The Prisoner’s Dilemma
There was once a game show on British television called Goldenballs, a cut- throat game of bluff and back-stabbing, hosted by Jasper Carrott, a comedian best-know for making snide remarks about foreign commercials. Goldenballs, though, was compelling TV.
After a few rounds of skulduggery, all but two contestants were eliminated, and- as game shows like to do- they were pitted against each other to play for a jackpot. Each player was given two golden balls: one with the word ‘Split’, and the other with the word ‘Steal’.
Should they rat on the other or not?
If both players chose ‘Split’, it was all sunshine and roses: each would take home half of the jackpot. If one player chose ‘Steal’ and the other chose ‘Split’, the stealer would take home the entire jackpot, while the splitter would leave with nothing. If both picked ‘Steal’, though, both would leave empty-handed.
While the thrill of Goldenballs was watching two non- mathematicians trying to convince each other that they were lovely people and not about to do something as devious and nasty as try to steal the jackpot before doing exactly that, the final round was based on a famous problem of game theory: The prisoner’s dilemma, first posed by Merrill Flood and Melvin Dresher in 1950.
The traditional set-up involves two members of a criminal gang, both of whom are arrested on flimsy evidence.
Each is offered a choice: they may exercise their right to remain silent, or they may incriminate the other. If both remain silent, they will both be convicted of a minor charge, and probably get a couple of years in prison.
If one prisoner rats and the other stays silent, the grass will walk free, while the silent party can expect a long prison sentence. If, on the other hand, each criminal incriminates the other, both will get moderately long sentences. The usual language is to ‘cooperate’ by staying silent or to ‘defect’ by stitching up their so-called friend.
It is similar to Goldenballs in that the best outcome all round is for both prisoners to cooperate- but in both cases, there’s a twist.
If you were playing either game, your best option is defect. If the other person chose to cooperate (or split), you would get a lighter sentence (or a bigger jackpot) by defecting (or stealing).
If they chose to defect, you would get a moderate sentence by defecting, or a very long sentence by remaining silent- so you’d do better to defect in that case, too. In Goldenballs, if your opponent steals, you’re getting nothing in either case; but if you steal, too, you get to watch the grin on their face vanish, which is wroth any number of jackpots, if you ask me.
The game changes if you extend it to multiple rounds when a strategy of tit-for- tat, or doing what the other player did last time, tends to outperform simple defection- but that’s story for another write.
Meanwhile, the prisoner’s dilemma isn’t a toy problem for use in game shows and what-ifs, it can be used to model many situations in which competition and cooperation are options-from doping in sport, to spending on advertising, to freeloading, to nuclear arsenals.
Thank you for reading.😊
references: britannica, cracking maths
