## Holding a country to ransom

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 $n$ people, it is possible to devise a democratic system in which $n^\alpha$ of those people control the decisions, where $\alpha=\log 2/\log 3$. 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 $n=3^k$ for some positive integer $k$. (It is easy, but slightly tedious, to modify what I am about to write to take care of more general $n$.) 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 $2^k$ 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 $n=3^k$, then $2^k=n^{\log 2/\log 3}$, 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 $n=1+3+3^2+\dots+3^k$. But this is within a constant of $3^k$. Similarly, if we start with some $n$ that is not of that precise form, that again affects the estimate by just a constant factor.

So the conclusion is that in principle $Cn^{\log 2/\log 3}$ people can mess up a country with population $n$. 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”.)

### 18 Responses to “Holding a country to ransom”

1. Thomas Says:

I figured out how to construct a system in which n^a people can control the whole population when a=log(3)/log(6). Its also a multilevel representative democracy (and yes, everyone on each level has an equal amount of power).

The top level has 1 president, who can decide the legislation any way they want. There are four senators, who each vote for who to make president, and the majority vote (3 out of 4) wins.

The country is split into six parts, each having a sub-president (who is elected in the same way). Each sub-president can vote for a specific two senators. Therefore each senator has three people who can vote for them. The majority vote (2 out of 3) decides.

The first, second, and third sub-president can vote for the first senator. The first, fourth, and fifth sub-presidents can vote for the second senator. The second, fourth, and sixth sub-presidents can vote for the third senator, and finally, the third, fifth, and sixth sub-president can vote for the fourth senator. You can check that each sub-president has equal power.

This goes the same way down through the levels. Now, it turns out that you only need the fourth, fifth and sixth president to decide the president in the next level, because those sub-presidents control the vote for three of the senators in the next level, and those three control the vote for the next president. Therefore, any group of size Cn^(log(3)/log(6)) in the population can control the country if they distribute themselves correctly.

2. Fernando Says:

Of course, allowing a minority to block legislation is part of the design of the American system, built in on purpose. Could one compute how large that minority has to be?

• Thomas Says:

I created a system last night such that if you have seven people at the bottom level, then some groups of three people can control the entire country, but no particular person is stronger than another (there is an automorphism of the group for each pair of people that takes one person to another).

However, it is incredibly complicated, requiring four levels, and the next level up from the bottom has 28 people so I doubt anyone would ever use it. It has really interesting symmetry though.

3. Thomas Says:

I wonder, if you have n people on the bottom level, what is the minimum amount of people who can control the country (in the bottom level) given that everyone in the bottom level has equal power.

4. Sune Kristian Jakobsen Says:

Here is an old poem (or grook, as he called them) by the mathematician and poet (and many other titles) Piet Hein:

His party was the Brotherhood of Brothers,
and there were more of them than of the others.
That is, they constituted that minority
which formed the greater part of the majority.
Within the party, he was of the faction
that was supported by the greater fraction.
And in each group, within each group, he sought
the group that could command the most support.
The final group had finally elected
a triumvirate whom they all respected.
Now, of these three, two had final word,
because the two could overrule the third.
One of these two was relatively weak,
so one alone stood at the final peak.
He was: THE GREATER NUMBER of the pair
which formed the most part of the three that were
elected by the most of those whose boast
it was to represent the most of the most
of most of most of the entire state —
or of the most of it at any rate.
He never gave himself a moment’s slumber
but sought the welfare of the greater number.
And all people, everywhere they went,
knew to their cost exactly what it meant
to be dictated to by the majority.
But that meant nothing, — they were the minority.

5. william e emba Says:

How many people voted for Gerald Ford? All Bush 43 needed was a majority of 5 votes.

Regarding the current shutdown, how many people voted for Eric Cantor? The shutdown exists mostly because the House snuck in a rule at the last minute that granted Eric Cantor sole power to allow budget-related votes to proceed.

More generally, Republican Speakers have generally followed the “Hastert rule” (introduced by his predecessor Gingrich, actually), which requires support from a “majority of the majority” before a vote is allowed. Perhaps the Republicans wouldn’t be so beholden to it if were called the “Lenin rule”. As it is, Boehner has not insisted on the Lenin rule no matter what, but there is no progress possible under the current “Stalin rule”.

6. Marcus Pivato Says:

A multilevel representative democracy very similar to the one you describe is the subject of the article “Pyramidal Democracy”, which appeared in the Journal of Public Deliberation in 2009. The article is freely available here:

The particular problem you describe is discussed in Section 12 on page 22 of the article. While theoretically possible, I think this problem is highly unlikely to occur in practice, because it would require an extremely well-organized and wide-ranging conspiracy.

I should also point out that the main purpose of such a “pyramidal” political structure is not primarily as a system for *voting* (for which purpose it would be woefully inefficient, and vulnerable to various pathologies), but rather, as a system to facilitate *deliberation* (something which is notably absent from our current political system). This emphasis on deliberation should make “takeover by fanatics” less likely.

(Incidentally, none of this has anything to do with the Tea Party shutdown of the U.S. government. The shutdown has more to do with shameless gerrymandering of Congressional districts, the takeover of the Republican party by right-wing crazies, and other peculiarities of the American political system, as alluded to by William E Emba above.)

• gowers Says:

But how did right wing crazies manage to take over the Republican party? There are plenty of less crazy Republicans, but if you have a system where a party tends to vote in a block and how they vote is decided by a majority of the representatives, and whether a representative retains his or her job is decided by party activists, then we do have something like a multilevel democracy, which can explain why a smallish group has such a big influence. Also, gerrymandering is a way of making the distribution of influence more efficient in the way I described at the end of the post.

7. Marcus Pivato Says:

Hmmm. The last post ate the link I sent. Let’s try again. The article is available here:

http://www.publicdeliberation.net/jpd/vol5/iss1/art8/

8. Ori Gurel-Gurevich Says:

Of course, one can do much better – the exponent can be 1/2, e.g. if the voters are arranged as the edges of a 2 dimensional torus and the decision according to whether there is a crossing or not. You cannot get a better exponent, if you require the voting scheme to be transitive, due to birthday paradox argument – otherwise you’d have two disjoint conflicting witnesses.

9. Giles Warrack Says:

“If you hold one-half of one-third of the reins of power in Washington, and are willing to use and maintain that kind of discipline even if you will bring the entire temple down around your head, there is a pretty good chance that you are going to get your way.” Norman Ornstein of the American Enterprise Institute (quoted in Simon Johnson and James Kwak’s excellent book “White House Burning”)

10. anon Says:

This is the type of calculation that gives “mathematically possible” a bad name. It has no relevance to the US situation, which is much more complex, nor to anything else in the real world.

• Ori Gurel-Gurevich Says:

This is the type of comment that gives “talkbacks” a bad name. It has no relevance to the post, which is much more complex, nor to anything else in this blog.

• anon Says:

I remind you that Gowers wrote: “Here is a quick thought about the mathematics of the US shutdown, “

11. anon2 Says:

There is a fun real-world application for this kind of “multi-level democracy”-design (how often do you get to write the constitution of a country?).

Think stocks and investors voting on the board and management of a company. In every level the majority of stock-holders get to decide on the management of company A, which in turn can vote for all voting stock which company A owns in company B.

You see the possibilities, especially since the (directed and weighted) “ownership graph” is not acyclic.

There is actually some research in Italian organized crime and Korean chaebols using pyramid-like structures to defraud investors or retain control despite of accepting outside investors.

Otherwise, the concept of “doughter companys” is a regular example, which seems not to be frowned upon.

Indeed, if the ownership graph is sufficiently acyclic, parts of it can completely decouple (i.e. an empty set of stock-owners can control an entire subgraph). In such a case management has de-facto taken over the involved companies and investors are screwed. Cynical people may speculate that this has already happened on a large scale.

However, as far as I know (glancing at ownership data published by SEC- unfortunately the data is shitty to parse and I have other things to do, therefore only a data-inspired gut feeling) complete decoupling of major parts of the world economy has not happened so far (too much stock in posession of actual humans, as opposed to corporations).
This does of course not mean that “de facto decoupling” (sufficient majority to make loss of control highly unlikely) has not happened.

12. Majorities in public and in politics | Stats Chat Says:

[…] problem gets: Tim Gowers (prompted by the US government shutdown) does the mathematician thing and derives the extreme case. And the problem is exacerbated by the fact that politicians aren’t as knowledgeable about […]

13. Und bei uns? | UGroh's Weblog Says:

[…] um alle politischen Entscheidungen zu kontrollieren. Wie dies geht ist in dem Artikel T. Gowers auf seinem Blog beschrieben. […]

14. John Beattie Says:

I seem to recall an article somewhere, possibly in The Listener, saying that the Militant Tendency used more or less this mechanism to control votes in Liverpool City Council in the late seventies.