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.

Common Knowledge of Rationality + Consistent Alignment of Beliefs = Common Priors

Heap and Varoufakis summarize the last assumption of game theory's axiomatization of rational behavior in the "consistent alignment of beliefs" axiom: no instrumentally rational person can expect another likewise rational person who has the same information to develop different thought processes.

This is usually justified by the Harsanyi doctrine: when two rational people examine the same information, they must draw the same inferences, and independently come to the same conclusion.

Robert Aumann fiercely defended this principle in his article "Agree to Disagree" (1976) and his earlier article "Subjectivity and Correlation in Randomized Strategies" (1974).

Aumann argues, if you assess it is going to rain tomorrow with 75% probability and I assess it will rain tomorrow with 33% probability, then we must have different information and we should update our probabilities accordingly until we converge on some shared probability estimate. That is, through dialogue, we (as rational actors) will arrive at a conclusion we both agree upon.

When we combine "consistent alignment of beliefs" with the common knowledge of rationality, we end up with common priors (i.e., a source of beliefs). The connection is this: if you know you are rational and you know your adversary is rational and (using consistent alignment of beliefs) you know your thoughts about what your adversary might be doing have the same origin as your thoughts about your own actions along the same line as your adversary's thoughts, THEN you adversary's actions will never surprise you. Beliefs are consistently aligned in the sense, if you actually were able to know your adversary's plans, you wouldn't want to alter your beliefs about those plans. Conversely, if your adversary knew about your planned actions, then your adversary wouldn't want to alter their beliefs they hold about your prospective actions which underpin their planning about their future actions.

Observe this dialogue needs to happen in "real" (i.e., historical) time and not in "logical time" (in the sense of the length of a logical derivation of hypothetical dialogue). Without such actual dialogue, there's no need to come to any agreement. Scott Aaronson has shown (arXiv:cs/0406061) such dialogue can be done in finite time and, in some sense, "efficiently".

One of the problems with this, the inference of common priors from the premises on the Common Knowledge of Rationality coupled to the consistent alignment of beliefs argues the dialogue occurs in "logical time".

The problem with this is for "one shot games", where interactions between the players occur only once and in the absence of communication, there is literally no opportunity for such dialogue.

Prior Beliefs

We need some "initial beliefs" for our rational actors to have, so as to avoid an infinite regress in reciprocal expectation of actions pursued. We saw how rational actors update their beliefs via Bayesian updates, but we need some "initial prior" to start the process. Without common priors, we can get senseless results.

But the choice of prior probability distributions in Bayesian analysis can impact the posterior distribution considerably. The field of "Reference Priors" uses information theory to measure how the choice of prior distribution affects the posterior probability. The choice of priors has a rich history and while it is true "objective" (or "noninformative") priors have "minimal impact" on the posterior, but that is not the same as "zero impact". Noninformative priors can lead to improper posterior, which is dangerous. How we choose a prior seems to be a hotly contested topic (does the choice of priors "matter"? What is an appropriate way to do it?) which Andrew Gelman has written extensively on.

Even if we restrict ourselves to only "stable" priors, I'm not sure this is much progress.

Revenge of the Nerds German Philosophers

One thing which the German philosophers Kant and Hegel pondered was the self-conscious reflection of human reason upon itself. Can our reasoning faculty turn on itself and, if it can, what can it infer? Phrased more relevantly, when reason knowingly encounters itself in a game, does this tell us anything about what reason should expect of itself?

Hegel's Phenomonology of Spirit (or more generally, his philosophy of Spirit) addresses this train of thought (and more). Further Hegel takes Reason reflecting on reason as it reflects on itself as part of the restlessness which drives history. Outside of history, for Hegel, there are no answers for the question of what one's reason demands of others' reason. History provides a changing set of answers.

Also worth mentioning is that game theory uses "reason" akin to Hume's usage in his famous passage We speak not strictly and philosophically when we talk of the combat of passion and reason. Reason is, and ought only to be the slave of passions, and can never pretend to any other office than to serve and obey them. Reason is a tool to help achieve the ends of subjective passions. Hegel rejoins in his lectures on the History of Philosophy, in chapter 2 on Hume in particular, In itself reason thus has no criterion whereby the antagonism between individual desires, and between itself and the desires, may be settled. Thus everything appears in the form of an irrational existence devoid of thought; the implicitly true and right is not in thought, but in the form of an instinct, a desire.

Kant's Critique of Pure Reason via his Transcendental Dialectic investigates Reason's excesses. For other Kantian repudiations of game theoretic "reason", see O'Neil's Constructions of Reason (1989), e.g., page 27 et seq.

Conclusion

So we finally have answered the question posed so long ago: beliefs are formed by taking into account common knowledge of rationality coupled to consistent alignments of beliefs. This bootstraps a rational actor's belief system by considering that actor's rational adversary's beliefs which have already solved the riddle of what is the original actor's belief system.

And if that sounds circular...that's because it is...

References

• Shaun Hargreaves Heap and Yanis Varoufakis, Game Theory: A Critical Introduction. Second ed., Routledge. (This is the axiomatization scheme I am following.)
• John Searle, Rationality in Action. MIT Press, 2001. (This provides a different set of axioms for rational behaviour, equivalent to the axioms of game theory, and discusses implicit assumptions & its flaws.)
• S. Morris, "The Common Prior Assumption in Economic Theory". Economics and Philosophy 11 (1995) 227–253. Eprint.
• John Harsanyi, "Games with Incomplete Information Played by 'Bayesian' Players: Part 1, The Basic Model". Management Science 14, 3 (1967) 159–182. Eprint
• Robert J. Aumann, "Agreeing to Disagree" (PDF). The Annals of Statistics 4, 6 (1976) 1236–1239. doi:10.1214/aos/1176343654.
• Scott Aaronson, Common Knowledge and Aumann’s Agreement Theorem [blogpost]
• Scott Aaronson, "The Complexity of Agreement". Proceedings of ACM STOC (2005) pp. 634–643, eprint arXiv:cs/0406061

Common Knowledge of Rationality

Game theorists sought a solution to initial belief formation by treating beliefs as purely subjective assessments of a situation or matter, which could degenerate into permitting almost any belief to (and thereby action from) instrumentally rationally actors. The resolution to this was another axiom: the common knowledge of rationality.

As a rational actor, it would be prudent to stipulate your adversaries are rational actors themselves. Consequently, they would also stipulate you are rational, too. This Common Knowledge of rationality turns out to be a tacit axiom of game theory.

As an example of this, in the episode "Peak Performance" in "Start Trek: The Next Generation", the android Commander Data analyses Commander Riker's strategic abilities in a memorable scene:

DATA: I have several examples of Commander Riker's battle technique. At the Academy, he calculated a sensory blind spot on a Tholian vessel and hid within it during a battle simulation. And as a lieutenant aboard the Potemkin, his solution to a crisis was to shut down all power, and hang over a planet's magnetic poles, thus confusing his opponent's sensors.

TROI: And from these specifics, what general conclusion can you extrapolate?

DATA: Only twenty-one percent of the time does he rely upon traditional tactics. So, the Captain must be prepared for unusual cunning. Counsellor, Commander Riker will assume we have made this analysis, and knowing that we know his methods, he will alter them. But, knowing that we know that he knows that we know, he might choose to return to his usual pattern.

Data would continue in this manner ad infinitum had he not been interrupted, and it is precisely what the common knowledge of rationality states. We could formally generalize this thus:

1. each person is instrumentally rational
2. each person knows (1)
3. each person knows (2)
4. each person knows (3)
5. ...and so on ad infinitum.

How does this help? By itself, it has a fundamental problem which Heap and Varoufakis illustrate in the following example.

Suppose you have a desire to be "fashionable" when deciding what clothes to wear. But this requires taking into account that other people want to be "fashionable" too. So you need to take into account what clothes they will wear, when deciding what clothes you will wear (in order to realize your desire to be fashionable).

However, other people want to be "fashionable" too, and they will select what to wear based on the expectations of what other people (including you) will wear.

So you need to account that what clothes they will wear depends on what they think you will wear, which affects what clothes you are planning to wear. But other fashionistas, knowing this, will adjust what they wear accordingly. Knowing that you know that they know you know, you now can adjust accordingly.

And so on. This process doesn't really stop, unless we add another assumption. The belief formation doesn't "settle down" without adding an assumption about common priors, which is what Bernheim's "Rationalizable Strategic Behavior" (1984) and Pearce's "Rationalizable Strategic Behavior and the Problem of Perfection" (1984) do.

[History: D. Lewis introduced the concept of "common knowledge" when analyzing a philosophical problem in the book Conventions (1969), but Robert Aumann imported the concept to economics in his 1976 paper "Agreeing to disagree" (Annals of Statistics 4 (1976) pp. 1236–1239).]

References

• Shaun Hargreaves Heap and Yanis Varoufakis, Game Theory: A Critical Introduction. Second ed., Routledge. (This is the axiomatization scheme I am following.)
• John Searle, Rationality in Action. MIT Press, 2001. (This provides a different set of axioms for rational behaviour, equivalent to the axioms of game theory, and discusses implicit assumptions & its flaws.)
• John Geanakoplos, "Common Knowledge". Journal of Economic Perspectives 6, 4 (1992) pp. 53–82
• John Geanakoplos, "Common Knowledge". Chapter 40 in Handbook of Game Theory with Economic Applications vol 2 (eds. R.J. Aumann and S. Hart), North Holland 1994, pp. 1437–1496. SemanticScholar.
• Pierre Lescanne, "Mechanizing Common Knowledge Logic using COQ". Annals of Mathematics and Artificial Intelligence 48 1-2 (2006) pp 15–43, eprint.
• Pierre Lescanne, "Common knowledge logic in a higher order proof assistant?" arXiv:0712.3147

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.

Agents are Instrumentally Rational

Game Plan: We'll introduce the notion of "instrumental rationality" as an ordering of alternatives with some technical condition. Then we'll discuss measures of "preference" via utility functions. Then we conclude by discussing maximizing utility under uncertainty.

Loosely put, individuals who are instrumentally rational have preferences over various "things" (e.g., baby-back ribs are preferred to chicken, and chicken is preferred to bread). Such individuals are deemed "rational" for picking actions which satisfy those preferences. The only constraint is that preferences are ordered in some suitably "weakly coherent" way (e.g., ribs are still preferred to bread if there is no chicken).

The convention is to call these "things" preferred as "Alternatives".

Definition 1. Let an actor be choosing between countably many possible different alternatives x₁, x₂, x₃, …. An actor is called Instrumentally Rational if the actor has preferences satisfying the following conditions:

1. Reflexivity: No alternative x_i is less preferred than itself.
2. Completeness: For any two alternatives x_i and x_j, either (1) x_i is strictly preferred over x_j, (2) x_j is strictly preferred over x_i, or (3) the actor is indifferent between the two alternatives.
3. Transitivity: For any alternatives x_i, x_j, x_k, if x_i is no less desired than x_j, and if x_j is no less desired than x_k, then x_i is no less desired than x_k.
4. Continuity: For any alternatives x_i, x_j, x_k, if x_i is (strictly) preferred to x_j, and if x_j is (strictly) preferred to x_k, then there exists some "composite" of x_i and x_k (call it y) which is equally as desired as x_j.

Remark 1 (On Continuity). There are two ways to interpret the continuity axiom. The first perspective is to think of y as a "basket" containing "bits" of x_i and "bits" of x_k. For example, if x_i is "18 ribs", x_j is "half a roasted chicken", and x_k is "10 rolls", then there is some composite ("9 ribs and 5 rolls") which is equally as desirable as half a chicken.

The other perspective is to think of y as a lottery, where the actor obtains x_i with probability p (0 < p < 1) and x_k with probability 1 − p. The continuity axiom then says there is some p for which the actor is indifferent between the lottery y and the alternative x_j.

Remark 2 (Ordering, Utility Functions). The first three axioms taken together implies the actor has a well-defined preference ordering (in the mathematical sense). When the continuity axiom is added, the preference ordering may be "represented" by a utility function (i.e., a function assigning to each alternative x_i some real number U(x_i) reflecting the "utility" or "desire" for that alternative). An actor making choices to satisfy his or her preference ordering can be viewed "as if" maximizing his or her utility function.

Now, discussions of "utility" of an alternative should not be confused with the philosophy of Utilitarianism. A utility function just assigns some numbers such that the ordering induced by it is the same as the actor's preference relation. That is to say, U(x_i) > U(x_j) if and only if x_i is strictly preferred to x_j. The numbers represented by U(x_i) are measured in utils, which is Agent-dependent and measures that Agent's preference for the given alternative.

Ordinal Utilities, Cardinal Utilities, Maximizing Expected Utility

Definition 2. If we assign utility "arbitrarily" but in a manner consistent with the preference ordering (e.g., for any alternatives X and Y such that X is preferred to Y, we assign the utilities such that U(X) > U(Y) but otherwise the quantities remain arbitrary), then we call such utility the Ordinal Utility.

Here we must stress again there are two important points of assigning utility in a manner which captures only the preference ordering (and nothing else).

First, ordinal utility does not describe the agent's "intensity of desire" for an alternative. The "strength of preference" is not captured by this notion. So how much more I want ribs than chicken is not adequately described by this notion, just the fact that I really want ribs right now (and not chicken, much less bread).

Second, ordinal utility cannot be compared "across agents". The ordinal utility I assign to a full rack of baby-back ribs cannot be compared to anyone else's ordinal utility for, say, Lasagna. We can only compare my ordinal utility for baby-back ribs against my ordinal utility for Lasagna.

Dealing with Uncertainty

My local BBQ joint smokes 1 pig per day, and when it's all sold, there's no more. If I am hungry, should I go before the lunch rush or afterwards?

Here we must talk of Prospects, outcomes and their associated probabilities.

For our particular situation, there is a decision I must make (go before the lunch rush or after) and two outcomes (there is food left, or they ran out of food). One prospect is given by the possible outcomes to a given choice of going before the lunch rush (go before lunch AND they have food, go before lunch AND no more food; p, 1 − p). The other prospect is given by the decision to go after the lunch rush (go after lunch AND they have food, go after lunch AND no more food; q, 1 − q).

Observe, each decision has different possible outcomes, but the probabilities for the outcomes on a given decision must sum to 100%: something must happen when I take a decision.

We now need to consider preference ordering over prospects.

Definition 3. Suppose a person must choose between actions with uncertain outcomes, in the sense that: each action has various possible outcomes associated with it, each with some probability. We call this action a Prospect and represent it by a pairing of the possible outcomes with their respective probabilities (y₁, y₂, ...; p₁, p₂, ...) where the outcome y_i occurs with probability p_i, and the probabilities sum to 1 = p₁ + p₂ + ... (since there must be an outcome to the action).

Remark. There is a "nested structure" to prospects, in the sense that y_i might be an "atomic outcome" (e.g., "there will be food", "there will be no food", "it will rain", "the world will end", etc.) or another prospect (imagine "I flip a coin; if it is heads, then I do this action, but if it is tails then I do some other action").

An actor's Preferences over Prospects are called Consistent if the preference satisfies axioms (1), (2), and (3) of Definition 1, and:

4. Continuity: Consider three prospects y_i, y_j, and y_k, and suppose the first is preferred to the second and the second is preferred to the third. Then there exists some probability p such that the prospect (y_i, y_k; p, 1 − p) is equally as preferable to y_j (compare to the second interpretation forwarded in Remark 1).
5. Preference increasing with probability: If y_i is preferred to y_j, letting y_m = (y_i, y_j; p₁, 1 − p₁) and y_n = (y_i, y_j; p₂, 1 − p₂), then y_m is preferred to y_n only if p₁ > p₂.
6. Independence: For any three prospects y_i, y_j, and y_k, if y_i is preferred to y_j, then there exists a probability p such that the prospect (y_i, y_j; p, 1 − p) is no less desired than (y_i, y_k; p, 1 − p)

Given this notion of "consistent preferences over uncertain prospects", how can we develop a notion of instrumental rationality?

Maximizing Expected Utility

The first step is to introduce the notion of Cardinal Utility, which assigns to a given outcome y_i the intensity for an agent's preference for that outcome u(y_i).

The second step is to consider the Expected Utility of a Prospect y = (y₁, y₂, ...; p₁, p₂, ...) as the sum E_u[y] = u(y₁)p₁ + u(y₂)p₂ + ..., which is the expected value of the "random variable".

Now, an agent with cardinal utility u(-) is considered instrumentally rational if it picks the action whose prospect has the maximum expected utility.

Example 1. If I go to my favorite BBQ restaurant before the lunch rush, the prospect looks like ("get food", "no food"; 0.95, 0.05). If I leave after the lunch rush, the prospect looks like ("get food", "no food"; 0.1, 0.9).

My cardinal utility function looks like u(get food) = 10, u(no food) = −30.

The expected utility for going before the lunch rush is then

E[before]	=	u(get food)×0.95 + u(no food)×0.05
	=	10×0.95 − 30×0.05
	=	9.5 − 1.5
	=	8

The expected utility for going after the lunch rush is then

E[after]	=	u(get food)×0.1 + u(no food)×0.9
	=	10×0.1 − 30×0.9
	=	1 − 27
	=	−26

Since 8 > −26, it is rational to go before the lunch rush to try to get food.

Next time, we'll discuss flaws with this notion of instrumental rationality, both logical and empirical.

References

• Shaun Hargreaves Heap and Yanis Varoufakis, Game Theory: A Critical Introduction. Second ed., Routledge. (This is the axiomatization scheme I am following.)
• John Searle, Rationality in Action. MIT Press, 2001. (This provides a different set of axioms for rational behaviour, equivalent to the axioms of game theory, and discusses implicit assumptions & its flaws.)