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Game Theory - Distribution Of Pirates' Gold Coins Puzzle

The distribution of 100 gold coins in 5 pirates is an interesting mathematical puzzle which is based on game theory. There are 5 pirates A, B, C, D, E who got 100 gold coins one day. They must decide how to distribute those coins among them. Finally, they decide to distribute the coins using the following procedures:

Pirate A will propose a distribution plan and others will vote for this plan. The plan will get approved if at least half of the voters agree. The private who raised the plan will not attend the vote. If the plan gets approved, the coins will be disbursed based on this plan. If the plan is not approved, the pirate who raises the plan will be thrown into the sea and die. Then the next pirate will make a new proposal to run this game again. If you are the first pirate who will make the initial proposal, what kind of distribution plan will you make? This game is based on the following conditions. All privates are evil, which means they will try to get the max amount of coins and will try to kill other pirates if possible. Those pirates will also do their best to survive even if they don't get any coins. Moreover, those pirates are clever enough to choose the best strategy to survive and get the max amount of coins.

The answer will be released after 30 seconds from now. You can pause the video if you need more time to think of the answer. Some hints will be shown up after 15 seconds to help you resolve this puzzle.

Here is the answer. Let's consider this case. If only pirate D and E are left and the others A, B and C already died. Whatever proposal pirate D makes, E will reject it. Rejecting D's proposal will enable pirate E to kill D and get all coins. So, if D doesn't want to die, he must unconditionally accept pirate C's proposal. As for pirate C, he knows C will support his proposal even if he gives zero coins to him. Since pirate D will vote for him, he doesn't need to care about pirate E's result. So, for pirate C, the best plan which can maximize his gain is 100 for himself and give D and E zero coins. for Pirate B, since he will be clever enough to know what C will propose, what he can do is to offer pirate D and E a small amount of coins which will push pirate D and E to vote for him. Pirate B will make a proposal of distribution as 98 to 0 to 1 to 1. In this plan, he will give pirate C nothing and offer D and E one coin each. If D and E reject pirate B's plan, after pirate B is killed, they will get nothing from C's proposal. They will be clever enough to know that they will get more coins from pirate B's plan. Similarly, pirate A will also know all about this logic. So, he can propose a plan of 97 to 0 to 1 to 2 to 0, or 97 to 0 to 1 to 0 to 2. This plan will give up pirate B and offer one coin to pirate C and 2 coins for pirate D or E. Compared with pirate B's plan, pirate C and D or E will get more coins, and thus they will support pirate A's plan and make the proposal approved. Finally, pirate A will survive and get the max number of 98 coins.

From the first glimpse of this puzzle, it looks like pirate A is in the most dangerous position since his proposal will be easily rejected by others and he will be killed. However, if he can make his proposals in others' shoes, he can make the best plan to get him most of the coins. The key point is to identify what others' proposals will be and create a better one by offering some extra benefits for those unimportant players.

The similar logic also happened in our daily life or in our workplace. This can explain why the boss of a company will get a better relationship with ordinarly employees than with other executives. Similar to the position of pirate A, the boss will be easier to get along with ordinarly employees by offering a small amount of favor to them, similar to the one or two coins to pirate D and E.

Now we can make a small modification on the original rules. If we change the voting rule to allow the proposer to attend the voting, which means the pirate who submit the proposal can also vote for himself. What would be the answer for this puzzle? You are welcomed to leave comments under this video.

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Image from Pixabay: https://pixabay.com/zh/illustrations/...