An interesting logic problem
that runs as follows:
(of different ages) have 100 gold coins
to divide amongst themselves. They decide on the following approach to determine how much each pirate receives:
The eldest pirate proposes an allocation. All pirates (including the eldest) then vote on the proposal. If the majority accept the proposal then the coins are divided in the way suggested. If not, then the eldest pirate is executed and the new eldest amongst the remaining pirates proposes a new allocation. If the votes are tied then this is enough for the proposal to be accepted.
Assuming that the pirates are motivated primarily by survival, then to a lesser extent by greed and finally to the least extent by sadism (i.e. they'd prefer to receive a gold coin and see someone get executed than just receive one coin earlier, but would prefer one coin to none and an execution; and obviously would prefer 0 coins and surviving to 100 coins and being executed), and act in a logical way, what is the maximum number of coins the eldest pirate can get?
Although the problem revolves around the eldest pirate, in order to solve the problem it is preferable to work from the youngest and work up to the case of eldest. Thus let the youngest pirate be A and the eldest pirate E.
A knows that if only he and B survive the earlier rounds, then B will get all 100 coins as his vote in favour counts as a majority. He also knows that B will prefer this outcome above all others and so will vote against any proposal made by C - even if C offers all 100 coins to B, he'd rather see C get executed and then take the 100 coins than just take the coins from C. As C will obviously vote for his own proposal, A therefore will vote in favour of any proposal by C which offers him just a single gold coin, as by voting for he receives a coin whereas by voting against he receives 0.
In order for his proposal to be accepted, D needs one other to vote with him. By offering 100 coins to C he can secure his vote, as this is more than the 99 he will receive if D is executed and he is forced to buy A's vote. However, B's vote can effectively be bought for 1 gold coin as he knows that if D is executed, A and C will split the coins between them and so he receives nothing. An offer of 0 to B is insufficient to secure his vote as he'd prefer to see D get executed and then receive 0 from C than simply accept 0 at this stage.
Thus in the case of 4 pirates, D receives 99 coins and B receives 1, whilst A and C receive 0. In order to get his proposal accepted E (the eldest in this case) requires the vote of 2 others, and it will obviously be easiest to get these votes from A and C who know they will get 0 coins if they vote against. In order to over-rule the sadism motive for A and C he only needs to offer them a single coin each.
Hence the eldest pirate can survive and keep 98 of the gold coins by offering 1 to A and C.
Krimson observes: I am not entirely convinced. It seems to me that this is a problem that is best analysed in terms of game theory. If we look at the case when there are three pirates left, doesn't the fact that A can execute C give him quite a says lot of leverage in negotiations with C, so that A would get more than 1 coin? After all the premises of the problem indicate that execution is associated with a hefty negative pay-off. It seems to me that this is the kind of negotiation game which has no analytic solution.
Which seems entirely reasonable if your objective is actually finding out what people would do in such a situation. However, I think this objection basically illustrates the dangers in modelling humans by mathematics in the first place. Given the rather convoluted conditions outlined, I think the whole point of this exercise is in thinking through its logic - if a puzzle is interesting, I'm happy to find a solution within its parameters, regardless of whether that translates to a realistic outcome (which I guess drops me in the pure maths camp where the real world is of little concern). Perhaps if we instead proposed 5 computer programs arguing over resources...