Ideas in Game theory: Jan-ken-po Win-win Zero-Sum Competition Prisoner's Dilemma Cheating Tragedy of the Commons Gorging Cooperation Deception I'm OK Johari Window Attachment Politics Traps Altruism Recursion Coopetition Values Rules References Links.
They say "life is a game." What this metaphor means, is that in life there are rules, and your role is to play by the rules, attempting to maximize your gains and minimize your losses. When you utter the phrase "life is a game," it is usually in response to some dissatisfaction with the rules such as a personal setback, and is a way of distancing yourself from the emotional impact or the reality of things. A perfect example is the Oscar winning Italian film "Life is Beautiful," where a father convinces his very young son that being in a concentration camp is only a game. Jan-Ken-Po Children love to play games. They seem to spontaneously make up rules, and change them at will, just to have fun. As an introduction to game theory, consider this favorite game of children worldwide:
American: Paper Scissors Stone Chinese: Ching Chang Polk Japanese: Jan Ken Po Korean: Ra Sham Bo
Here are the rules: opponents face each other with a closed right hand fist. Both opponents then chant the three words together in rhythm moving their fists down each time. On the third time, the opponents suddenly open their hands into one of three signs: paper : hold out all five fingers scissors: hold out the second and third fingers, close the thumb over the last two fingers (like the victory sign). stone: keep your hand closed in a fist. Do not open it. In English, the chant sounds like:. . ."paper, scissors, stone, - paper, scissors, stone - paper, scissors, stone (on stone you suddenly display one of the three signs) In Chinese, it sounds like: ching, chang, polk, - ching, chang, polk, ching, chang, polk. In Japanese: jan, ken, po - jan, ken, po - jan, ken, po ... In Korean: Ra, sham, bo - ra, sham, bo ..... This is how to determine the winner: PAPER wins over STONE because it can wrap around the stone. STONE wins over SCISSORS because it can break the scissors. And SCISSORS wins over PAPER because it can cut paper. If both players make the same sign, its a tie, and then you do it over. If you want, you can play Jan-ken-po right now! The game is often used in decision making, such as who will make the first move. I often see kids playing this game on the side of a swimming pool: if you lose, you take one side step closer to the cold water, until you have to jump in! There is some strategy to the game, because if your opponent displays some kind of pattern, such as using one sign too frequently, or a tendency to follow a set pattern, this can be taken advantage of. Also you cannot reveal your hand too soon, or your opponent might be able to switch. Zero-Sum Games The subject of game theory has been investigated by the best minds. The brilliant mathematician, John von Neumann,1 2 , inventor of the modern computer and co-developer of the atom bomb, investigated the case of Zero-Sum games, where one person's win is another person's loss. The amount of resources that you are competing for is fixed, so the gains added to the losses sum to zero. For example, chess is a zero sum game. It is impossible for both players to win. Non-Zero-Sum Games However, Monopoly, (if it is not played with the intention of having just one winner) is a non-zero-sum game: All participants can win property from the bank. If all players cooperated, they all can get richer although this is not the true intent of the game. John Nash won the Nobel Prize for extending game theory analysis to the case of non-zero-sum games. It has been widely influential in the most weighty matters, such as economics, military strategy, and politics. To get a better understanding of game theory, let's simplify Jan-ken-po by removing the rock, so you can only display one of two signs, either paper or scissors. You can then succinctly represent the outcomes in the form of a table:
You can ask, what are the whole number solutions to this equation, and you can see that:
and
are both solutions. It was Fermat's custom to mark his books with comments, and here is what he wrote about this equation: "On the contrary, it is impossible to separate a cube into two cubes, a fourth power into two fourth powers, or, generally, any power above the second into two powers of the same degree: I have discovered a truly marvelous (demonstration of this general theorem) which this margin is too narrow to contain." In other words, there is no integer solution to the equation, if the exponent is greater than 2. Wiles, as a 10 year old child, read Eric Temple Bell' short essay on Fermat, and became smitten by this equation, and decided that he was going to solve it. 40? years later he did. But, the most amazing thing, is that during the time he was solving it, as a Professor at Princeton University, he spoke to no one of his labors. Why, because, if he were to disclose partial results, then someone else might beat him to the solution, and get all the fame! So, he did everything in complete secrecy. Finally, he needed to have someone check his proof, someone who was not only knowledgeable but who could also keep a secret. He found him, but couldn't very well visit him everyday to explain the math. People would know something was up. So, he concocted a graduate level course, and invited his friend over to audit the course. Graduate students signed up for it, but since they didn't know what he was up to, it was just a bunch of boring equations, extremely difficult, I might add, so all of them eventually dropped out. It was just his friend and himself, and when they were satisfied that the proof was correct, they announced it at a meeting. The applause was thunderous, and the next day, he was written up in the New York Times. It is very rare for a mathematician to be mentioned in a newspaper. Well, there is much more intrigue to this story and you can read the book "Fermat's Enigma" for the details. So, Wiles opted for the win-lose option, and fortunately he won. And it was a close call, because a mistake was found in the proof, and for more than a year after the formal announcement, he and other mathematicians raced to fix the error. It was a nightmare for Wiles, who delayed the formal publication of his work, because he knew that to publish a flawed manuscript would release an onslaught of questions by would-be gapfixers, and he knew this distraction would destroy his own hopes of fixing the proof, and while giving others valuable clues. But not all mathematicians choose this option. An equally famous mathematical prodigy by the name of Paul Erdos (pronounced air-dish)(1913-1996), chose the win-win option. He freely shared his time, and results with anyone who wanted to collaborate, a measure of that being the Erdos number. He loved mathematics more than anything else, never acquiring the baggage that encumbers us mere mortals: he never bought a home, married, had children, drove a car, never even had a job, and he was not interested in sex. Erdos' generosity was legendary. Everywhere he went, he was taken care of. Mathematicians around the world sought his advice, and would gladly pay his expenses if he would drop by to help them. Like grandmaster chess players who can play 10 people simultaneously, people would gather at his hotel room, and Erdos would go around the room, helping a half dozen mathematicians simultaneously. At Bell Labs, where I used to work, I heard stories. One researcher built an addition to his home just for Erdos! In anticipation of Erdos' visit, he would stock the refrigerator with bags of grapefruit, Erdos' favorite fruit. Any money Erdos earned from his lectures, he donated to worthy causes, often helping young mathematicians with much appreciated monetary gifts early in their careers. Often he posed tantalizing problems, and offered a reward to the first to solve it. He set the award amount according to the difficulty of the solution. As a consequence he published an astounding 1500 papers during his lifetime, almost 500 of them written with collaborators. In contrast, the celebrated von Neumann only wrote 154 papers. Nash fewer still, but the reason why, is another fascinating story. Politics of Liberal vs Conservative In US politics, a liberal is a "cooperator" someone willing to share, and even be taken advantage of if there is a chance the common good will benefit. For example, liberals are willing to pay more in taxes to help the homeless because they believe that the homeless are good people, and will use these funds wisely to help themselves get back on their feet. A liberal will also favor cutbacks in military spending, taking this risk with the hope that the other side will make fewer missiles also. Conservatives, on the other hand, can be viewed as "defectors", in that they believe one should enjoy the fruits of ones labor alone, and not share them with anyone else. They worry that taxes may be squandered, so they favor lowered taxes, so that income can be kept under one's individual control. They worry that another nation will exploit a unilateral arms reduction. They fear being laughed at, for making the "sucker payoff" to welfare cheats, and arms treaty violators. Traps When one plays to win, one can be drawn into a trap. For example, Wiles was unusually self-assured about his ability to reach his goal. Basically, he spent his entire life proving Fermat's Last Theorem. But he was also in a trap. It is like waiting for the bus. You wait, it doesn't come, you wait some more, and then you wait just a little bit more to justify the time already spent waiting. This is the hook that ruins gamblers, who make one more bet to justify the losses already incurred. The late Jeffrey Z. Rubin (1941-1995), Professor at Tufts University suggests that traps can be avoided if you set limits in advance regarding the level of your commitment and involvement, and more importantly, stick to it! Also, avoid asking others for advice. Like waiting for the bus, if others are waiting there too, even if they are complete strangers, then you are more likely to wait a little too long. And watch out if you have a need to impress. While everyone wants to be loved and respected, people get entrapped when they know they are being judged, and feel they have to prove something about themselves. Influence of Game theory in Psychology: I'm OK, You're OK The psychiatrist Thomas Harris (19??-1995) wrote a book on interpersonal interaction entitled "I'm OK, You're OK." The following table invented by Frank Ernst, called the OK Corral, was inspired by this book, for which he got an Eric Berne Memorial Award :