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.
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.
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
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:
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
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 :