## Maths Illustrated

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. Mr. Crowe's character shares with students his intimate view of mathematics as an "art form" after entertaining them with a story of a fly and two bicycles. (Follow the link to learn how von Neumann reportedly solved the problem. Mr. Crowe's character urges students to look for the problem's essence.) The monologue is unscripted! Mr. Crowe's insight doesn't just inform his portrayal of John, it permeates his revision with Ron Howard of the script. As a result the movie's story makes sense to me. In a recent interview, Mr. Crowe describes acting as "a very mathematical job".

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.

Did you notice the (curious, praiseworthy and blameworthy) details enumerated below?

Math consultant Dave Bayer was one of the professors in the second instance of the movie's pen ceremony. (Just a glimpse.)

Mr. Crowe's character John says "papers in hand, Mr. Bayer" to a student in a stream exiting his classroom. (Nod to Dave.)

The author of the advanced calculus text used in Alicia's first scene is Bayer. (Missed this. Dave had to tell me.)

Translation by one (1947 to 1948) and dilation by a factor of two doesn't quite correct the movie's early time line. (Why rush?)

John's office blackboard displays the same math when Parcher invades as when Sol and Bender loitered. (But not inbetween.)

John endures ten weeks of insulin coma therapy; Nash endured six. (Six is a perfect number. Why change it?)

The classroom scene shows what John and Alicia are about and why they connect. (Bravo. Perfect elaboration of the real story.)

Distracted by Charles, John doesn't notice that pigeons don't scatter when Marcee charges through. (Noticed on 6th viewing.)

An enlarged infinity symbol represents John's bicycle path. (Bravo. Sylvia Nasar's biography of Nash calls it a figure eight.)

John decodes 67 46 9 0 as 67 degrees and 46.9 minutes. (Bravo. David Sloan found Starkey Corners at longitude 67°47'20".)

The line "we've developed several ciphers" sticks out like a sore thumb in the Pentagon scene. (Why not "the data's clean"?)

John sounds like a math moron in his Nobel speech. (Movies should be unreal - Nash didn't give a speech - but not unrealistic.)

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