A Beautiful Goof

A couple weeks ago, I was buried up to my armpits in Game Theory (nothing to do with Quake or World of Warcraft, trust me), Decision Theory, Cryptology and a dozen other fascinating topics. After a couple days non-stop writing/reading/studying/coding on those topics, I felt I really needed a two-hour break. Yet, feeling guilty about leaving my books for a minute, I compromised by downloading renting that award-winning movie about John Nash. I figured if I was not studying, at least watching a biopic on one of the pioneer in the field of Game Theory wouldn’t be straying too far off.

As it turns out, the movie is not as bad as I’d expected (which is not to say it is any good). Russel Crowe is as convincing as you would imagine a hunky Australian actor playing a nerdy US mathematician to be. All along, you half-expect Crowe to draw a gladius and slice open his mathematical studies nemesis. Instead, you see him mumbling and x-ray-visioning his way into mathematical stardom and bona-fide paranoid schizophrenia.

I won’t bore you with minor details of technical implausibility: this is Hollywood we are talking about, after all. Although I really got a kick off alleged cryptanalytical genius Nash seeing “flashes” of instantly decoded patterns, mere seconds after laying eye on those damn russkys’ dastardly unbreakable code: trust me, no matter how gifted, nobody “sees” such results, they are very painfully and slowly extracted through endless sequences of operations (I hate to break it to Rain Man fanatics out there, but even Mr. Turing had to use a very big computer for that, and he wasn’t exactly dimwitted).

One striking inaccuracy, though, was the mandatory illustration of Nash’s most significant achievement and cornerstone of modern Game Theory: the eponymous Equilibrium. The concept is rather cleverly illustrated by a (likely apocryphal) story involving a bunch of hot girls (led by one “very hot blonde”) showing up in the Princeton pub where Nash and his fellow beer-drinking mathematicians are sitting, resulting in the horny pack of scientists all at once setting their sight on the blonde bombshell. That’s when Nash has an illumination and explains that such a strategy would be very sub-optimal, since we all know brunettes are way wilder in the sack… Erm, no… since if all competed for the blonde’s attention, they would likely hinder each other’s effort, be all turned down, then get rejected by the other girls who wouldn’t appreciate being second dip, resulting in everybody’s dissatisfaction… Instead, he suggests, an optimal equilibrium in that matchmaking game would be attained if everybody picked one of the lesser hot girls and left the hot blonde alone. Get it? Nash Equilibrium, all that…

Except it doesn’t work.

In that scene, the suggested Nash Equilibrium assigns a guy to each of the “average-looking” brunette and none to the “highly-desirable” blonde. And so, every single guy has a personal incentive to try and move up to the blonde woman, perceived as more desirable (for the sake of reasoning here, we will assume he doesn’t know that part about hot blondes being the boring ones in the bedroom)…

Which is precisely the opposite to a Nash Equilibrium (a situation where none of the player has a strictly positive incentive to change their strategy).

In even plainer term, consider that the equivalent to an onscreen Albert Einstein proudly scribbling his “famous E=mc3 formula” on a blackboard.

And this is why you probably should not rely on Ron Howard and Russel Crowe for your mathematical needs.


  1. It seems you’re correct. I had to get out the store bought (really) DVD and watch that scene again after reading your post here.

    I enjoyed that particular movie, these days it’s hard to find any movie with reasonably good acting and a reasonably developed plot that doesn’t include scene after scene of extreme and or gratuitous violence. Beautiful Minds had a rather sad ending, similar to real life.

    With respect toward that scene you mention and the lack of Nash Equilibrium, the only scenario I can possibly conceive is that among those wishing a partner there is a perfect one-to-one pairing, and each partner of each pair is completely happy with that particular person for the rest of their natural lives. Given my limited observations of human nature, that scenario doesn’t seem at all realistic, though it seems to occasionally happen with a few isolated pairs of partners.

    Do you have an alternate scenario that does fit that particular scene, is a Nash Equilibrium, and is also believable?

  2. klk

    Well, the movie’s modeled after real life, a real life. I still found many “technical” aspects to be rather contrived, even by Hollywood’s rather low standards.
    I am told the book (on which the movie was based) is both much more faithful to the truth and a very good read.

    Regarding the pairing game, I think you are digging too deep into it, in that to me there is only one utility function alluded to: how “hot” is the girl each guy can find. And still according to the movie, the successful pairing with the blonde bombshell unquestionably yields the highest utility score. No matter what, she is always seen by every guy as a more preferable choice over whichever girl they might end up picking.

    As for what the real Nash equilibria of that game are: there are many. Basically, any combination that involves one guy with the blonde girl and each other guy with someone else: taking in account Nash’s narrative comment that “competing for the blonde’s attention could only result in both competitors to be turned down”, none of the “brunette-assigned” guy would reach a better utility result by attempting to hit on the blonde girl (the guy paired with her, would obviously have no reason to “trade her down”)… Those would therefore be Nash Equilibria in so much as nobody would have any incentive to move out (you could intuitively get to that result, by realizing the solution has to be symmetrical, since players are supposedly identical in their chances).

    Of course, there being many Nash Equilibria means you basically can’t solve the game (easily in a satisfying way) without additional elements.

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