The pirates and the treasure (ENG/RUS)
Ten pirates found 50 gold coins and want to split it. All pirates have relative ranks and the pirate with the highest rank divides the treasure. Then all pirates vote on the proposal and if at least 50% of pirates vote for it, the money is split; if however less than 50% of the pirates support the proposal, then the proposer is killed and the next most senior pirate makes a new proposal and the entire process is repeated. When the pirates vote, each of them makes his decision based on three factors (listed from the highest to the lowest priority):
1. Each pirate wants to survive.
2. Each pirate wants to maximize his profit.
3. Each pirate would rather vote to kill other pirate, if it doesn't affect his own well-being.

A. How will these ten pirates split 50 gold coins?
B. What's the largest number of pirates that can split 50 gold coins without killing a single pirate? How would they do it?

Russian version:

Десять пиратов нашли 50 золотых монет и хотят их поделить. Все пираты имеют относительный ранг и старший (по рангу) пират делит сокровище. После этого все пираты голосуют, и если хотя бы 50% пиратов голосует за предложение старшего, то оно утверждается. Если же менее 50% пиратов поддердивают это предложение, то старшего убивают и его место занимает следующий по старшинству пират. Таким образом процесс дележки начинается заново. Пираты основывают свое решение голосовать "за" или "против" на трех факторах (перечислены в порядке убывания важности):
1. Каждый пират хочет остаться в живых.
2. Каждый пират хочет получить максимальное количество монет.
3. При прочих равных каждый пират хочет убить другого пирата.

А. Как эти десять пиратов разделят 50 монет?
Б. Какое наибольшее число пиратов может разделить 50 монет и при этом обойтись без убийств? Как они это сделают?

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All comments: 8
Anatoli Sizykh: Rich dead or poor alive.
I can see few outcomes, depends on which factor has more power:

1) If all the pirates rather stay alive, when they would split money evenly after first vote.

2) If higher ranked pirates want to get a bit richer, then they would take a risk to convince others to split money between eight pirates and apparently kill two lowest ranks.

3) If desire to kill prevails, then pirates will keep voting against every proposal until second lowest ranked takes all the money.
2010-02-28 07:02:45    
Armin Shamshiri: to Anatoli Sizykh:
The priorities come in the exactly same order as I stated them:

1. Each pirate wants to survive - that's the most important factor for each pirate; pirate would rather live with 0 gold than die with a million.

2. Each pirate wants to be as reach as possible.

3. Pirate will vote "kill" if and only if his life and his money will not be in danger as a result of such voting.

**************************************************************

"1) If all the pirates rather stay alive, when they would split money evenly after first vote."


Of course all pirates want to stay alive, but why would they split the money evenly? It is not even always possible, what if there are 9 or 11 pirates? And why would the leading pirate decide to give others so much money?

************************************************************

"2) If higher ranked pirates want to get a bit richer, then they would take a risk to convince others to split money between eight pirates and apparently kill two lowest ranks."

Every single pirate wants to be rich. By the rules of the problem only the pirate who splits the money can be killed and only if more than 50% votes for it. So in case with 10 pirates, the most senior one can be killed if at least 6 pirates vote for that. Two least senior pirates can not be killed ever, because the rules simply do not allow it.

*********************************************************

"3) If desire to kill prevails, then pirates will keep voting against every proposal until second lowest ranked takes all the money."

Desire to kill prevails only if the pirate can be sure that his own life and money will not depend on whether he votes "yes" or "no". If the pirate knows that if he votes "no" then he would die himself a few rounds later and if he votes "yes" he will survive, then the pirate would certainly vote "yes" to save his life.
2010-03-01 01:03:22     Last updated: 2010-03-02 14:03:54    
Anatoli Sizykh: They did not vote to kill
"1) If all the pirates rather stay alive, when they would split money evenly after first vote."


Of course all pirates want to stay alive, but why would they split the money evenly? It is not even always possible, what if there are 9 or 11 pirates? And why would the leading pirate decide to give others so much money?

If all the pirates smart enough to understand that if majority is not satisfied with the proposal, they can be killed.

………………

"2) If higher ranked pirates want to get a bit richer, then they would take a risk to convince others to split money between eight pirates and apparently kill two lowest ranks."


Every single pirate wants to be rich. By the rules of the problem only the pirate who splits the money can be killed and only if more than 50% votes for it. So in case with 10 pirates, the most senior one can be killed if at least 6 pirates vote for that. Two least senior pirates can not be killed ever, because the rules simply do not allow it.

Here is assumed that eight senior pirates realize the risk of been killed if the lower ranked majority keeps rejecting every proposal. While second lowest rank has no risk to be killed in any outcome (unless he votes NO for his own proposal).

Please note: pirates do not vote to kill- they vote how to divide the money, and at the moment of the vote nobody knows other guys intentions.

………………


"3) If desire to kill prevails, then pirates will keep voting against every proposal until second lowest ranked takes all the money."


Desire to kill prevails only if the pirate can be sure that his own life and money will not depend on whether he votes "yes" or "no". If the pirate knows that if he votes "no" then he would die himself a few rounds later and if he votes "yes" he will survive, then the pirate would certainly vote "yes" to save his life.

Even if the pirate votes “yes” it does not guarantee him to stay alive. On the other hand- if the pirate votes “no”, he has a chance to make a proposal which can satisfy the majority and live wealthier.


Kind Regards,

Anatoli Sizykh

2010-03-01 19:03:39    
Armin Shamshiri: to Anatoli Sizykh
Please note: pirates do not vote to kill- they vote how to divide the money, and at the moment of the vote nobody knows other guys intentions.

Each pirate knows that all pirates vote according to the three influencing factors. Pirates are greedy and evil, but they are also completely logical, so they follow these three factors no matter what. And thus each pirate can also try to deduce how other pirates will vote. He doesn't "know" how others vote, but he can calculate it. And that's the trick for the leading pirate - to find a way to split gold in such a way, so that he can both stay alive and keep as much gold as possible.

So if you think that the best strategy for #1 pirate is to give everyone 5 gold, then there should be some reason why 5 gold and not 4 or 3 or 6 or some other number.

Even if the pirate votes “yes” it does not guarantee him to stay alive. On the other hand- if the pirate votes “no”, he has a chance to make a proposal which can satisfy the majority and live wealthier.

It is true that in some cases voting "no" can be more beneficial to the pirate than voting "yes". It is, however, unclear whether voting "no" is always good for each pirate.
2010-03-02 14:03:33     Last updated: 2010-03-03 05:03:25    
Anatoli Sizykh: "fair" proposal
split money in the following order (from highest to lowest rank):
10)46
8),6),4) and 2)- each gets 1 gold coin
the rest of the pirates get no gold.
2010-03-04 20:03:56    
Armin Shamshiri: to Anatoli Sizykh
Good job :)
How about part B?
2010-03-05 10:03:32    
Anatoli Sizykh: How much is "too much"?
the maximum number of pirates- 102;
leader and 101) will get nothing but life;
100),98), 96) etc. (even numbers)- each gets one coin.
2010-03-05 21:03:55    
Armin Shamshiri: to Anatoli Sizykh
102 will work, but it is not the largest possible number.
2010-03-07 15:03:26    

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