After yesterday’s primary in Indiana, both Ted Cruz and John Kasich have dropped out of the race, leaving Donald Trump as the presumptive nominee for the Republican party. This was disappointing to those of us who were hoping for some contested-convention drama, but sensible, since Trump is now virtually certain to go into the convention with the 1237 pledged delegates required to secure the nomination on the first ballot.

At various points during this primary season, I’ve calculated what the delegate counts would have been under alternative allocation schemes. The Republican party uses a process that varies from state to state. In general, the early states tend to allocate delegates more proportionally among candidates, while later states tend to be more winner-take-all. By contrast, the Democratic party uses a uniform method: proportional allocation among the candidates receiving more than 15% of the vote (applied both at the state and congressional-district levels).

Under the actual scheme, Trump currently has about 1013 pledged delegates (plus, possibly, some supporters among the formally unpledged delegates from Pennsylvania). He only needs to win about half of the remaining 445 delegates to reach the threshold, and he would be almost certain to do so, even if Cruz and Kasich stayed in the race.

However, if the Republicans used the Democratic scheme (applied only at the state level, for simplicity), Trump would have 912 pledged delegates, and would need 73% of the remaining delegates to prevent a contested convention. Under proportional allocation with no threshold, he would have 799 delegates, and would need to win over 98% of the remaining delegates.

One consequence of (and perhaps rationale for) using winner-take-all allocation in a drawn-out primary season is that it drives out challengers, forcing consensus around the leading candidate and allowing them to begin their general-election pivot earlier.

Another consequence is that a pesky plurality of “voters” can drive the nomination of an openly bigoted narcissist, subverting the ability of the party elite to install one of the usual cryptically bigoted narcissists. That’s a consequence that will either be hilarious to Democrats in November or hilarious to extraterrestrial archaeologists who visit the smoldering remains of our planet thousands of years in the future.

Methodological notes:

1. For these projections, I used some made-up numbers for the Virgin Islands, Colorado, and Wyoming, which have 75 delegates among them, but did not hold popular votes. For the proportional allocation numbers (with and without thresholding), I gave Trump 36 of those, Cruz 25, and Kasich 14. You could argue for different made-up numbers, but it would not alter the basic story.
2. I quit assigning delegates to candidates after they suspended their campaigns. In certain states, due to early voting, under a strictly proportional allocation scheme, some candidates would have earned a few delegates after quitting the race. More significantly — but even harder to control for — under a different allocation scheme, candidates would have made different decisions about whether/when to drop out, how/where to spend their time and money, and so on.

Today, both the Republican and Democratic parties will be holding their primaries in New York. What should we expect from the Republican primary?  Briefly, the most likely outcome is that Donald Trump will walk away with more than 80 of New York’s 95 delegates. The exact number will depend on the share of the vote that Trump gets, both statewide and in the various congressional districts. However, there are 27 congressional districts, which is a large enough number that we can predict the distribution of results, without worrying about the details of individual districts. That, along with New York’s delegate allocation rules, gives us the following curve:

538’s final polling average has Trump at 52.1%. We can adjust this based on the relationship between Trump’s polling numbers and his final performance, but the adjustment turns out to be pretty small. Best guess is for Trump to get 51.9% of the vote, which would give him about 84 delegates.

That’s the headline. If you’re interested in the details, keep reading.

The red line is the predicted relationship and the black dot is the specific prediction. The blue line is an alternative predicted relationship based on a lower variance among congressional districts (details below).

Delegate Allocation

New York has a total of 95 delegates for the Republican primary. Fourteen of those are allocated based on the statewide results. The other 81 are allocated based on the results in the various congressional districts.

Statewide Delegates

If one candidate gets more than 50% of the statewide vote, they get all 14 of the statewide delegates. If not, delegates are allocated proportionally among the candidates receiving at least 20% of the vote. That’s what creates the discontinuity in the graph above. If Trump gets over 50%, he’ll get all 14. If he gets just under 50% (and Cruz and Kasich both get over 20%), he’ll probably get seven of them.

That graph assumes that both Cruz and Kasich get at least 20%. In the unlikely and very specific scenario where, say, Cruz gets 19.9% and Trump gets 49.9%, Trump would get about 5/8, or about 9 of the statewide delegates. That’s not a big difference, but it is sort of interesting that, in New York, it might be that a strategic anti-Trump movement would want to split their votes between Cruz and Kasich to ensure that both get over the threshold. That runs counter to the conventional wisdom in most states, which says that anti-Trump voters should rally behind whichever non-Trump candidate is more viable.

Congressional-District Delegates

Three delegates are allocated on the basis of the results in each of New York’s 27 congressional districts. Here are the rules that are relevant to the current election:

If one candidate receives more than 50% of the vote, they get all three delegates.

If not, the leading candidate gets two delegates, and the second-place candidate gets one.

Given Trump’s commanding lead in the polls, it seems likely that he will win most or all of the congressional districts. His delegate haul will depend primarily on the number of districts in which he gets over the 50% mark.

We can guess at this if we assume that the individual congressional districts are Normally distributed around a mean given by the statewide result. What we need, then, is a standard deviation for that distribution.

If we look at earlier Republican primary results, we find that the standard deviation among congressional districts clusters around two values. Most of the states for which results by congressional district are available have standard deviations of around 4, including Missouri, Mississippi, Texas, Tennessee, Oklahoma, Arkansas, and Alabama. But Wisconsin and Georgia both had standard deviations of around 7. My instinct is that New York is more like Wisconsin and Georgia, and the curve shown above uses a standard deviation of 7.

If we use a lower standard deviation, like 4, we get the blue curve in the figure above. The upper part of the curve gets a bit steeper. If Trump gets over 50% statewide, reduced variance among congressional districts means that he falls below 50% in fewer of those districts.

The lower part of the curve is fairly insensitive to the standard deviation. If Trump is below 50% on average, higher variance means that he gets over 50% in more congressional districts. However, it also means that he is more likely to come in second in other congressional districts. On the whole, it’s a wash. The curves above assumes that Trump comes in second in any CD where he receives less than 37% of the vote.

Going into today’s Democratic primary in Wisconsin, the polls have Sanders with an edge of 2-3 percentage points over Clinton. The obvious expectation would thus be that Sanders will win by about two or three points. Except that the polls have been wrong this primary season. I’m not talking about Michigan, where polling missed the election outcome by 20 points — or not just about Michigan anyway.

Previously, I did a simple linear regression on the primary results to date, comparing the final polling averages and projections to the actual outcome. I’ve updated that analysis with the more recent results, although the result is more or less the same:

When we compare the final polling averages to election outcomes, we find two things. First, the regression line has a slope greater than one: in the states where Clinton was expected to win big, she mostly outperformed expectations; in states where Sanders was favored, he tended to overperform. This sort of effect might come from a variety of places. For example, it could be that late-deciding voters tend to go with the candidate they think will win, or it could be that voters supporting a candidate very likely to lose become demoralized and stay home. Or, probably, lots of other things.

Second, there is an offset of about 5 points. That is, in close races, Sanders does about five points better on average than the final polling numbers would predict. I am fairly certain that this offset is due to a mismatch between the voter turnout models used by pollsters and the reality this election. Specifically, this election seems to have significantly higher turnout among younger voters compared with previous years. And those younger voters heavily favor Sanders. So, when a pollster constructs their final numbers using a model based on 2012 turnout, they underestimate the number of young Sanders supporters.

The final projection from 538’s Polling Plus model has Sanders winning 50.2% of the vote in Wisconsin to Clinton’s 47.2%. Plugging that into the regression formula gives a Sanders margin of victory of 8.3%.

In the Democratic primaries, delegate apportionment tends to track pretty close to the raw vote totals. So this would project Sanders to win about 46 of the state’s 86 pledged delegates.

In order to go into the national convention with half of the pledged delegates, Sanders needs to win a little over 56% of the remaining delegates. An eight-point win today would give him about 54% of today’s delegates, which would leave the race in pretty much the same state it has been in for a while: Sanders has a shot at overtaking Clinton, but you would not get anything close to even odds on it.

One methodological note: the regression presented here does not include any of the results from the last two rounds of primaries and caucuses — most of which Sanders won by large margins — because there was little to no polling data for those states in the run-up to their elections.

One final note: the conventional what-passes-in-political-punditry-as-wisdom is that Sanders does best in caucus states, while Clinton does better in primaries. That may be true, but all of the points included here are from primary states, with the exception of Nevada. (If we exclude Nevada, the projection changes only slightly: to Sanders by 8.8%.) So, even if there is a primary versus caucus effect, the fact that caucus states received so little polling means that is does not have much effect on this analysis.

Today, March 15, the Democratic primary will be allocating 691 delegates based on voting in Florida, Ohio, Illinois, Missouri, and North Carolina. Yesterday, I posted a rough calculation suggesting that Sanders would need to win about 351 of those delegates in order to be on track to win half of the pledged delegates before the convention.

[NB: If he wins more than that, it by no means guarantees that he will win the nomination. Similarly, if he wins fewer, it does not guarantee that Clinton will win. Which is to say, whomever you support, go vote! Also — maybe even more importantly — figure out who sucks the least in your local down-ballot races, and vote for them!]

So what is actually going to happen today? Well, if we look at the polling averages at aggregators like Real Clear Politics or 538, they suggest that Clinton will carry all five states, with big wins in Florida and North Carolina, and narrower margins in Ohio, Illinois, and Missouri. But is taking the polls at face value the best approach? (Spoiler: No)

We’ve now had enough primaries that we can reasonably compare polling averages and actual outcomes. The following three graphs plot the advantage held by Clinton in the final polling averages or projections before each primary versus Clinton’s actual margin of victory. So, Sanders victories show up as negative numbers. These plots do not include states like Colorado and Minnesota, where there was very little polling before the primaries, and also leaves out Vermont and Mississippi — because you start to get non-linear behavior in landslides.

First, for Real Clear Politics’s polling averages:

The red line is a linear regression, and the blue line is what you would expect if polling were accurate. Michigan is a big outlier, but the rest of the results lie reasonably close to the line (overall R² = 0.84).

Looking at the blue curve, it seems clear that the polls systematically underestimate the margins of Clinton’s victories in the states where she wins big, and they underestimate Sanders’s performance in states where the competition is close.

The slope of the red line is 1.46, and the intercept is –6.58. One interpretation of the >1 slope would be that undecideds tend to go with the winner, because they shake out proportional to the rest of the voters and/or due to a bandwagon effect. The negative intercept, on the other hand, suggests a systematic underestimation of Sanders’s support. Given the very pronounced difference in the typical ages of Sanders and Clinton supporters, and the high turnout of young voters in this primary, I’m inclined to think that this reflects a mismatch between polling firms’ likely voter models and reality.

Whatever the reasons, the red line does give us a way to estimate likely outcomes based on the polling data. Here’s what that looks like:

Here, again, the values indicate Clinton’s expected margin of victory. This regression would predict narrow wins for Sanders in Missouri and Illinois, a narrow win for Clinton in Ohio, and huge wins for Clinton in Florida and North Carolina. This outcome would give Clinton about 399 delegates and Sanders 292.

We can do the same thing for 538’s polling averages. The difference is that RCP uses a simple average of some number of the most recent polls, while 538 uses a continuous weighting scheme to account for poll recency as well as weights reflecting sample size and performance history of individual polling firms. The result is qualitatively similar, however:

Here the slope is 1.4, offset –6.33, and R² = 0.89. This predicts results of

Compared with the RCP analysis, this predicts a smaller margin for Clinton in North Carolina, but predicts that she will win Illinois. The predicted overall delegate haul is nearly identical, though: Clinton 403, Sanders 288.

Finally, we can look at 538’s “Polls Plus” estimator, which includes information about things like endorsements:

Here, the slope is much lower, 1.24, indicating that this method has done a better job of predicting the magnitude of Clinton’s previous wins. The offset is similar, at –6.44, and the fit is the best of the three, with R² = 0.91. Predicted results:

Projected delegate count: Clinton 401, Sanders 290.

This would, of course, fall quite short of Sanders’s target of 351.

So, in order not to lose even more ground to Clinton, Sanders would need to a substantial swing. Is that possible? The results in Michigan clearly indicate that it’s possible, although the results from all of the other states suggest that it’s not very likely.

There are a couple of places a substantial deviation could come from. First, if actual voter prefernce has been changing rapidly over the past week, the polling averages will naturally lag behind that change. Even individual polls are typically conducted over the course of a few days. So, if Clinton’s recent statements about Nancy Reagan, or the Chicago protests, or the Death Penalty, or Libya have alienated any Democrats, that may not be fully reflected in the polls.

Second, in states with open primaries, Democratic voters may cross over, particularly in light of the increasingly urgent anti-Trump movement. If those crossover voters are substantially more likely to be Clinton supporters than Sanders supporters, that would create a shift. A friend in Michigan told me, anecdotally, that she knows a number of Clinton supporters who did just this, partially due to the polls, which indicated that Clinton would win the state easily.

Adding a ten-point swing (e.g., due to 5% of voters switching from Clinton to Sanders) to the 538 Polling-Plus projections would give Sanders victories in Ohio, Illinois, and Missouri, and would produce a delegate count of Clinton 366, Sanders 325. This, incidentally, would be very close to 538’s uncorrected delegate targets.

Sanders would need a swing of about 17.5 points in order to reach the delegate target of 351, which accounts for Clinton’s current lead.

Of course, even if there is a swing, it is unlikely to be uniform across the states. Which means that it is finally time for tea-leaf reading!

I wanted to get down my own predictions, which are going to start from the 538 Polling Plus correction described above, but then use some intuition based on eyeballing the polls for trends and outliers. Here goes:

Florida: Clinton +30, Delegates: Clinton 139, Sanders 75
There’s a modest trend in the past few days that is not captured in 538’s average, but it may not have much effect, due to high rates of early voting in Florida.

Ohio: Sanders +1, Delegates: Sanders 72, Clinton 71
There’s again a sharp recent movement toward Sanders, and, if crossover votes do take away preferentially from Clinton, this is the state where we should see the biggest effect.

Illinois: Sanders +5, Delegates: Sanders 82, Clinton 74
Here, there are polls from March 7 and earlier, which all have Clinton leading by 20 to 40 points. Four polls with more recent data give Clinton an average lead of +2, and the three that are entirely from the last week give Clinton an average lead of less than 1 point.

Missouri: Sanders +8, Delegates: Sanders 38, Clinton 33
Very little polling data here, but, again, evidence of a recent shift towards Sanders.

North Carolina: Clinton +22, Delegates: Clinton 65, Sanders 42
Maybe a recent shift, but probably not more than a point or two.

Total: Clinton 382, Sanders 309

This would put Sanders still short of his targets, but, if he can actually claim victory in Ohio, Illinois, and Missouri, that will probably be enough to maintain his plausibility as a candidate. And so we will reconvene for the next round of primaries!

In the Republican presidential primary system, different states apportion their delegates among the candidates in a variety of ways. In the upcoming contests in Florida and Ohio, all of the state’s delegates are pledged to the candidate who receives the most votes state-wide. In Nevada, delegates are allocated proportional to the vote total (you get one of the 30 delegates for each 3.33% of the vote you get).

But a lot of the states are much more complicated. In South Carolina, three delegates are assigned to the leading vote-getter in each of the seven congressional districts, and the remaining 29 go to the winner statewide. Trump won all 50 this year by leading the pack in each congressional district.

A number of the states impose a minimum threshold. For example, Massachusetts and Kentucky do proportional allocation of their 42 and 46 delegates, respectively, among all candidates receiving at least 5% of the state-wide vote.

Several states add a winner-take-all threshold. Vermont does proportional allocation of its delegates among candidates who receive at least 20% of the state-wide vote. But if any candidate gets more than 50%, they receive all 16.

In general, the effect of these rules is to push delegates towards the winning candidates — and drive inviable candidates out of the race. This happens gently at first, as many of the early contests are more proportional. Then, as the season progresses, things take on more of a winner-take-all flavor.

Here’s what the allocation bias looks like in the Republican primaries and caucuses that have taken place through March 8 (data from The Green Papers):

The brownish-orangish line is what you expect from strictly proportional allocation of delegates, and the points plotted indicate allocations from individual state-wide (and Puerto Rico-wide) contests. Red is Trump, Blue is Cruz, Purple is Rubio, and Black is Kasich.

One thing you can see from the plot is that Trump seems to be the greatest beneficiary of the current allocation system. This makes sense, of course, since he’s the frontrunner, and the system is basically designed to drive a consensus around the frontrunner. And, if the frontrunner were anyone other than Donald Trump, the Republican leadership would probably be very pleased with how it was working.

As a little thought experiment, here’s what the current delegate score would look like if all of the Republican primaries and caucuses used proportional allocation without a viability threshold (besides whatever minimum percentage is required to get one delegate):

That’s not to say that things would have worked out this way under that allocation scheme, since a different scheme would have led to different reporting, different campaign strategies, and so on. But, it’s a nice simple way to quantify the effect of structural properties of the primary system on the outcome.

In that spirit, what this tells us is that about a fifth of Trump’s delegate total — and about half his lead over Cruz — can be chalked up to Republican delegate allocation math.

We could ask the same question for the Democrats, but it is not nearly as interesting. All of the Democratic contests follow the same formula: proportional allocation of delegates among candidates exceeding 15% of the vote. About a third of the delegates come from applying this formula to the state-wide vote, and about two thirds from applying it individually to each congressional district.

That system also punishes low-performing candidates, but it does not reward high-performing ones in the same way. There are no winner-take-all states or triggers (unless you win more than 85% of the vote, guaranteeing that no one else reaches 15%).

So, the Democratic system leans a bit more towards proportional overall. But much more important is the fact that there are only two competitive candidates, both of whom are rarely in danger of failing to meet that 15% threshold.

Red points represent Clinton, and Blue represent Sanders. The only real outliers are Vermont, where Sanders got 86% of the vote, and Mississippi, where Clinton topped 85% in two of the state’s four congressional districts.

I’m not advocating for any particular delegate allocation scheme here. We know there’s no perfect voting system. I just hope to contribute in my own small way to the enormous pile of regrets plaguing Republican party leaders as Trump sits atop his throne of skulls forcing them to fight to the death.

In the wake of Supreme Court Justice Antonin Scalia’s death, Republicans have been climbing all over each other like a less well intentioned pile of zombies in an effort to most loudly claim that President Obama has no right to appoint his successor. Most of the arguments have focused on the fact that we have now entered the final year of Obama’s presidency. As you will recall, back in 2012, the ballots for president clearly stated that the results would only be construed as representing the will of the people for the next three years.

Obviously, these arguments fail any non-disingenuous reading of the constitutional and historical evidence (and contradict arguments previously made by many of those same Republicans), but, you know, the constitution, like the bible, is sacred, infallible, and beyond scrutiny — except when it turns out to be politically inconvenient.

But at the most recent Republican debate, everyone’s favorite cingulocidal maniac Ben Carson offered a different argument:

Well, the current constitution actually doesn’t address that particular situation, but the fact of the matter is the Supreme Court, obviously, is a very important part of our governmental system. And, when our constitution was put in place, the average age of death was under 50, and therefore the whole concept of lifetime appointments for Supreme Court judges, and federal judges was not considered to be a big deal.

Carson is correct that the “average age of death” used to be under 50. In fact, it did not exceed 50 until sometime in the early 20th century. However, as anyone with any educational background in public health or medicine might be expected to know, the dramatic gains in life expectancy have come mostly from reductions in early-life mortality, due to things like sewers, vaccines, and antibiotics. So, unless the Founding Fathers were appointing toddlers to the Supreme Court (spoiler: they weren’t), life expectancy at birth is pretty irrelevant. Here are a couple of graphs (generated here):

The gray line is life expectancy at birth from 1850 to 2000. The orange and red lines are life expectancy from age 60 for women and men, respectively. Since people are not typically appointed to the supreme court until they are in their 50s, this is actually the relevant data.

So, it is true that someone appointed to the supreme court today might be expected to live longer on average than someone appointed in the 19th century, but only by about ten years. But does that mean that justices given lifetime appointments to the supreme court serve longer than they used to? Not so much. Here are a couple more graphs, constructed from this data:

In the top panel, each diagonal line indicates the term of a single Supreme Court Justice, running from the date and age of appointment to age date and age of death or retirement. The black lines are justices who died in office, red lines are justices who resigned or retired, and blue lines are the eight justices currently serving.

In the bottom panel, each dot represents a single justice. “Mid-Term Year” is the halfway point of their tenure (middle of the line in the top graph), and duration is how long they served (length of the line in the top graph. The line is a ten-point moving average. Current justices are not included.

Notice that justices were not often dying by age 50, even in the early days. There are a couple of interesting trends, though.

First, there’s a transition as we get into the 20th century, when it becomes much more common for justices to retire, rather than die in office. So, while the upper limit on the age we might expect a justice to live to might have increased by about ten years, the upper limit on the age at which they leave the court has not changed substantially in 200 years.

Second, after an initial shake-out (during which many of the justices did not have any sort of legal credentials), the long-term trend from 1820 to 1950 is towards shorter average term lengths (declining from around 20 to around 15 years). Starting with the second half of the 20th century, the trend has been towards longer tenures, with a recent average closer to 25 years. However, if you look at the scatter plot, you can see that this increase is mostly due to the absence of any short-term justices since 1970.

So, it is true that we should probably expect that the next person appointed to the Supreme Court will be there for the next twenty to thirty years, but terms of that length have been around since the beginning.