Today is your lucky day! You are the captain of a 10-member pirate ship, and together with the rest of your crew, you have found treasure: 100 indivisible gold coins! Now, the question arises as to how the treasure should be divided. Though it might be tempting as the captain to keep all the gold for yourself, there is a very strict pirate code. You must propose the first division, and every crew member gets to vote on your proposal. If your proposed division receives at least half of the votes, it is accepted. However, if your proposal is not accepted, you must, unfortunately, walk the plank. The second-highest-ranking pirate is then allowed to make a proposal under the same conditions, and so on. So, what is it that you propose in order to avoid the plank?
Before we dive deeper into a division strategy, it is important to know a few things. Not only do pirates live by a strict pirate code, but they also share the same values. They have the following strictly ordered list of priorities:
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Self-preservation: Staying alive is valued above all else.
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Greed: A pirate wants as much gold as possible.
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Bloodthirst: If all other results are equal, pirates enjoy seeing others walk the plank.
Furthermore, pirates are perfectly logical and rational and share the common knowledge that all other pirates are also perfectly logical and rational. Hence, they can rely on this reasoning when deciding their actions.
At first, it might seem that a lot of the treasure must be given away to secure votes. If you decide to keep most of the gold, why wouldn’t the other pirates vote "no" and make you walk the plank? However, as will become clear, each pirate considers not only the current proposal but also the proposals that might follow.
This problem is known as the Pirates' Treasure Problem. The solution has its roots in game theory and can be effectively analyzed using recursion and backward induction.
For ease of notation, we number the pirates 1 through 10 from the lowest to the highest rank. If there is only one pirate, they will simply keep all the coins. Naturally, they will vote "yes" to their own proposal, and thus they will be both alive and rich. This same reasoning applies to the situation with two pirates left: Pirate 2 will propose to keep all the coins for themselves and vote "yes". Pirate 1, in this case, is left with nothing.
Things become more interesting when a third pirate is still alive. Pirate 1 knows (due to the rationality of themselves and the other pirates) that if Pirate 3’s proposal is not accepted, Pirate 3 will have to walk the plank. After that, only two pirates will remain, and Pirate 1 knows they will receive nothing, as Pirate 2 can take everything for himself. To win Pirate 1’s vote, Pirate 3 only needs to allocate 1 coin to Pirate 1. Pirate 1 will vote "yes," as 1 coin is better than 0 coins. With this, Pirate 3 has two votes (including their own) and can keep the rest of the coins.
What happens when there is a fourth pirate? Pirate 4 needs two votes for their proposal to be accepted. Using the same reasoning as above, Pirate 2 knows that if they do not vote "yes," they will receive 0 coins if Pirate 3 makes the proposal. Hence, Pirate 2 will agree to accept 1 coin. With Pirate 2’s vote and their own, Pirate 4 has enough votes. Pirate 4 does not need to give Pirate 1 or Pirate 3 any coins and can keep the remaining treasure for themselves.
Pirate 5 knows all this, and in line with the reasoning above, will propose to give one coin to Pirate 1 and 3. They prefer this 1 coin over having 0 coins if there are 4 pirates left and vote “yes”. Pirate 5 can keep the rest for himself. This reasoning continues for all 10 pirates. The division of the coins for each scenario is shown in Table 1.
When proposing a distribution for the gold, you only need to give away 4 coins! Pirates 2, 4, 6, and 8 each receive 1 coin, and you can keep the rest for yourself. Not bad at all!
There are some interesting aspects to this problem. Even though it might initially seem that you need to give away a large share of the coins, in reality, you can keep the vast majority! This solution depends on the pirates' rationality and their understanding of game theory principles. Each pirate evaluates their vote by considering the outcomes of future proposals if the current one is rejected. They prioritize their survival and seek to maximize their gold while ensuring no better alternatives exist in future rounds. This recursive reasoning demonstrates the importance of backward induction in decision-making. Furthermore, the strategy highlights how a seemingly hierarchical system can lead to surprising outcomes, with even the lowest-ranking pirates influencing the captain’s decisions. Clearly, the Pirates' Treasure Problem is a fascinating example of strategic reasoning!