Mathematics & Political Coalitions

Mathematics and Political Coalitions SH: Hello everyone, I am Sam Hansen SW: And I’m Sadie Witkowski. SH: And you are listening to Carry the Two, a podcast from the Institute for Mathematical and Statistical Innovation aka IMSI. SW: The podcast where Sam and I talk about the real world applications of mathematical and statistical research. SH: This is the third episode of our season all about the mathematics of democracy and politics. Last episode we talked all about the apportionment of representatives and the electoral college. This time around we are going to dig into what mathematics can tell us about the structures of political coalitions, especially hidden ones SW: Ooook, I am going to definitely need more explanation from you on this one Sam. With voting and apportionment I had a good sense of where we would be going but I got nothing for political coalition structures SH: That’is very understandable, unlike voting and your representatives political coalitions are not something that comes up everyday. Which is why I have three different examples to share with you of how mathematicians and data and computer scientists and psychologists are approaching them SW: Great, you want to get right into it? SH: I do. But first let’s meet the person who is going to walk us through the work AM: Hi, my name is Andrea Mock. I currently work as a data scientist at Aura and received a bachelor's degree in data science from Wellesley College. SH: The work that Andrea is here to talk to us about was a collaboration with our friend Ismar Volic, guest on the previous two episodes and professor and chair of the mathematics department at Wellesley college SW: Ismar is everywhere this season! SH: He really is! In this case Andrea and Ismar were building on previous work done by Abdou and Keiding who introduced the idea of using simplicial complexes to model political coalitions SW: Umm, simplicial what now? SH: Simplicial Complexes. These are collections of simplexes SW: …You know that doesn't help right? SH: I do but, how about we have Andrea give it a try AM: So it's basically a structure made from vertices and edges, and that's how we describe objects in different dimensions. SW: Ok, I am starting to get it. But maybe an example would help? SH: Don't worry, Andrea has you AM: A zero simplex is one point. A line segment, so two points connected by a line, is a one simplex. And then a two simplex is a higher dimensional simplex, which is three points connected by edges. So I'd say that the way we count things, we start with one point being zero, a line being one, and then two simplex is a triangle, and the three simplex would be a tetrahedron SW: And if a 3 simplex is a tetrahedron is a 3 dimensional shape made of 4 triangles connected together, so is a 4 simplex would be a four dimensional object made of 5 tetrahedrons connected together SH: Exactly, and that shape is known as a 5-cell or hypertetrahedron, but honestly that isn't that important. What is important is how Andrea and Ismar used these complexes, or sets, of these simplicies to model political structures AM: And when we think of a simplicial complex, it consists of a bunch of vertices and edges. And if we look at political structures, we can describe these in a similar way where political structure is made up of agents. So this could be political parties, voters, members of a legislative body, or along with a collection of subsets of these agents, which are called viable configurations. So this could be different parties that are able to form coalitions SW: Alright, I think I can see how the models look but how are they identifying these coalitions, the viable configurations as she called them SH: That is where identifying the complete simplices come in AM: In this case, we say the vertices that we see in a simplicial complex are these agents, so political parties, individuals, voters. And then the actual simplices are the viable configuration. So the different agents that are able to form coalitions that are able to work together. SW: So once they defined how to use simplicial complexes to model political structures, what did that allow them to do? SH: Quite a lot actually. They used existing theorems from the study of simplicial complexes and homology to investigate a lot of things, especially how stable different configurations of coalitions are AM: If we're talking about stability, it tells us how many subsets of simplices exist, so how many coalitions are possible in the system if we're thinking about it in a political systems type of way. SW: I get it! Since the points, or political agents, in a simplex are connected to each other then when you look at a full complex,, the more simplices there are the more coalitions are possible and the more stable a political structure SH: Precisely, and they also looked at what could make the structure more stable SW: Did they find anything? SH: They did AM: In general, we say that introducing a mediator adds stability, so it's able to create connections throughout the system to make it more stable. SW: By mediator, do they mean an external party that comes in when there is a problem, and tries to help solve it? Like when the UN gets involved to try and find a solution to a trade conflict between two countries before it escalates to violence SH: Exactly SW: So mediation really does work in some cases! SH: Most of the time yes, but it is important that the mediator has connections to all of the agents in the system. Because introducing a mediator can backfire if the mediator is less connected and ends up making only a subset of agents more connected, isolating them from the rest of the system SW: Like a mediator in a property dispute knowing the person on one side but not the other: then they look biased. That makes sense, in fact I can definitely think of cases where a self-appointed partisan mediators just makes things worse SH: For sure, but at least they were able to show that well connected mediators increase stability. Another thing they looked at is what happens when two parties that did not agree come to a compromise to work together AM: For example, you have different parties that then, end up having some sort of coalition. Previously, they weren't talking to one another, but now there is a connection between the two. They're not completely separate from one another. There's some configuration where they're able to work together with. And then if we think about the stability of the system, it would increase as we merge things together. SH: One really cool thing about using simplicial complexes to do this work is that once you have two parties, or coalitions, working together you can actually merge them together into a single point, because as Andrea says AM: You can think about this as one party delegating power to another. SW: And if one party delegates its power to another then you don't need the first party in the model anymore SH: Which then simplifies the model and makes it easier to analyze visually SW: Wow, that is cool. Really the whole visually modeling these structures is cool, which makes it unfortunate that we’re currently in an audio only medium right now SH: Andrea agrees with you. Maybe not about the audio part, but about it being cool. AM: I think this topic in general helps you visualize how coalitions are formed, how different parties might sit to one another and I think this was just a really cool way to visualize this. SW: So where does she hope this work will go in the future? SH: In the paper Andrea and Ismar outline a number of different directions this work can go mathematically, but she also shared during our interview her hope that this would go beyond mathematics AM: One hope would be that in some cases, you know, this could be used to map out different political structures and in a playful manner that not you know just not just mathematicians can take a look at but maybe your everyday you know person who is interested in you know dabbling a little bit in some some math topics. Ad Break SH: The next two mathematical examinations of political structures are both going to focus on identifying groups and coalitions within the US House of Representatives SW: So what, they used math to show that Republicans work with Republicans and Democrats work with Democrats? SH: Trust me, it goes deeper than that and to show you how I spoke first with Gunnar Carlson GC: I'm a professor of mathematics emeritus at Stanford University, but also an entrepreneur in a company called Blue Light AI. SW: And what was Gunnar’s interest in the House of Reps? SH: Well it started out more generally than that. He was interested in how groups of people form, specifically when GC: Sometimes it also happens that, well, there are groups that form and maybe you don't even quite recognize them. SH: And with the work of him and his collaborators he thought GC: We have the mathematics that could sort of capture those kind of groups as well SW: Ooh, identifying groups that are not obvious sounds like it could be really useful SH: Especially when you have a large data set and are trying to understand its dynamics. But first they needed to test their mathematics SW: I am guessing this is where the House comes in? GC: And the House of Representatives seemed like a very natural thing then. You know, there's a lot of kind of zero, one, yes, no kind of data there. Very simple data type. And, you know, one would expect to find certain interesting things. And we did. SW: So what are these interesting things? SH: We'll get to it but may I have a moment to talk about how they found them first? SW: I don’t mind a methodological detour SH: Thank you! What they did was take the voting records of members of the House for each year from 1992-2011 and for each year they created a data set where a member was represented by their collection of votes, 1 for yay, -1 for nay, and 0 for abstain, present, and all the other possibilities. SW: Gotcha, so each member of the house is a really long ordered sequence SH: Yep, and then they took this data and used the MAPPER algorithm, which was developed by Gurjeet Singh one of Gunnar's collaborators, and it’s a tool that simplifies collections of high dimensional points into two dimensional maps of points that are clustered together if they are similar, even directly connected by edges if they are really similar, and not connected or clustered if they aren't SW: And this clustering is what showed the interesting things? SH: Indeed. Though the first thing you see is what you expect GC: What is the first thing that comes out of the course is all Republican and Democrat. And that's not a surprise. SH: And while it is not a surprise, it is also a good sign. Any guesses why? SW: Umm, because it reflects what we know: that the two parties often vote with one another? SH: Precisely, and if they didn’t cluster together that would have been a good sign that something with the algorithm was not exactly working SW: Of course, but this is obvious! You still haven't shared the interesting part! SH: Ok, here it is GC: There was a group that tended to be more, how shall I say, maybe bipartisan and whatnot. It was a small group and it came out very nicely in the clustering. SH: Something Gunnar says would not have been obvious without creating these visual maps GC: The visualization allows you to find those, you know, which would be very hard to find otherwise. I mean, if you just if you yourself are just sort of, you know, scouring through that data, that's pretty hard to do. SH: And with these visualization there are often clusters, which you can analyze based on their color, with the traditional red for Republicans and Blue for Democrats, as well as where the clusters are GC: The big clumps were kind of more spread out and larger, and you could actually see, you know, the movement from very conservative to more middle of the road. And then in another one, you might see also, you know, the extremely left wing and then moving to the middle of the road. SH: There are also small clumps, sometimes even single points on their own GC: These are kind of anomalies. Mostly you have people acting as a team in strong lockstep, but there are some people that we found that just don't behave in such a lockstep way. SW: Neat, that is interesting. You can actually see some of the “mavericks” of the House SH: Right? Gunnar and his collaborators work also showed that the amount of bi-patisanship change from year to year including some years there was little to no bipartisanship. They also showed that partisan groups were often made up of distinct internal coalitions, some of which would be bi-partisan and other not, with the parties in some years being incredibly fractured SW: Hmmm SH: In fact, let me just show you the maps SW: Whoa, those are really cool. Dang, look at all those patterns. We're going to have to share this on the pod’s notes. We will include a link to this for sure in the show notes. SW: But looking at these though, their definitely seems to be a difference in how they look in the earlier versus the later years SH: That is something that Gunnar also brought up GC: If you look at 2002, for example, you can see that the blues are divided up into two or three groups and that one of the flares coming off that blue is actually red. In other words, that's a group of blue people who voted with the red. But on the other hand, as you go to 2011 or, you know, or even 2009, you see the heavy partisanship. SH: And if you remember back to these years, the House was very split. Major legislation like the Dodd-Frank Act only was passed with only 3 Republican yays and the Affordable Care Act was passed with no Republican support at all SW: Yeah, I remember those times. But I guess that is another example of the mathematics working though, since we do know that partisanship has been on the increase. I do wish we could know more about these bi-partisan clusters SH: Well then final investigation into political structures that I want to share with you is going to make you real happy SW: That is what I like to hear! SH: This time we are joined by a psychologist ZN: I'm Zachary Neal. I'm a professor of psychology at Michigan State University. SH: And a data scientist SA: Hi, I'm Samin Aref. I'm an assistant professor teaching Streaming Data Science at University of Toronto. SW: So which of the two thought to study the structure of the House of Representatives SH: That would be Zachary ZN: I first came to this as a substantive researcher in psychology and sociology and a little bit in political science, interested in how legislators interact. And the problem is that we can't directly ask legislators what their networks look like because they're busy and they'd probably lie to us. And so we need an indirect way to do that. SH: Samin on the other hand had a different interest SA: Developing my methodological research, I needed interesting data. So I sent an email to a computational sociology mailing list, asking pretty much everyone on that mailing list if they have a certain type of data. And Zach was, you know, one of the people who responded to me and sent me some very interesting data on bill co-sponsorship in the U.S. Congress. SH: At first Samin just used the data in his own research, but over the next few years they kept running into each other at conferences and eventually they decided to work together. SW: And the data they were using was from bill co-sponsorship? SH: It was, which as Zachary says is a tricky dataset ZN: But there are a whole bunch of challenges working with that kind of data. The sort of most straightforward challenge is just observing two legislators co-sponsoring one bill doesn't mean they're collaborating. Just observing they sponsor 10 or even 100 bills might not necessarily indicate they're collaborating. SW: Weird, so how did they end up overcoming these challenges SH: With some really interesting math of course! ZN: We first start by counting the number of times a pair of legislators co-sponsored a bill. So this could be, you know, maybe they've co-sponsored five, they worked together on five bills. Sometimes people work together on huge numbers of bills. Next, we have to find the expected distribution of that value. So how many times would we have expected this particular pair of legislators to have worked together? SW: The expected number of bills they should have worked together on? SH: Yeah, and it will be different for each representative because they all have different legislative behavior. For example ZN: So some legislators are just sort of legislative busybodies. They get involved in everything that comes across their desk. And so observing lots of co-sponsorships is not particularly interesting for them. It's not surprising because they get involved in everything. SH: And then there are the party leaders like the speaker of the house or the whips. Who tend to quite busy with responsibilities that do not include bill sponsorship ZN: So observing them engaging in co-sponsorships is really noteworthy because they do almost nothing. So when they do something, it catches your eye. SH: Not to mention how the content of bills changes sponsorship behavior. Some draw legislators in ZN: Some bills that get introduced in Congress are things like commemorative coin acts or the Americans love puppies bill. And everybody gets involved in those. And so observing people co-sponsoring the Everybody Loves Puppies bill provides no information about what people are actually doing. SH: And others drive legislators away ZN: Then we also get bills that say restrict access to reproductive care. The legislators that get involved in those bills, those attract many fewer legislators. Those are very risky, partisan bills. And so who signs onto those actually provides us lots of information. SW: And with that data they are able to estimate how often two legislators should have been co-sponsors? SH: They are, and once they have they look for two things ZN: What we’re looking for is pairs of legislators that have an unusually high number of partnerships. We call those political alliances or in signed graphs, that's a positive tie. But we're also looking for pairs of legislators that have an unusually low number of partnerships. So those we call political antagonisms or in signed graphs, those become the negative ties. SW: Hold on, wait a minute. Signed Graphs? Guessing these aren’t autographs on a bar chart [laughs] SH: No, no they’re no. A signed graph is a graph or network made up of nodes that are collected by edges. What then makes the graphs signed is that each edge is marked with either a positive or a negative sign, depending on if they are between two politicians who are allied or two politicians who are antagonists. SW: Ok, you can continue now SH: Thanks. I really appreciate your permission. Now that they had their signed graph, they wanted to use it to identify coalitions. The first thing they wanted to do was test to see if their model made sense ZN: Because we know Congress is polarized. And so we know if the method is working, we ought to see a big clump of Republicans and a big clump of Democrats. And so it provides us a way of validating the method. SH: So to validate their method they assumed that there were two coalitions in the House and split the network based on that assumption ZN: We sort of knew what we ought to find, and that's exactly what we did find. SW: That must have been a great feeling, always wonderful when your method works like it should SH: 100%, but they didn't stop there ZN: So we think these positive-negative ties are in the right places. Is there something that a simple sort of two-party polarization model might be missing? And that's where some of Samin's models came in to help us detect these more hidden things that are less obvious. SW: And how do Samin's models detect these less obvious things? SH: By letting the data speak for itself instead of using traditional party based analyses SA: So, at the system level, we have the U.S. Congress. At the component level, we have the individual legislators, but we typically need a unit of analysis that is somewhere in between, somewhere between the system and the components. So for that kind of meso-level unit of analysis, depending on our assumption, we may use a data-driven method or we may just go with the more traditional and well-established method of thinking of political parties. SW: I get it, let the data tell you what's in it instead of making a bunch of assumptions before you even start. SH: Exactly, and assuming that parties are the main driver of who co-sponsor bills together is just not something that the data shows SA: We compare the idea of just partitioning the legislators based on their party affiliation, we see that there will be hundreds of ties that would be inconsistent with that idea. Because the idea is pretty much based on thinking that all the collaborations are partisan collaborations. But actually, in early 80s, there were many, many examples of bipartisan collaborations. That's why this idea of just partitioning the legislators based on their party affiliation is very inconsistent with the data. SW: Then what does the data show SA: There's actually a third coalition that is made up of ideologically fluid legislators, I mean, who would change from a session to another, but you know, the third coalition is there, and this third coalition is made up of super effective legislators. SW: So just like with Gunnar, we see more bi-partisanship than we assume? SH: They do both agree on that point. Though they both also agree that their studies show that things are getting much more polarized over time SW: As unfortunate as that is, I can’t say I’m surprised. Did they share anything about where this third coalition comes from? SH: They did, but first it is important to note that, that other during sessions where the party in power changes, the third coalition is still more closely aligned ideologically with the party in power than the other party. A factor that actually comes from how this third coalition develops ZN: One of the patterns, at least in this context, that we consistently observe, is that the party in power has many more negative ties than the minority party. And we suspect that what we're seeing here is the majority party, because they have control, they also have some flexibility to have some infighting, to have some disagreement, because they control the agenda. And so they have that flexibility, whereas the minority party doesn't have those degrees of freedom. The minority party needs to circle the wagon, needs to remain cohesive if they're going to remain relevant at all. SW: It is interesting to think that the party in power is more likely to have internal negative ties SH: Right? And it turns out those negative ties are the THING that make the third coalition SA: When we look at the fraction of negative edges, we see that this third coalition is considerably distinct, considerably different from the other two coalitions because they have much higher fraction of negative edges. SH: And Samin is not exaggerating when he says that the fraction is distinct SA: In some of the more recent sessions, some of these legislators in this third coalition have something around 30 percent of their ties being negative ties, where for legislators in the other two coalitions, the number is less than 5%. So that's why, as you suggested, the negative ties are very fundamental to this composition of the coalitions that we see in these networks. SW: 30% to less than 5%, yeah you can call that distinct. SH: There was something else that was quite intriguing about that third coalition ZN: What's interesting is that the members in this third coalition are likely not representatives that are at the top of mind for everyone. These are not the most outspoken, most media visible representatives, but they're the ones that are actually getting work done, getting bills passed behind the scenes. SW: It is always the quiet ones… SH: Always. Zachary thinks this says something about the importance of elections, especially the less flashy ones ZN: Yes, Congress is becoming more polarized, but electing local representatives matters. matters, because even if they're not the ones, you know, in the front of CNN and MSNBC and Fox News, they're getting to Congress and doing work on behalf of their constituents. And so what they do actually does have an influence in spite of the polarization. SW: That is a take away I can get behind, and a good reminder that there are elected representatives out there that are putting the work in SH: Something I forget all the time I am sad to say. Now there is one last thing I want to share with you. It is a message from Samin to us all that I really hope we can all heed SW: Yeah, let’s hear it SA: I just want to remind them of the benefits of bending the assumptions every now and then, if you work in an interdisciplinary context. *Music* SH: Don’t forget to check out our show notes in the podcast description for more Andrea, Gunnar, Samin, and Zachary, including links to their work that we discussed on this episode SW: And if you like the show, give us a review on apple podcast or spotify or wherever you listen. By rating and reviewing the show, you really help us spread the word about Carry the Two so that other listeners can discover us. SH: And for more on the math research being shared at IMSI, be sure to check us out online at our homepage: IMSI dot institute. We’re also on twitter at IMSI underscore institute, as well as instagram at IMSI dot institute! That’s IMSI, spelled I M S I. SW: And do you have a burning math question? Maybe you have an idea for a story on how mathematics and statistics connect with the world around us. Send us an email with your idea! SH: You can send your feedback, ideas, and more to sam AT IMSI dot institute. That’s S A M at I M S I dot institute. SW: We’d also like to thank Blue Dot Sessions for the music we use in Carry the Two. SH: Lastly, Carry the Two is made possible by the Institute for Mathematical and Statistical Innovation, located on the gorgeous campus of the University of Chicago. We are supported by the National Science Foundation and the University of Chicago SH: Hello everyone, I am Samm…Oh I almost said Samuel. Wow, that is, that is real hardwired SW: [laughter] A ghost from a past SH: That is not the right clip at all SW: Not the same one. Like, that’s new SH: Quite a lot actually. They used existum theorem. Together: Existum? SW: Existum theorem. SH: Well then. [Burp] SW: That was beautiful. SH: [laughs] Well now. SH: I mean, it was so important that there were sirens going on. SW: Going out on a siren is really something for a political podcast. [laughter]