Multi-Dimensional Politics
Medicare, taxes, abortion, gun rights, voter rights, welfare, immigration - these are some of the hotter issues in US politics today. These are just a subset of the seemingly endless list of political issues, ranging from what services the government should provide, to what substances people can put into their body, to how our country should interact with others, and much more. Some citizens care deeply about politics and follow most or all of the issues in discussion. Many have a handful of policies they take strong stances on, and others they don’t care about. Some are single-issue voters, while others don’t care at all about politics. To anyone who cares about more than one issue, the two party system is doing them a disservice, as their opinion on one issue shouldn’t be assumed to inform their opinion on the other issue, but that’s what a two party system mandates. Further, the two party system is systemically problematic to everyone, not just those with opinions on multiple issues, as I explain below.
In America, people often think of political standing as a point along a spectrum - more liberal-minded people to left, more conservatively minded people to the right.
Figure 1: 1D Political Spectrum
However, this is a huge oversimplification of a much more complex system. In recognizing this, some people express political standing as a position in a 2D plane, with separate axes for social freedom (LGBTQ rights, abortion, drug laws) and economic oversight (taxes, corporate regulations, union rights), since opinions on these axes are independent of one another. Here’s a diagram of the 2D “spectrum”, with labels for which quadrant the two main US political parties generally inhabit:
Figure 2: 2D Political Spectrum
Additionally, the upper-left quadrant is generally libertarian, and the lower-right is generally authoritarian. I used “generally” in the past two sentences, because the 2D depiction, while better than the 1D depiction, is still an oversimplified model, so opinions won’t fit perfectly into it. For example, when it comes to gun ownership, Republicans are farther towards the “More Social Freedom” side than Democrats, so clearly, lumping all social issues together is an oversimplification. This also doesn’t give much thought to immigration or foreign policy.
If we wanted to create an accurate graphical representation of political opinion, we would need a different axis for each independent issue. This would result in an n-dimensional graph, where n=the number of political issues. Of course, there are far more than 3 political issues, so depicting such a model would be difficult in our 3D world. However, by using a simplified 3-dimensional political ecosystem, we can show the shortcomings of a 2 party system in a way that scales to n-dimensional ecosystems.
Let’s say that we have a country where there are only three issues:
Personal tax
Immigration
Voter rights
In reality, it doesn’t matter what the three issues are; I just chose three that are more of a spectrum of opinions than a binary ‘for/against’. For example, abortion and LGBTQ rights are fairly black-and-white, while personal tax opinions have a larger grey area - income vs. sales vs. wealth, middle class rates, rates on the ultra-wealthy, loopholes/deductions, etc.
Let’s also assume this country has a 2 party system, and you want to vote for the candidate that most closely aligns to your beliefs on all three issues. Using our 3D model of political opinions, the way to find the party closest to your beliefs is as follows.
Mark your position in the graph, P, and the positions of the two parties, A and B.
Draw a line through the two party points, AB
Draw a segment from P to a point, P’, on AB, such that PP' is perpendicular to AB
The party point, A or B, that P’ is closest to represents the party your opinions most align with
Here’s an example, where the person whose opinion is shown by P is more closely aligned with B than A (proof at bottom for those curious, Appendix 1).
Figure 3: 1D Spectrum Projection
Before going forward, I should note that it’s a simplification to think that the closest point is definitely the one you’d vote for, but that doesn’t actually negate the result. Someone may value one issue more than the rest, such that they may side with the further overall party, provided they’re closer on that particular issue. To handle that, we could scale the entire 3D model along each axis depending on how much they care about that issue. For example, doubling every value along the “Immigration” axis would make differences in immigration opinions count twice as much, and then we can do the exact same exercise.
Getting back on track, what we see from the diagram above is that the two party system has reduced a 3D model to a 1D model. AB in that diagram is equivalent to the USA’s left-right spectrum, and all that really matters is one’s projection onto that line, P’. If this is the case, that means the distance from P to P’ represents lost opinion, as the further apart they are, the more dissimilar they are. The amount someone has to compromise in a given election is shown by the distance from P to the candidate they voted for, but the point here is that, regardless of candidates, as long as there’s a 2 party system, peoples’ opinions will be projected onto a single line. In this way, the lost opinion I’m referring to is a symptom of the system itself.
If we were to introduce a third party, C, to our example, then we would find the closest party in an analogous fashion. However, instead of projecting onto AB, we would need to project onto a plane that all three fall on, ABC. In doing so, almost everyone will have less lost opinion, because the projection onto ABC will almost always be closer than the projection onto AB. The simple reason for that is that AB itself falls on ABC. This means the projection onto ABC can’t be any worse than the projection onto AB, and if the projection onto ABC doesn’t happen to land exactly on AB (which is possible, but has probability=0) that means the projection onto ABC is shorter. This same argument can be applied infinitely many times, with more and more parties added. With a shorter projection, the system has less inherent lost opinion, which means people are more well represented.
While reducing the amount of lost opinion is beneficial in its own right, perhaps the biggest advantage of the multi-party system comes from the dynamics of the parties themselves. To look into that, let’s go back to the 1D representation of a 2 party system. Let’s also ignore the Electoral College and consider an election scheme where popular vote is what determines the winner, as it makes for a much simpler example, and the Electoral College would still have analogous results. If you place all the voters along the spectrum and find the median voter, that would give you the point along the spectrum that splits the electorate in half. The 2 party equilibrium, then, would be any arrangement where the midpoint between the parties’ positions along the spectrum coincides with the median voter’s position. That way, half the electorate favors each party. However, this can be achieved by an infinite number of position pairs. Here are two equilibria:
Figure 4: Possible 2-Party Equilibria
As you can see, the bottom example has more extreme political parties, but it’s still in equilibrium since both parties are the closer of the two to half of the population. Having a 2 party system with parties so far apart makes compromise nearly impossible and leads to more of a tug-of-war, where the party in power spends much of their time undoing what the previous had done, rather than following a more sustainable trend of advancement. This also leaves a large portion of the electorate - the moderates - without someone they particularly agree with.
If we introduce more parties, it becomes harder for this radicalization to happen, as all of them would have to move apart together, and even if that did happen, a new party would arise to represent the moderates.
Let’s scale the example in Figure 4 to a 3 party system. To do so, we’ll need to move from the 1D political spectrum back to the 2D political plane, as there’s guaranteed to be a plane all three lie on. We’ll also need to recognize that the benchmark to meet is not as clear with more than 2 choices - it’s possible that 34% of the vote is enough to get the plurality and win, if the other choices get 33% each, but 34% also doesn’t guarantee a win. We’ll have to settle for the more vague notion of “getting the most votes”. Finally, to represent the border between each parties’ voters, we can’t simply look at the midpoint on the spectrum. Instead, we’ll use a Voronoi diagram, which is a mathematical representation that shows all the points on a plane that are closer to each set point on that plane. For example, here is a possible Voronoi diagram of the radicalized political spectrum from above:
Figure 5: 2-Party Voronoi Diagram
The green section is everywhere that’s closer to B than A, and the purple is everywhere that’s closer to A than B. With just two points, Voronoi diagrams aren’t that interesting. Let’s try adding a third party in orange:
Figure 6: 3-Party Voronoi Diagram, Extreme Party C
This is much more interesting, and shows that C, which is only slightly more moderate than A & B, has the plurality if we assume even distribution of voters on this plane, and even more so if we assume voters are more dense in the middle. This will force A & B to move inwards more if they want to get elected. If C actually had a moderate platform, the effect would be more pronounced:
Figure 7: 3-Party Voronoi Diagram, Moderate Party C
Here, C not only has a plurality, but a majority. This is again assuming standard distribution, which of course is a simplification, but even if opinions are clustered in certain locations, that would just change where the parties would need to position themselves to stay relevant. The point here is that the parties couldn’t push outward with confidence that people on “their side” would still prefer them - if there’s more than two parties, the parties need to be wary of becoming too extreme and pushing voters to a more moderate party.
An unaddressed question you might have at this point is why a party in the 2 party system wouldn’t just move more to the center, tipping the scales to their side and forcing the other to follow suite. Well, there is an aspect of this. Most of the positive press around Joe Biden this election cycle has been for that reason - appealing to moderates. On the other hand, two of the other top candidates, Bernie Sanders, and Elizabeth Warren, are more liberal than Barack Obama and Hillary Clinton. We’ve seen so far that Sanders and Warren are much better at energizing and growing support than Biden. The same was true for Trump in 2016, on the other side of the spectrum. This is likely because the moderates on one side are less likely to be disgruntled than the more extreme people when the other side is in power. This means those non-moderates, who want their party to move further from the center, are likely the more involved ones who donate, volunteer, and vote in the primary race, which is what determines where the party ends up positioned in the general election.
The final point I want to make about the multi-party system is on compromise. I mentioned earlier how some political issues are a fairly binary “for/against” decision, while others have a large grey area. While grey area allows for more than two stances to be taken on an issue, there are rarely issues that have more than two extreme positions. By this I mean that any alternative stance you take is a middle ground between the two extremes. For example, I think everyone’s position on immigration falls somewhere between “let everyone who wants to come in” and “don’t let a single person in”. The significance of this is that it’s unlikely that all three parties’ stances will be compatible, which means the two that most agree will be forced to make compromises with each other in order for anything to get done. By definition, no two parties would agree on all the issues, or they’d be the same party, so compromises would have to come between different parties for different issues. This would make the “us versus them” mentality that’s plaguing US politics right now detrimental to the parties championing it, since you’d need others’ cooperation to get anything done.
Before I go, I’ll leave you with some data from other countries to show that this isn’t just good on paper and bad in practice: The Economist Intelligence Unit has a metric called Democracy Index, aiming to measure the strength of each country’s democracy. In the most recent list, the 17 strongest democracies all have a multi-party system, with Malta’s 2 party system coming in at 18. The USA is the second 2-party system, at 25, and 42 of the top 50 have at least 3 dominant parties.
Appendix 1: Proof of Projection onto 1D Axis
Here is the diagram from above of P’s projection onto the line through A and B.
Figure A1: 1D Spectrum Projection
You might be confused by this being a 2D diagram, but when you establish three points in space, as we did with P, A & B, there is always a plane that all three fall on. The diagram above is simply looking directly at that plane, regardless of where that plane is relative to the “opinion axes”. To find the political party closest to our opinions, we want to find the shorter of PA and PB. This can be done using the Pythagorean theorem - both are the hypotenuse of a right triangle where one of the legs is of length PP', so the one with the shorter second leg must have the shorter hypotenuse. That second leg is determined by how close P’ is to the party point (A or B). Thus, the party P’ is closest to is also the party P is closest to.
