Reprinted from the University of Toronto Bulletin
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Games Where the Winner Doesn't Take All
Judith Knelman

It's commonly believed that so-called zero-sum games like Monopoly, poker and bridge, in which what one player wins represents the loss to his opponents, are an imitation of life. Success means someone else's failure, a feast someone else's famine. Survival of the fittest means it's you or the other guy: to keep on top of the competition, you have to deprive others of what you all want.

Anatol Rapoport, professor emeritus at the University of Toronto, director of the Institute for Advanced Studies in Vienna until December and now back here to do research and teaching in peace studies at University College, returned to Toronto each summer to teach a course in social psychology that demonstrates, among other things, the folly of this notion. In 1970 he came to the University on a dual appointment in mathematics and psychology, and now he combines the two approaches in a statistical analysis of how people tend to resolve conflict. By means of a program of strategy he has worked out for a simple game called the Prisoner's Dilemma, which looks something like tick-tack-toe and takes even less time to play, he is able to show that life is not a zero-sum game at all. Not only is it not necessary for the winner to take all: it is impossible. The winner does best by sharing and never attempting to put one over on the opponent. To win, you quietly follow the other person's lead, never trying to outmaneuver him except in immediate retaliation.

Life, says Rapport, is a mixed motive game in which the interests of people partly coincide and partly conflict. To get what they want, they have to co-operate. They must trust each other consistently and be prepared to share the rewards available.

The game, which was discovered and circulated in the early 1950's, has aroused a tremendous amount of interest in academic circles, he says, because it demonstrates an important moral lesson: that the meek shall inherit the earth. When it is played in a situation that simulates society or evolution -- a tournament environment wherein every player uses his own peculiar strategy consistently against every other player and then against himself -- those who co-operate do much better than those who try to trick their opponents.

"Think of two scorpions in a bottle", he suggests. "If neither attacks, both will survive. If one attacks, the other retaliates, and both die. An even worse situation for the scorpions develops when one has to plot its strategy for survival on the assumption that the other may attack at any time".

The game worked out to represent the prisoner's dilemma mathematically gives each prisoner two alternatives. Each is told that if both keep quiet they will both get a sentence of two years, but if one rats he will get off free while the accomplice will get the five years. The catch is that if they rat on one another both will get four years. If each is sure that the other will keep quiet as well, that is the best course for both. But can they trust each other?

Prisoner's Dilemma
Your Accomplice
Keeps QuietRats
YouKeep Quiet(-2, -2)(-5, 0)
Rat(0, -5)(-4, -4)

The dilemma of the game is in the circumstance that it is in the interest of each prisoner to implicate the other whether or not the other cooperates. If the other keeps quiet, he will still get a two year sentence, while telling on the other gets him off free. As betrayal by both results in a four year sentence while keeping quiet could result in a five year sentence, it's best to rat no matter what the other does. However, if neither rats, both only get two years.

Robert Axelrod, a political scientist at the University of Michigan, where Rapoport taught before coming to the University of Toronto, decided to extend this problem to a tournament using computers to find the best consistent strategy for this sort of dilemma, which regularly confronts individuals and governments, in the form of potential rewards rather than punishments. The goal is to do as well as possible in your dealings with others over the long term. Rapoport won over all the other experts with the shortest, and simplest program submitted. TIT for TAT, which shows you that you do not have to deprive others in order to succeed yourself. His strategy is to co-operate or defect according to the lead of the other player. Even the most successful of the rival programs came to grief when they had to play against themselves, but TIT for TAT did nothing to hurt itself. It demonstrates the golden rule, do unto others as you would have them do unto you.

Method of Play
Player A
CooperatesDefects
Player BCooperatesX,Y
Defects

If Player A chooses the left column and
Player B chooses the top row, the top
left box prevails. Player A gets the value
of Y in that segment, and Player B the
value of X.

You play the game over and over again with the same partner, so that what happens in one game influences what happens in the next. You also play it over and over again with other people, just as you interact more than once with a large group of people in your everyday life. The idea is to accumulate the highest overall score. It is not necessary to vanquish individual rivals in order to do this.

One of you chooses the boxes in a horizontal row, the other in a vertical column. Neither knows until the game is over what the other's choice was but it's obvious that if you are playing for money you will want 10 points rather than -10 and may have to settle for 5. You get the number of points contained in the square that overlaps both choices, the row chooser getting the number of points on the left and the column chooser the number on the right in that square.

This is one example of the game being played for money.

5, 5-10, 10
10, -10-1, -1

If you were sure that the other would choose the column or row that contained the 5's, you might opt for that one -- except that you might be tempted to outmaneuver your opponent by opting for the 10. The hitch is that the two of you have to play again, and he wouldn't trust you the next time.

"It is a very neat way of testing to see how people co-operate and compete," says Rapoport. Many experiments have been conducted -- including some at the Ontario Science Centre -- to see how long it takes people to realize that it pays to co-operate, how long one player can stand to be the loser in an effort to induce the other to be co-operative, etc. Rapoport says that people in the test situation of a long series of plays (usually about 300) seem to learn at the beginning not to co-operate but to try to beat their opponents or to defend themselves. Then they both start losing. About half start out co-operating, and after an initial decline of co-operation 70 per cent of the men have fixed on co-operation, but there is not the same recovery for women, of whom only about 35 per cent end up co-operating. That seems to be not because women are less co-operative but because they pay less attention to a diagrammatic representation of the game. When men play without seeing the diagram they do as poorly as women or worse.

The research has obvious implications in many areas from domestic (the parent-effectiveness method seems to be based on it) to international. Rapoport uses it to plead publicly for nuclear disarmament: in fact, he was recently brought from Vienna to Toronto to give a symposium on the university's responsibility in the promotion of peace sponsored by Philosophers for Peace, Physicians for Social Responsibility, Science for Peace, the Student Christian Movement, University College and the University of Toronto Campaign for Nuclear Disarmament. He thinks that like two scorpions in a bottle we are doomed if we do not trust our rivals. And even if our trust is not justified he points out -- if the other side does not disarm and we do -- we may actually be safer than if we remained armed, since once we are no longer a threat they would have no need to attack us.

"I have no use for either superpower", says Rapoport. "I very much admire the small democratic countries that are not powerful." Canada, he says, is "sensible". It has the advantages of the U.S. without succumbing to the excesses.

In his own life, Rapoport has avoided competition as much as possible. He gave up a career as a concert pianist in Europe and the U.S. in the 1930's, because he did not enjoy the competitive aspect of it. In 1938 he enrolled in the graduate program in mathematics at the University of Chicago and received his Ph.D. two days before Pearl Harbour -- on Dec. 5, 1941. Then he became a captain in the U.S. Air Force. "At that time I believed it was absolutely necessary to crush the Axis," he says. "Now I'm not sure. We expected that the destruction of the fascist power would bring global co-operation, but that certainly did not happen."

As in the game, the secret of success lies in the correct definition of the problem. "You make your choice by asking not "How do I do better?", says Rapoport, but "How do we do better?" You have to trust each other to co-operate. Then the answer is obvious."