## Pirate Game Analysis – Intuitive vs. Theoretical Results

The pirate game is famous multi-player game that can apply to different mathematical and economic models.

The game has the following rules: suppose 5 rational pirates, A, B, C, D, and E, try to distribute 100 golds. Pirate A is superior to B, B is superior to C, C is to D, and D is to E. The most senior pirate proposes the distribution, and then all pirates vote on whether they agree or disagree with such distribution. If half or more pirates agree, then the distribution stands and the game ends with the proposed distribution. Otherwise, the most senior pirate gets thrown into the sea, and the game restarts with the next most senior pirate proposing a distribution.

In general, pirates base their decisions on three factors in the following order: survival,  max payoff, throw more people overboard.

Our intuition may be that since survival is the most important factor, pirate A would distribute himself very few coins to make sure enough pirates agree with this distribution.

However, the theoretical results are far from our intuitive results. Let’s start with two pirates left on board, D and E. D will distribute himself 100 coins because even if E disagrees, D’s proposal stands. Now we expand the game to three pirates. If C wants to maximize his payoff, he cannot rely on D because D would have got 100 coins in a two-pirate game, which means C will hurt D as long as C gets some coins, so C has to rely on E to agree with his proposal. Rational pirate C knows that rational pirate E knows he would get nothing if C is thrown and D takes over, so all C has to guarantee E is one coin. Therefore, the 3-pirate game has the following result:

Now we extend the game to four players. B needs at least one agreement from C, D, or E to survive. B does not want to pull C to his side because if he does, he’ll have to give C more than 99 coins. B can get D to agree with his proposal, because if B is thrown, D would not get any coins, so as long as B gives D one coin, his proposal stands. B does not want E either, because he will have to give E two coins for E to agree with him, thus B’s payoff is not maximized. The 4-pirate game result can be shown in the following picture:

Similarly, when we look at the 5-pirate game, we see that A wants support from C and E by guaranteeing them one coin, because they would get no coins if A is thrown overboard. As a result, A will propose that A gets 98 coins, B gets 0, C gets 1, D gets 0, and E gets 1:

Now that we have solved the theoretical results of the 5-pirate game, we can extend the game to more pirates while keeping the number of coins still be 100. When there are 200 or fewer pirates, the most senior pirates need to give every other pirate a coin, and the rest to himself.

In all the cases above, we have discussed how other pirates’ vote depend on payoff and throwing more pirates overboard. Now let’s look at how other pirates’ vote can depend on their own survival. Now we look at cases with even more pirates. The 201st pirate needs at least 101 agreements to survive, so he has to give pirates 199, 197, …, 1 one coin each, and leaving himself no coins. Similar with the 202nd pirate, who gives pirates 200, 198, …, 2 one coin each to survive.

The 203rd pirate cannot get enough agreements, so he cannot survive no matter how he distributes the coins.

The 204th pirate knows that pirate 203 will support him in order to survive, so all he has to do is to distribute the coins to every other pirate and leave himself nothing.

The 205th pirate cannot count on the 203rd and 204th pirates who can survive without supporting him, so he will be thrown.

Pirates 206 and 207 cannot get enough agreements, but then pirate 208 can, with 205, 206, and 207’s support.

Now we get a list of pirates who can survive: 1-200, 201, 202, 204, 208, 216, … 200+2^n.

In conclusion, the pirate game is very interesting model of game theory. When there are enough coins to distribute, the most senior pirate cares about his payoff and so are other pirates below him. When there are way more pirates than coins, the senior pirate is not powerful anymore and has to worry about his survival and so are other pirates below him.

https://omohundro.files.wordpress.com/2009/03/stewart99_a_puzzle_for_pirates.pdf