In the riddle of the pirates and the 100 coins, they are there five pirates very intelligentperfectly rational and very greedy that they must divide a chest containing 100 gold coins. The captain is the first to propose a strategy of dividing the treasure and all the pirates, including him, can vote. If the proposal is rejected, the captain is thrown overboard and it will be up to his successor to make a new proposal, and so on.
To survive and return home with as many coins as possible, the captain will have to exploit the common knowledgethat is, the awareness that “everyone knows that everyone knows”, and applying one impeccable logic. Let’s see together what strategy he should propose.
The riddle of the pirates and the 100 gold coins
After months of searching the crew of the Eastern Star found the Captain’s chest Barbarossa. Now Abel, Beatrix, Conrad, Darius and Eleanor have to share 100 coins that they found inside the chest. To do this, they follow the pirate code: the captain can be the first to propose a strategy for dividing the spoils and the crew (including the captain) must decide whether or not to approve the captain’s plan.
If the proposal And accepted by the majority, or if there is a tie, the coins they come divided according to the captain’s strategy. If instead the majority votes againstCaptain Abel comes thrown into the sea and the command passes to Beatrix, who can submit a new proposal. Voting starts again with the remaining crew, following the same rules. If Beatrix’s plan is also rejected, she will end up the same way, leaving command to Conrad, and so on arriving at Darius and finally Eleanor, until a proposal is approved or until there is not a single pirate left alive.
All crew members are greedy, bloodthirsty and selfish: They don’t trust each other and don’t try to agree or collaborate. Their priorities, in order, are:
- remain you live;
- get the greater number possible of coins;
- throw into the sea anyone who doesn’t bring them a advantage.
All pirates, however, are also extremely thentelligent and brilliant in logical deductions and are aware that others are the same: they reason impeccably and perfectly predict the behavior of others.
Which should be the Captain Abel’s proposal so as not to be thrown into the sea and take as many coins as possible?
The solution to the riddle
As a first approach it might occur to us that Abel should try to corrupt the crew giving them most of the gold, to increase the chances of his plan being accepted. If we think about it for a moment and use logic, however, we discover that Abel can keep up to 98 coins for himself. Let’s see the reasoning step by step starting from the simplest case: the one in which two pirates remained.
What happens if there are two pirates
If only two pirates remain (Darius and Eleanor) the captain is Darius and it is up to him to propose the division of the treasure. In this situation, Darius it simply can propose to keep all 100 coins for himself. Since the proposal passes even in the event of a tie, Eleanor’s vote would not be enough to stop it.
Consequentially, if only two pirates remain, the captain keeps all the gold.
What if there are three pirates
If three pirates remain (Conrad, Darius and Eleanor), it’s up to Conradwho in this case is the captain, make your proposal. Everyone knows that if there were only two pirates left, Darius would keep everything and Eleanor would get nothing. So, everyone knows that Eleanor wants to avoid Darius becoming captain at all costs and that will vote in favor of any proposal you assign them at least 1 coinbecause it would still be better than nothing.
Conrad’s proposal, therefore, is to hold 99 coins for himselfgive 1 to Eleanor and 0 to Darius. The vote results:
- Conrad: favorable (Obviously)
- Eleanor: favorable (gets 1 coin instead of 0)
- Darius: opposite (left empty-handed)
With two out of three in favour, the proposal passes. As a result, Darius does not want Conrad to become captain, because in this scenario he would not receive any coins.
What if there are four pirates
If we find ourselves with four pirates, it is Beatrix who proposes the distribution of the treasure. To get her plan approved, Beatrix needs at least one more favorable votebecause the proposal passes even in the event of a tie.
Beatrix knows that if she were eliminated, the next captain would be Conrad, and in that situation, as we have already seen, Darius would not receive any coin. For this reason, his proposal is to keep it to himself 99 coinsgive one to Darius and zero to Conrad and Eleanor.
At that point, the grades would be:
- Beatrix: favorable (Obviously)
- Conrad: opposite (left empty-handed)
- Darius: favorable (gets 1 coin instead of 0)
- Eleanor: contrary (left empty-handed)
The result is 2 in favor and 2 againstthen the plan passes and Beatrix survives.
Consequently, neither Conrad nor Eleanor want Beatrix to become captain, because in this situation they would not receive any coins.
What if there are five pirates
And so, if there are five piratesperfectly rational, selfish and ruthless, Captain Abel can exploit the preferences and fears of others. To pass his proposal he will need three votes favorable.
Everyone knows that if Abel were thrown overboard, the next captain would be Beatrix, and Conrad and Eleanor would receive zero coins under his command. So Abel will make this proposal: it holds for himself 98 coinsgives 1 to Conrad and 1 to Eleanor.
Beatrix and Darius, who are left with nothing, will obviously be against it, but Conrad and Eleanor will vote in favor, passing the proposal. Abel thus obtains the necessary votes and is kept 98 gold coins.
The surprising result of this riddle depends not just on how pirates vote, but on whether each of them is aware of what the others knowthe strategies they could adopt, and the perfect rationality of all participants. This shared level of awareness, called “common knowledge,” allows everyone to accurately predict the reasoning of others.
If you want to play again, try asking yourself: what would have happened if there had been 6 pirates? What if there were 100?
