Here is a quick thought about the mathematics of the US shutdown, not to be taken too seriously (the thought I mean — the shutdown obviously is to be taken seriously). It’s for the benefit of anyone who is puzzled that the Tea Party can have such a large influence, and more generally how a political system can be stable when almost nobody likes it. I’m going to prove that in a country of people, it is possible to devise a democratic system in which of those people control the decisions, where . For example, in a population of 100,000,000, all you need is a band of fanatics with about 112,000 people — or approximately 0.1% of the population. Although we do not have such a system and the distribution is unlikely, the systems and distributions we do have still allow a minority to have undue influence, and for similar reasons. What I’m about to describe is the extreme case.
The system I have in mind works as follows. It’s a multilevel representative democracy. Suppose for convenience that for some positive integer . (It is easy, but slightly tedious, to modify what I am about to write to take care of more general .) Suppose that the country is divided into three “super-constituencies”, each of which gets a vote in the top-level decision-making body (known as the triumvirate). Suppose that decisions in that body are passed by a majority vote. A group of people that wants to control the country can do so as long as it can control at least two votes in the triumvirate.
How are the members of the triumvirate chosen? They are elected by another triumvirate one level down. The representative in the top-level triumvirate is representing the views of the three people in the triumvirate one level down, and is worried about stepping out of line, since then he/she risks being deselected by the three people in the level-2 body.
So if a merry band of fanatics wants to control a representative in the top-level triumvirate, it is enough to control at least two of the representatives in the second-level triumvirate that selects the top-level representative.
Of course, we can iterate this argument. So how many people do we need to control the country? We need two at the top level, and therefore four at the second level, and so on. Therefore, we need at the bottom level. (Note that the representatives do not have to be fanatics themselves — if they don’t vote in the way that the fanatics want, then they get deselected by the people one level down, losing all those lovely perks that go with a high-level job in politics.) If , then , so we’re done.
One might want to make small adjustments to the bound to allow all the different levels of influence to be disjoint. So then . But this is within a constant of . Similarly, if we start with some that is not of that precise form, that again affects the estimate by just a constant factor.
So the conclusion is that in principle people can mess up a country with population . If you have more people than that, then the main thing you want is a system with a few levels of groups within groups — not necessarily formal at every level — and a distribution that is not too concentrated and not too diffuse. (If it is too concentrated, then you’ll end up wasting a lot of votes on controlling representatives who are already controlled, but if it is too diffuse, then you won’t control anybody except at very low levels. In the extreme case, what you want is to be arranged in what can be viewed as a discrete approximation to the Cantor set: in less extreme cases you still want to be somewhat “fractal” and “Cantor-like”.)