In Our Time is a fantastic programme on Radio 4 covering ideas of culture, history, philosophy, religion and science, with a full archive available if you have a few hundred spare hours. In a recent episode Melvyn and the gang (The Right Honourable The Lord Bragg and three professors) discussed game theory, a bit of a whistle-stop tour as In Our Time has to be, but plenty of food for thought.
One of the most interesting things they discussed was the ultimatum game. In the ultimatum game there’s a sum of money (say 100 gold coins) and two players (let’s call them Geoff and Jeff, to avoid confusion). Geoff proposes a division of the money between the two of them, Jeff can then either accept the proposal and take what was offered, or reject it in which case neither player gets anything.
On a purely rational basis Geoff could offer Jeff one gold coin while keeping 99 himself, as faced with a choice of one coin or nothing Jeff should take the money and be grateful. Would you, in Jeff’s position, accept that offer? Or would you tell Geoff in irrational but highly anatomically detailed terms precisely where he could shove his single coin? If the ‘gold’ coins were chocolate money, would your answer be different than if they were 24 carat doubloons? The game has spawned a lot of research, experimentation and variations, and a bit of idle wiki-link-following led to a rather fun Puzzle for Pirates based on a broadly similar premise:
There are 5 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, including the proposer, then vote on whether to accept this distribution. If the proposed allocation is approved by a majority or a tie vote, it happens. If not, the proposer is thrown overboard from the pirate ship and dies, and the next most senior pirate makes a new proposal to begin the system again.
Pirates base their decisions on three factors. First of all, each pirate wants to survive. Second, given survival, each pirate wants to maximize the number of gold coins he receives. Third, each pirate would prefer to throw another overboard, if all other results would otherwise be equal. The pirates do not trust each other, and will neither make nor honor any promises between pirates apart from the main proposal.
A sensible option at first glance would be for Pirate A to offer most of the money to the others, lest he get chucked overboard and sent to Davy Jones Locker, me hearty, arrrrr etc. He doesn’t need to do that at all; with the tweaked rules it’s a neat logical brainteaser with a solution, click through to Wikipedia if you’d like the details and explanation. Well worth bearing in mind, I’d say, if you’re in a group of five exploring a dungeon and you need to propose a way of splitting up a pile of cash at the end…