abs to switch between different example strategies.
One thing these new strategies do not obviously do is to dethrone Tit for Tat on its own turf. We cannot be certain till more work has been done, more tournaments have been run, but it is certainly possible that Tit for Tat (or generous Tit for Tat, or Win-Stay, Lose-Shift) will keep its crown. Stewart and Plotkin found a Press-Dyson strategy that did slightly better than generous Tit for Tat in a single simulated tournament, but there’s no really good reason to think that advantage will prove reliable. We just don’t know, yet. This generous strategy, or at any rate a similar one, is also available to play with in the interactive toy.
The goldfish and the elephant
If you have played with the computer opponent, and read how its strategies work, you may be puzzled or surprised to see that the computer ignores all but the immediately previous move when deciding what to do. How is it that such a short memory is sufficient? And how is it that we, who can remember the whole game, cannot use this additional knowledge to gain an advantage over the forgetful machine?In the jungle, there lived an elephant. This elephant was the most memorious of all the elephants, and she could remember everything that had happened since the birth of the world in every tiny detail. Sometimes on a hot afternoon the elephant would cool herself by the watering hole, and sometimes she would pass the time with a goldfish who lived there in the water. The goldfish was always surprised to see the elephant, for he never could recall anything that had happened more than a few short moments before, and he had no idea that the elephant was there almost every day.
To pass the time, the elephant taught the goldfish to play a game. In the jungle they knew it by a different name, but it was the game we know as Prisoner’s Dilemma. After some weeks the elephant found that the goldfish knew how to play, though he could not remember having been taught or even having met the elephant before. Memory is like that.
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