Some Pirates and Their Treasure

Studying the behavior and reasoning that humans have is a fascinating field, and trying to understand how situations unfold based on rational thinking is a highly relevant and interesting field of study.  It does happen to merge mathematics and the social sciences quite beautifully as well, and here I wanted to talk about one of the most fundamental examples in this area (which doesn’t happen to require much mathematics, but is intriguing nevertheless). This is the dilemma of the pirates and their gold, which if you haven’t seen already, will make you see the intricacy of game theory.

Upon a island in the Caribbean, 5 pirates have a sack of 100 gold coins which they ransacked from a nearby innocent town.  The most senior pirate is obviously in charge, and gets to decide the distribution of coins, but if more than half of the five pirates (which include the senior one) disagree, there is mutiny, and this senior pirate is killed, after which the next most senior pirate takes the same role.  These pirates are intrinsically clever and greedy, and therefore rational in every way.  What is the distribution that the senior pirate will employ to maximize his earnings, and also ensure his survival?  Think about it, and remember to think about it rationally.  I have laid out the analysis below this inspiring picture of a jolly pirate.

pirate7.jpg

Yet again, we will bring it down to the simplest situation and see what we can derive from there.  To make things easier, the most senior pirate will be Pirate 5, and next is 4, and so on until 1.

In the case of two pirates, Pirate 2 can simply allocate all the gold to himself, because even after doing so, at least half of the pirates are in favor of the allocation.

Pirate 2: 100        Pirate 1: 0.

Moving onto to three pirates, we can realize that Pirate 1 is in favor of anything at this point, because if Pirate 3 is killed and it is left to Pirate 2, Pirate 1 will be left with nothing based on the logic we just used above.  So Pirate 3 can give one gold to Pirate 1, and keep the rest, gaining Pirate 1’s vote of approval and getting 99 coins at the same time.

Pirate 3: 99        Pirate 2: 0        Pirate 1: 1

With four pirates, we can see that Pirate 2 is desperate for anything, as if Pirate 4 is killed and Pirate 3 is left in charge, Pirate 2 will be left with absolutely nothing (we just saw this with 3 pirates above).  So Pirate 4 can give one coin to Pirate 2, and so now two out the four votes are in favor, so this distribution is fulfilled.

Pirate 4: 99        Pirate 3: 0        Pirate 2: 1        Pirate 1: 0

Lastly, to our final situation of five jolly pirates, using the same reasoning we understand that Pirates 1 and 3 will get nothing if Pirate 5 is murdered and Pirate 4 is left under control, and therefore will be happy with anything.  So Pirate 5 simply appeases them with 1 coin each, and gains their votes, reaching a majority of 3 votes out of 5.  And so we have our final allocations.

Pirate 5: 98        Pirate 4: 0        Pirate 3: 1        Pirate 2: 0        Pirate 1: 1

If this is not what you thought it would be, join the club.  Analyzing purely rational decision-making in today’s information driven world couldn’t be more useful, and can help you make everyday choices and understand how situations unfold.  Now, to the moral question… where and when does this kind of rationality lead to injustice in our world? Oh and also, make the most of today, it only comes every 4 years…

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