This is the name given to an illustration of the most important mathematical work of John Nash. Another name is 'the centipede game' based on the appearance of one way of diagramming the game.
Imagine that two men have been arrested for a minor crime for which the police have overwhelming evidence. The police believe the same two men committed a major crime, but lack evidence for a conviction. The expected sentence is one year for the minor crime, ten years for the major.
In separate rooms, the police offer each prisoner the same deal. If he confesses to the major crime and testifies against the other, his testimony will not be used against him, and he will not be prosecuted for the minor crime. In other words, he will get one year less in jail than he would if he stayed silent, but the other will get ten more. This is true no matter what the other does.
For each prisoner's selfish interest, it seems better to confess. For the two together, it is better that each remain silent. This is what is called a dilemma. Formally, there is no dilemma - each prisoner should confess.
Most people instinctively reject this result, while accepting each step in the argument. In real life, no decision is isolated. Even on the assumption that life in jail is completely controlled by the guards, someday you hope to be back in the community you are leaving. Your reputation there will depend on whether you confessed, and you might well be willing to spend a year in jail rather than be known as a 'snitch'. Your instincts were right.
To model life more realistically, similar games are run repeatedly. Before each turn, each player is told how the other acted at his last turn. I think this book is the one I read, with more detail than this web page about computer simulations of this kind of situation.
In the simplest possible example, two players take turns, deciding whether to be mean or generous. In a generous turn, you do something that costs you a dollar, but is worth two dollars to the other player. The other player knows your turn before he takes his next turn. After each turn, there is a one-in-a-hundred chance that the game will end. You have no communication with the other player, except through the turns. Your object is not to 'beat' the other player, but to end up with as much money as possible.
For example, suppose you leave work every day, stop at McDonalds for takeout, and make sure you arrive home in time for the 6o'clock news. You find your neighbor does exactly the same thing. You decide that each of you will buy two meals on alternate days, so the other will save the time and gas of the detour to McDonalds. Whenever it is your turn to buy you have to decide whether to do what you promised, or to save money by not buying the meal for you neighbor.
It seems reasonable to believe that if you do not buy the second meal, your neighbor will be less likely to buy your next meal. The arrangement benefits both of you, saving time and money, and also you find it's more pleasant to watch the news together and discuss it afterwards. If one day it was his turn, but he fails to turn up with a meal, you have a problem. If you buy him a meal next day as if nothing had happened, will he be tempted to sponge off you without reciprocating? If you don't, the whole beneficial arrangement may collapse. If you question him about his failure, he may become offended, he may lie, or in some other way the damage may be increased.
In such a situation, it can be demonstrated that the best strategy is one called 'tit for tat'. Here you are generous if the other player has yet to take a turn, and otherwise you do whatever the other player did last.
So far, I have described the work of John Nash and others. Below, I will describe my own efforts.
One assumption we have made so far is that each player judges correctly whether the other has been generous. In our McDonalds example it strains the imagination to imagine any ambiguity, but we will try. Later we will discuss a more complex situation in which miscommunication is frequent. Imagine that one day your neighbor arrives to eat the McDonalds you bought for him. You have eaten yours already. When he opens the package, he tells you he discovered no food. You know you paid for two meals and acted in good faith. Did McDonalds stiff you, or did your neighbor eat the food in the kitchen, then lie about it to try to cheat you out of another meal? Clearly the way you both handle this will have a big influence on whether your arrangement survives. If your neighbor agrees to buy the next day as if nothing had gone wrong, perhaps you could try cheating him occasionally now you know he will accept it. If you agree to buy next day, he has the same temptation if, in fact, he lied.
Under the mathematically simple conditions I described earlier, I have shown that cooperation breaks down if miscommunication occurs on at least four percent of turns. If there is a dispute at least once every three weeks (21 days) then go back to just buying your own food at McDonalds.
If there is a possibility of miscommunication, you cannot afford to be as harsh as 'tit for tat'. This policy, if the other player also adopts it, results in a series of mean turns after the first miscommunication, ending only when there is a 'lucky' miscommunication (you buy only one meal, but McDonalds accidentally gives you two). Usually, you cannot predict the exact frequency of miscommunications, nor do you always understand the exact value of a generous act. Under such circumstances, a reasonable general strategy is to randomly forgive one-third of the other player's apparently mean turns, and retaliate against the other two-thirds.
To invent plausible miscommunications is beyond me (witness my earlier attempt). This forces me to use a famous public example. It is famous partly because so many people have passionate beliefs about the morality of the situation. I am not attempting to discuss the morality here, merely to stress that miscommunications do occur, that the results can be tragic for all concerned, and that game theory can diminish the tragedy if it is understood and acted upon.
The following approximations may not be accurate, but they will suffice for illustrating game theory. There are about ten million Jews in the world, and Israel considers itself to represent them. There are about one billion Moslems in the world, and most of their leaders consider themselves to some extent opponents of Israel. Israel also considers them generally as active or possible opponents: for example Iran is clearly an opponent, although it is not Arab.
Israeli policy toward its Moslem neighbors varies, but the approximation I will use for analysis is that whenever a Moslem kills a Jew, Israel kills ten Moslems. The Moslems who feel most driven to respond may lack the technical and organizational capacity to kill ten Jews for every Moslem, so I will approximate their policy by assuming they obey game theory, killing two Jews for every three Moslems killed. Since there is no central Moslem authority, Moslems have less ability to adjust their policy, so this discussion will center on Israeli policy, not to judge it morally, but to ask whether its actions can lead to its proclaimed wishes, or indeed to any result that Israel could be plausibly considered to regard as satisfactory.
Assume there have been no incidents for some time: then a settler vehicle accidentally kills three Moslems. The Moslems initially do not realize it was an accident, so they kill two Jews. The Israelis then kill twenty Moslems. By this time the Moslems have realized the initial killings were an accident, but they still have eighteen 'extra' dead Moslems to avenge, so they kill twelve Jews. The Israelis kill 120 Moslems, and the Moslems kill 80 Jews. It is easy to see that the violence will continue to escalate indefinitely. For all the obvious approximations I have made, at least this prediction seems confirmed by events. Even if the Israelis succeed in maintaining their ten-to-one kill ratio, the end result of current policies, carried out without modification, can only be ten million dead Jews, one hundred million dead Moslems, and nine hundred million Moslem survivors. I have never met anyone, Jew or Gentile, who regards this as an acceptable outcome, but it seems the inevitable result of Israeli and Moslem policy.
If the Israelis could contrive a kill ratio of one thousand to one, the end result would be one billion dead Moslems, one million dead Jews, and nine million Jewish survivors. Whether this is technically and politically feasible for Israel, or whether they would consider it an acceptable outcome, I leave for others.
If the Israelis were to reduce their response to the game-theoretic two-for-three, let us rerun the scenario above, with the more pessimistic assumption that the Moslems never realize the first deaths were accidental. The Israelis kill three Moslems accidentally. The Moslems kill two Jews. The Israelis kill one Moslem. The Moslems probably kill one Jew. The Israelis possibly kill one Moslem. There it ends, at least until the next accident.
Clearly this is still not ideal - in general for everyone who dies in an (unavoidable) accident two will still die in retaliatory strikes, but it seems hard to avoid the conclusion that greater understanding of game theory would benefit the Middle East.
A policy intelligently designed to produce peace without genocide must meet two criteria: first it must make enemy attacks unprofitable (which the Israeli policy does achieve) and it must result in something resembling peace if the 'enemy' adopts a mirror-image strategy (which the Israeli policy does not achieve). A two-for-three policy accomplishes both.