The Pirate Game

Antonius still seems confused.

"Fine, let me show you the bright side of Math… Come with me." George Sphrantzes pulled the latter a side into a quiet corner of an age old Gothic style pavilion built here three centuries ago by the Komnenian emperors, from its still firmly standing pillars one can see the dark burnt marks of the malicious destructions by the flames of war not long ago.

Antonius sat there on the rock bench, while George Sphrantzes hurriedly went to scatter around the surrounding bushes and plains for five rocks ranging from big to small and arranged them onto the table in one straight line.

Antonius is baffled as he cannot understand how five pebbles can be related with Mathematics. "What is this? You want to play pebbles with me?"

George ignored his question and carried on arranging the pebbles. A while later he rose his head and started his lecture. "Now, Lord De'Ricci, I remember that you used to master the craft of… armed maritime privateer, right?"

"Aye, I am a pirate." Antonius never denies his past.

"Now, imagine that these five rocks are the crew on a pir…privateer ship, and all of them are logical thinkers." George pointed towards the biggest rock on the table. "The biggest one over here being the boss who is in charge of all the crew, the second biggest rock being the vice captain, the subsequent rocks taking up various roles on the ship ranging from big to small depending on its size…."

"It is hard for pirates to be a logical thinker in first place."

"They managed to ransack a merchant ship and acquired a loot of a hundred ducats…."

"That's pitifully sum of money."

"… So they have came up with a plan, the highest ranking pirate shall present to the rest a plan of how to divide the gold first. All the pirates, including the proposer, shall have a chance to vote on this proposal, and the proposer shall have the right to decide on the results if he has half of the total approval…."

"We don't really vote on pirate ships, very rarely we would do that sort of things."

"…."

George Sphrantzes cleared his throat and continued. "… If the proposal is passed with the support of at least half of all the pirates, like three supporters out of five, two out of four, the gold is divided according to the proposal, but if the proposer did not manage to gain more than half of the votes, the proposer is thrown overboard leaving the rights to the next high ranking pirate to call for a vote…"

"How can they throw their captain over board for a hundred ducats? That is absurd!"

"…"

"Thus, the pirates shall make their proposal according to three things, the first priority is, of course, survive without being thrown overboard. Secondly, to find a way to maximise his profit from the share of coins. Thirdly, throw the others in front of him over board to make sure he gets a promotion in rank. Now, if you are the captain, what would you propose?"

Antonius tried hard to think for a while frowning there scratching his head staring soullessly at the five rocks on the table. It took him a while before re-opening his mouth. "I don't know? Maybe I shall choose not to get any gold and split the rest equally among the others to gain their support?"

George Sphrantzes laughed. "The answer is that the captain can gain up to a maximum of ninety eight ducats, giving the third pirate and the last pirate one ducat each."

"Very drool, George, very drool." Antonius rolled his eyes.

"Now, imagine this, lord De'Ricci." George stood up shoving the two smallest stones to a side. "If there is only pirate four and five left, pirate four will give him a hundred ducats and give the fifth one zero, because he himself counts as half of the vote and is able to make the final decision."

"Alright?"

"If, there are three pirates left, three, four, and five, the third pirate knows that if he gets thrown overboard, the fourth pirate shall give the fifth pirate zero ducats making him gain totally nothing, thus he can choose to give the fifth pirate only one ducat to gain his support. Thus, the final result shall be the third pirate gaining ninety nine ducats, the fifth pirate gaining one ducat while the fourth gaining nothing."

"Okay?"

"If there are four pirates left on board, which is the second, third, fourth and fifth, and the second pirate knows all of this, he will only need to give the fourth pirate one ducat getting half of the total support, while giving the rest of the two completely nothing, getting a total of ninety eight coins for himself."

"What about the fifth pirate? Don't the second one need to give him something at least also?"

"Good question." George thought for a moment organising his language. "Because the fifth pirate knows that he will only get one ducat according to this theory from both the second pirate and the third pirate, thus with the same amount of profit he shall prioritise on throwing the second pirate over board first."

"What if the third, fourth, and fifth pirate decided to unit together and rob the first and second pirate of all the coins?"

"…. That is something we will not consider in this theory, if you choose to scruntinise on other surprise elements the entire theory will never be logical, because seriously speaking why don't you say that the fifth pirate is a titan and thus he can just take all the coin himself? Anyways…"

"… Thus, if the first pirate, using all of his logical thinking, knows about all of this, he can choose to have the third and fifth pirate to support him by offering them one ducat each…."

"Why not the fourth one?"

"Because the fourth pirate can choose to dump the first pirate into the ocean and get the same amount of coins from the second pirate, if the first pirate wants to gain his support he need to give him two coins to change his mind, by which he will not be able to maximise his profit"

"I see." Antonius frowns his eye brow as he fiddles around with the five stones on the table. "But… How does this whole pirates thing help me in the first place even if I understand the entire theory?"

"Don't be so impatient…" George Sphrantzes patted Antonius on the back of his hand. "Listen to me…"