Misplaced Patriot

I've been watching the "Game Theory" lectures by Ben Polak on the Yale Open Courses site, and he gives an example game that's a bit depressing.

He deals with an abstract game of racial balance. Suppose you have two (large) sets of people of equal size, N green people and N blue people. Suppose that there are also two neighborhoods of equal size that these people can live in.

The nature of game theory is to start with a postulate about what result makes people happy. In his example, he takes an optimistic view of human nature - that all green and blue people would prefer to live in a neighborhood that is split 50%/50% between blue and green. The only racial discrepancy is that a green people prefer to live in green-majority neighborhood rather than blue-majority neighborhood, if the mix is unequal. So if a green person had to choose between a neighborhood that is 40% green and a neighborhood that is 60% green, he will prefer the 60% green. Similarly, blue people prefer to be in the majority in their district rather than in the minority.

This game has three Nash equilibria. A Nash equilibrium is a solution where nobody could improve their situation by changing their decision. Two of the cases are segregated - all the greens are in neighborhood A and all the blues are in neighborhood B, or visa versa. In both these cases, nobody has any desire to change - they cannot improve their happiness by moving, because moving means becoming the minority in the other neighborhood.

The other equilibrium is when each neighborhood is exactly 50% green and 50% blue. Then nobody benefits from switching because they all prefer to be in 50%/50% neighborhoods, and moving would mess that up.

Unfortunately, this last equilibrium is "unstable." If you jostle it a little, say, making the blue/green mix 53%/47% in neighborhood A (and hence 47%/53% in neighborhood B) then a green person in A thinks, "Ideally, I'd love to convince 3% of the greens in B to switch places with 3% of the blues in A, to get back to 50%/50% mix, but, unfortunately, I have no such power over them. However, if I move from A to B, I will be happier than I am now, because I prefer to be in the majority if there is an unequal mix.) Similarly, the blues in neighborhood B have reason to want to move to A. So what happens is a horrible surge towards segregation.

Now, of course, this is only a theoretical game. Still, it's remarkable that, even with an optimistic view where living in the 50%/50% neighborhood is considered ideal by everybody, you get something like a mutual "white flight" just by assuming that people prefer to be in the majority rather than the minority.

Now, most segregation is, of course, not of this kind. But this game can be applied to, say, school choice in public schools. If a school system allows parents to pick schools for their kids, without the system enforcing racial balance in each school, you might well get a self-segregation in the system, even if everything is fair, even if all parents agree they'd prefer perfect racial integration.

On the other hand, consider a converse case. Let's say you have N men and N women. A simple model here might make people prefer to be in the sexual minority in a neighborhood, and even prefer being the only man or woman in a neighborhood. (Okay, that's a bit extreme.) In this case, oddly, the people rush towards purely mixed neighborhoods. In this case, nobody wants to be in the majority, so the rush is towards equality.