In this penultimate post taken from my Intro to Political Analysis class notes, we’re talking about coalitions—and how to predict which coalitions will form. 

The fundamental building block of politics is the mass political party. The first mass political party was formed in the US. After the “corrupt bargain” of the election of 1824, Jackson’s supporters formed a mass political party, fully recognized as the Democrats by 1840.

But we won’t be really talking about the US, because we don’t have coalitions, because of our party system. Party systems (as with any systems, according to systemic theory) can be categorized as the one, the few, and the many.

The one would be places like China and Cuba, with only one party.

The US’s two-party system would be considered “the few.”

And the many is what you see in parliamentary countries (like most of Europe.) This is the kind of system we’ll focus on.

So why do different countries have different systems? Duverger’s Law, which states that the number of parties is based on the government, and how that government manages votes.

Single member districts (a.k.a. winner take all, first past the post, plurality) systems result in a two-party system.

Proportional representation—the percentage of votes determines the percentage of seats—results in a multi-party system.

And an authoritarian government results in a single-party system.

So, why parties? The aim is to gain control of government through majority rule, and you can’t do that with individuals—you have to have a mass of individuals, acting in cooperation. In a two-party system, getting a majority is always going to happen for one of the two parties. Let us never speak of them again.

In a multi-party system, one party can still get a majority, but many times no party has a majority. In such cases multiple parties need to join together to form a coalition. When they do this they write a manifesto, a contract which determines what the coalition will do, and which member-parties control what.

To predict coalition-forming, it may seem like looking at ideology is the best way to go—whichever parties have similar beliefs will form a coalition. But this isn’t always the case, and ideology is messy, so we’ll be using cold, hard, formal quantitative analysis—specifically, the weighted voting game. To use the weighted voting game we need at least three pieces of information:

1. The quota—the number of votes needed for the coalition to win. Usually this a simple majority, though a two thirds majority is also common.
2. The number of parties.
3. The number of votes each party controls.

So the weighted voting game is modeled like this:

q : V₁, … V_k

Where q is the quota, each V is the number of votes of a specific party, and k is the number of parties.

For an example, let’s look at the UK House of Commons Election in 2010 (their most recent election resulted in the Conservative party having a majority of votes, so no coalition was needed.)

There are 650 seats in the House of Commons, and a simple majority is the number of votes needed, so our quota is 650/2 + 1—326. The Conservative party won 306 votes, Labour won 258, and the Liberal Democrats won 57. So it would be noted like so:

326 : 306, 258, 57

Each party has two possibilities—they can be in the coalition, or they can be out of the coalition. So the number of possible coalitions is 2^k. In this instance it’d be 2³—8. Here they are.

1. Labour (258 votes)
2. Labour and Liberal Democrats (315 votes)
3. Labour and Conservative (564 votes)
4. Conservative (306 votes)
5. Conservative and Liberal Democrats (363 votes)
6. Liberal Democrats (57 votes)
7. Labour, Conservative, and Lib Democrats (650 votes)
8. No one (0 votes)

Of course some of these coalitions aren’t really likely, like the coalition of no parties (the “caretaker government”) or the coalition of everybody (the “grand coalition.”) We’ll ignore those, and look for minimum winning coalition to determine what coalition is likely to form. A minimum winning coalition has two properties:

1. Winning—it has ≥ the number of votes required to have a majority.
2. Minimum—There are no surplus parties (parties not needed to have a majority) in the coalition.

With the first rule we eliminate most of them, and with the second rule we eliminate the coalition of everyone (because either Liberal Democrats or Labour could be removed without the coalition dropping below the quota), and we’re left with these two coalitions:

1. Labour and Conservative (564 votes)
2. Conservative and Liberal Democrats (363 votes)

Minimum winning coalition does not mean minimum votes, it just means minimum number of parties, which is why it’s possible to have multiple “minimums.” To determine which of these is more likely, we can look at which has the least parties. Coalitions with fewer parties are more likely to form, because ideology is still somewhat important, and coalitions can’t be pulled in ten directions at once. That doesn’t help us here though, because both of these coalitions only have two parties, so next we look at which one has the least seats. Again, it’s better that power be consolidated, and easily managed—so we arrive at our minimum winning coalition with the fewest parties and fewest voters—the Conservatives and the Liberal Democrats. And this is the coalition that formed, so score one for quantitative formal analysis, right?

Now this is a pretty simple example—in other countries there are more parties who more evenly divide the vote, and there might be a minimum winning coalition with the fewest parties, and a different min-win coalition with the fewest votes. And at that point things get tricky. Looking at the ideologies of the the member-parties, and which coalition is more closely aligned could help. Also looking at how spoils are split—if they’re split evenly between voters, the one with fewer voters will win. If they’re split evenly between parties, the one with fewer parties will win. Maybe. Use your judgment.

Next week is the last post in this series—median voter theorem, and the best system of democracy (spoilers, there is none.)

“Please, just sit in your chairs and hold onto your brains if this is too much.”
-Professor Dion