primeideal (![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png)
primeideal) wrote2020-08-05 05:08 pm
primeideal) wrote2020-08-05 05:08 pm
Entry tags:
Dimensionality Reduction and Politics
Two ideas that have been swirling around in my head but I haven't found the right words to combine, until now:
Some of the mathy/data science things I'm learning about fall into the linear algebra category. Linear algebra is something that does give us an answer to the "when are we ever going to use this anyway." Let's say you're Netflix. You have lots of data about which people have watched which movies (maybe how they rated them, too), and you'd like to be able to recommend movies to people. You can build a matrix* where each row represents a person, and each column represents a movie, and there's a 1 in that row and column if the user has watched the movie. The problem with this is that it's very high-dimensional--there are thousands of movies and millions of users, so that's going to be difficult to manipulate. Additionally, the grid is "sparse"--most people haven't seen any given movie, so most of the entries will be zero. What you'd like to do is simplify the data into a more usable form.
One trick is called "singular value decomposition." You can "factor" the matrix, let's call it M, into the product of three other matrixes, U*G*F. U measures the relationship of users to genres--if someone liked Blade Runner, and also liked Star Wars, their relationship with the "science fiction" genre will be high. G measures the genres themselves, and is a "diagonal" matrix which is easy to work with--the largest numbers in the diagonal represent the most clearly defined genres. And F measures the films' relationship to genres; something like "The Princess Bride" might be a 70% match for the "romcom" genre but also a 50% match for the "fantasy" genre.
Now, you can make G smaller by only taking the top 10 genres (or some other number), and then cut off the extra rows and columns of U and F accordingly. Instead of M=U*G*F, you now have M (approximately) u*g*f, where little u, g, and f are much smaller and easier to work with. Success! The tradeoff is that this is only an approximation--some users' tastes will fall outside clear genres, and some movies might be difficult to categorize. But if you're a big-data company, that's the sacrifice you have to work with.
So why is this on my mind (other than that I should be learning some coding stuff for work)? Political science, of course.
Duverger's law says that, if you have a first-past-the-post system (most votes wins, like the way people vote for members of Congress in the US, but not like party-list parliamentary systems), you should expect a two-party system to emerge. Why? Because if you try to bring in a third party, you're likely to split the vote and see someone you really dislike win instead.
This means that the political divide in these countries seems to boil down to a one-dimensional spectrum. In the US, if you're more to the "left," overall, you'll probably vote for the Democratic candidate. If you're more to the "right," you'll probably vote for the Republican.
However, there are really lots of different issues people have opinions on. Political compass/Nolan Chart type diagrams try to view two dimensions, one correlating to economic freedom and one correlating to social freedom. A stereotypical Democrat might prefer fewer drug laws and higher taxes on the rich to fund schools. A stereotypical Republican probably prefers neither. A Libertarian might want both fewer drug laws and lower taxes, and so which party they vote for depends on which issues they feel more strongly about (or if one of the candidates is a complete lunatic, hypothetically speaking).
In NationStates (which is great, by the way), the non-economic axis is further subdivided into "personal freedom" (same-sex marriage, religious freedom) and "political freedom" (voting rights, censorship of anti-government views...), which means nations can inhabit one of 27** types of 3x3x3 nation categories with low/medium/high freedoms in personal/political/economic freedoms. And so on.
And it doesn't stop there--any issue has the potential to be partisan, and many people's opinions aren't correlated with their parties on every issue. So if I (hypothetically speaking) agree more with the Republican candidate 20% of the time, more with Democratic candidate 70% of the time, and have very moderate views 10% of the time, I'm going to vote for the Democrat, not 7 times out of 10, but 10 times out of 10.*** Because even if I were to help the Republican get elected sometimes, they wouldn't magically only carry out the policies I like.
In other words, politics is dimensionality reduction! We take our complex, multidimensional, not-always-agreeing beliefs in a multidimensional space, and project them down onto a one-dimensional line (or maybe more if you have a party list), and then go "okay, guess I'm voting for the Democrat, then."
The problem is when you try to project back. Given a single voter on the Democratic side of the spectrum, can you work backwards to say "well, she must agree with me about hate speech and abortion rights and police reform, because we voted for the same person"? No, you can't! Because she might have based her votes on the candidates' policies on climate change or nuclear disarmament or wealth taxes, and it's not obvious that any of these things have anything to do with each other. If you're politically engaged, it's easy to say "well someone who believes X also believes Y, because the Republican platform is in favor of both X and Y," but for many people--even many politically informed people--that just isn't how it works. And so trying to find which n-dimensional point turned someone into a "moderate Democrat" is a perilous exercise.
*not The Matrix, which I did watch on Netflix
**there are actually 28; if your nation is a "Queendom" and would otherwise be a "Father Knows Best State" you actually get to be a "Mother Knows Best State" :D
***I feel like there was a paper about how simpler is better ("mixed strategies" is relevant here) but I can't find it.
Some of the mathy/data science things I'm learning about fall into the linear algebra category. Linear algebra is something that does give us an answer to the "when are we ever going to use this anyway." Let's say you're Netflix. You have lots of data about which people have watched which movies (maybe how they rated them, too), and you'd like to be able to recommend movies to people. You can build a matrix* where each row represents a person, and each column represents a movie, and there's a 1 in that row and column if the user has watched the movie. The problem with this is that it's very high-dimensional--there are thousands of movies and millions of users, so that's going to be difficult to manipulate. Additionally, the grid is "sparse"--most people haven't seen any given movie, so most of the entries will be zero. What you'd like to do is simplify the data into a more usable form.
One trick is called "singular value decomposition." You can "factor" the matrix, let's call it M, into the product of three other matrixes, U*G*F. U measures the relationship of users to genres--if someone liked Blade Runner, and also liked Star Wars, their relationship with the "science fiction" genre will be high. G measures the genres themselves, and is a "diagonal" matrix which is easy to work with--the largest numbers in the diagonal represent the most clearly defined genres. And F measures the films' relationship to genres; something like "The Princess Bride" might be a 70% match for the "romcom" genre but also a 50% match for the "fantasy" genre.
Now, you can make G smaller by only taking the top 10 genres (or some other number), and then cut off the extra rows and columns of U and F accordingly. Instead of M=U*G*F, you now have M (approximately) u*g*f, where little u, g, and f are much smaller and easier to work with. Success! The tradeoff is that this is only an approximation--some users' tastes will fall outside clear genres, and some movies might be difficult to categorize. But if you're a big-data company, that's the sacrifice you have to work with.
So why is this on my mind (other than that I should be learning some coding stuff for work)? Political science, of course.
Duverger's law says that, if you have a first-past-the-post system (most votes wins, like the way people vote for members of Congress in the US, but not like party-list parliamentary systems), you should expect a two-party system to emerge. Why? Because if you try to bring in a third party, you're likely to split the vote and see someone you really dislike win instead.
This means that the political divide in these countries seems to boil down to a one-dimensional spectrum. In the US, if you're more to the "left," overall, you'll probably vote for the Democratic candidate. If you're more to the "right," you'll probably vote for the Republican.
However, there are really lots of different issues people have opinions on. Political compass/Nolan Chart type diagrams try to view two dimensions, one correlating to economic freedom and one correlating to social freedom. A stereotypical Democrat might prefer fewer drug laws and higher taxes on the rich to fund schools. A stereotypical Republican probably prefers neither. A Libertarian might want both fewer drug laws and lower taxes, and so which party they vote for depends on which issues they feel more strongly about (or if one of the candidates is a complete lunatic, hypothetically speaking).
In NationStates (which is great, by the way), the non-economic axis is further subdivided into "personal freedom" (same-sex marriage, religious freedom) and "political freedom" (voting rights, censorship of anti-government views...), which means nations can inhabit one of 27** types of 3x3x3 nation categories with low/medium/high freedoms in personal/political/economic freedoms. And so on.
And it doesn't stop there--any issue has the potential to be partisan, and many people's opinions aren't correlated with their parties on every issue. So if I (hypothetically speaking) agree more with the Republican candidate 20% of the time, more with Democratic candidate 70% of the time, and have very moderate views 10% of the time, I'm going to vote for the Democrat, not 7 times out of 10, but 10 times out of 10.*** Because even if I were to help the Republican get elected sometimes, they wouldn't magically only carry out the policies I like.
In other words, politics is dimensionality reduction! We take our complex, multidimensional, not-always-agreeing beliefs in a multidimensional space, and project them down onto a one-dimensional line (or maybe more if you have a party list), and then go "okay, guess I'm voting for the Democrat, then."
The problem is when you try to project back. Given a single voter on the Democratic side of the spectrum, can you work backwards to say "well, she must agree with me about hate speech and abortion rights and police reform, because we voted for the same person"? No, you can't! Because she might have based her votes on the candidates' policies on climate change or nuclear disarmament or wealth taxes, and it's not obvious that any of these things have anything to do with each other. If you're politically engaged, it's easy to say "well someone who believes X also believes Y, because the Republican platform is in favor of both X and Y," but for many people--even many politically informed people--that just isn't how it works. And so trying to find which n-dimensional point turned someone into a "moderate Democrat" is a perilous exercise.
*not The Matrix, which I did watch on Netflix
**there are actually 28; if your nation is a "Queendom" and would otherwise be a "Father Knows Best State" you actually get to be a "Mother Knows Best State" :D
***I feel like there was a paper about how simpler is better ("mixed strategies" is relevant here) but I can't find it.