22 April 2008

Delegate math in the Pennsylvania primary

In tonight's Pennsylvania primary, the structure of the delegate allocation heavily favors to Obama, according to cnn.com video coverage.

The argument is the following: there are 55 at-large delegates which are assigned proportionally to the popular vote in the entire state. There are also 103 delegates divided up among the 19 Congressional districts, with more heavily Democratic districts receiving more votes. (For example, the 2nd district -- mine -- gets nine delegates, which I think is the most of any district nationwide. That's basically the western half of Philadelphia.) The 9th district gets the fewest, with three; numbers for other districts are here.

Now, the formula that assigns the delegates (I can't find it right now) basically says that the number of delegates that a district gets is proportional to the number of Democratic votes in the last few elections.

So assuming turnout is stable, the outcome really isn't any different than it would be if all the delegates were assigned "at large" -- up to rounding errors from the fact that delegates are quantized, but I don't believe the rounding errors break consistently one way or the other. (Roundinf error often do.)

For a small example, consider a hypothetical state with two districts. The first district historically has a turnout of 45,000 Democrats and gets three delegates; the second district historically has a turnout of 105,000 Democrats and gets seven delegates. In addition there are five at-large delegates.

Now say the turnouts in the election are the same; and 65% of Democrats in the first district vote for Clinton, and 65% of Democrats second district vote for Obama. So the first district breaks 29,250 to 15,750 for Clinton, and the second 68,250 to 36,750 for Obama. The state as a whole goes 84,000 to 66,000 for Obama -- 56% to 44%.

Then Obama gets 4.55 delegates in the second district, 1.05 in the first, and 2.8 in the state as a whole -- guess what! He gets 8.4 delegates, 56% of the total of fifteen. (Rounding those, Obama gets nine.)

Something similar is true for the state as a whole.

If anything, the district-based allocation helps Clinton relative to allocation based purely on the popular vote, because new voters tend to break for Obama (or at least they have in previous contents), and districts with a lot of new voters will be slightly undercounted.


Anonymous said...

I think you're missing a crucial point here. In districts with fewer delegates you need a higher margin to pick up any delegates.

Suppose you had two districts each with 4 delegates and the same voting patterns in each district. Any outcome between something like 35 and 65 results in an even split.

On the other hand suppose there's just one 8 delegate district. Then a much smaller lead (say getting in the high 50s) picks you up a 2 delegate lead.

Michael Lugo said...


what you're saying is that what I referred to as "rounding error" -- which I dismissed as unimportant -- may actually be quite important.

You might be right. If I get a chance I'll think about it.

Ben Allen said...

Mathematically, this rounding issue is the difference between E(X) and E(rounded X) for a random variable X. If X is evenly distributed between 0 and 1, the two expected values should be the same. But if X has a central tendency, the rounding will exaggerate that central tendency.

The other interesting thing is that turnout was anything but stable. The number of voters in this primary far exceeded the usual turnout. I don't know if representation based on typical (rather than actual) turnout favored one candidate or the other, but it certainly had the potential to.

Kevin C. said...

A poster on the UsElectionAtlas Forums took a look at this issue nationwide, essentially recalculating what the results would have been if delegates were awarded "at large".

The conclusion was that Obama would be 28 delegates further in the lead (a swing of 14 from Clinton to Obama).

His post (along with a map of where each candidate gained) is here

Anonymous said...

I have a feeling that in a "winner takes all" contest, Hillary would have been the nominee by now.

thomson2008 said...

If Clinton wins 56% of the vote in Pennsylvania, she moves just over the 55.7% viable candidate threshold for a 19-16 at-large delegate victory, but under the 57.5% threshold for a 12-8 victory in PLEO delegates. So, I think the most likely statewide delegate split is actually 30-25 in favor of Clinton. This is the same as Giordano's low-end projection, and different than his 33-22 projection.
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