You have five pirates, ranked in seniority from 1 to 5 and 100 gold coins. Starting from the highest ranked pirate, an offer is made on how to divvy up the coins. If the offer receives 50% or higher acceptance, then the offer stands. If it doesn't, then that pirate is killed and the next most senior pirate makes an offer. What does the highest ranked pirate offer to the other pirates in order to maximize his share?
You have to attack this one backwards. Use a pen and paper if they give it to you to map it out visually too if they allow you too. Remember, for these type of questions, its more abt how you think than getting the right answer.
To make this more clear, I will label pirates: P1, P2, P3...with P5 being the least senior. Attack this backwards: if it got down to the last 2 (P4 and P5), P4 would simply offer himself 100 coins and P5 0 coins, since he only needs himself to accept the offer.
A step before that, P3 is aware of the fact that P5 will get nothing if he does not accept this offer, so P3 will offer himself 99 coins, P4 0, and P5 1 coin. P5 will accept this because he knows that if he does not, in the next round, he will get 0 coins.
A step above that, P2 knows that P4 knows that he will get offered nothing if it makes it to the next round, so he also offers himself 99 coins, and P4 1 coin, and P3 and P5 nothing. P2 and P4 will accept this.
At the first level, P1 knows all of this. If it gets to the next round, P3 and P5 will be offered nothing. So for the 50% or more acceptance rate to be accepted, he offers himself 98 coins, P3 1 coin, and P5 1 coin, since at the next stage P3 and P5 will be offered nothing.
It is easiest to map this out visually and run through the scenarios from bottom up.
P1 2 COINS bonuses
P2 1 COIN
P3 1 COIN
Greed and fear would usually affect the logical thinking of people. So if P1 were to offer P1 (98), P2 (0), P3 (1), P4 (0) and P5 (1), there is a high risk that P2 and P4 will incite P3 and P5 to be greedy and reject the offer. This is the classic example of high reward and high risk.
A more conservative approach would be to give everyone a share on the gold, with the amount increasing at an increasing rate according to the seniority. Square root 100 by 5 we have 2.52. using this amount, we find the cumulative amount of gold starting from P5. Eg 2.52 to the power of 1/2/3/4/5.
P5 = 2.5
P5 + P4 = 6.3
P5 + P4 + P3 = 15.9
P5 + P4 + P3 + P2 = 39.9
P5 + P4 + P3 + P2 + P1 = 100
Thus P1 should offer the gold as P1(60), P2(24), P3(10), P4(4), P5(2)
pirates dont share gold coins. they either kill each other and run away with the gold coins
28(P1) 26(P2) 24(P3) 22(P4). Get rid of most junior pirate.
P1 - 50, P2- 25 , P3 - 12, P4 -6, P5 -3 dividing in the ratio of 50%.
so we get remaining four coins. 4 is shared to P1 & P2 as bonus
so that every senior pirates are satisfied >=50% of their juniors pirates.
finally P1 - 52, P2- 26 , P3 - 13, P4 -6, P5 -3
K.m.Kalyana Sundaram kalyan1978 at gmail dot com
It seems to me that this is a question to see how much risk you are prepared to take to maximise returns. The risk for failure is pretty severe, i.e death. Therefore you need two other pirates and yourself to agree. P4 and P5 have no risk of being killed, as if it gets that far then P4 just gives himself 100%, votes for it and gets the needed 50% vote.
P2 is incentivised to vote down the first offer, in any event.
Therefore i'd offer P4 and P5 33% each and take 34%. I really need this offer to succeed, else I'm dead.
The first and second ranked pirates are at the most disadvantage position. Assuming they are all greedy pirates, no matter what offer these first 2 pirates make, the last 3 can still vote 'nay', better still the first 2 will die and last 3 can divide all the gold coins. But if get down to 3 pirates left, the last one will get nothing. So the most senior pirate should offer this way:
P2 gets 50 because that is what he can get if P1 is killed. p2 and p3 each get 50coins.
By this, P1 can save his own ass and have some gold to spend.
P1 - 98 coins
P2 - 0 coins
P3 - 1 coin
P4 - 0 coins
P5 - 1 coin
Since P5 will accept any offer due to P4 having the ability take 100 coins when it is his turn. P2 will only need his own and P5's approval for acceptance. Thus rationally speaking, it is in P3's best interest to accept whatever P1 gives as P2 has the ability to gain acceptance without P3's permission. P1 just needs P3 and P5's support.
Pirate 1 should keep 96 coins for himself. offer 1 coin to pirate-4 and offer 3 coins to pirate-5.
i worked it backwards. lets say that all proposals have been rejected and now only pirate-4 and 5 survive. So pirate-4 can keep all 100 coins to himself and still satisfy the 50% criteria. under this pirate-5 gets 0 coins.
So it is in the interest of pirate-5 to save pirate-3. pirate-3 in turn has to offer pirate-5 something better than 0 i.e. atleast 1 coin.
this scenario is when pirate-1 and 2 are dead. to save himself pirate-2 would try to win the vote of pirate-5 to meet the 50% criteria. So naturally pirate-2 would offer atleast 2 coins to pirate-5.
same goes for pirate-1 who in turn to save his life would offer pirate-5 3 coins and to meet the 50% criteria would also offer pirate-4 1 coin.