Wednesday, August 09, 2006

weeping and nashing of teeth aka he got game

This is very non-Grenadaesque entry about love, romance and game theory which I started to write on over a month ago but didn't pick up again until now. One of the more oddly thought-provoking scenes from the film A Beautiful Mind takes place in a bar where the mathematician John Nash and his classmates are admiring the same blonde woman from across the room:

Nash: If we all go for the blonde, we block each other, and not a single one of us is goin'’ to get her. So then we go for her friends, but they will all give us the cold shoulder because nobody likes to be second choice. But what if no one goes for the blonde? We don'’t get in each other'’s way, and we don't insult the other girls. That'’s the only way we win. That's the only way we all get laid.

Nash (continuing): Adam Smith said, the best result comes from everyone in the group doing what'’s best for himself, right? That's what he said, right? Incomplete. Incomplete! Because the best result would come from everyone in the group doing what'’s best for himself and the group.

Hansen: Nash, if this is some way for you to get the blonde on your own, you can go to Hell.

Nash: Governing dynamics, gentlemen... governing dynamics. Adam Smith... was wrong.

I think there is a stereotype/fetish out there that Black men are especially attracted to thick white women and vice-versa (e.g. Lose weight, or be reduced to dating Black men! Oh, the horror. . .). To explain this phenomena some people point to different "African" standards of beauty, class issues, racial self-hatred and other factors. And those are all part of the picture. But something which occurs to me that another way to think about this situation is to say that on both sides, the relevant players have learned to "go for the brunette". Men judge women by their appearance. Women judge men by their status and earning power. And what happens in terms of relationships is that the different actors learn to swim in the part of the pool where they can handle the competition and certain equilibrium points are reached. First tier with first tier. Second tier with second tier. etc. However athletic, handsome, funny, etc. a black man is going to be "second tier" in terms of status. (e.g. consider all the personal ads where people openly declare a racial preference). However kind, intelligent, talented, a women who doesn't fit into anorexic standards of beauty is going to be "second tier" in terms of appearance. Thus the stereotype.

An article which shares some other insight on the issue of body image and romance among Black Americans asks the question Is the Size of Beauty Changing in the African American Community? and is also worth a look at.

The last brief remark I wanted to share on this general subject of relationships and game theory (which actually kind of inspired the whole entry) relates the concept of Nash equilibrium to romantic situations.

For those who don't know what a Nash equilibrium is, imagine a game of at least two players. Each player chooses a particular strategy. In a simple game, the strategies would be something like "always choose rock" or "randomly choose paper half the time and scissors half the time". A more complex game would have strategies like "be coy and play hard to get" or "be the bad boy since girls like that". So if you can imagine that all the players in a particular game have chosen a strategy, a Nash equilibrium would be a situation where no player has anything to gain by unilaterally changing that strategy. So a Nash equilibrium is a kind of stable place where the players would tend to adopt certain patterns of behavior which worked and stick to them.

When I began this entry and was thinking about the lives of my friends, I had initially wanted to make the claim that post-romantic break-up situations didn't seem to have Nash equilibria. I looked at the weirdness and foolishness they seemed to be going through where one person or another always wants a little more or a little less than the other person is willing to give.

But for his dissertation Nash proved that if you include what he calls mixed strategies every finite game has a Nash equilibrium. And as I'm writing now, more than a month after making my initial conjecture, it turns out that in spite of my early pessimism, the specific situation I had in mind actually has settled to a more or less stable point. And in my own way I've found some small but still surprising empirical confirmation of Nash's result, showing that mathematics occasionally has something relevant to say about our social lives. Pretty deep.