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Decision theory

The **pirate game** is a simple mathematical game. It illustrates how, if assumptions conforming to a homo economicus model of human behaviour hold, outcomes may be surprising. It is a multi-player version of the ultimatum game.

## The Game[]

There are five rational pirates, A, B, C, D and E. They find 100 gold coins. They must decide how to distribute them.

The Pirates have a strict order of seniority: A is superior to B, who is superior to C, who is superior to D, who is superior to E.

The Pirate world's rules of distribution are thus: that the most senior pirate should propose a distribution of coins. The pirates should then vote on whether to accept this distribution; the proposer is able to vote, and has the casting vote in the event of a tie [since this is the right of the proposer]. If the proposed allocation is approved by vote, it happens. If not, the proposer is thrown overboard on the pirate ship and dies, and the next most senior pirate makes a new proposal to begin the system again.

First of all, the pirates want to survive. Secondly, the pirates want to maximize the amount of gold coins they receive and, thirdly, they like throwing other pirates overboard.

## The Result[]

It might be expected intuitively that Pirate A will propose that the allocation shall be 20, 20, 20, 20, 20. However, this is not the theoretical result.

In the game theoretic analysis, the allocation offered by Pirate A that would be accepted (assuming all pirates are rational and are capable of understanding the scenarios that will occur when they accept/reject offers) is: Pirate A: 98, Pirate B: 0, Pirate C: 1, Pirate D: 0, and Pirate E: 1.

This is apparent if we work backwards: if all except D and E have been thrown overboard, D proposes 100 for himself and 0 for E. He has the casting vote, and so this is the allocation.

If there are three left (C, D and E) C knows that D will offer E 0 in the next round; therefore, C has to offer E 1 coin in this round to make E vote with him, and get his allocation through. Therefore, when only three are left the allocation is C:99, D:0, E:1.

When B makes his decision, he knows this; he must therefore make sure that he is not thrown overboard. He does this by offering 1 to D. Because he has the casting vote, the support only by D is sufficient. Thus he proposes B:99, C:0, D:1, E:0.

A, as a rational agent, knows that this is the allocation of coins if he is thrown overboard. He therefore offers A:98, B:0, C:1, D:0, E:1.

Hence, the allocation which gives the most to A but will nevertheless be accepted is:

A: 98 coins B: 0 coins C: 1 coin D: 0 coins E: 1 coin

## Extension[]

The game can easily be extended to up to 200 pirates. Ian Stewart extended it still further in the May 1999 edition of Scientific American (in particular to 500 pirates).

## References[]

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