A Beautiful Mind

A Beautiful Mind is a 2001 American film based on the life of John Forbes Nash, Jr., a Nobel Laureate in Economics for his work on game theory. The film was directed by Ron Howard and written by Akiva Goldsman. It was inspired by a bestselling, Pulitzer Prize-nominated 1998 book of the same name by Sylvia Nasar (Faber & Faber, 1998). The film stars Russell Crowe, along with Jennifer Connelly, Ed Harris, Christopher Plummer and Paul Bettany.

Here are some references to Go from the movie (minor spoilers for those who have not yet seen the movie):

• Nash, while in Princeton, is lured to play by some of his friends-to-be. He loses, much to his amazement, as he says: "I had the first move... my play was perfect." According to Game Theory, in a complete knowledge game, he should've won, or at least achieved a draw. He then concludes "The game is flawed"! (quote fixed by Benjamin Geiger) (Note, Black plays the same stone twice - an earlier scene shows a stone in a place where Nash later plays, resulting in self-atari. They're also apparently playing with plastic stones.)

• Later on in the movie, as Nash returns to Princeton to try and fight his... problem, his old friend points at a goban while they're walking in the park, and invites him to an another game.

If you watch the DVD, there are several deleted scenes which include Go.

• In one, he totally trashes a Go game being played by two other math people when he throws down the game board to his newly invented (and "unflawed") Hex, right on top of their Go game!

Actually, he slides the board, knocking most of the stones off of the goban onto the floor. He then goes on to say, "Revised Conflict parameters nullify outcome potentials. An ideal move sequence guarantees first player victory, that means if you win, you deserve to win. I prefer chess."''

• In another, he stares introspectively at a Go board, puts down a stone, and keeps thinking.

Soundtrack:

• Not really related to the video itself, but in the movie's soundtrack, there is a piece called "Playing a Game of Go," which is a quite beautiful, calm, haunting melody that I will be sure to have going on Winamp next time I play online.

Book:

AvatarDJFlux: I haven't seen the movie, but I've been reading Nash' biography by Sylvia Nasar (A Beautiful Mind, Faber & Faber, 1998) from which the movie was inspired. Go is mentioned six times in 388 pages, and of course there's no trace of the scenes above described.

-- Janne Jalkanen

If you watch the DVD, there are several deleted scenes which include Go.

In one, he totally trashes a Go game being played by two other math people when he throws down the game board to his newly invented (and "unflawed") Hex, right on top of their Go game!

Benjamin Geiger: Actually, he slides the board, knocking most of the stones off of the goban onto the floor. He then goes on to say, "I prefer chess."

In another, he stares introspectively at a Go board, puts down a stone, and keeps thinking. Correctly, he places a stone on an intersection.

-- Ben Shoemaker

AvatarDJFlux: I haven't seen the movie, but I've been reading Nash' biography by Sylvia Nasar (A Beautiful Mind, Faber & Faber, 1998) from which the movie was inspired. Go is mentioned six times in 388 pages, and of course there's no trace of the scenes above described. Even if they are true, it is just ridiculous that anyone could pretend to play "perfectly" at Go. Even top professionals, at their stratospheric level, admit to make some mistakes every now and then.... ;-)

Benjamin Geiger: The movie has very little to do with the book; the creators of the movie took quite a few liberties with the plot. (It's still a good movie, in my opinion.)

dnerra: Of course this is ridiculous, and in the movie this episode is meant to characterize Nash: how he mistakenly believes he could understand the world (uhm, just Go here :-) ) "perfectly" just from abstract mathematical analysis.^[1] This in contrast to his more successful attempt of modelling reality with his Nash equilibrium. By the way, if you haven't seen the movie yet -- well, it was nice, but nothing extraordinary I'd say, so no need rush to the theatre :-)

nobody: Also, it alludes to some of Nash's work in game theory - specifically, he proved that the player making the first move in Hex theoretically wins (even if the specific winning lines of play are computationally too difficult to determine). Perhaps the suggestion is that Nash believed other zero-sum deterministic games (like go and chess) are like this, or that Nash's game theory was inadequate to cover all such "conflict situations" and Nash was frustrated by this, or any number of other things (such as dnerra's suggestion). Really, it's just a film maker's way of illucidating the personality and concerns of a character in the film.

Of course, even if he played perfectly, game theory does not state that he should have won. It is true that Go is a complete-knowledge game, so (assuming perfect play from both sides) what the theory says is that we have one of three cases: either the first player always wins, or he/she always loses, or he/she always draws. And at the moment we don't know (most likely we never will) which case is true for Go. Even if anyone is that naive to believe his plays to be perfect, he still could not conclude that he is guaranteed to win.

Side note: Taking in account komi, I would not bet on either possibility (White or Black always wins in a perfect game). Draw is highly unlikely.

As a comparison, in chess the perfect game is also a black victory, OR an white victory OR an draw, and we don't know which, but in that case the most likely situation is that a perfect game always draws.

-- Marco Tarini

In Go, if the winning party were always White, then Black should start with a pass move. This forces either a draw (White passes) or a Black win (since Black now moves second). With non-integer komi, draws are impossible. Hence, the best possible result for White after a Black pass is to for White to pass to gain a draw. Thus White can never win, assuming that both parties play a perfect game. The result of the perfect game must thus be either a draw or Black win. Does my logic compute? :-) -- Janne Jalkanen

Dieter: No. You are assuming that komi switches sides after a pass, which is not the case.

Janne Jalkanen: Actually, I thought that two passes would mean a draw, with no stones on board, komi or no komi. Then I started wondering: If both players pass immediately, what is the result? Are all points on board dame or would they have to be played out? Or would both players passing be a loss to both players? The result probably varies according to the rule set used, yes?

Dieter: I was referring to Hence, the best possible result for White after a Black pass is for White to pass to gain a draw. This is not true, because White now has the advantage of komi AND moving first. You seemed to apply symmetry where there was none. Probably I don't understand the point.

Illich: No, the point Janne Jalkanen made was correct. Maybe I can rephrase it (with my not so good english):

What we want to prove: "In no komi Go games, Black either wins or draws."

Proof by contradiction:
(1) Contradiction: Assume that after perfect play White wins
(2) If Black knows this, Black passes for the first move.
(3) This leads to two follow-ups:
(3a) White also passes, the game is drawn.
(3b) White plays. Then Black can use White's perfect strategy from point 1 and so Black wins.

Both 3a and 3b are contradictory of 1. So the original sentence is proved.

For games with komi, it cannot be proven - it is simple a matter of the size of komi: if it is bigger than a certain amount, White will win after perfect play.

Bill: Illich, Janne Jalkanen did not say "no komi", he said, "With non-integer komi, draws are impossible." So he was assuming a komi.

One thing I did not like about the Go in the movie is that, at the end of the game, Nash's opponent slams the stone down and the move is greeted by a lot of ooh's and aah's from the audience, and the game immediately is over at that point. (He captured a single stone if I remember correctly) I was expecting him to say 'checkmate!', just like in movies where chess is depicted, but luckily that didn't happen. Of course, the 'perfect play' statement made up for it. --Garf

It appeared to me as if Nash's opponent had managed to catch Nash in shortage of liberties and pick off a good sized group of stones. -- Storm Crow

Benjamin Geiger: From closer inspection, it's a group of at least 42 stones.

My favorite Go-related moment in the film takes place when Nash is challenged to a game. 'Scared?' his opponent asks him.

'Terrified.' he responds, 'Petrified. Mortified. Stupefied. By you.'

-- Regyt (quote fixed by Benjamin Geiger)

[1] Now that I read this again, immediately a certain Go player comes to my mind, not living too far away from where I live... Am I the only one? :-). Dieter: hmmm, would that be the one chasing the "pen of God" with which he will write the perfect Go book?

I think that the game they played was taken from a 1971 game between Rin Kaiho and Ishida Yoshio (1971-06-10.sgf from GoGoD). Looks like they tried to finish off the game badly...

Moves 218 to 218  

Gul Are you sure of this board position? If i remember correctly the board position shown in the movie is impossible, several dead groups showing. I will look at this scene again soon....

alexx I paused the DVD and the game before the one Nash plays looks like:

White resigns  

ilan: The movie is typical of the Hollywood "high concept" approach: Producers noted that "Shine" had won an Oscar and made Money, so they looked around for another story of talented later life redemption. The movie has nothing to do with the book, which is an accurate description of how unpleasant Nash was when he was "sane" and how unpleasant his colleagues were when he went "insane". I found the book so depressing, knowing many of the people involved, that I had to stop reading it. The go part of the movie is yet another heavy handed Hollywood attempt to convey "genius". Compare with the variations on Salieri that Mozart comes up with in "Amadeus", essentially a mishmash of pyrotechnic variations from his best known works. Hmmm, that movie also won an Oscar and made Money...

(AJ) Actually the formation is

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Dieter: Even though I have long vowed not to go and see Hollywood movies because they never fail to disappoint, disgust and offend me, I accept that a movie about a mathematician has to twist or simplify the truth if they do not want their audience to be mathematicians only. I think "A beautiful mind" strikes a better balance between truthfulness and appeal than the awful "Good Will Hunting" where the alleged higher mathematics displayed merely high school operations.

• kokiri seems a bit harsh to me - i seem to remember going to see it during my finals only to find that the proof on the board in the first scene was something to do with banach spaces that i'd literally put down in order to go to see the film. Maybe you're just better at maths on the continent. Banach space, urgh, painful just to remember them...

• • hk The film "It's my turn" from 1980 is famous in certain circles for showing a proof of the "Snake lemma" from category theory, so they do exist.

• ilan: Agreed. Some of the hardest mathematics is "elementary," that is, only using High School mathematics. In fact, for a long time, people did not believe there could be such a proof of the Prime Number Theorem, and this achievement helped Atle Selberg win the Fields Medal and Paul Erdos win the Cole prize.

As for the Go parts, if they were to shed a different light on his tormented genius, non-players too could have done with deeper explanation of the game. Go will not be very succesful as the epitome of complexity, if people are largely ignorant of it.

Anyway, I'm convinced the popularity of Go in the West will not come from within the Go community. The bridges are built through otherwise unrelated areas of interest, such as movies and mangas.

ilan: I think I can explain the inclusion of Go in the movie, and in particular the scene described above. The point is that American movies and television want to portray the human element in game playing and therefore these minimise objective criteria. For example, in the numerous TV programs about the adversarial trial process, one often hears "I will win because I am better than you," but rarely "I will win because the facts will prove me correct." I am waiting for the day when some American show will have the order: "Figure this out!" answered by: "Actually, I just proved that the problem is formally intractable, so there is nothing that can be done, in a precise sense." So, the purpose of the Go scene is to mark a departure from American film mythology, and this in order to convey to the audience the concept of Platonic mathematical ideal, that is, the belief by most mathematicians that their results are independent of human existence, in some sense.

nobody: "A Beautiful Mind" is a film about people, not Go. The game is simply a stage prop that suggests the concerns of the characters - mathematics, game theory, conflict, etc. Don't be too upset that it isn't portrayed in a realistic way.

In general, Hollywood has a tough time conveying intellectual excellence (the best one I can think of is "Real Genius") usually because the actors aren't that bright and this can't be hidden, and the writers haven't excelled in any such activity, so have a strange love/hate relationship with them. This failure is most easily identified when a character is described as extremely intelligent, that is because if, in a movie, someone has to say X, it is because the film was unable to convey X directly.

But Dieter, I don't think you should dismiss the entire corpus of American film making, especially not the movies dealing with game playing. Off hand, I can think of at least two movies that are fairly accurate in conveying the essence of games. The Hustler does this extremely well for pool (as opposed to the sequel "The Color of Money", which, like "A Beautiful Mind", had nothing to do with the book). The recent movie "Rounders" is also a fairly good depiction of Poker. Both these movies have some rather subtle features about the game: In The Hustler, the challenger plays defence in a given position by leaving the position essentially the same as he found it, and his opponent then plays an incredible shot. The idea is to show that the second player saw a shot the other didn't. In Rounders, the big poker player is a Russian emigre, which I believe is in reference to the JFK Cold War statement: "They play chess, we play poker."

DJ: Ilanpi, now that you mention Rounders (isn't it the one with John Malkowitch and Matt Damon?): I saw that movie long ago, and in a real-life scene taken from a real Poker championship I thought I saw Jimmy Cha in his version of Poker champion (and not as American go pro 4D...).
Can you confirm this?

ilan: DJ, it was Johnny Chan, see http://www.homepokergames.com/chan.php and the scene was from the 1988 World Series of Poker. The complete list of players of the championship table (No Limit Texas Hold 'Em) is available in the book http://www.jetcafe.org/~npc/reviews/gambling/championship_table.html which also gives a detailed analysis of the exact hand depicted in the movie. A partial list of participants is available online: http://gaming.unlv.edu/WSOP/annual/1988.html

I should also mention the movie "Kingpin" which is essentially the Hustler story transposed to Bowling. This movie may disgust and offend you, but that is its exact intent. Anyway, I liked it.

-- http://img488.imageshack.us/my.php?image=beautifulmindgo350sp.jpg

Here's from the first game by Nash.

gtwade: Am I seeing this right? By the way, thanks to whoever provided the link. But, in comparing the three images I notice that the board changes. In the top image, it looks like there is a huge group of black stones surrounded by white that will be taken with a single move, in fact, the move shown in the bottom image. However, in the middle image, a black stone is conspicuously missing, adding a liberty where there was none before. It almost seems like someone who knew go constructed the first image, and something happened to it, then the people making the shot had to reconstruct it, but got it wrong. In any event, as the board stands in the first image, the white stone played would have been 'the only move.'

Ren: Yup, you're seeing it right, they mixed up the sequence. Also looks like The Great John Nash is about 24 kyu ;) Seriously though, I'm betting the confusion is mostly due to camera workings and editing. You really can't assume that they know how to play.

Warfreak2: How about this?
http://img218.imageshack.us/img218/7017/blindspotgq2.jpg
Black's huge dango is dead anyway, better to tenuki to the circled spot, those white stones can be cut and captured! There would be a double atari for white, of course, but it is shortage of liberties... except that all of a sudden, just like black, white has an extra liberty (at A) in the next shot, too!
Oh wait, white can catch black in a connect-and-die. This is what happens at 3AM.

tapir: Because of this movie I didn't start playing in 2002 but four years later.

(AJ) Actually the formation is:

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I looked up everything and wrote it all down. The thing with the bottom left corner was never shown and thus I got as much as possible. If white places down a stone at 6 Across and 9 Up (from bottom of board, and this is what his rival did), Black would lose 55-57 stones, depending on White's follow up (seeing as there are 2 trapped black stone, when hite plays there).