5 pirates discover 100 gold coins. the pirates are ranked by
seniority, pirate A is the most senior, while E is the least
senior. Then they decided to split the loot.
The most senior pirate will propose a distribution, den
everybody votes, including the most senior pirate. if at least 50%
of the pirates accept the proposal den the gold will be divided as
as proposed. if not the most senior pirate will be killed and the
process starts again with the next senior pirate, until a plan is
approved.
the pirates are extremely rational and noe how others will vote
in every scenario.
staying alive is their top priority, followed by getting as much
gold as possible, and finally, if given a choice between otherwise
equal outcomes, to have fewer pirates on the boat.
the most senior pirate thinks for a moment and then proposes a
plan that maximizes his gold, and which he knows the others will
accept.
a) how does he divide up the coins?
b)what plan would the most senior pirate propose on a boat full
of 15 pirates?
Pirate 5 being the most senior knows that he needs to get 2
other people to vote for his solution in order for him not to be
executed. so who can he get to vote for him, and why would they
choose to vote for him? if you start thinking that pirate 4 will
never vote for him, because he would rather have 5 die and then be
in charge and take it all for himself, you are on the right track.
but it gets more complicated.
lets consider if there were only 1 pirate. obviously he would
take it all for himself and no one would complain.
if there were 2 pirates, pirate 2 being the most senior, he
would just vote for himself and that would be 50% of the vote, so
he's obviously going to keep all the money for himself.
if there were 3 pirates, pirate 3 has to convince at least one
other person to join in his plan. so who can he convince and how?
here is the leap that needs to be made to solve this problem.
pirate 3 realizes that if his plan is not adopted he will be
executed and they will be left with 2 pirates. he already knows
what happens when there are 2 pirates as we just figured out.
pirate 2 takes all the money himself and gives nothing to pirate 1.
so pirate 3 proposes that he will take 99 gold coins and give 1
coin to pirate 1. pirate 1 says, well, 1 is better than none, and
since i know if i don't vote for pirate 3, i get nothing, i should
vote for this plan.
now we know what happens when there are 3 pirates. so what
happens with 4? well pirate 4 has to convince 1 other person to
join in his plan. he knows if he walks the plank then pirate 3 will
get 99 coins and pirate 1 will get 1 coin. pirate 4 could propose
giving pirate 1 two coins, and surely pirate 1 would vote for him,
since 2 is better than 1. but as greedy as he is, pirate 4 would
rather not part with 2 whole coins. he realizes that if he gets
executed, then pirate 3's scenario happens and pirate 2 gets the
shaft in that scenario (he gets zero coins). so pirate 4 proposes
that he will give 1 coin to pirate 2, and pirate 2 seeing that 1 is
better than 0 will obviously vote for this plan.
a common objection is that pirate 2 is not guaranteed to vote
for this plan since he might hope for the case when there are only
2 pirates and then he gets all the booty. but that is why i said
that the pirates are extremely intelligent. pirate 2 realizes that
pirate 3 is smart enough to make the optimal proposal, so he
realizes that there will never be 2 pirates left, because 3 doesn't
want to die and we just showed that 3 has a winning proposal.
so lets sum up at this point
once you see the pattern it becomes very clear. you have to
realize that when a pirate's plan does not succeed then that means
you are in the same situation with one less pirate.
1. pirate 1 needs 0 other people to vote for him. so he votes for
himself and takes all the money. 2. pirate 2 needs 0 other people
to vote for him. so he votes for himself and takes all the money.
pirate 1 gets 0. 3. pirate 3 needs 1 other person to vote for him.
he gives 1 coin to pirate 1 for his vote - if we are reduced to 2
pirates, pirate 1 gets 0 so pirate 1 knows 1 is better than none.
pirate 3 takes 99. pirate 2 gets 0. 4. pirate 4 needs 1 other
person to vote for him. he gives 1 coin to pirate 2 - if we reduce
to 3 pirates, pirate 2 gets 0 so pirate 2 knows 1 is better than
none. pirate 4 takes 99. pirate 3 gets 0. pirate 1 gets 0. 5.
pirate 5 needs 2 other people to vote for him. its clear now that
the 2 people he needs to convince are the 2 who get shafted in the
4 pirate scenario - pirate 3 and pirate 1. so he can give them each
1 coin (which is better than 0 - what they would get otherwise) and
keep 98 for himself.
what happens if there are 15 pirates? pirate 15 needs 7 other
people to vote for him, so he recruits pirates 13,11,9,7,5,3, and 1
with 1 coin each and keeps 93 coins himself. those pirates will all
vote for him because they know that they get 0 coins if he dies and
pirate 14 is in charge.
Saw this question before in a website.
