hat.
And so they played, hour after hour. The elephant could remember everything, of course: every move they had ever played, the precise way the sunlight had glinted on the water and the leaves had rustled at that moment she unclenched her trunk to reveal the black nut or the white nut. The poor goldfish could remember only the move they had just played, and then only by concentrating as hard as he could.
Still, the elephant found increasingly she did not win as often as she would have liked, and the goldfish’s play seemed to be improving, though his memory plainly was not. One afternoon she realised with shock that she was losing, not every turn but more often than not, though she thought hard each turn about everything either of them had ever done and the goldfish did not even recall that he had narrowly escaped being eaten by a crocodile just five minutes before.
That evening when the sun had set, the elephant sat down to think about what had happened. How was it possible that all her memories should profit her so little against this forgetful fish? Slowly she began to understand it. This forgetful fish, she thought, he can remember nothing. Only the move we have just played, and sometimes not even that. And the history of the game – this long history that I remember so vividly – does not exist to him. His choice is informed only by what he can remember. And the thing that I am up against is only the tiny stone that he shoots so deftly from his mouth, and this thing – this choice of his – is unlinked from the past. All those memories of mine cannot help me to predict what the fish will do, for they cannot change what he will do. Only the turn just played is real to him, so that only can matter to me.
Sadly, the elephant lay down to sleep. The elephant has no advantage of the goldfish in this game, and likewise you have none over the goldfish-memoried machine. This observation, that a longer memory confers no advantage against an opponent with a short one, is one of the key points of Press and Dyson’s work and appears as Appendix A in their paper.
About the name
Oh, if you are wondering why it’s called the Prisoner’s Dilemma, that is because of a story that has been told to explain this game since shortly after its discovery. The story goes that two men, having engaged in a joint criminal enterprise, are apprehended by the law and interrogated separately. Each prisoner is given the opportunity to reduce his own sentence by testifying against the other – but that testimony will increase the prison sentence handed down to the other man, and if both testify then there will be evidence enough to put both behind bars for longer than if neither had. In this version the payoffs are prison sentences rather than cash prizes: it has the same paradoxical quality that each prisoner individually is always better off by defecting, but collectively they would be better to invoke their right to remain silent.
On the topic of names, did you know that the phrase Tit for Tat is the origin of the rhyming slang ‘titfer’, meaning a hat?
- Conversation on The Edge website with William Poundstone, Bill Press & Freeman Dyson, and Karl Sigmund & Martin Nowak.
- Technical paper with all the details, by Press and Dyson.
- Discussion paper by Stewart and Plotkin.
- Mike Shulman attacks “hype and misinterpretation” of the Press-Dyson result in a long blog post that includes a clear but detailed explanation of the Press-Dyson theorem.
- The aforementioned interactive toy.