The N-Party Problem
Jan 21, 2012

The N-Party Problem

There is a famous problem in classical mechanics, a branch of physics, called the n-body problem. While interesting in its own right, the problem can be used as an analogy that is illustrative towards politics; in particular, the issue of predicting long term trends of political systems with two or more major political parties. 

The n-body problem from physics is as follows. Given a number of bodies under the influence of gravity (such as the sun and and planets), a physicist might like to be able to predict the future orbits of these bodies. For just two such bodies (a one planet solar system, say) this problem is solved and the orbits can be predicted. For n>2 bodies, however, there is no nice analytic solution to this problem in general and astronomers are forced to use numerical approximations to compute complicated orbits. What is interesting about this n-body problem is that a relatively simple looking situation becomes very tricky as soon as we have three or more bodies to consider. 

Now let us consider political systems with various numbers of political parties in them. The US has a very simple two party system. There are no other meaningful parties and the two parties oscillate every couple of elections between who is in power, and they have done this for a very long stretch of time. Much the way the dynamics of two gravitational bodies is quite simply and they just oscillate around each other, so too is the dynamics of two political parties quite simple and they alternate terms in power. 

What is most remarkable about a two party system like that in the US, is that averaging over time the support for both parties is quite close to fifty-fifty. There are ups and downs, of course, but no party can be massively ahead in support form the other for a prolonged period of time. Note the analogy with a 2 body system oscillating around a fixed center of mass. That this dynamic is true is quite interesting and demands an explanation. 

If we measure parties in terms of a fixed, continuous name then the parties have remained the same. If, however, we measure parties in terms of a set of policies and values that they espouse then there has been enormous, incredible changes over the last century. For example, today there is a fight over gay marriage, yet only 45 years ago marriages between whites and blacks were still banned in sixteen states. Yet those old battles are long since forgotten and the parties of today simply accept the status quo and fight on the new issues. 

Politics is a complicated process with many factors, but it is conventional and not entirely unreasonable to refer to a political 'spectrum' which we can think of as being represented by a one dimensional variable and different people - and parties - are located at different points along this spectrum. As society has changed, so too has the location of the parties along this spectrum (perhaps it is better to think of the spectrum changing underneath the parties, but never mind). For simplicity, we can assume that people to the one side or other of both parties will support the party they are closer to on the spectrum. 

The fundamental dynamic that keeps the parties moving, on average, in tandem along the line with roughly equal support is the following. Should a party find itself somewhere along the spectrum such that a minority of people support it, it will move in the necessary direction (as in closer to the other party) on these issues as it tries to secure more people who will vote for it. For example, as public sentiment about allowing blacks to marry whites increased, parties chased after this sentiment by adopting it so as to not lose any share of public support. We call this process 'appealing to independents' in the nomenclature of US politics, but what it really means if shifting the policies of the party towards the center, whatever that may be at any given moment. This process is what mathematicians call stable in the sense that while the tumultuous nature of politics may often upset this balance, there is a continual dynamic that pushes back to this 50:50 support equilibrium. 

Just as in the gravitational case, where the bodies oscillate but gradually succumb to gravity and get closer and closer to each other, so to do the political parties get closer to each other over time. As a view passes into being held by a reasonable majority of the population, the parties will tend towards being nearly identical on this issue. People often claim that the Democrats and Republicans are simply two sides of the same coin and there is not much difference between them. Personally, I believe that while there are lots of similarities, there are still important differences that make real differences to peoples lives. Regardless, this explains the underlying dynamic of why they are indeed so similar and we should expect them to be so. 

We have seen that a 2-party system, much like a 2-body gravitational system, behaves in a predictable and stable fashion. What about political systems with more than two large parties?

In Canada, there are three dominant parties that can be represented by having three different, changing, locations along the political spectrum (ignoring the Bloc Quebecois which we will get to later). For much of its history, there has been some variant of conservatives versus Liberals acting much like a two party system. However, the existence of the NDP on the far left makes predicting long term outcomes very difficult. The Liberal party can hope to expand support either to the left or the right of the spectrum and, conversely, either the Conservatives or the NDP can poach support from them. Recently the NDP surged into the Official Opposition for the first time in Canadian history federally. When, or even if, this was ever going to happen is not something easily predicted. 

This extra degree of freedom that comes from having multiple parties and thus parties being able to move in both directions along the political spectrum to try and gain support results in a fundamentally more complicated political system that does not have the kind of long term, structural stability that a two party system has. The more parties a system has, in general the more complicated and unpredictable it becomes. In some countries, the ever changing combinations of coalitions and shifting perspectives between multiple parties becomes increasingly difficult to predict with any certainty even in its short term evolution, let alone any longer trends. 

The only trend that may be possible is that, like the two party system, over time we expect the parties to get closer together on many issues as they all seek to expand support. Old issues that parties were divided on become resolved as society in general tends towards a large majority opinion. Of course, new issues and new conflicts create more space between the parties, but the continual shrinking of space between parties that comes from this dynamic as old issues get resolved tends to shrink the gap between parties. In the last 15 or so years, many Westminister parliamentary systems have seen centuries old parties of the center, like the Canadian Liberals, decline considerably in support. The relatively smaller differences between all three parties now don't seem sufficient to justify three distinct ranges on the political spectrum. 

Unfortunately (at least from an analytical perspective) a location on the one dimensional political spectrum is too simplistic to capture the core identity of many parties, especially in countries with large numbers of issues. Other parties often break down in a couple of ways. First, there are the "identity" parties, which represent some minority demographic in the population. For example, the Bloc Québécois represents the interests of the francophone Quebec demographic and many countries will have parties dedicated to particular religious, ethnic, or geographic groups. Secondly, there are "issues" parties like the Green party or Marijuana party which focuses most of a specific issue in society. Thirdly, there are the ideology parties like the Communist or Libertarian parties that push various significantly fringe ideologies in society. 

Many of these remain on the fringes and are largely irrelevant to the broader political direction of a country, but in cases like the Bloc Québécois they have at times managed to get significant power. Their success depends on the relative importance of their particular identity or idea or ideology to people in general. As time goes by, things like ethnic identity sometimes drop in importance to other issues such as the size of government. And sometimes not. The relative importance of these parties outside the traditional spectrum thus fluctuates with the zeitgeist of the times and how relatively important these issues are to segments of the electorate. 

Disclaimer: there has been, and remains, a fundamental asymmetry between the positions of parties and of the electorate at large. This is because of the asymmetric influence that an assortment of vested interests have in the political system. On some issues, like gay marriage (which doesn't have any form of established industrial lobby that profits off it one way or the other), the views of the parties get drawn relatively close to that of the population. On other issues, like financial regulation, the views of society at large are often less important than that of the financial interests. However, the scope of this post is looking at long term changes in the political system. We can accept that there is this pressure which stilts the political spectrum from being purely a matter of reflecting public opinion, but nonetheless the issue of changing public opinion remains - at least in the long term - of critical importance. 

In much the same way that going from two gravitational bodies to three or more changes the ability to predict long term outcomes of a system from something quite easy to something very complicated, so too does this analogy hold for political systems with n parties in them. A two party system has a very steady and simple dynamic with a lot of predictive validity. Three or more parties make for a much more complicated and unpredictable political system. 

Thoughts on this post? Comment below!

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Stephen Johnson said...

In your analogy, would the laws of physics equate to the electoral system in use?
FPTP directly causes a two party system.
Change the electoral system and the planets in the universe might better reflect public opinion.

bazie said...

The analogy probably breaks down at this point, but perhaps one could loosely think of FPTP as a sort of "stronger" force that more closely binds the largest two bodies into their oscillating orbit. But anyways you are definitely right that various forms hybrid proportional representation or transferable votes or whatever else make it more likely to have more parties.

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