t variant may fare better than plain Tit for Tat, the so-called Generous Tit for Tat in which an opponent’s defection is forgiven 10% of the time. Then there was a suggestion in 1993 that “win-stay, lose-shift” fares slightly better than generous Tit for Tat most of the time. (I am grateful to James Junghanns for drawing my attention to this, in the comments below.)
Still, it is reassuring to find that merely playing the game repeatedly with the same people is enough to favour strategies that resemble decent human behaviour, rather than the stark sociopathic logic that seems to reign when playing only once, and this is a comfortable place to rest.
But we can rest no longer! In a paper published earlier this year, Bill Press and Freeman Dyson outline a series of startling mathematical results that change forever our understanding of the game. And, if the game is a model of human affairs, does that change our understanding of the world more generally too?
Press and Dyson uncovered a class of strategies for the iterated Prisoner’s Dilemma – that is, the game where the Prisoner’s Dilemma is played repeatedly with the same other player – that have strange, almost unbelievable properties. One will force the other player’s score, on average, to be a certain number; and nothing she can do will change that score, however she chooses to play. Another type, which Press and Dyson dub “extortionate”, is able to take advantage of the other player’s self-interest. Against an extortionate opponent, the more you allow yourself to be taken advantage of the better you will score. So a purely self-interested player, who will modify his play only to increase his own score, can be taken for a ride by an “extortionist” who is able to use the other player’s self-interest against him.
All this is heady stuff, and it is hard to say precisely what the broader implications are. I have made a little interactive toy where you can pit yourself against both these types of strategy, if you would like to see how they work in practice. Click the tabs 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 t