The Pirate Game is an extension of the Ultimatum Game. While the Ultimatum Game involves 2 players and a few gold coins, the Pirate Game involves 5 pirates and 100 gold coins making offers to one another. Describing the problem and explaining the solution will take up too many words, but unless you understand the problem and the solution, the rest of this post will not make sense, so I must ask you to read the Wikipedia article before proceeding to the next paragraph. The Wikipedia article has the solution as well, but try to figure it out yourself first. It’s more fun that way. Most importantly, understand the assumptions (The wiki entry calls them “factors”) involved, because that’s the focus of this post.

We are now in the second paragraph of this post and I will assume that you have heeded the warning in the previous para. So, without further ado, here is the assumption I want to relax: “the pirates do not trust each other, and will neither make nor honor any promises between pirates.”

To understand why this assumption is important, let’s revisit the solution at the  point where we are down to three pirates, C, D and E. The canonical solution says that C cannot offer any deal to D that leaves C alive, because D knows that he can get 100 coins plus a dead C, which is a better deal for him than 100 coins and a live C. And C knows that if he dies, E is going to get nothing under D’s regime, so C can get away with offering one coin to E.

But… this solution seems unsatisfactory. D and E are left with nil and one coin respectively. If D could credibly promise to offer more than 0 to E, they could conspire to refuse C’s offer and divide the 100 coins between themselves. The exact proportion could be anything as long as E got more than one. Of course, as soon as C gets wind of the deal, he will try to save his neck by offering a better deal to either D or E. He may also try to offer a deal to both D and E to ensure that neither initiates a conspiracy with the other.

In the strict version of the game, these deals are precluded by the rule that the pirates do not trust the other pirates to keep any promises they make. The relaxation of this rule turns the problem into one with an indeterminate solution. If the lack-of-trust assumption is in place, the three player pirate game results in a definite solution, with C getting 99 coins, D getting none and E getting one. Relax the assumption, and while we cannot definitively solve the problem, we can safely say that D and E will get a better deal than they would get under the strict version. Now, note that if you’re down to two players, there is no difference between the strict version and the one with the assumption relaxed, as there is no longer anyone to make deals with. E will be badly off in the two player version, i.e. he is in a better position when he has an intermediary to play off against the one who has custody of the gold coins than he is in when he has to negotiate directly.

In other words, the three player game is a pretty good model of reality, where E is the proletariat, C is the bourgeoisie government and D is the communist movement wanting to overthrow C. It explains why the threat of a revolution worked to the benefit of the proles in some cases, while in others the threat was managed by buying off the revolutionaries. It also explains why an actual revolution invariably ends up as a bad deal for the masses.

But if we know that achieving a revolution to overthrow C is bad for E, it’s a safe bet that C also knows this, so why would the prospect of D and E colluding ever be a credible threat to C? To show credibility, E may need to demonstrate a willingness to irrationally go against his interest. If C is afraid that E and D will kill him even if it’s against E’s interests, it’s cold comfort for him to know that E will be worse off as  a result. One way for C to have this fear is if another E in another game has overthrown his C. The C in our game will be afraid that something similar may happen in his own game. We may note that something very similar happened in Western democracies vis-a-vis communism. Another way is to set up the game so that the overthrow of C doesn’t turn it into a two-player game. For example, if D turns into C and vice versa, the next iteration is also a three player game. This is of course what happens in a multi-party democracy.

I am sure we can find many other real life situations where the three-player Pirate Game is a good model. What about the four-player game? I haven’t thought through it in detail, but here is one situation where it may be a useful model:

I’ll let others do the analysis required, but I have ab strong hunch that as you add more pirates to the game, the possibility that E gets a good deal reduces significantly