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.

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?