for 1,3,5...199 and pirates 203 and 204 will be vote yes)
3. YES (100 coin for 1,3,5...199 and pirates 262245-524488 will be vote yes)[/COLOR]

## 1. How can 5 pirates divide 100 coins evenly?

The solution lies in the logic behind the distribution of coins. Each pirate must get a fair share of the coins, otherwise they will vote against the distribution and no one will get anything.

## 2. What is the best strategy for the pirates to get the most coins?

The first pirate should propose a distribution where he gets 99 coins and the other pirates get 0. The second pirate will vote against this as it is unfair. The third pirate will then propose a distribution where he gets 98 coins and the first pirate gets 1, which will be rejected by the first pirate. This continues until the fifth pirate proposes a distribution where he gets 96 coins, the fourth pirate gets 1, and the first and second pirates get 0. This will be accepted and the fifth pirate will get the most coins.

## 3. What happens if one pirate decides to be greedy and take all 100 coins?

If one pirate decides to take all the coins, the other four pirates will vote against the distribution and the first pirate will end up with nothing. This is why the first pirate must propose a distribution that is fair to all, otherwise they risk getting nothing.

## 4. Can the pirates come up with a solution where all five of them get an equal number of coins?

No, it is not possible for all five pirates to get an equal number of coins. The closest they can get is a distribution where the first pirate gets 98 coins, the second gets 0, the third gets 1, the fourth gets 0, and the fifth gets 1. This is the most equal distribution possible.

## 5. What is the significance of this famous problem in the field of mathematics and game theory?

The 5 logical pirates problem is a classic example of game theory, where mathematical principles are applied to strategic decision making. It demonstrates the concept of a Nash equilibrium, where the best outcome is achieved when each player acts in their own self-interest. The problem also highlights the importance of negotiation and compromise in finding a mutually beneficial solution.

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