The first step towards applying rational behavior to Congressional politics is to consider a body of voters deliberating on a proposed bill. The bill is up for a passage vote (i.e., a vote considering whether to enact it or not), so a given voter has two choices: yea [enact] or nay [do not enact].
We model each voter as independent rational agents who possibly interact. But the real question I'd like to address in this post is: How do we model the bill, the question?
Example 1. Consider a ballot initiative for giving a raise to school teachers. The initiative will pay school teachers $x per year. Ostensibly x could be any real number.1 Strictly speaking, it would be a subset of the real numbers, since we'd have to truncate real numbers to 2 digits after the decimal point. Each voter has a belief about what the pay should be, and this could be determined subjectively. Some may believe school teachers should be volunteers or charity funded, and thus would prefer x to be 0. Others may believe teachers deserve a living wage and thus prefer x to be closer to, say, $45000. This "preferred wage" each voter has, we call the voter's Ideal Point.
The choice the voter faces is between $x and whatever the current wage $wcurrent. We need to give each voter a utility function U mapping any given proposed wage to that voter's "utility". More precisely, it measures "how far off" a proposed wage is from that voter's "ideal wage". The exact interpretation and mathematical properties of the utility function is the topic for a future post, today we're interested only in the issues.
The one-dimensional real line containing the proposed wages $x versus $wcurrent is the domain of the utility functions of the voters. This "space of possible school teacher wages" is the Issue Space of the proposed measure. (End of Example 1)
Dimensional Reduction. We could divide up any piece of legislation into policies. Our previous example could have simultaneously included a change in taxes to fund the increase in school teacher wage, and we'd have 2 ostensible dimensions to consider: the tax rate, and the school teacher wage.
For a real piece of legislation, such a naive translation of a bill into policies may result in a combinatorial explosion of dimensions in the issue space.
What (apparently) happens is, we bundle policy dimensions into (hopefully coherent) world views which we classify as the Political Spectrum. In some sense, we implicitly perform a kind of Principal Component Analysis to reduce the proposed policies implemented in a given bill down into a lower-dimensional "Policy Space". This is done informally, and we do it all the time when we say, "Oh, this bill is a liberal bill", we just boiled down all the policies into one-dimension (the left/right spectrum).
There is no exotic geometry to the policy space, it's usually N-dimensional real space for N around 2.
Definition 1. A bill's Issue Space is the space of all possible implementations of the proposed policies contained in the legislation's text.
The Policy Space is a "coarse-grained" N-dimensional real space, in the sense that any legislation or proposed policy can be located as a point in that N-dimensional space.
Warning: This distinction between "policy space" and "issue space" is one I am making at present. In the literature, the terms are used interchangeably to refer to the "coarse-grained" lower-dimensional space. Following suite, I will have to respect tradition, and in future posts use the terms interchangeably unless otherwise explicitly stated.
Model Refinement. If we take this seriously, then we just need to model actors (rational voters) using (i) their ideal point and (ii) their utility function (preferences). Well, we also need to model:
- the institutional factors ["rules to the voting game"],
- if voters interact with each other and how it'd affect their behavior, and
- how voters get and process information.
Empirical Concerns. We also need to determine how many dimensions there are to the policy space. We could, ostensibly, have a large number dimensions (say, N = 26 dimensions or something), but that's just a wild guess. As far as I am aware, there is no rigorous way to measure the dimensionality of the policy space.
I also wonder about the geometry of the issue space (is there curvature? What about symmetries?) as well as its topology (is it connected? Compact? Does it have nontrivial homotopy groups or homological ring?). This wouldn't really impact much, except the geometry may have surprising results in voter behavior.
Further, we have to come up with some model of voter utility functions. There are two popular choices, namely a Gaussian and a quadratic polynomial, both functions of "distances" between the voter's ideal point and the proposed legislation location in policy space. The "distance" is measured using a voter-dependent metric (how "painful" it is to stretch that distance away from the voter's ideal). I'll discuss this more in a future post on ideal points.
References
I don't really have any, since this is glossed over in the literature to get to voter preferences in spatial voting models.
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