Games, Dilemmas, and Traps

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.


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:

I show paper I show scissors
You show paper Tie I win, you lose
You show scissors I lose, you win Tie

In game theory, instead of saying win or lose, a mathematical weight can be assigned to the value of each outcome. This is called the payoff matrix. In this case, we could imagine a third party bank making payments according to the table. The object of the game is to make the most money. For example, here is one payoff matrix representation of the above game:

I show paper I show scissors
You show paper 0,0 3,0
You show scissors 0,3 0,0

The first number represents what the bank pays me at the end of the turn, and the second number represents what the bank pays you. So, if I show scissors and you show paper, then the bank pays me 3 dollars, while you get nothing. However, if I show paper and you scissors, then the bank pays you 3 dollars while I get nothing. If we both show paper, then is a tie, and the bank pays nothing and similarly if we both show scissors. Now, the weights can be assigned in whatever way you like, which makes the game much more interesting to play. For example:

I show paper I show scissors
You show paper 2,2 3,0
You show scissors 0,3 -1,-1

The strategy of playing becomes different depending on the weights. Also, by assigning different weights, the interpretation become different. For example, the above matrix of values could then be interpreted as:

I show paper I show scissors
You show paper win,win win,lose
You show scissors lose,win lose,lose

Competition vs Cooperation:

We have all heard of trying to find the win-win solution, where both sides win. So for example, we can relabel the table's rows and columns with decisions from real life:

Cooperate Compete
Cooperate win,win win,lose
Compete lose,win lose,lose

In other words, if we both cooperate, we might be better off than if we both competed! It depends on how you assign values in the matrix. For example, we could be two different airline companies deciding on how to set the airfares. Let's put some values back into the payoff matrix:

I raise fares I keep fares same
You raise fares 2,2 3,0
You keep fares same 0,3 0,0

So, I might raise the fares on my flights, with the hope that you also raise them. If we cooperate, we both might make more on each ticket. However, I am also tempted to keep them the same, in which case, I get more passengers and therefore more revenue, and you lose by getting less. The opposite could also happen. If we keep our airfares the same, then we keep the status quo. Obviously, reality is much more complex, since if we both raise airfares, we might be accused of price-fixing by governmental authorities, and then end up losing. But we could also both lose if demand fell off for all flights. So, it is important to be able to make good predictions and have an accurate model of what the payoff matrix values are or will be.

Prisoner's Dilemma

Certain payoff matrices lead to situations where it is difficult to make a decision. The story of the Prisoner's Dilemma (an anecdote originating with Albert W. Tucker) is one of them: we are two burglars, and we just got caught by the police near the scene of the crime. The police interrogate us, and then after working us over, we return to our individual prison cells. Unfortunately, we cannot talk to each other. Then the policeman comes to my cell and proposes the following deal: "if you cooperate with me, and confess, you will go free, and your partner gets the max: 5 years." This policeman also gives the same deal to my partner. Then we sit in our individual cells, trying to decide what to do. We are both tempted to snitch, because there is a chance we will go free. However, there is the well known thief's code of honor: "never snitch on the other." This is because the police rarely have all the evidence to convict, and as long as I don't implicate you, and you don't implicate me, the sentence will be light, in this case, 1 year in jail on the charge of carrying a concealed weapon. However, if we both confess and snitch on the other, we both will go to prison for 3 years. And if this happens, when we are in prison, we will think to ourselves, well, it served us right because we both broke the thief's code of honor.

So what will you do. Clearly it is in both of our interests to keep silent. But how can I be sure that you will not doublecross me? And similarly vice versa? After all, how honorable is a thief? My silence might cost me 5 long years, a long time to feel regretful, since you got off scot free. So this is the dilemma and here is what the matrix looks like:

I keep quiet I snitch
You keep quiet We both serve 1 year I go free, you get 5 years
You snitch You go free, I get 5 years We both serve 3 years


Cheating is basically the same as snitching. I found the following game with the following payoff matrix on the web:

Cooperate Cheat
Cooperate 3,3 5,0
Cheat 0,5 1,1

Try your hand and see if you can beat the computer: Play Prisoner's Dilemma now!
After trying to beat the computer, peek at my hint webpage on how to improve your score.
Here is another place to play, where you can even select the strategy the computer uses.

Tragedy of the Commons

The same dilemma we faced as prisoners, we also face in many real life situations. The term "tragedy of the commons," was coined by Garret Hardin, in an 1968 article appearing in Science . Imagine living in a small community, in cottages which front onto a large common area, such as a park. We each have a cow, and usually we let the cow graze in our yards, from which we get milk. We use the milk on our oatmeal, and turn the excess into cheese, which we also eat, but also barter for other items, such as shoes, clothes or utensils. Now, one day, I get the bright idea, "I know, I will sneak out in the middle of the night and bring my cow onto the common park area to graze. Then it will produce more milk, and I can make more cheese, and then I can buy more things or have a bigger family." Well, things actually turn out even better: the cow has a calf, and I have a delicious veal stew, or you let it grow up so you can have another cow. As long as I can keep this a secret, then I am better off. However, its hard not to brag, and so I tell you my best friend, and so you do this too. Well, if a few people do this, no one really notices, and we can get away with it because the cost is shared by everyone else. However, what happens when the the secret is out, and the whole town gets the idea? Soon everyone is bringing their cow onto the common area to graze, even during the day! Now what happens? The park becomes a useless muddy field.

The analogy applies to so many real situations in the world today. A few of us drive cars, which belch soot and carbon dioxide into the atmosphere. No one notices. However, when we all do it, then we have serious air pollution, an ozone hole, and serious degradation of the environment. Similarly for all the worlds major problems. A big one is the global population explosion. Hardin used the tragedy of the commons to argue for the repeal of the United Nations' specific guarantee of reproductive freedom.

Links to Tragedy of the Commons dilemmas


Another example of the dilemma is gorging. Here is what one table might look like:

I'll be fair I'll be selfish
You'll be fair We both eat fair share I gorge, you eat fair share
You'll be selfish You gorge, I eat fair share We both gorge

We believe in competition, which often means we look out for ourselves first, and be selfish. Some of us are especially good at it. In fact, so good that currently the wealthiest 1% of the world's population owns as much as the bottom 95% Want to know who these people are? They are listed online in Forbes magazine right here! We live in a "winner-take-all-society." How can this happen? Well, the idea of division of labor is involved, in combination with our highly automated manufacturing society.

Let's consider one possible example. I play the violin, and I get a great deal of satisfaction volunteering for my community orchestra. However, let's say I had the idea of becoming a concert violinist. Often parents get this idea for the child, and force their child to practice ungodly numbers of hours. Typically, for someone planning to enter the profession, just to play in the second violin section in a professional orchestra would require about 7500 hours of practice before entering college. Its 10,000 hours if you are planning to play in the first violin section. That's 1000 hours per year for 10 years, or 3 hours per day from age 7 to 17! But lets say your are gifted, and you manage get to the absolute pinnacle of achievement, to become the best solo violinist, with a recording contract with a major label. You then become the "winner take all" artist! Your CDs will sell the best, because most people can only buy a few recordings, and most likely they will spend it on the best interpretation, rather than on the second best, even though that recording is almost as good, indeed probably indistinguishable. And because CDs can be stamped for just pennies a piece in high volume, once the recording has been made, its almost pure profit to sell. Similarly for software, and now you know how the richest man alive made his wealth.

Fermat's Last Theorem

The competition that I speak of pervades every field. For example, recently, in 1993, Andrew Wiles proved Fermat's Last Theorem. This was the holy grail in pure mathematics, a problem so simple to propose that a child can understand it, but unproven for 350 years. Anyone to solve this problem would win instant fame. Here is the story. The French mathematician Fermat (1601-1665) while reading Bachet's Diophantus, a book of arithmetic using only whole integer numbers, came to the section regarding Pythagoras' equation:

x2 + y2 = z2

You can ask, what are the whole number solutions to this equation, and you can see that:

32 + 42 = 52


52 + 122 = 132

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.


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 :

I'm OK, You're OK I'm OK, You're not OK
I'm not OK, You're OK I'm not OK, You're not OK

I'm OK, You're OK, the healthiest way to interact, encouraging words
I'm OK, You're not OK: Arrogance, putting others down, using discouraging words.
I'm not OK, You're OK: Learned helplessness
I'm not OK, You're not OK: We are both very sick!

Note that all options in game theory have a certain probability of being selected. As I write this, I'm thinking about the school bombing in Littleton, Colorado (20 April 1999) where 14 students and 1 teacher were killed in a rampage by two students. They just got fed up of being told that they were not OK, and said, you are not OK either and then....kaboom!

Johari Window

Known to Self Not Known to Self
Known to Others Open Blind
Not Known to Others Hidden Unknown

To better understand my brief comments below, read my webpage on the Johari Window. first. The Johari Window is model of interpersonal interaction. Each of us can be represented by a window. Let's examine mine. The "open" window represents things I know about myself and that you also know about me. This corresponds to the win-win situation in game theory, because when we have certain knowledge in common, our friendship is more valuable. The "hidden" window represents things that I know, but that you don't know about me. When I reveal information about myself, it goes into the open window, potentially increasing the value of our relationship. But there is a risk, depending on the kinds of information revealed. As I reveal information, you may take this information and use it against me. For example, I may reveal a personal weakness and seek your advice. So, initially, its a "I lose, you win" situation. However, now consider the "blind" window. These are things that you know about me, that I don't know about myself. By my taking the risk to reveal a personal problem, you may be able to provide insights that I am not aware of myself, thereby helping me. Your expression of care is a win situation for me, but you lose, in that you may have to expend a certain amount of energy and time helping me. But as trust grows, we both can move into the win-win situation of human interpersonal growth, since I would want to return the expression of care back to you. It's like the fuzzy story. Finally, the "unknown" window represents things I don't know about myself and you don't either. Initially, in a relationship, this is the large window, because we don't know anything about each other. But as we conduct the exchanges of information via the blind and hidden windows, these enlarge, making the open window larger, and the unknown window smaller. It is exciting to discover new things about ourselves. On the other hand, if, in the above example, the information I reveal is used against me, for example, you laugh at me, then the unknown window becomes larger, and it is a loss, because I would be hurt, and this could even lead me to hurt you.

Attachment Theory
Model of Self
Low dependence High dependence
Model of Other Low avoidance Secure Preoccupied
High avoidance Dismissing Fearful

Links: Attachment

Private property

John Dewey believed that the anarchy of the present competitive profit economy be replaced by a planning society in which production is democratically controlled for the good of all. In other words, we all control the means of production together . For example, the idea of the internet was researched by the government, and the first ones were built by the government, a communication network designed with the idea of survivability from a nuclear war. It was financed with taxes collected from all of us. But if you look what happening now, you can see that a few companies are trying to capitalize on the idea, by attempting to control the information that we see. Most people, when they launch their browsers, use the default webpage provided by a few companies, notably Microsoft and AOL (Netscape). You are then bombarded with slanted news and advertisements, and like so many sheep, you are herded and slowly fleeced. Why should a few companies try to usurp control of this resource, and derive large profits at we the people's expense?


In fact, even this webpage, if accessed through some search engines, becomes surrounded by advertisements! They are fleecing even me a little, although they return the favor by at least indexing my webpage. For example, just click on the search button below and

"Ask Jeeves!"  

What happens is that you get my "What is a Wordmap webpage listed somewhere on their search results listing. Select to view this webpage, and you will see it is surrounded by their advertisement. They are making money off of me. But next, select the "Game Theory" link on the wordmap, and you get back to this page. Click on "recursion" in the wordmap and you'll jump down right back here. But then click on the search button again. You will be surprised what happens! You get the wordmap webpage again, and two advertisements! This process is called recursion, and guess what, I've been fleeced one more time. At least they allow you to get rid of the advertisement when you click on "remove frames" in their banner ad. I will shorten this recursion, once this webpage gets indexed by them. Well, try it!! Click on the search button!


Rules, Values, Laws and Knowledge

I am trying to construct a set of values, rules to live by that corresponds to the win-win situation. One embryonic attempt is my malama webpage. What I am going to say next will require many webpages to explain, so don't feel bad if you don't understand. I'm interested in the application of mathematical principles to the construction of legal and social systems. Game theory is one approach, since values can be assigned to the outcomes, and can also be assigned to the probability that one choice would be made over another. There may not be a set of optimal rules applicable to everyone. The best we may be able to do is the same as in physics: of the many mathematical theories available, physics adopts the one which explains the data the best, subject to the law of parsimony. For example, Euclidean geometry was used by Galileo and Newton, for classical mechanics. Einstein had to adopt curved space geometries to explain relativistic effects. Yet, the entire edifice did not have to be replaced, because mathematics has a deep idea called self-consistency. Only Euclid's parallel postulate was swapped and replaced with another axiom, but the proofs are essentially identical. Similarly, in the way we adopt laws or social conventions. For example, physics has the idea of conservation of energy: energy is neither created nor destroyed. In economic systems, an identical analogy can be applied, money is neither created nor destroyed. So if you are wondering why your small town's economy is not doing so well, look at where the money is flowing. Do you spend your money at a local merchant, who then spends its profits in your community? Or at a big box retailer that sends its profits somewhere else? Perhaps love and attention are the same. Do you spend it on your children and spouse, who are most likely to return it to you, or do you spend it somewhere else? Maybe its like the fuzzy story. The best book to read about work in this area is: "The Great Disruption," by Francis Fukuyama (1999).

I'm also interested in the fundamental physical law/theories, knowledge/epistomology and the creation and destruction of knowledge.


Avinash K. Dixit and Barry J. Nalebuff. (1991) Thinking strategically : the competitive edge in business, politics, and everyday life, New York : Norton 393 pages.
Bell, Eric Temple (1937) "Men of Mathematics," NY: Simon and Schuster, pg 71.
Berne, Eric"Games People Play,"
Brandenburger, Adam M. & Nalebuff, Barry J.(1996) "Co-opetition:1. A revolutionary mindset that combines competition and cooperation. 2. The Game Theory strategy that's changing the game of business.," NY:Doubleday 290 pp.
Childs, John L. (1951) "The Educational Philosophy of John Dewey," in The Philosophy of John Dewey, ed Paul A. Schilpp, NY: Tudor Pub.p. 443.
Frank, Robert H., Cook, Philip J. "The winner-take-all society : how more and more Americans compete for ever fewer and bigger prizes, encouraging economic waste, income inequality, and an impoverished cultural life," New York : Free Press, c1995.
Fukuyama, Francis (1999). "The Great Disruption: Human Nature and the Reconstitution of Social Order," Simon and Schuster, NY, 336 pgs.
Fukuyama, Francis (1999). "The Great Disruption," The Atlantic Monthly, v. 283(5), pp. 55-80.
Hardin, Garret (1968) "The Tragedy of the Commons" Science, v162(1968):1243-1248.
Harris, Thomas, (19), "I'm OK, You're OK,"
Poundstone, William (1992), Prisoner's Dilemma
Nasar, Sylvia (1998) "A Beautiful Mind: a biography of John Forbes Nash, Jr.," NY:Simon & Schuster.
Rubin, Jeffrey Z. (19??) "Caught by Choice, the psychological snares we set ourselves," The Sciences, v
Schechter, Bruce (1998) "My Brain is Open: The Mathematical Journeys of Paul Erdos," NY: Simon and Schuster, 224 pp.
Singh, Simon (1997) "Fermat's Enigma," NY:Walker 315 pages.
Hoffman, Paul (1998) "The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth," 302 pp. NY: Hyperion

External Links:

Introductory Sketch of Game Theory
Levine's Zero-Sum Game Solver

Last updated 18 August 1999

Copyright © 1999 by Duen Hsi Yen, All rights reserved.


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