A beautiful equilibrium — "A
Beautiful Mind" is the first film that deals with a profound social issue that
touches everyone and yet that few people mention or even understand — a Nash
equilibrium. The film also focuses on the mental illness of mathematician John
Nash but only because Nash won the 1994 Nobel Prize in economics for a two-page
paper called "Equilibrium Points in N-Person Games" that he published in the
Proceedings of the National Academy of Sciences in 1950 when he was just 20 years
old. There would have been no film without the Nobel Prize.
Yet the
filmmakers failed to correctly explain a Nash equilibrium. And a mere flash of
text at the film's start only hints at why a review article in the September 1999
issue of the Journal of Economic Literature said that the impact of Nash
equilibrium in the social sciences "is comparable to that of the discovery of the
DNA double helix in the biological sciences."
So what is a Nash
equilibrium?
Nash equilibrium shows how selfish competitors should act
given how their competitors act. A Nash equilibrium has a simple mathematical
definition. Here is how Nash described it in words in his 1950 paper called
"Non-Cooperative Games": "Each player's mixed strategy maximizes his payoff if
the strategies of the others are held fixed. Thus each player's strategy is
optimal against those of the others." So each player does the best he selfishly
can given the competitive context of his competitors doing the best they can. The
competitors constrain one another's selfishness. And no competitor has an
incentive to change his strategy once all the competitors are in a Nash
equilibrium.
Jean-Jacques Rousseau suggested a hunting example in his 1755
"Discourse on the Origin of Social Inequality" that has led to something called
the deer game. Suppose you are one of four starving people in the forest. You
have just two options: You can try to catch a rabbit or a deer. But you can catch
a deer if and only if all four of you try to catch one. How each person behaves
depends on how the others behave.
Suppose the other three persons catch
rabbits. Then the best you can do is catch a rabbit too because you have no
chance to catch a deer. This rabbit hunt is a Nash equilibrium because each
person does his selfish best given what the others do and because no one has an
incentive to switch hunting strategies.
Thus does order arise from
competitive struggle.
Now suppose the other three persons try to catch a
deer. You still eat if you catch a rabbit but then they can't catch a deer. They
will catch a deer if you help them and then you can all have a feast. So it is in
your selfish interest to try to catch a deer. The deer hunt is also a Nash
equilibrium because each person does the best he can and has no reason to change
his hunting strategy. The latter point is essential. A player cannot have any
incentive to switch strategies.
The movie gets this backward when it
concocts a "blonde game" in a bar. Each young man in the bar wants to pick up a
blond woman rather than a brunette. Then a blond beauty walks in with several
brunettes. The Nash character (Russell Crowe) conceives the Nash equilibrium in
this fictitious scene. He claims that no man should pursue the blonde because
they can't all have her (and this will insult the brunettes). So he claims that
the optimal strategy is to pursue only the brunettes.
But each man will
want to switch from his brunette to the blonde if all the other men have
brunettes. So this is not a Nash equilibrium. The film's logic says that children
will pick up only the pennies on a sidewalk and not the hundred dollar bill lying
next to the pennies because they can't all have the bill. Our own selfishness
says otherwise.
Studies of ultimatums have shown that we can be so selfish
that we become envious and we don't achieve Nash equilibrium. Suppose I have a
hundred dollars and I offer you a share of it. The rules let us keep our shares
if you accept my offer. But neither of us get anything if you reject my offer.
Then I should offer you as little as possible and you should accept anything I
offer. But more than half of players reject an offer less than twenty dollars
even though accepting even one dollar is better than nothing.
Nash
equilibrium can also explain the darker side of behavior. I published a paper I
wrote as a student about outlaws who grow and steal marijuana plants (you can
download it from my USC web page). It pays to steal if there are many more
growers than thieves because growing pot is so risky. But it pays to grow if
there are too many thieves because a grower has some chance of harvesting
something while thieves find little to steal and other thieves will steal from
them. Players adjust their strategy mixes of growing and stealing until they
reach Nash equilibrium (whereas legalization lets growers organize and use the
police).
Yale economist Stephen Morris applied Nash equilibrium to
political correctness in a 2001 issue of the Journal of Political Economy.
Political correctness deals with not telling the truth because of fears to one's
reputation. An advisor may lie to her boss if she fears some words or opinions
will harm her reputation. The extreme case leads to a "babbling equilibrium"
where the advisor's advice is no better than flipping a coin and so her boss
ignores her. This can apply to advisors from stockbrokers to astrologers to
political consultants.
John Nash deserved his Nobel Prize — and a
more accurate movie. — Bart Kosko
| R.W. Bradford is
editor and publisher of Liberty |
|
Axis of stupid — How can we explain
George W. Bush's call for action against Iran, Iraq, and North Korea, the "Axis
of Evil"? The notion that Iran, Iraq, and North Korea form any kind of axis is
preposterous on the face of it, as has been pointed out by any number of
observers. Iran and Iraq are traditional and very deadly enemies and North Korea
seems to exist in an isolated fantasy world comparable only to that of José
Gaspar Rodríguez Francia ("el Supremo"), whose rule of Paraguay in the early
19th century prohibited people and goods from entering or leaving the
country. I suspect that Bush's otherwise goofy pronouncement is a product
of his discovery of the wisdom of Randolph Bourne's maxim "War is the health of
the state."As head of state, he naturally wants it to be healthy. We've had a
nice little war to vanquish the Taliban with hardly a single American casualty.
That war made the state more popular, enabling the president to increase taxes
and the power of the state. But that war is over. For the state to continue to
grow more health, new opponents must be found. What Iran, Iraq, and North
Korea have in common is governments that are immensely disliked (and with good
reason) by most Americans — disliked enough, I suspect Bush believes, that
Americans will be happy to wage war against them. I am not so sure that
Bush is right about this. Americans were good and mad about the terrorist attacks
of Sept. 11, and accepted the view that Osama was behind them and that the
Taliban were harboring Osama. But during the last ten years, neither Iraq nor
Iran nor North Korea has done the U.S. any more harm than did el Supremo, and
none show any inclination to try to do so. Historically, Americans have never had
much enthusiasm for wars against countries that do not threaten us, especially
once there are casualties. And — unlike the Taliban — Iraq, Iran, and
North Korea are certainly capable of causing casualties. When I was a
teenager, I read Latin American history voluminously. I remember one episode to
this day. Some time in the 1860s, a mob in one of Bolivia's capitals stormed the
British embassy, grabbed the ambassador, stripped him naked, and paraded him
through the streets so that locals could spit on him. The news was brought to
Queen Victoria along with a proposal for Britain to recover its honor by invading
the Andean country. Victoria rejected the proposal, choosing instead to issue a
command that Bolivia no longer appear on maps issued by the British
government. For all I know, the story is apocryphal. But even a fable can
contain an important lesson. — R.W. Bradford
| Stephen Cox is a
professor of literature at UC-San Diego. |
|
| Word Watch It is always
interesting to see the effect of public events upon our language. The case of
Sept. 11 and its aftermath is especially interesting, because the effect has been
— what shall I say? — curiously abstract. Take the name of the
event itself: "Nine Eleven." It's impossible to think of a less inherently
meaningful designation of an historical event. It's like calling America's day of
independence "Seven Four." The closest analogue is the famous "days" of the
French Revolution (e.g. "9 Thermidor"), but they don't consist entirely of
numbers. There are several reasons for the abstractness of the name for
what-happened-last-September. The terrorist attacks took place in several
locations, so a name can't be built on "New York." And if you called what
happened "The Terrorist Attacks," you would miss part of its meaning. As
perceived by the American populace, the event was a cause of regret (for the
great loss of life), but it was also a cause of celebration (of the courage of
the victims and rescuers). A less successful naming is "Ground Zero," for
the site of the former World Trade Center. The name responds adequately to the
problem of locating a concise phrase for a long, messy concept. "The Site of the
Former World Trade Center" clearly wouldn't do. "Ground Zero" suggests, however,
that the place is like a bomb site — any old bomb site — which it
clearly isn't. The phrase is too open-ended, and at the same time too erroneously
specific, since the obvious associations of "Ground Zero" are with World War II.
"Ground Zero" will always be nothing more than default terminology.
"Fighters" is a locution that did not begin with Sept. 11 but has been immensely
popularized by its aftermath. There are no more "soldiers" left in the world;
they are all either highly specific "American Special Forces operatives" or
highly generalized "Islamic fighters." Use of "fighters" responds, no doubt, to
the difficulty of deciding whether those guys in Afghanistan are really in an
army or what, dude, but it also responds to the movement for social equivalency
that gave us "worker," as in "sex worker" and "home worker" (for "prostitute" and
"housewife," respectively), not to mention "shooter" (for "murderer," or
"terrorist" or "assassin" or "crazed postal worker" or "policeman"), the bottom
line of which is simply "anyone who shoots a gun." Much more inspiring
are two foreign additions to our vocabulary, both of them abstract enough to have
immense expressive possibilities. One is the syllable "bin," from "Osama bin
Laden," which has now acquired the meaning of "fanatic in some creepy, foreign
way"; thus, "Johnny bin Laden" (John Walker Lindh) or Janet bin Reno (Janet
Reno). The other is "Taliban," a word that Americans gladly adopted without even
caring to have it translated. All they needed to know was the abstract concept
— some bunch of self-righteous weirdos trying to boss everyone else around.
Since nobody cares what the word originally meant or how it was originally used
or even whether it was supposed to be singular or plural, one can say with
impunity, "He is a Taliban," but also, "The Taliban fight on." And the satiric
potential is enormous. It's easy to see that everyone from NOW to MADD to those
grasping relatives of yours can quite amusingly and accurately be called the
Taliban. The word isn't quite as useful as some words popularized by earlier wars
("spam" immediately comes to mind), but it will do, and do nicely, when those
relatives come around. — Stephen Cox |
| |
