Maths Illustrated
John Nash, played by Russell Crowe in A Beautiful Mind,
was awarded a Nobel prize for his work on noncooperative games. To
view an example that illustrates Nash's equilibrium theorem for math students and movie fans, first
download Acrobat Reader.
This page was referenced in my movie review in the April 2002 issue of Notices of
the American Mathematical Society. If you are unable to read the
pdf version of the review at the Notices website, you
can read an html version of the review here. If you would like to
hear how John Nash understands his life, watch the segment on Misconceptions about Mental Illness in his interview for the
PBS documentary A Brilliant Madness. Also online is an audio recording of
the May 2002 conversation with his biographer and the math consultant on A Beautiful Mind.
I promised to explain the game theory in John's response to Alicia's question "I believe in deciding things will be good luck,
don't you?". Holding her handkerchief, he replies "No. I don't believe in luck. But I do believe in assigning value to things."
John wins a Nobel prize for his concept of equilibrium in noncooperative games. If the noncooperative game is a
two-person zero sum game, the strategies in a Nash equilibrium are optimal strategies. Von Neumann showed that
optimal strategies exist for every two-person zero sum game by proving that every
such game has a unique "value" (the expected payoff to one player and loss to the other, when both play optimally). John assigns an emotional worth to the handkerchief as he assigns a numerical value to a zero sum
game. He believes in his affinity for Alicia as strongly as he believes in his intuition for game theory.
Nash proved that every general sum (noncooperative) game has an equilibrium:
a collection of (mixed) strategies, one for each player, such that no player can improve his (expected) payoff by changing his
(mixed) strategy unilaterally. Examples like the Prisoner's Dilemma show that
equilibrium strategies should not be called optimal, contrary to
Keith Devlin's presentation on National Public Radio
and Akiva Goldsman's scene for A Beautiful Mind.
A Nash equilibrium is not necessarily a "best solution" nor does it necessarily give the "best result". (The links above are to
the 7 minute audio clip and 90 second video clip from which I quote.)
Nevertheless, at such an equilibrium, no player is motivated to change his (mixed) strategy since he cannot force other players to
change theirs.
In the Prisoner's Dilemma, what's best for the two players is
for them both to refuse to testify (and each spend 1 year in jail), but this is not a Nash equilibrium since if one were to testify
he would be rewarded with a reduced sentence (0 years). At the Nash equilibrium they both testify (and
each spend 2 years in jail), since if only one were to refuse to testify he would be punished with a longer jail sentence (3 years) based
on the other's testimony.
To use the governing dynmaics scene (just choose a format to
view the 90 second clip) to motivate an
analysis of Nash equilibria in a noncooperative game, do not consider John's
four friends as the only players. John says "If we all go for the blonde, we block each other and not a single one of us
is gonna get her", so John may be considered one of the players. He suggests that "no one goes for
the blonde", but the visual shows only his four friends pairing up with the other women; as Martin suspects, John plans to pair himself
with the blonde. The fact that he doesn't do so at the end of the scene should be attributed to the fact that he suddenly decides to
rush off to work out the details of his Nobel-prize winning idea. The details of the two-player version of this noncooperative
game reveal that an equilibrium need not yield a "best result" for any of the players.
(See the sidebar in my review.) Contrary to
John's statement at the end of the scene, the theory of noncooperative games assumes that each player does what is "best for
himself", regardless of what is best for "the group". Altruistic behavior can develop in ongoing social, political,
and economic interactions despite this assumption, as explained in
Axelrod's book on the iterated
Prisoner's Dilemma.
Mr. Crowe's understanding of the nature of mathematical discovery is revealed by his insight "Nash's mind is much more the way we think of an artist's mind, rather than a scientist's mind," quoted in production notes published with the Newmarket shooting script.
Math consultant Dave Bayer's contributions to A Beautiful Mind are described in an article in Science. He was asked to make the "mathematics reflect Nash's descent into mental illness and his slow emergence". The line "the zeros of the Riemann zeta function correspond to singularities in spacetime" is inspired! It makes perfect sense in the story, as do most math details in this movie. (The governing dynamics scene, dialogue in the Pentagon scene, and Nobel acceptance speech are exceptions.) To help folks who know the significance of 3.14 make sense of the string of inequalities on John's forehead in the photo below, Dave Bayer explains that the young John Nash used Greek letters playfully. (See, for example, page 18 of Nash's PhD thesis, reprinted in The Essential John Nash.) Bayer is also an expert on a game Nash created, now known as Hex. Hex is more interesting than Tic Tac Toe, but almost as easy to learn! To play against a computer opponent, download Hex for the PC or download Hex for the Mac; to play with your children or friends online, alternately click on hexes in a Hex board online.
