American Politics as a Game: Roadmap

I'm going to present 2 games coupled together which seem to capture American politics. I don't think any of this is new or innovative, it just weaves together theories and models into a single tapestry described by 2 games. This post will just present a "big picture" of what's going on and what my interests are as far as topics which I'll be writing about in the future.

The two games are the election game and the legislation game, or "Getting to Congress" and "What do I do here in Congress?" I will give conceptual explanations for the games involved, not writing down any formal rules. Since the 2020 election is on everyone's mind, I will give more detail to the election game than to the legislation game.

Election Game

The first game is the Election Game, which involves candidates trying to "persuade" voters to cast their ballot for them. So we have at least two "types" of players in this game: candidates and voters, ostensibly they interact, the candidates can see (after the fact) how each other interact with voters (e.g, hold rallies, give stump speeches), and the voter's cast their ballots on Election Day. Whoever gets the most votes wins the election.

For the sake of simplicity, the only election games I will be considering will involve legislators and the Presidency, held every 2 years (so the Presidency is involved only "every other" time the election game is played). Again we can refine this to distinguish House members running for election from Senators running for election, and further involve the Governorship races as well as state legislator races. But for simplicity, we start small then successively refine the game.

(I confess, most of my thinking has taken inspiration from John Jackson's The American Party System: Continuity and Change over Ten Presidential Elections (2015), though Jackson is sometimes ambiguous in his terminology, and doesn't frame his model as a game as such. I borrow his terminology for "party elites", "voters", and "candidates", though the latter two are just the vernacular usage one would expect. I extend his model to include "activists".)

Refining the Game

We can refine this game arbitrarily much, adding new player types ("party elites" which fund the candidates, recruit them, etc.; "activists" which operate the "Get Out The Vote" [GOTV] efforts, which form the pool of recruits for party candidates; etc.). This involves modeling political ambition, to some degree.

We can also consider further intragame aspects. For example, we can imagine in a state, two factions vying for power among the political elites within the same party. This was what happened, e.g., nationwide in 2010 with the Tea Party. Coalition management becomes an issue if we take factions seriously.

But we can also consider inter-game aspects. Senator Mark Hanna [wikipedia] (R-OH) who was able to control the party machinery for Southern Republican parties, ostensibly he would be both a candidate and a "party elite", though in different states. As Edmund Morris put it in Theodore Rex (pp.38–39)1 For more on this, see Horace and Marion Merrill The Republican Command: 1897–1913 (1971) pp.74–75 for Hanna's politicking in the 1900 national convention which secured his position a kingmaker with Southern delegates, and Richard Sherman's The Republican Party and Black America from McKinley to Hoover, 1896-1933 (1971) pp.19–20. Herbert David Croly's Marcus Alanzo Hanna: His Life and Work (1912) pg.298

The South was Hanna's chief source of political strength. No matter that he himself represented Ohio. No matter either that the Republican Party in Dixie was so weak that in some state legislatures it had no seats at all. What did matter was that the South was disproportionately rich in delegates to national conventions. Hanna's expert cultivation of these delegates, and his control of party funds as Chairman of the Republican National Committee, had guaranteed the two nominations of William McKinley. In his other role, as Senator in charge of White House patronage, he had been a rewarding boss, showering offices and stipends upon the faithful. As long as the South continued to send delegations of these blacks north every four years, Mark Hanna would remain a party kingmaker.

We could also consider the situation where we want to model political ambition: several members of the House want "bigger positions". The governorship and a senate seat both have opened up. Each of these legislators have to weigh their own ambitions against the likelihood one of their opponents would win the seat.

Voters

The voters appear to be irrational actors. Or, at least, that's what Campbell, Converse, Miller, and Stokes have found in their book The American Voter (1960). This doesn't mean we cannot model them. It just means how they determine their vote is not by a utility function.

We can take the converse perspective, and try to model voters as rational, but this opens up a huge can of worms (as far as modeling is concerned). How do we determine their utility function?

If James Carville is right, and voters determine who to vote for based on "It's the economy, stupid", then we need to model the economy. I posit economists are incapable of this (see, e.g., Hill and Myatt's The Economics Anti-Textbook or Keen's Debunking Economics for details) and more importantly this would be too distracting from the bigger issue modeling how voters choose who to vote for.

We could try to model the utility functions as exogenous quantities (i.e., not explained by the model, but just supplied by empirical observation or statistical modeling). But this feels underwhelming, and not better than just using statistics ab initio when modeling voter behavior.

My personal belief is that, it is plausible legislators are rational actors (in the game theoretic sense) because they are foist into an unnatural situation (being forced to run for re-election every so often). But voters are not constrained in such a manner. There is no compelling reason to believe voters would behave any differently voting than doing anything else, in which case voters are swayed by the cognitive biases we all experience.

Party Elites

This is a very sinister name for a lackluster type of player in the games. Once upon a time, we could imagine these players as the cigar chomping bosses picking candidates in smoke-filled rooms. But since the McGovern-Fraser reforms of 1972, the cigar chomping bosses have lost their power gradually over the past half century or so.

The "party elites" refers to the boring bureaucrat who has to decide "how to divide the dollar" among candidates they want to endorse, and who to encourage to run for office. Anyone can become a "party elite" in this brave new post-McGovern-Fraser world.

The goal for party elites is to recruit and back candidates who will implement policies the elites desire. This is simple enough, until we start modeling ideology (think: Tea Party versus Establishment Republicans; Progressives versus Establishment Democrats). Then Party Elites contend over the party machinery, in some appropriate sense, responsible for dispensing funds to candidates and recruitment.

In some sense, there is indirect communication between voters and party elites mediated through elections. This would impact which faction among party elites has "power", i.e., greater say in how to allot funds and who to endorse or recruit.

Legislation Game

Once elected, legislators need to play the Legislation Game of introducing bills, trying either to block or to pass them, all before the next election. We can refine this game in quite a few ways, but first perhaps we should clarify terminology.

At the federal level, Congress works in Sessions or 2-year intervals to introduce bills, work on them, and pass them. That's the name of the game: passing (or blocking) legislation. We can view this as a "repeated spatial voting game" coupled to a few other games (Chicken, Divide the Dollar, etc.). Our interests is specifically modeling contemporary legislation, not producing some dynamical system which explains how we got from 1789 to here.2 A historic note: we take for granted bills are identified by one of a half-dozen standard types [e.g., HR, S, SRes, etc.] and a number and the congress number. This didn't start until the 14th Congress, according to the data provided by the Library of Congress. Before then, it is difficult to determine the bill numbers, and seemingly post hoc to assign any identification to those early bills. Eugene Nabors's Legislative Reference Checklist: The Key to Legislative Histories from 1789-1903 is a blessing to researchers, even today, since that patient scholar went through the early bills and assigned numbers to them, and identified bills with the resulting statutes.

Even in its simplest form, the origins of bills is rather elusive. Just like voter preferences, we could model it as exogenous and not worry about "where bills come from": it comes from us, by hand! Or we could model it endogenously, there is some mechanism within the model responsible for legislators creating a bill. But without modeling bill drafting at all, well, why on Earth would legislators meet?

Assuming, somehow, legislators draft and introduce bills, we are confronted with the degree of realism we want to approximate. Bills are assigned to committees. The committees may or may not even schedule hearings for the bills, depending on the attitude of the committee chair. Assuming the committee holds hearings and eventually approves it, the committee (usually) files a report detailing their findings, and the bill is either referred to more committees or the chamber's presiding officer (like the committee chair) may or may not schedule time for debate. There are mechanisms to force a bill to a vote, but again that's rather complicated.

We can refine this legislation game, extending its core concept to incorporate strategic voting (voting against one's interest to feign interest in something else), amendments, include a new type of player ("lobbyists") which could make the legislator's dynamics with party elites more intriguing. I need to research this area more before committing myself to anything, I'm not even sure there are adequate game theoretic models of lobbyists.

Further, presumably legislator behavior changes relative to when they are up for election next. A senator can play the legislation game thrice before playing the election game, whereas all members of the House must alternate between the election game and the legislation game. Does this impact behavior for House members compared to Senators? Do their utility functions change if they change chambers?

Concluding Remarks

I've only outlined the two "subgames" in American politics relevant to elections, but have not described how they are coupled together. Presumably voters care about what their representatives do, which guides the utility functions for the legislation game. Presumably party elites care if their elected candidates are faithfully implementing the policies promised. The interactions between legislating and elections need to be further explored (or explored at all).

We also have not discussed the other branches of government. Presumably we could model the President as a 1-person chamber (that's what veto power allows the President to do, after all) which can draft legislation for the other chambers (it's what the White House Office of Legislative Affairs does and has done since Eisenhower created it). Presumably budget considerations could be modeled, since the Budget and Accounting Act of 1921 specified the Executive branch needs to propose the budget.

I'm hesitant about modeling the Judicial branch, however. In practice, two lawyers try to persuade a judge. That's what a Court Case is. But the means by which persuasion is accomplished is decidedly not a "game" (it cannot be accurately modeled using game theory). Further, the Judicial branch interprets laws which the Legislature has passed and enacted, which is hard to model. We could handle a case-by-case (sorry for the pun) modeling philosophy, but there is no elegant "one size fits all" model as for the legislature above.

I also want to warn against trying to transform the model presented here into a "unified theory of Congress", since there's still quite a bit exogenous to the model. Laws are proposed to respond to prevailing problems and conditions, which are not modeled within this "coupled game". Although this model proposed may "tie together" various disparate games strewn throughout the literature, providing a more cohesive and appealing model, it is not the "unified theory" you are probably hoping for (beyond explaining legislator behaviour given exogenously observed bills and perturbations).

But we have, I think, successfully integrated a number of theories and models into one coherent model. We have woven together political ambition, spatial voting, election campaign behavior, and power dynamics at various levels. Ostensibly this could be extended to include state legislatures, governorships, as well as the Presidency. But we only have a hand wavy description of the games, we don't actually have a proof that "When restricted to x, we recover the political ambition game" (or any similar such proposition). This would be interesting to pursue, perhaps.

What I am interested in, however, is whether we could provide conditions describing "party systems", i.e., periodic shifts and realignments in the ideology of the parties. If so, how long does a party system last? Under what conditions will a party realignment happen? Can they be avoided? How long does a realignment take? Can this be empirically tested?

References

• John S. Jackson, The American Party System: Continuity and Change over Ten Presidential Elections. Brookings Institute Press, 2015.

Voters

• Angus Campbell, Philip Converse, Warren Miller, and Donald Stokes, The American Voter. Unabridged edition. University of Chicago Press, 1980.
• Warren E. Miller and J. Merrill Shanks, The New American Voter. Harvard University Press, 1996.
• V.O. Key, The Responsible Electorate. Belknap Press of Harvard University Press, 1966.
• Peter F. Nardulli, Popular Efficacy in the Democratic Era: A Reexamination of Electoral Accountability in the United States, 1828-2000. Princeton University Press, 2005.

Estimating Legislator Ideal Points

We briefly introduced the idea of issue spaces as a formalization of the political spectrum. Now we want to figure out where legislators are on that political spectrum.

Game theory models political actors as an Ideal Point in an issue space equipped with a utility function on that issue space. But how do we estimate (unobservable) ideal points? One strategy is to try to use votes, which is the basis for the (i) Item-Response and (ii) NOMINATE families of algorithms.

Basic Idea

The policy space consists of s dimensions. Legislator i has his/her utility function for voting yes (y subscript) on measure j be a function of the "distance" from the legislator's ideal point to the proposed legislation's point. We use a slightly generalized version of the Pythagoren theorem, where the "distance" first dilates the coordinates by the subjective weights the legislator places w_k for each dimension k in the policy space:

$$l_{ijy}^2=\underset{k=1}{\overset{s}{\sum <!--\; \sum \; -->}}{w}_k^2{d}_{ijyk}^2$$

A legislator's utility function is then some "suitably nice" function of these distances, u(l). Well, this isn't quite the end of the story, because we're dealing with statistical regression, we just described the "deterministic part" of the utility function. We also have the "random noise" ε

$$U(l_{ijy})=u(l_{ijy})+\mathrm{\epsilon <!--\; \epsilon \; -->}$$

We fix u to be either a Gaussian function or a quadratic polynomial. The only condition is that "it looks like a frown" (it has a global maximum at the legislator's ideal point, and is strictly decreasing).

How to progress? Well, we can represent the probability of voting "yea" in terms of the utility function (and how this is done varies model-by-model), then estimate the parameters (the w_k for each legislator, and each legislator's ideal point, and each motion's location in the policy space) using something like maximizing the likelihood or expectation maximization.

NOMINATE models

The utility functions are Gaussian functions. If there are s dimensions to the policy space, legislator i has his/her utility function for voting yes (y subscript) on measure j, where w_kd_k measures the "cost" for deviating from the legislator's ideal point in the k^th dimension of the issue space,

$$u_{ijy}=\mathrm{\beta <!--\; \beta \; -->}\exp <!--\; \; -->[\underset{k=1}{\overset{s}{\sum <!--\; \sum \; -->}}{w}_k^2{d}_{ijyk}^2/2]$$

Observe the exponent is just the l "cost" for the legislator to support the measure. Well, this isn't quite the end of the story, because we're dealing with statistical regression, we just described the "deterministic part" of the utility function. We also have the "random noise" ε, giving the utility function U as

$$U_{ijy}=u_{ijy}+{\mathrm{\epsilon <!--\; \epsilon \; -->}}_{ijy}$$

Note: we can similarly define the utility for voting "nay" by considering instead the location of the status quo in policy space, computing the distance to that point for the legislator. This is precisely u_ijn the utility function for voting "nay". The stochastic term ε of the utility function is assumed to follow an "extreme value distribution", which lets us write the probability legislator i votes for outcome y on roll call j as:

$$Pr(\text{Yea})=P_{ijy}=\frac{\exp <!--\; \; -->(u_{ijy})}{\exp <!--\; \; -->(u_{ijy})+\exp <!--\; \; -->(u_{ijn})}$$

The exact details of this variant of the NOMINATE algorithm may be found in "Scaling Roll Call Votes with wnominate in R", and it works for a single session of congress.

The models describe estimating the ideal points for a finite set of legislators within the same session of Congress. But how do we handle "progress"? I.e., how ideal points evolve over time (across multiple sessions of Congress)? The legislator's ideal point is then a polynomial in t (sessions since joining), supposing the legislator has served T terms (thus far in his life):

$$x_{it}=x_{i0}+x_{i1}{P}_1\left(\frac{t-<!--\; -\; -->1}{T-<!--\; -\; -->1}\right)+\cdots <!--\; \cdots \; -->+x_{i\mathrm{\nu <!--\; \nu \; -->}}{P}_{\mathrm{\nu <!--\; \nu \; -->}}\left(\frac{t-<!--\; -\; -->1}{T-<!--\; -\; -->1}\right)$$

Where P_k is a Legendre polynomial, and the x_it are more parameters to be determined. Why use a Legendre polynomial? It's unclear to me, presumably for its completeness relation (any function on the domain 0 < x < 1 can be adequately approximated by "enough" Legendre polynomials). This is the DW-NOMINATE variant.

Problems

Although ubiquitous in the literature, there are some problems with the NOMINATE scores.

First, the dimensions are not as clear to interpretation as its proponents claim. The first dimension is always interpreted as the "partisanship" dimension, but there's no clear way to glean that other than guessing.

Second, it poorly describes how someone's views evolve over time. This is important if we wanted to discuss, e.g., "party realignments" (Are the Republicans from the 1990s "the same as" the Republicans in 2019?).

Third, NOMINATE requires a lot of data before it can produce decent results. This has probably improved since the original algorithm, there are so many now it's hard to keep track.

Fourth, it's not Bayesian. This is unfortunate from a performance perspective. If I have just computed the NOMINATE scores for legislators based on the first session of congress, then 6 months into the next session I want to update those scores...I have to recompute everything from scratch. This isn't as terrible as the previous problems, but it is irritating.

Item-Response Models

The basic idea is to take advantage of votes as if they were responses to a survey, then use the already developed Item-Response Theory. The basic idea, as applied to ideal points of legislators, is to consider roughly a probit model for the probabilities that a legislator will vote "yea":

$$Pr(y_{ij}=1)=\mathrm{\Phi <!--\; \Phi \; -->}({\stackrel{\to <!--\; \to \; -->}{\mathrm{\beta <!--\; \beta \; -->}}}_j\cdot <!--\; \cdot \; -->{\stackrel{\to <!--\; \to \; -->}{x}}_i-<!--\; -\; -->{\mathrm{\alpha <!--\; \alpha \; -->}}_j)$$

where Φ is the CDF for the Normal distribution.

This can be reinterpreted as an Item-Response model used (apparently) in educational testing, where β_j is the "item discrimination parameter" and α_j is the item difficulty parameter. Clinton, Jackman, and Rivers' "The Statistical Analysis of Roll Call Data" (2004) was the first to approach ideal point identification using Item-Response theory, at least so far as I can tell from the literature.

This led to a multitude of variants: emIRT improved performance, for example; while Martin and Quinn's work on Supreme Court justices ideal points produced innovative algorithms which are Bayesian and dynamical (take a "random walk" in the issue space, as it were).

This turns out to be superior for analyzing the dynamics of ideal points. Specifically for the questions of party realignment, Caughey and Schickler (2014) caution us to use a dynamic IRT model. Although computationally intensive, progress has been made (easily bundled, e.g., with the idealstan R package).

Problems with Item-Response Models

First, Item-Response models are scale-invariant — we can rescale the coordinates for the policy space however much we want. So the numeric values themselves may not matter for ideal points insomuch as their relationship to each other.

Second, for policy spaces which are not 1-dimensional, item-response values are rotation invariant. For 1-dimensional policy spaces, item-response doesn't know whether to order values from most liberal to most conservative or vice-versa.

But both these problems can be solved using semi-informative priors in the Bayesian approaches.

The third problem, perhaps more grave, is we are restricted to certain dimensions due to computational constraints. The NOMINATE algorithm could handle 8 dimensions, no problem; but item response algorithms struggle with determining ideal points in more than 2 dimensions within a sensible period of time.

Conclusion

If you are interested in an overview — a "big picture" of congress — without concern about nuance, the NOMINATE scores may be good enough.

Although it produces a decent approximate ideal point for legislators, it fails to adequately capture how a legislator evolves over multiple sessions. This makes it less than ideal for making any claims about party realignments.

Further, it fails to capture issue-specific nuances for each legislator. Presumably higher dimensionality fixes the problem, but giving, say, 16 numbers worsens the intuitive picture for a single legislator. It is unclear if the Item-Response families suffer the same problem. (See arXiv:1209.6004 for details.)

References

• Nolan McCarty, Measuring Legislative Preferences. This review fleshes out more sordid details underpinning the general notion of "ideal points" than I have written about.

NOMINATE algorithms

• Keith Poole and Howard Rosenthal's Congress: A Political-Economic History of Roll Call Voting. Oxford University Press, 1997.
• Devin Caughey and Eric Schickler, "Substance and Change in Congressional Ideology:NOMINATE and Its Alternatives".
• Keith Poole, Jeffrey Lewis, James Lo, Royce Carroll, "Scaling Roll Call Votes with wnominate in R". Journal of Statistical Software 42, 14 (2011).
• Royce Carroll, Jeffrey B. Lewis, James Lo, Keith T. Poole, "Measuring Bias and Uncertainty in DW-NOMINATE Ideal Point Estimates via the Parametric Bootstrap". Eprint.

Item-Response Algorithms

• Joshua Clinton, Simon Jackman, Douglas Rivers, "The Statistical Analysis of Roll Call Data". American Political Science Review 98, 2 (2004) pp. 355-370. Journal page.
• idealstan: an R Package for Ideal Point Modeling with Stan
• Andrew D. Martin and Kevin M. Quinn, "Dynamic Ideal Point Estimation via Markov Chain Monte Carlo for the U.S. Supreme Court, 1953-1999". Political Analysis 10 (2002) 134-153. (See esp. section 3 for the basic model.)
• Imai, Kosuke, James Lo, and Jonathan Olmsted, "Fast Estimation of Ideal Points with Massive Data." American Political Science Review 110, No. 4 (2016), pp. 631-656. Eprint
• Sean M. Gerrish, David M. Blei, "The Issue-Adjusted Ideal Point Model". arXiv:1209.6004

Issue Space: A Primer on Spatial Voting

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 $w_current. 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 $w_current 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:

1. the institutional factors ["rules to the voting game"],
2. if voters interact with each other and how it'd affect their behavior, and
3. 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.