## Andy vs. the Ideal Point Model of Voting

Last week, as I walked into Andrew’s office for a meeting, he was
formulating some misgivings about applying an ideal-point model to
budgetary bills in the U.S. Senate. Andrew didn’t like that the model
of a senator’s position was an indifference point rather than at their
optimal point, and that the effect of moving away from a position was
automatically modeled as increasing in one direction and decreasing in
the other.

Executive Summary

The monotonicity of inverse logit entails that the expected vote
for a bill among any fixed collection of senators’ ideal points is
monotonically increasing (or decreasing) with the bill’s position,
with direction determined by the outcome coding.

The Ideal-Point Model

The ideal-point model’s easy to write down, but hard to reason about
because of all the polarity shifting going on. To recapitulate from
Gelman and Hill’s
Regression
book (p. 317), using the U.S. Senate instead of the Supreme Court, and
ignoring the discrimination term, the ideal-point model is

```Pr(y[i]=1) = invLogit(alpha[j[i]]-beta[k[i]])

= 1/(1 + exp(-(alpha[j[i]]-beta[k[i]]))).
```

y[i] is the vote by senator j[i] on bill k[i]. The value alpha[j] is
the ideal point of senator j and the value beta[k] the position of
bill k. The probability of a 1 vote versus the difference
alpha[j[i]]-beta[k[i]] looks as follows.

But what does a 0 or 1 response mean? In Gelman and Hill’s words, the
vote is “coded so that a ‘yes’ response (y[i]=1) is intended to
correspond to the politically ‘conservative’ outcome, with ‘no’
(y[i]=0) corresponding to a ‘liberal’ vote.”

To identify the parameters, we can center the alphas around 0 and
assume that liberal senators will have smaller alpha[j] and conservative
senators larger alpha[j] values. On the same scale, more conservative
bills will have larger beta[k] and more liberal bills smaller beta[k].

The Problem: Directional Monotonicity

Gelman and Hill correctly state that “if a justice’s [senator's]
ideal point is near a case’s [bill's] position, then the case [bill]
could go either way, but if the ideal point is far from the position,
then the justice’s [senator's] vote is highly predictable.” This is
easy to see from the graph. It’s the specifics of how the votes get
more predictable that’s problematic.

Given a vote on bill k with position beta[k], the further the
senator’s position is to the right (i.e., larger alpha[j]), the more
likely they are to vote conservative (i.e., y=1), and the further a
senator is to the left (i.e., smaller alpha[j]), the more likely they
are to vote liberal (i.e., y=0).

Holding a senator’s ideal point alpha[j] steady, bills further to
the left (i.e. smaller beta[k]) are more likely to get conservative
(i.e., y=1) votes and those further to the right more likely to get

The obvious thing to plot here is the expected number of votes
predicted by the model versus the position of a bill. The following
graph is a simulation. I simulated a mixture of 57 Democratic and 43
Republican senators (using Norm(-1.5,1) and Norm(2,1) sampling
distributions, respectively). I then centered them so they’d have a
mean of zero. The senators’ positions are shown on the x axis as
circles, colored blue for Democratic and red for Republican senators.

The red curve shows the expected number of 1 votes given the
position of a bill. It’s colored red because a conservative bill is
coded with a vote of y[i]=1 being for the bill. Thus the red curve is
the number of votes for a conservative bill based on its position.

Although only 51 votes are required to pass a bill, 60 are required
to block an opposing filibuster, so we have drawn a horizontal line at
60 votes. It intersects the curves at the least extreme positions at
which a bill is expected to pass even under threat of filibuster.

Things flip for a liberal bill, where outcome y[i]=0 is considered a
vote for the bill. The blue curve shows the number of votes for a
liberal bill based on its position. Thus the coding of a bill makes a
difference to the position at which it’ll pass as well as the
direction of movement which increases likelihood of passage.
Specifically, for a bill coded as conservative, the more liberal its
position, the more likeliy it is to pass. For bills coded liberal,
the more conservative its position, the more likely it is to pass.

Andrew’s right — the model doesn’t make sense. Making a bill more
liberal or more conservative has the same effect on senators of both
parties, making them either more or less likely to vote for the bill,
depending on how the coding is chosen. Flipping the coding actually
changes the point at which a bill will pass, and seems redundant given
that a bill’s position, beta[k], is also coding a bill’s
liberal/conservative orientation.

A More Concrete Example

For concreteness, consider the US\$ 800+
2009
stimulus bill
, a liberal bill which squeaked through the
U.S. Senate. We will suppose that voting for it is “liberal”, so
coded as 0. Presumably larger stimulus packages are also more
liberal, corresponding to smaller estimated beta[k] (see what I mean

Now consider any old senator j with any old ideal-point alpha[j], no
matter how far right or left. What happens to their voting
preferences as the bill beta[k] moves to the left or right? If the
bill moves to the left, beta[k] decreases, so alpha[i]-beta[k]
inreases, and the vote is more likely to be 1 (against the stimulus).
Thus no matter what the senator’s ideal point is, they’re more likely
to vote against the bill as the target amount becomes more liberal
(rises). If the bill moves to the right, presumably corresponding to
a smaller stimulus package, beta[k] increases, so alpha[i]-beta[k]
decreases, so the vote is more likely to be 0 (for the stimulus).

If we recode the outcome so that y[i]=1 rerpesents a vote for the
stimulus package, we’re left with the same problem in the opposite
direction.

Pardon me for whipping out the Latin, but I am back in
academia, and this sure feels like a
reductio
.

A Better Approach?

A better approach would presumably model a senator’s preferences for
bills based on proximity to their ideal point rather than difference.
Then, the further the bill got away from the senator’s ideal in either
direction, the more likely they’d be to vote against it, rather than
assuming their preference for the bill goes up in one direction of
change and down in the other relative to their indifference point.

A hypothetical bill with a position to the left (or right) of all
the senators will still act monotonically, but now with the expected
directional effect, namely increasing expected votes as the bill
approaches the senator’s ideal points. Points in the middle act in
different directions on different senators, moving closer to some and
further from others based on their ideal points.

In addition to hierarchical structure on the ideal points and
multiplicative discrimination terms (for senator and/or the bill), as
Gelman and Hill discuss, we could also add some basic predictors in
addition to senator and bill identity, such as for the party of the
senator, the party of the senator(s) who introduced the bill, the
chance of filibuster, pork going to the senator’s state, and all the
the other things that make the U.S. senate so much fun.

1. numeric says:

Any modeling of a bill (particularly in the Senate) that does not include the probability of being pivotal is suspect–a Senator's "ideal" point is pretty flexible, if they are pivotal, because they can always add something to the bill to make it more advantageous to the lucky
pivotal Senator (and to the disadvantage to other Senators, usually). In other words, the bill is endogenous, not exogenously presented to the Senators,
so should be modeled as a multi-player game. Using
the stimulus example, Specter was pivotal and used that
to get an extra \$10 billion for NIH.

On another matter, reading Gelman's critique of Axelrod's
movement of evolutionarily stable strategies into the political science realm, I was struck by how little understanding of the actual mechanics of warfare in that period was displayed. Most casualties in this war came from artillery file followed by random "overhead" machine gun fire and gas attacks (the British General Staff put "wastage" at 7,000/men day). Actually seeing, and being seen, by an enemy, except during attacks, was rare (reports from the British of never seeing a German until the Somme were common). So firing one's rifle (the only activity under the direct control of the individual soldier) at an enemy was a very small part of the conflict, except, of course, during an attack, where if you didn't stop them, they would be in your trench and kill you (or vica versa if you were carrying out the attack). I would recommend The Great War and Modern Memory and Goodbye to All That for descriptions of what faced the fighting man during that conflict (the 7,000/day comes from The Great War and seems high, but considering that most were wounded and could return to the conflict, and that the average tenure on the line for the British was 5 months before being killed or wounded, and that half the British Army was under 19 at the end of the war, it probably isn't far off).

A much better example of a TFT strategy in warfare was during the American Civil War, where, with one army attacking another and overrunning a position, leading to hand-to-hand combat, the choice was whether to use bayonets or the rifle as a club. The bayonet was not only more immediately lethal but quicker (one can parry and thrust in one maneuver, whereas with a club one has to lift and then strike). So the optimal strategy was to bayonet, no matter what the opponent did. On the other hand, if both sides only clubbed, there was a much greater chance of surviving (on both sides). This seems to fit the prisoner's dilemma paradigm perfectly, and a repeated game TFT would call on both sides to club, as the issue (control of the position) would not change whether one used bayonets or clubbing, but the carnage would be much less with clubbing.

2. I think you might want an unfolding model here. As I recall, these assume an ideal position and the probability of agreement decreases as you move away either to the right or to the left.

Jim Roberts at Georgia Tech has published extensively on this topic.

3. James says:

But, the relevant comparison is the value a legislator attaches to a bill relative to the value she attaches to the status quo, no? If we assume utilities are concave, a far right (left) legislator values a bill that shifts the status quo slightly to the right (left) more than does a moderate. In which case, the monotonicity assumption makes sense.

Of course, this neglects the possibility that extremists vote against a moderate bill in the hopes that a bill more to their liking will be introduced at a later date.

4. James says:

Sorry, I jut reread and wanted to revise my comment.

Wouldn't the problem with a proximity model be that both extremes would be predicted to vote against moderate bills? This seems theoretically dubious, given that the comparison all legislators are making is (presumably) to some status quo. So an extremist to the left (right) would be expected to value a bill that moves the status quo slightly to the left (right) quite highly.

5. Andrew Gelman says:

Numeric:

My critique of Axelrod was methodological. I do not claim any expertise on trench warfare and so I pretty much relied on Tony Ashworth's book, which was the secondary source that Axelrod has relied on.

6. Anonymous says:

"if a justice's [senator's] ideal point is near a case's [bill's] position, then the case [bill] could go either way, but if the ideal point is far from the position, then the justice's [senator's] vote is highly predictable."

Sorry this is off base. If a voter's ideal point is right on top of a bill they are going to vote for it in the standard spatial model. It's when their position is equidistant from the sq and alternative that they become unpredictable I. The standard spatial model. The gelman-hill version is a special case of the standard model where the position of the bill is to the right of all voter's ideal points. That is a strang situation and that is why the model seems suspect. But it's not an interesting or general case.

7. Anonymous says:

Sorry I meant to say the gelman-hill model is a special case of the standard model where the proposal is right of all voters and the sq is left of all voters. Again, strange, esp. with an endogenous agenda.

More to the point in spatial models of ideal point voting, given locations of sq and alternative, the cutpoint (point of indifference Andy doesn't like) is isomorphic to any indifferent voter's ideal point. If you're right of the cutpoint on a conservative bill you become more likely to vote for it because your ideal point is to the right of an indifferent voter's. Makes a lot of sense to me.

8. Bob Carpenter says:

Thanks for the refs. I think everyone's talking about the model from:

Clinton, Jackman, and Rivers. 2004. The Statistical Analysis of Roll Call Data. Am. Poli. Sci. Rev..

It incorporates both relative position and distance, by modeling a voter's position as well as the position of a yea-vote and nay-vote on a bill. The predictor is based on the difference between two distances, the Euclidean distance from the voter to a yea-vote and the Euclidean distance from the voter to the nay-vote, plus Gaussian noise.

That makes much more sense.

9. Bob Carpenter says:

Oops, that link was to a draft. Here's one to the final paper:

Clinton, Jackman, and Rivers. 2004. The Statistical Analysis of Roll Call Data. Am. Poli. Sci. Rev.