Pirates with PhDs
This post is about one of my favorite brainteasers. I love it for two reasons:
- It has a surprising but understandable solution.
- It makes you consider what it means to be rational and logical.
The puzzle has a logical solution, but wanted to test whether actual humans would come to that solution by simulating the puzzle in real life with seven participants. But first, if you haven’t heard it before, you should spend some time to solve it yourself.
The Puzzle
Seven pirates have 100 gold coins to split amongst themselves. They have chosen an interesting way of splitting the pot.
- The first pirate gives a proposal for how many pieces of gold each pirate will receive.
- Then, everyone (including the first pirate) votes yes or no on this proposal.
- If 50% or more of the pirates approve the proposal, it’s accepted and the process is over.
- Otherwise, pirate 1 is killed and pirate 2 proposes a new way of splitting the gold and pirates 2–6 vote on the new proposal.
- If the vote fails again, pirate 2 is killed and pirate 3 proposes a solution. It continues like this until a vote passes.
You can assume:
- These are no normal pirates! Not only do they plunder the high seas, but they also have PhDs in mathematics and you can expect them to behave hyper rationally.
- Each pirate wants to stay alive and get as much gold as they can. They have no ethical concerns about killing other pirates.
How should pirate 1 propose splitting the gold to maximize their gold?
The answer is below this awesome gif of a cat pirate.
Solution
Pirate 1 should propose giving 1 gold to pirates 3, 5, and 7 and keep 97 gold for themselves. This may surprise you, so let me explain.
First, let’s look at the simplest version of the problem. If there were just two pirates left (#6 and #7), then pirate 6 would just keep 100 gold for themselves. Since there are only two votes, pirate 7 cannot override this decision. So, pirate 7 knows that if it gets to this point, then they will get nothing.
If there are three pirates remaining, pirate 5 could offer pirate 7 just one gold for their vote. Since one gold is better than none, pirate 7 will vote yes, and the vote would pass 2 against 1. So, if it gets to three pirates, it will be split 99–0–1. We can continue this pattern for four pirates. Pirate 4 only needs one vote to get 50% of the votes, so can offer just 1 gold to pirate 6. Pirate 6 would accept since they would get nothing if it got down to three pirates. If we follow this logic all the way to seven pirates we get:
2 pirates: 100–0
3 pirates: 99–0–1
4 pirates: 99–0–1–0
5 pirates: 98–0–1–0–1
6 pirates: 98–0–1–0–1–0
7 pirates: 97–0–1–0–1–0–1
This solution seems too imbalanced to be correct, but it’s the logical conclusion of all pirates acting rationally and trying to maximize their own gold. There’s no way for pirates 2–7 to get any more gold alone by voting a different way. In game theory, this is known as a Nash Equilibrium — no player has anything to gain by changing strategies. Changing the results would rely on collusion between two or more pirates (and pirates aren’t the most trustworthy bunch
The Experiment
The correct answer is logical, but perhaps real humans wouldn’t reach the same conclusion. So, I got seven friends together and asked them to do the puzzle in real life. They played the game three times:
- In the first game, there would be public discussion, but no private messaging.
- In the second game, there would be no public discussion or private messaging — only voting.
- In the third game, players could private message each other.
Everyone playing the game knew and understood the correct answer to the problem. They were all smart and rational people playing to win. The rules were the same as the riddle (except of course, we didn’t kill anyone if their vote failed; they just didn’t get any gold).
Hypothesis
As I mentioned earlier, the decision to vote yes or no by each player reaches a Nash Equilibrium, so no player is incentivized to vote differently. However, pirate 1’s strategy doesn’t produce a stable Nash Equilibrium. The equilibrium is stable if it is resilient to small mistakes. In this case, if pirate 3, 5 or 7 make a mistake and vote the wrong way, it has dramatic implications for pirate 1. So, if I were pirate 1, I’d mitigate this risk by:
- Giving an additional pirate some gold
- Giving each pirate a bit more than 1 gold (around 5 should do it)
The less discussion there is, the more power the first pirate will have. So in round 2, I thought that pirate 1 could probably get away with taking 97 gold. But in rounds 1 and 3, I thought that they’d only get away with about 80 gold and splitting the rest amongst four other pirates. Regardless, I thought pirate 1's vote would always pass, they would take the majority of the gold, and the games would be short.
Oh, how I was wrong…
The Games
In game 1, pirate 1 quickly proposed a vote that exactly followed the puzzle before too much could be discussed (keeping 97 gold for themselves). And then, everyone voted no.
Suddenly, the game had completely changed. Now, instead of the proposing pirate having all the power, they had all the fear. After pirate 1’s vote failed, pirate 2 ended up successfully proposing a split which only gave them 30 gold. They were now convinced that the equilibrium was extremely unstable and needed to give significant gold to three other pirates to get just two votes.
In game 2, there was no discussion allowed. Pirate 1 clearly felt the fear of being in the first position and proposed an insane 10 gold for themselves and 15 gold to each other pirate. In this case, pirate 1 is actually getting less gold than the other players. I thought they could have easily asked for more, but the vote only passed with the minimum required four votes.
In game 3, there was a public discussion as well as private messaging. This game was nuts and I must leave out a lot for succinctness. First, pirate 1 proposed a 37/21/21/21 split which failed. The player who turned down 21 gold ended up getting nothing at the end of the game, and hence, the theme of the last game was hubris. In several cases, players backchanelled with other players and voted no thinking they’d end up with a better deal. Then, they’d see their plans backfire and end up with nothing.
In the second round, pirate 5 was granted 30 gold, but had an offer for 31 gold if they voted no. Again, hubris took over and they voted no, only to end up with nothing at the end of the game. The game finished in the 4th round with a 60/40 split between pirates 4 and 7.
Conclusions
In all three games, pirate 1 never got the 97 gold the brainteaser promised. In fact, it was the worst position to be in (two deaths and 10 gold). Fear and hubris quickly overtook the cold logic of the puzzle to produce a dynamic game in which every player thought they had a shot at winning more. The number of pirates also made reasoning through strategy difficult. With 3–4 players, the optimal strategy quickly becomes clearer, but with seven pirates, it seemed like anything could happen. Hence, the game relied much more on short-term tactics, fear, hubris and a bit of luck.
The theoretical solution to the problem is logical, but are the pirates really behaving rationally? The problem assumes that the pirates are emotionless, greedy algorithms who grab as much gold as they can. But people may be willing to take a risk if they don’t value a single gold coin highly. If the value of a coin is low, then the difference between 1 coin and 0 coins may be inconsequential. In this case, pirates may be tempted to vote no on the first round to see if other pirates feel similarly.
The theoretical pirates also appear heartless and have no concept of fairness. In the real life experiment, players voted no and forfeited gold to punish an ‘unfair’ result. This persuaded other pirates to ‘behave’ and propose more even splits. This bias toward fair outcomes suggests that aren’t always happy with the logical, mathematical solution. Sometimes, the optimal solution is not a human solution.
Thanks to the seven pirates — Brad Svenson, Hari Arul, Camilo Cabrera, Jason Zhang, Charles Huyi, Patrick Costello, and Dawson Zhou.
The game was a lot of fun to play — let me know if you try the game yourself!
Thanks to Alice Avery And Dawson Zhou for helping me review this post.
