Lesson 2: Modeling Preferences

How Much?

It’s widely thought that during the past thirty years, the U.S. has experienced severe and increasing levels of political polarization. Commentators point to this polarization as a cause of the U.S. government’s failure to respond effectively to a series of crises, including the 2008 recession, global climate change, rising economic inequality and the 2019 COVID pandemic. And of course political polarization seems to be one of the causes of the 2021 insurrection at the U.S. Capitol and the smaller scale but still frequent incidents extremist political violence since then.

How bad is it? How does the polarization occurring in the U.S. at present compare to the political conditions that drove the most significant outbreaks of political violence in previous periods of U.S. history, such as the U.S. Civil War, the violent suppression of the labor movement in the 1920s, the regime of white supremacist terror that prevailed in southern states from the 1880s through the 1960s, and the wave of riots, bombings and assassinations that roiled the country in the late 1960s and early 1970s? How does it compare to the political conditions that caused social and political conflicts to erupt in significant and long-lasting violence in other places in the world, such as Northern Ireland during the “troubles” of the 20th Century, latin american countries such as Chile and El Salvador in the 1970s and 1980s, and the Balkans in the 1990s? If we understood polarization better – both its relative severity and its causes and consequences – perhaps we could do something about it. Perhaps that understanding would point the way to institutional reforms that would reduce polarization or short-circuit it as a cause of violence and democratic backsliding.

We cannot build a better understanding of political polarization unless we can gauge its relative severity across time and space. For instance, if we have a hypothesis about what causes it, the only way to test that hypothesis is by comparing the circumstances in which it is relatively severe to the circumstances where it is less severe. Moreover, any ideas we have about its effects on political violence can only be tested through similar comparisons: I.e. comparisons of the level of political violence in circumstances with relatively severe polarization to the level of violence in circumstances with relatively less severe polarization. In other words, an impression that polarization is worsening in some general sense is not sufficient. We have to be able to measure how much polarization is occurring in one place, time or political institution relative to another.

In this and the next couple of lessons, you’ll learn your first set of PPT tools by exploring their use by political scientist attempting to measure and explain variation in the severity of polarization. In this lesson in particular, you’ll learn the basic PPT model of individual preferences, along with an indispensible model of preferences and conflict over public policy called the spatial model. Once you have these under your belt, you’ll move on in Lesson 3 to learning how to use utility functions to model preferences along with the PPT approach to modeling individual choice. Finally, in Lesson 4, you’ll begin learn how to model uncertainty and choice under uncertainty.

It’s Complicated

The first step to developing a measure of the severity of political polarization is to be clear about exactly what we mean by the term. Political scientist Nolan McCarty (2019) explains that the recent rise of polarization in the U.S. amounts to the combination of at least three distinct phenomena, each occurring to different extents among different subgroups of U.S. “political elites” – meaning officeholders, candidates for political office and the small portion of the public that pays close attention to politics:

Policy Polarization
A group experiences policy polarization to the extent that there are large portions of the group who express support for positions towards each of the extremes of a given policy issue. McCarty offers policies towards abortion as an example. Policies at one extreme of this domain would make abortion illegal under all circumstances. Policies at the other extreme would make abortion legal under all circumstances. Between these extremes are an array of policies that would make abortion illegal in some circumstances and legal in others. A group of persons experiences policy polarization on this issue to the extent that there is a large proportion of the group who support policies towards the “illegal-under-all-circumstances” extreme and a large portion who support policies towards the opposite “legal-under-all-circumstances” extreme. Presumably, persons with views closer to either extreme find compromise or bargaining with those with views towards the opposite extreme unacceptable or impossible, and thus to the extent that there are large portions of a group with views towards the extremes, political conflict over the issue within the group sharpens. The polarization the U.S. has experienced consists in increasing polarization of this form among political elites on each of a variety of policy issues.
Ideological Consistency
Ideological consistency occurs when a person adopts positions on a range of policy issues that are all mutually determined by or consistent with a single ideological worldview or dogma. For instance, one such ideological worldview is economic conservatism. In this view, markets are almost always superior to government as a mechanism for allocating resources. A worldview like that can be interpreted to require specific positions on every policy issue, including trade, labor, environmental, education, tax and healthcare policy. When ideological consistency is limited to a handful of eccentrics with little political influence, it’s no big deal. But during the past thirty years in the U.S, at least two distinct groups of political elites have emerged, each group made up of adherents to a particular ideological dogma. Conflict between these groups seems unresolvable because the ideological consistency of their members causes every member of each group to disagree with every member of the opposing group on all policy issues all at the same time. This means that they cannot resolve conflict on any one issue by cooperating with each other on other issues, because there are no issues on which they agree. In effect, the growth in the numbers of political elites with ideologically consistent policy positions causes policy polarization on almost all issues simultaneously.
Partisan Divergence
Partisan divergence is a combination of processes through which persons’ party identification becomes more predictive of their policy positions, and persons feel disdain, fear and animosity towards those who do not share their party identification. The result is increasing uniformity of policy positions within parties, increasing divergence of policy positions between parties, and high levels of expressed inter-party animosity.

Polarization, then, is not just one thing. It is at least three different things, each happening at different rates and magnitudes among different sub-groups of political elites. And that makes gauging the overall severity of polarization in any one place, time or political institution complicated. To see why, imagine we sought to characterize the degree of policy polarization in one political institution, let’s say the U.S. House of Representatives, on one particular issue, let’s say abortion policy. Knowing that polarization on this issue amounts to growing portions of persons supporting each of the extreme positions on that policy, suppose we determine the proportion of the House’s 435 members who express support for the “illegal-in-all-circumstances” position and the proportion who support the opposite “legal-in-all-circumstances” position. This would be an excellent measure of policy polarization on abortion in the U.S. House. Indeed, if we could determine these proportions among U.S. House members for each of a number of years, we could then say whether and how much policy polarization on abortion in the U.S. House has grown over time.

We encounter a complication, however, when we consider what this measure would tell us about the level of overall polarization in the U.S. House, as opposed to the level of policy polarization on the single issue of abortion. The House members we identify who express support for each of the extreme positions on abortion will vary in the extent to which they are polarized on other policy issues and in the extent to which they are caught up in the two other aspects of polarization – i.e. ideological consistency and partisan divergence. For instance, some will be ideologues whose position on abortion is one of dozens of policy commitments that all put them at odds with enemies from an opposed ideological faction. But others will have arrived at their position on abortion without regard for whether it is “consistent” with their views on other policy issues according to one or another ideological dogma. These non-ideological members will be able to find common ground with their opposites on abortion policy on other policy issues.

Thus our measure of policy polarization on abortion can only tell us about the severity of overall polarization if we combine it with measures of policy polarization on all other issues, measures of polarization driven by ideological consistency and measures of partisan divergence. It’s unlikely we could measure all of these subtle aspects of attitudes and positions of U.S. House members for any one year, let alone for the multiple years needed to gauge change in the severity of overall polarization over time. And even if we could measure all of these aspects of polarization simultaneously for the members of the House, we’d be left with an intractable conceptual problem: Most likely, we’d find polarization on some issues or in some aspects to be severe, and find polarization on other issues or in other aspects to be less severe. Thus we’d need to decide how much an increase in one aspect of polarization should count for a decrease in any other aspect.

The Spatial Model

You’ve seen this story before. Political outcomes result from the aggregation of the actions of large numbers of persons. Persons differ immensely from one another in their circumstances and behavior. Thus political outcomes are extraordinarily complex and cannot be known exactly in every detail. You learned in Lesson 1 how political scientists respond to this difficulty. They build models – representations of politics that are partial in that they depict only some aspects of what would otherwise be overwhelmingly complex phenomena.

The spatial model is a PPT model that can represent conflict over public policy in almost any political process. It’s especially powerful as a tool for modeling polarization. As you master it, you’ll discover that it can be used to partially represent all three aspects of polarization – i.e. policy polarization, ideological consistency and partisan divergence. But it’s easiest to learn it as a model of policy polarization.

Recall what we mean by policy polarization: A group experiences policy polarization to the extent that there are large portions of the group who hold views towards each of the extremes on a policy issue. Implicit in this is the notion that policies on the issue in question are arrayed across a space. Some policies are located near the middle of this space, while others are on located at the extremes. In the simplest version of the spatial model, first applied in political science by Anthony Downs (1957) and Duncan Black (1958), we represent this space as a line, called a “policy space” or sometimes a “policy continuum”:

Having drawn a line to represent the space in which policies towards an issue are positioned, we can then depict policies on that issue as points on this line. We depict the notion that two policies on an issue are at opposite extremes by placing them at the opposite ends of the line, and depict the notion that a policy lies between these extremes by placing it somewhere on the line between the two ends.

For instance, consider again policies that set the terms on which persons may legally terminate a pregnancy. The two extremes on this policy are “legal under all circumstances” and “illegal under all circumstances”. Since these are the two extremes, we place them on opposite ends of the policy space, like this:

Any policy that lies between the extremes is placed somewhere between the ends of the line. For instance, a policy under which abortion is legal in all circumstances during the first trimester of pregnancy and illegal under all circumstances after that would be depicted as positioned somewhere between “legal under all circumstances” and “illegal under all circumstances” like this:

When we represent just one policy that lies between the two most extreme policies, it doesn’t matter exactly where on the line between the two extremes we place that policy. However, when we want to represent two or more policies that each lie between the extremes, we must be deliberate about how those policies are ordered along the line relative to one another. The basis on which we order those policies is subtle. But it is absolutely crucial for understanding everything that follows, so make sure you follow the logic.

When we place any pair of policies in a particular order on the policy space, we are effectively representing those policies as ranked relatively to one another according to some criterion, with the policy placed to the right “higher” according to that criterion than the policy placed to the left. Crucially, the policies at the most extreme ends of the policy space are included in that ranking. Specifically, the one at the extreme right end of the space is at the very top of the ranking, and the one at the extreme left end at the space at the bottom. So, when deciding how to order the locations of two or more policies between the extremes, you have to account for the order of the policies at the extremes, and place the policies between the extremes in a way that is consistent with that order.

That’s all very abstract, so let’s see how it works with the example of abortion policies. We already identified our two most extreme policies, “legal under all circumstances” and “illegal under all circumstances” and placed them at the left and right ends, respectively, of our policy space, like this:

Now suppose that we want to place two additional policies in our policy space, each of which makes abortion legal under some circumstances and illegal under others, and thus each of which should be placed somewhere between the two extremes:

  • Abortion is legal during the first trimester of pregnancy under all circumstances and illegal after that.
  • Abortion is legal during the first trimester of pregnancy after the pregnant person participates in a counseling session about alternatives to abortion, and illegal after that.

How should these two policies be ordered relative to one another? Should “legal in the first trimester” be placed on to the left of “legal in the first trimester after counseling”? Or should “legal in the first trimester” be placed to the right of “legal in the first trimester after counseling”? Given the order in which we’ve placed “legal under all circumstances” and “illegal under all circumstances” above – i.e. with the former on the left and the latter on the right – we must place “legal during the first trimester” to the left of “legal during the first trimester after counseling”.

To see why let’s think through the implications of placing these two policies in the wrong order:

By placing (D) “Legal during the first trimester after counseling” to the left of (C) “Legal during the first trimester”, we have represented “Legal during the first trimester after counseling” as closer than “Legal during the first trimester” to “Legal under all circumstances”. Yet “Legal under all circumstances” is the least restrictive possible policy, while “Legal during the first trimester after counseling is” more restrictive than “Legal during the first trimester”. Since “Legal under all circumstances” is the least restrictive possible policy, and we’ve placed it at the left end of the continuum, every additional pair of policies we place on the line must be ordered relative to one another (and relative to any other policies on the line) from left to right in order of their restrictiveness. So, since “Legal during the first trimester” is less restrictive than “Legal during the first trimester after counseling”, the former must be placed to the left of the latter. So, placing the four policies (A) through (D) along the line in a way that correctly represents how each compares to all the others requires placing (C) to the left of (D). Further, we can add a label below the policy space that says “restrictiveness” with an arrow pointing to the right to help us keep track of how the policies placed on the line compare to one another:

Many people find the spatial model highly intuitive, because it represents concepts that are so widely used in talking about policy – i.e. “extreme” and “moderate”. So it can be easy to forget that the spatial model is a model, and to fall into the habit seeing it as an accurate and complete description of policy on any given issue. But the spatial model is a model, not a complete and accurate description! So it’s essential to learn to recognize important aspects of policy that any given spatial model does not accurately represent.

You learned above that the spatial model represents policies as ordered according to a single criterion. The spatial model developed above of policies towards abortion, for instance, orders those policies according to their “restrictiveness”. The fact that policies in that model must be ordered according to a single criterion prevents it from accurately representing important differences between policies.

Why? Because there are always multiple criteria that differentiate policies on any given issue. For instance, specifying the circumstances under which abortion is legal is only one of many ways through which U.S. states try to control women’s capacity terminate pregnancies. States also adopt policies that affect the supply of abortion services. For instance, anti-abortion activists in several states during the past decades have advocated for laws that would impose onerous operating standards on health facilities that provide abortions. Standards can be written that make it so costly to operate clinics that they become economically marginal or even non-viable. Such standards cause clinics to go out of business or to choose to stop offering abortion services. This reduces the supply of abortion independently of the circumstances under which undergoing or providing an abortion is legal.

So a complete account of abortion policy must order policies with respect both to their legal restrictiveness and their effects on the supply of abortion services. The model of the issue we developed above, however, only differentiates policies according to a single criterion. So it cannot capture differences between two policies that both put the same legal limits on abortion but that differ starkly in the operating standards they impose on abortion clinics.

What you’ve learned in this lesson so far is only the simplest kind of spatial model: a “one-dimensional” spatial model that represents differences between policies according to only one criterion. Political scientists sometimes want to think through the implications of multiple dimensions of conflict, and for that reason have developed spatial models that can represent differences between policies on any number of dimensions. Critically, these models represent more aspects of the policy issues they’re used to study than do one-dimensional models, but they are still models. There are inevitably important aspects of policy that they ignore or mis-represent.

Pause and complete check of understanding 1 now!

Individual Preferences

One key element of policy polarization is divergence between policies: Some policies lie at the extremes of an issue, while others lie between the extremes. The other key element is divergence between persons: Some persons support policies at one extreme or the other, while other persons support policies between the extremes. So far, you’ve learned how to use the spatial model to represent divergence between policies. We’ll now turn to the representation of divergence between persons.

The spatial model represents divergence between persons as arising from differences in individual policy preferences, meaning disagreements between persons about what policies they want the government to adopt or think the government ought to adopt. In this section, you’ll start to learn the general approach taken in PPT to representing individual preferences by learning how individual preferences are depicted in the spatial model.

All Preferences are Comparisons

To understand how PPT models represent individual preferences, it’s critical to recognize that the concept of “preference” only makes sense in a context in which a person is comparing multiple alternatives to one another. Thus preferences always amount to comparisons between alternatives.

Consider, for instance, a person’s preferences over what to have for breakfast. The statement “Lucinda prefers bagels for breakfast,” doesn’t just tell us how Lucinda feels about bagels. It tells us how Lucinda feels about bagels relative to other things she might eat. It tells us, for instance, that she prefers bagels to cereal, bagels to scrambled eggs, bagels to pastries, bagels to pancakes, etc. Moreover, “Lucinda prefers bagels for breakfast” is by no means a full description of Lucinda’s preferences over breakfast foods. It gives us no information whatsoever about what Lucinda might choose when there are no bagels in the kitchen, but there are, say, pancakes and pastries. “Lucinda prefers bagels for breakfast” tells us that Lucinda prefers bagels to pancakes and bagels to pastries, but it doesn’t say anything about how she ranks pancakes vs. pastries.

Thus, a complete depiction of a person’s preferences over a given set of alternatives must tell us how each of those alternatives compares for that person to each of the other alternatives. For instance, consider three alternative policies towards abortion, represented in the spatial model as follows:

We can depict any person’s preferences over these three policies as a collection of 3 comparisons:

  • Do they prefer (A) to (B), prefer (B) to (A) or see (A) and (B) as equally good?
  • Do they prefer (A) to (C), prefer (C) to (A) or see (A) and (C) as equally good?
  • Do they prefer (B) to (C), prefer (C) to (B) or see (B) and (C) as equally good?

For instance, here are representations of the preferences of three imaginary persons, Lucinda, Marisol and Niyam, over the three policies (A), (B) and (C):

Lucinda’s Preferences
She prefers (A) to (B), prefers (A) to (C) and prefers (B) to (C)
Marisol’s Preferences
She prefers (B) to (A), prefers (C) to (A) and prefers (C) to (B)
Niyam’s Preferences
He prefers (B) to (A), finds (A) and (C) equally good and prefers (B) to (C)
Representation of Preferences

Given a set of alternatives, a representation of a person’s preferences over those alternatives amounts to a collection of comparisons, with one comparison for each unique pair of alternatives. For each pair of alternatives (x,y), the comparison for that pair must be one of the following statements:

  • The person prefers x to y.
  • The person prefers y to x.
  • The person finds x and y equally good.

Single-Peaked Policy Preferences

Since the concept of preference is so general, we face an overwhelming variety of options when modelling individual preferences. When using a spatial model, we typically narrow things down by representing only a subset of possible preferences called single-peaked policy preferences.

Single-peaked policy preferences only make sense in the context of a spatial model in which each policy has a definite position in a policy space. For instance, in the one-dimensional spatial model you learned in the previous section, every policy has a position on a line like this:

Given a set of policies located on a line like the above, a person has single-peaked preferences over those policies if there is a location in the space called that person’s ideal point, and, given any pair of policies, the person prefers the policy in the pair that is closer in the space to her ideal point.

Single-Peaked Preferences

Given a set of policies, each of which has a definite location in a policy space, a person has single-peaked preferences over those policies if there is a location in the space called that person’s ideal point, and given any pair of policies (x,y)…

  • …the person prefers x to y if x is closer to her ideal point than y;
  • …the person prefers y to x if y is closer to her ideal point than x;
  • …the person finds x and y equally good if x and y are equidistant from her ideal point.

Let’s look at an example. Imagine three persons, Lucinda, Marisol and Niyam, who each have preferences over policies towards abortion. Imagine that abortion policies are arrayed along a line as in the spatial model and that Lucinda, Marisol and Niyam each have single-peaked preferences over abortion policies with their ideal points positioned as follows:

Now imagine three alternative policies towards abortion are positioned in the policy space like so:

Knowing that policies (A), (B) and (C) are positioned relative to Lucinda’s, Marisol’s and Niyam’s ideal points as above, and knowing that Lucinda’s, Marisol’s and Niyam’s preferences are single peaked, we know Lucinda’s, Marisol’s and Niyam’s preferences over policies (A), (B) and (C) are as follows:

Lucinda’s Preferences
She prefers (A) to (B), prefers (A) to (C) and prefers (C) to (B)
Marisol’s Preferences
She prefers (A) to (B), prefers (C) to (A) and prefers (C) to (B)
Niyam’s Preferences
He prefers (B) to (A), prefers (C) to (A) and prefers (B) to (C)

How did we derive these preferences? Let’s work through the process for “Marisol”. Marisol has single-peaked preferences with an ideal point located in the policy space like so…

…and we want to represent her preferences over three policies located in the policy space as follows…

By the definition of “Representation of Preferences” above, we know that Marisol’s preferences over policies (A), (B) and (C) will be a list of three comparisons answering the questions

  • Does Marisol prefer (A) to (B), prefer (B) to (A) or find (A) and (B) equally good?
  • Does Marisol prefer (A) to (C), prefer (C) to (A) or find (A) and (C) equally good?
  • Does Marisol prefer (B) to (C), prefer (C) to (B) or find (B) and (C) equally good?

Let’s start with the comparison between (A) and (B). Because Marisol has single-peaked preferences, we know that given any pair of policies, she prefers the policy in the pair that is closer to her ideal point. Examining the locations of policies (A) and (B) and Marisol’s ideal point in the diagrams, we can see that (A) is closer to Marisol’s ideal point than (B), like so:

So we know that Marisol prefers (A) to (B).

What about the comparison between (A) and (C)? Examining the distances between (A) and Marisol’s ideal point and (C) and Marisol’s ideal point…

…we see that (C) is closer to Marisol’s ideal point than is (C). Thus Marisol prefers (C) to (A).

Finally, the comparison between (B) and (C): This is an especially simple comparison to make, because policy (C) is between Marisol’s ideal point and policy (B). Therefore, the distance between Marisol’s ideal point and policy (C) must be a fraction of the distance between Marisol’s ideal point and and policy (B):

Thus (C) is closer to Marisol’s ideal point than (B), so Marisol prefers (C) to (B).

In order to use spatial models represent and explore questions about politics, you’ll need to become fluent in identifying the preferences of each of a set of persons over a set of policies from a diagram showing those persons’ ideal points and the locations of those policies. So the following check-of-understanding gives you several opportunities to practice. It also teaches you an technique called the midpoint rule which makes the analysis of spatial models much quicker, and that we’ll use repeatedly in the rest of this lesson.

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The Severity of Polarization

With the spatial model of policy in hand, we can now model variations in the severity of polarization. We’ll start in this section with a model of policy polarization and then move on to partisan divergence. Modeling ideological consistency will require some tools you won’t learn until the next lesson, so we’ll put it off until then.

Policy Polarization

Recall the definition of policy polarization: A group of persons experiences policy polarization to the extent that there are large proportions of persons in the group who express support for positions towards of the extremes of a given policy issue. Notice that in this definition, the extent of policy polarization in a group hinges on the distribution of preferences in the group as a whole, rather than the preferences of any one group member. So a model of policy polarization needs to represent a group of persons large enough that changes in the distribution of preferences in the group are easy to capture.

To model policy polarization, then, we’ll depict a group of 100 persons who each have single-peaked preferences over a policy space represented by a line. We’ll label each group member with a number, as in “group member 1”, “group member 2” and so on through “group member 100”. We’ll assign these numerical labels in the order that the group members’ ideal points are arrayed in the policy space from left to right, so the group member with the left-most ideal point will be group member 1, the group member with the second-most-left ideal point will be group member 2 and so on up to group member 100, who will have the right-most ideal point in the group. So a typical diagram showing the ideal points of the group members will look like this:

Since this is such a large group, most of the labels pointing to group members’ ideal points overlap with the labels pointing to other group members’ ideal points. That’s ok, because we’ll rarely need to distinguish any one specific group member from all the others in our analysis.

That said, it will be useful to be able to quickly see what proportion of the group members’ ideal points sit in one region of the space or another. So we’ll add an element to our diagrams that shows that. Specifically, we’ll divide the space up into 20 equally-sized intervals, then we’ll add a shaded bar over each interval, with the height of the bar showing the percentage of the group members whose ideal points lie in that interval, like this:

With this way of diagramming ideal points in hand, we can begin modeling variations in the severity of policy polarization within the group. Let’s start by depicting a stark contrast: Two groups that differ sharply in the level of policy polarization they are experiencing:

We’ve constructed this example in a way that emphasizes the fact that policy polarization amounts to large portions of a group expressing support for policies towards each of the two extremes. Notice that in the “Relatively Less Policy Polarization” group above, there is a substantial portion of group members with ideal points towards the rightward extreme of the policy space. But support for policies towards one extreme within a group does not by itself indicate severe policy polarization in the group. Policy polarization instead entails large portions of group members expressing support for positions toward each extreme. Thus the “Relatively More Policy Polarization” group has greater policy polarization than the “Relatively Less Policy Polarization” group because there are concentrations of ideal points in the latter group towards each of the policy space’s two extremes.

More generally, in the one-dimensional spatial model, policy polarization in a group occurs to the extent that:

  1. There are distinct concentrations of member ideal points at each of two modal locations;
  2. Greater proportions of group members ideal points are concentrated close to those two modal locations;
  3. Those two modal locations diverge away from one another and towards the extreme ends of the space.

To get a sense of what this looks like, play with the figure below, which lets you explore a range of different levels of severity of polarization within a spatial model

Pause and complete check of understanding 3 now!

Partisan Divergence

Partisan divergence is a combination of processes through which persons’ party identification becomes more predictive of their policy preferences, and persons feel disdain, fear and animosity towards those who do not share their party identification. The spatial model excels at representing one aspect of partisan divergence: the extent to which persons’ party identifications are predictive of their policy preferences. To capture that aspect of partisan divergence, we’ll extend the spatial model by assuming that each person in the model identifies as a member of one of two political parties, “Party A” and “Party B”. Then, we’ll add three elements to our graphical representation of the model…

First we’ll shade the label representing each person’s ideal point red if they are a member of Party A and blue they are a member of Party B, like so:

Second, we’ll show the distribution of ideal points across the policy space in three diagrams bundled together. The top diagram will show the distribution of the ideal points of the whole group, the middle will show the distribution of the ideal points of only the members of Party A, and the bottom will show the distribution of the ideal points of only the members of Party B.

Third we’ll mark the locations of each party’s mean ideal point. In the spatial model, a party’s mean ideal point is the average location of the party members’ ideal points. It gives a sense of the location of the ‘typical’ ideal point in the party. Thus comparing the locations of the two parties’ mean ideal points to one another gives a sense of where the ideal point of the typical member of each party is relative to the ideal point of the typical member of the other party and how far apart those typical values are.

An essential point that these three-part diagrams can make clear is that policy polarization and partisan divergence are independent phenomena. In other words, it is possible for the level of either to vary independently of the level of the other. In the above example, for instance, policy polarization is low – the ideal points of the group a clustered around a single modal location close to the center of the policy space. However, there is a noticeable level of partisan divergence. Party A members become more frequent relative to Party B members as we move to the right in the ideological space. Thus Party A’s mean ideal point lies to the right of Party B’s mean ideal point.

Conversely, here is an example with high policy polarization and low partisan divergence:

Policy polarization is evident in the bi-modal distribution of ideal points in the “whole group” diagram above. At the same time, there is no clear differentiation between the distribution of ideal points of the two parties. Members of each party are roughly equally represented at each location in the policy space, and thus the mean ideal points of the two parties are relatively close together.

You can play with the following diagram to get a sense of how the distributions of ideal points change as the severity of policy polarization and partisan divergence vary.

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Modeling Polarization in Legislatures

So far, you’ve learned how to use the spatial model to depict policy polarization and partisan divergence in a hypothetical group of persons. But to understand polarization in the United States, we need to explore its severity in particular groups of U.S. political elites at particular times. How polarized, for instance, are members of the U.S. House and Senate as opposed to, say, each of the U.S.’ 99 state legislative houses? (99 because 49 states each have two legislative chambers and 1, the state of Nebraska, has one chamber.) What about judges? How does polarization among U.S. federal judges compare to polarization among the judges who hear cases in each state’s judicial system? What about the media? How severe is polarization among journalists or among the media outlets that cover politics? In the remainder of this lesson, you’ll learn about a technique developed by Keith Poole and Howard Rosenthal (Poole and Rosenthal 1985), called NOMINATE,1 that uses the spatial model to represent the distribution of policy preferences in particular legislatures. You’ll then apply Poole’s and Rosenthal’s model, along with everything you’ve learned in this lesson about the spatial model, to assess policy polarization and partisan divergence in the U.S. House of Representatives.

Midpoints and Vote Matrices

Poole and Rosenthal developed NOMINATE to model preferences in legislatures that pass legislation using binary agendas. A binary agenda reduces any question the legislature must decide to a series of votes in which each legislator must support one of only two alternative pieces of legislation.

For instance, imagine a legislature that must choose the total amount of money the government will be permitted to spend in an upcoming year. After some discussion, three alternative levels of spending emerge that each have support from at least some members of the legislature: $10 billion, $20 billion and $30 billion. In principle, the level of spending could be chosen from among these three possibilities by a single vote in which each legislator indicates support for one of the three, with the alternative that gets the most votes declared the winner. In practice, most legislatures would use a sequence of binary agendas, with each agenda pitting only two of the three alternatives against one another. One such sequence, for instance, would be:

Vote A
$10 billion vs. $20 billion…
Vote B
Winner of Vote A vs. $30 billion…
Vote C
Winner of Vote B vs. loser of Vote A…

…with the ultimate level of spending set to the level that wins Vote C.

With this in mind, imagine a legislature with three members, labeled 1, 2 and 3. Suppose this legislature must choose a point in a policy space and legislators 1, 2 and 3 each have single-peaked preferences with ideal points as follows:

If this legislature uses binary agendas to adopt legislation, then every vote it takes will amount to a choice between two locations in the space. For instance, a vote between one policy “L” and another “R” will amount to choice between two locations like this:

Recall that in COU 2 you learned the shortcut called the midpoint rule for quickly determining persons’ preference orderings over a pair of policies in the spatial model: Given any pair of policies, every person with an ideal point to the left of the midpoint (the point exactly halfway) between those two policies prefers the policy on the left to the policy on the right, and every person with an ideal point to the right of the midpoint between those two policies prefers the policy on the right to the policy on the left. So, in the above example, the midpoint between the policies L and R is here…

…and thus persons 1 and 2 prefer policy L to policy R and person 3 prefers policy R to policy L.

When a legislature uses binary agendas to make decisions, every vote amounts to a choice between two policies. Thus, when we use a spatial model to represent votes in such a legislature, every vote is characterized by a midpoint – specifically, the midpoint between the locations of the two policies at issue in that (binary) vote.

Further, if we assume that every member of the legislatures casts a “sincere” ballot on each vote – i.e. each member casts their ballot on each vote for the alternative located closer to her ideal point in the space – then the midpoint between the two alternatives in each vote along with the locations of the members’ ideal points fully determine each member’s ballot on that vote. For instance, in the above example, when there is a vote between policies L and R and legislators 1, 2 and 3 each cast sincere ballots, legislators 1 and 2 will vote for policy L and legislator 3 will vote for policy R.

The idea of Poole’s and Rosenthal’s NOMINATE model is to use the record of a set of votes from a given legislature to infer (a) the midpoint between the two policies at stake in each vote and (b) the ideal point of every member of the legislature that cast ballots on that set of votes, under the assumption that all of those votes amount to choices between policies located along a line and every member of the legislature has single-peaked preferences on that line.

A vote matrix is the key to understanding how Poole’s and Rosenthal’s method works. For any given vote, a vote matrix records which members vote together on that vote and which members vote against each other. For instance, suppose that members 1, 2 and 3 cast sincere ballots on a vote between two alternatives with a midpoint located as follows:

Because 3’s ideal point is to the right of this midpoint, he will vote for the right-most of the two alternatives. Because 1’s and 2’s ideal points are to the left of the midpoint, they will vote for the left-most of the two alternatives. Therefore…

  • 1 and 2 will vote together
  • 1 and 3 will vote against each other
  • 2 and 3 will vote against each other

We can quickly summarize who votes together with one another and who votes against each other with a vote matrix, like so:

The matrix above has a row and column for each of the three legislators (1, 2 and 3). Each cell of the matrix contains a ‘T’ or an ‘A’ to indicate whether the legislators in that cell’s row and column vote ‘Together’ or ‘Against one another’.

Of course, the data we have to work with to model the distribution of preferences in any real legislature consists of nothing but the ballots cast on each vote by each legislator. “Policy spaces”, “policy positions” and “ideal points” are conceptual constructs we use to model aspects of the processes generating the votes we see in data. Poole’s and Rosenthal’s technique is a method for identifying the locations of legislator’s ideal points in a model that most closely corresponds to the votes we actually observe in a legislature’s records of votes.

So, Poole’s and Rosenthal’s technique starts by collecting a record of votes. For instance, check out the U.S. House’s roll call website. This lists all of the votes recorded in the U.S. House of Representatives during the past 30 or so years. Pick a vote at random and click the ‘View Details’ button. Scroll down to the “All Votes” table. Here’s a screenshot showing just the first few rows of that table from Roll Call 343 of the 117th Congress, 2nd Session:

A Vote from the U.S. House

You can see in any one of these tables that the U.S. House uses a binary agenda for every vote it takes. After all, every member has only one of two values – “yea” or “nay” – in her “Vote” column. Evidently, the House structures votes so that a “yea” is always a vote in favor of one of two alternatives and a “nay” is always a vote in favor of the other.

From any one of these records of “yeas” and “nays” we can construct a vote matrix. For instance, in the vote in the above screenshot, representatives Adams, Aguilar and Allred voted together and against representatives Aderholt and Allen. So the vote matrix describing just those five representatives’ positions on that vote is:

Once you have collected many many records of votes from a given legislature and converted each one into a vote matrix, the next step in Pool’s and Rosenthal’s technique is to use those matrixes to infer the order of the ideal points in the model that best fits the data. We’ll demonstrate this process of inference in the simplest possible example with just three legislators, labeled ‘4’, ‘5’ and ‘6’. Suppose these legislators participated in a binary vote, and the resulting vote matrix is:

We can infer the order of the midpoint between the two policy positions at issue in this vote and the ideal points of legislators 4, 5 and 6 through the following reasoning: We know that any pair of legislators who vote together must have ideal points on the same side of the midpoint. Conversely, any pair of legislators who vote against one another have ideal points on opposite sides of the midpoint. According to the above matrix, legislators 4 and 6 voted together. Thus 4’s and 6’s ideal points are both to the right or both to the left of the midpoint between the two policies at issue in this vote. On the other hand, 5 and 6 voted against one another, so one’s ideal point must be to the left of the midpoint while the other’s ideal point must be to the right. Similarly, 4 and 5 voted against one another, so one’s ideal point is to the left of the midpoint and the other’s ideal point to the right. All this implies that the midpoint between the two policies on the agenda in this one vote and the three legislators’ ideal points are ordered relative to each other in one of the following four ways:

At first, the variety in the above diagram can be overwhelming. It may not be clear what possibilities, if any, have been ruled out by the vote matrix. Look, however, at which legislator’s ideal point is positioned in between the other two ideal points in each possibility. In each, either legislator 4’s or 6’s ideal point is in the middle. Evidently, the vote matrix has ruled out the possibility that 5’s ideal point sits between 4’s and 6’s!

To narrow things down further, we need to observe more votes. For instance, suppose the same legislators, 4, 5 and 6, participate in two distinct votes, each one pitting two positions from the same policy space against one another, so that a single ideal point drive’s each legislator’s ballot on each vote. Suppose these two votes generate the following two vote matrixes:

The first (top) of these two matrices is exactly the same as the one we used in the example above. It has legislators 4 and 5 voting against one another, 4 and 6 voting together and 5 and 6 voting against one another. We already know that it implies that 4’s and 6’s ideal points sit on one side of the midpoint, while 5’s ideal point sits on the other, and thus it rules out any order in which 5’s ideal point sits between 4’s and 6’s. So that first matrix leaves us unsure whether 4’s ideal point or 6’s ideal point sits between the other two.

The second (bottom) of these two matrices clarifies things. In the vote it depicts, 4 and 5 vote against one another, 4 and 6 vote against one another and 5 and 6 vote together. Thus that vote requires that 5’s and 6’s ideal points sit on opposite sides of the midpoint in that vote from 4’s ideal point. And that rules out any order for the three ideal points in which 4’s ideal point sits between 5’s and 6’s. Together, then, these two matrixes allow for only two possible orderings of the three legislators’ ideal points – i.e. the orders labeled as “Possibility 1” and “Possibility 3” in the previous diagram:

Real legislatures have hundreds of members who participate in many hundreds of binary votes every legislative session. So the process of narrowing down the ordering for any set of real legislators’ ideal points is much more complex than what we’ve illustrated here. But the basic logic is no different from what you’ve seen above. The only part of Poole’s and Rosenthal’s technique that we haven’t demonstrated is their method for inferring the relative distances between legislators’ ideal points. This part of the technique relies on the fact that in real legislative voting, there will be no ordering of all the legislators’ ideal points that is perfectly consistent with all the votes that occur. For instance, there will inevitably be a few members who appear to have ideal points at opposite ends of the ordering based on their behavior on almost all votes, but who vote together and against the members who would otherwise seem to lie between them on a handful of votes. Poole’s and Rosenthal’s technique imputes distance between each pair of legislators’ ideal points on the basis of the frequency with which legislators in that pair vote in ways that seems to violate the ordering of their ideal points implied by most votes. They place legislators further apart from one another who vote “out of order” less often.

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Polarization in the House

You’ve come very far since you started out at the beginning of this lesson. You now understand how to depict policies and individual preferences in a spatial model. You’ve learned how to use a spatial model to depict two aspects of polarization – policy polarization and partisan divergence. Finally, you’ve learned part of the process through which Poole’s and Rosenthal’s NOMINATE technique models the policy preferences of members of a legislature on the basis of their voting records. With all of these tools in hand, you can now make progress on the problem we started with in this lesson: Modeling variation in the severity of polarization across different groups of political actors and across time.

Using Poole’s and Rosenthal’s technique, a team of political scientists (Lewis et al. 2022) have used all the votes ever taken in the U.S. Senate and House to construct a spatial model of the policy preferences of the thousands of persons who have served in those chambers from the first Congress of 1789-1790 up to the 117th Congress 2021-2022. The model specifies an ideal point in a two-dimensional policy space for each member Congress. You can explore and download the full model here.

For the purpose of this lesson, however, we will focus on the model’s depiction of preferences of members of the U.S. House of Representatives in just one policy dimension, starting with the 92nd Congress, which ran from 1971-1972. Observers of politics in the United States point to the early 1970s as the starting point of recent increases in the severity of political polarization. So, how and how much has polarization among the members of the House of Representatives changed since that time?

The following interactive chart shows the ideal points ascribed by the NOMINATE model to the members of the U.S. House of Representatives serving in each Congress from the 92nd (1971-1972) through the 117th (2021-2022). Each member’s ideal point in the chart is labeled with the state and district number the member represented while serving in the House, and colored red or blue according to the member’s party. To allow you to differentiate policy polarization from partisan divergence in the model, we use the technique you learned earlier in this lesson – i.e. for each Congress, we display the ideal points of all members serving in that Congress together, followed by the ideal points only the members of the Republican Party, followed by ideal points of only members of the Democratic Party. Use the controls at the top of the figure to see how the model’s representation of the distribution of House members’ ideal points has changed since 1971.

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References

Black, Duncan. 1958. The Theory of Committees and Elections. Cambridge University Press.
Downs, Anthony. 1957. An Economic Theory of Democracy. Harper.
Lewis, Jeffrey B., Keith Poole, Howard Rosenthal, Adam Boche, Aaron Rudkin, and Luke Sonnet. 2022. “Voteview: Congressional Roll-Call Votes Database.”
McCarty, Nolan. 2019. Polarization: What Everyone Needs to Know. Oxford University Press.
Poole, Keith T., and Howard Rosenthal. 1985. “A Spatial Model for Legislative Roll Call Analysis.” American Journal of Political Science 29 (2): 357–84.

Footnotes

  1. NOMINATE is an acronym for NOMINAl Three-step Estimation.↩︎