15 July 2008

Translating popular votes to electoral votes

By sheer chance, I came across the book Predicting Party Sizes by Rein Taagepera, a political scientist who was trained as a physicist. I was interested to run into a "theorem" (I'm not sure whether I can call it this, because the derivation in the book is rather heuristic) which states the following. Let V be the number of voters in a country like the United States which elects its president through an electoral college, and let E be the number of states in that country. Then let n = (log V)/(log E). For the United States at present, V is about 121 million (I'm using the turnout in the 2004 election), E is 51 (the District of Columbia is a "state" for the purposes of this discussion), and so n is about 4.7.

This quantity n is called the "responsiveness" of the system, and its rough interpretation is that if the party in control receives (1/2 + ε) of the popular vote, then it will receive (1/2 + nε) of the electoral vote, for small ε. More generally, let VD and VR be the number of popular votes obtained by the Democratic and Republican candidates, respectively; let ED and ER be their numbers of electoral votes. Then ED/ER is approximately (VD/VR)n. When VD/VR = 1 this reduces to the first statement.

Anyway, Nate Silver at fivethirtyeight.com showed the results of some of his simulations about a month ago and claimed that a one-percent swing in the popular vote corresponds to 25 electoral votes. It turns out that 25 electoral votes is 4.6 percent of the electoral college at a whole, so based on his simulations n = 4.6. I take this as evidence that Silver is doing something right. (n is also in this neighborhood for data from actual elections.)

12 comments:

Anonymous said...

"This quantity n is called the "responsiveness" of the system, and its rough interpretation is that if the party in control receives (1/2 + ε) of the popular vote, then it will receive (1/2 + nε) of the electoral vote, for small n"

Isn't the whole reason for studying this, to predict what happen when ε is very small? The ultimate outcome is binary, you either win or lose but voting is not. His theory seems to ignore all the people who vote for candidates that do not clear the threshold to receive electoral votes (nader, ect) and that it seems to ignore that victory in each individual states is binary (except NH and NE) with the result that: In 2000 I recall one candidate got "(1/2 + ε)" of the popular vote but received "(1/2 - nε)" electoral votes.... So i'm going to go ahead and guess this theorem doesn't hold for small values of ε which is precisely where it would be needed, since as ε increases in size electoral votes become irrelevant.


-The stupid kid.

P.S. so to my earlier question about which mean to use for expected value of % returns in the stock market in the Capital Assets Pricing Model... it turns out some guy named Benoit Mandelbrot already thought of this in the 1960's and he used the Levy distribution to model financial instruments, which doesn't have a mean, i don't think.... I think he may have do some work in geometry or something too.

Isabel Lugo said...

Well, from what I can tell it's an average. Obviously the number of electoral votes doesn't follow exactly from the number of popular votes.

Anonymous said...

Yes, but intuitively it doesn't seem right, where is this "1/2" comming from with regards to popular voting?

Presidential candidates often do not win the majority of the popular vote, but have to win a majority of the electoral college votes to be elected. With the result that: In the 1992 election, Ross Perot received 18.9% of the popular vote -approximately 19,741,065 votes but no electoral college votes.

After reading the description of the book, It seems like he's trying to predict the number of "major" parties a system is likely to produce. Though, i'm not sure how you really define "major" since marginal parties can have a huge influence on elections, (nader in florida for example) which are, after all, decided by a marginal of victory.

Anonymous said...

margin of victory....

Isabel Lugo said...

The usual convention in a lot of these analyses, including the section of Taagepera's book that I'm referring to -- and the one I was using here -- is to ignore votes by third parties.

Anonymous said...

Ok, ok, i give.

but, between ignoring "third" parties (the Republicans were once a "third" party), voters that don't participate (but could) and the winner take all nature of the states, it seems like the margin of error is greater than the average margin of victory.

But again, I think from the description I read, his goal was to give policy makers a tool to predict how electoral rules will effect organization of political parties... not to predict elections himself.

OK, now, i'm done.


-The Stupid Kid.

Joe said...

"Yes, but intuitively it doesn't seem right, where is this "1/2" comming from with regards to popular voting? Presidential candidates often do not win the majority of the popular vote..."

Assuming you mean those candidates who get elected...1876, 1888, 2000. That doesn't sound like "often" to me. This argument is not founded on facts; it probably arose after the 2000 election by some bitter people who had supported a candidate that won the popular vote but lost the electoral vote.

Niel said...

but, between ignoring "third" parties (the Republicans were once a "third" party), voters that don't participate (but could) and the winner take all nature of the states, it seems like the margin of error is greater than the average margin of victory.

Correct me if I'm wrong, but it seems that the percentage of voters that vote for third parties is insignificant. Not that it should (from an ethical standpoint) be insignificant, but rather that any vote-splitting due to third party candidates probably fits well into the margin of error for the number of votes in a given district.

Taking Florida in 2000 as the usual dramatic example, while Gore may have been able to carry it without Nader, the margin of victory would still have been small percentage-wise. (Particularly if you also try to account for the voters who apparently did not vote because the media reported the poll-closing time in Florida to be one hour earlier than it actually was, which predominantly affected the more right-wing panhandle.) The question is whether the margin of victory would have been outside the margin of error: if not, then Nader's effect on the Florida vote is more accidental than decisive.

Assuming a small number of third party voters, the model should still be reasonably good if you simply ignore those third-party voters.

Anonymous said...

@ niel:

"In the 1992 election, Ross Perot received 18.9% of the popular vote -approximately 19,741,065 votes but no electoral college votes."

19% of the vote if not insignificant. in for that given example, but I also means from an empirical, stand point, parties are "third parties" until they're not! So you ignore someone who got 50% as much of the vote as the sitting president, will you also ignore them when they get 75% as much, how about a 105%? where's the cut off? Since this system is supposed extrapolate to all countries, this assumption becomes even less tenable in parliamentary democracies, where parties fracture and merge all the time. And if you carry the same logic forward (er, backward) we would be assuming that the Federalists have a 50% chance of winning. The Republicans and Wigs should be ignored because at some point they were marginal third parties. Third parties become main parties, precisely by defying the odds and winning, which is a major component of the system, even if the third parties almost always lose.


@ Joe:

"Assuming you mean those candidates who get elected...1876, 1888, 2000. That doesn't sound like "often" to me. This argument is not founded on facts; it probably arose after the 2000 election by some bitter people who had supported a candidate that won the popular vote but lost the electoral vote."

and 1996 and 1992, and 60 and 48 (off the top of my head of close elections) It happens quite often, in fact 2004 is the only election i was around for and conscious of where the winner actually carried the popular vote.

-The Stupid Kid

CarlBrannen said...

anonymous, clearly Isabel meant "for small epsilon", as "for small n" doesn't make sense.

I.e. n is fixed. And for large epsilon, you end up with a landslide where one party gets all the electoral votes.

I've seen people complain about the responsiveness of the US system, but I think they fail to appreciate the advantages of having a system where the US government has a degree of stability not frequently seen in the world.

The two party system drives the parties towards the center. And the electoral system eliminates the need for worrying about vote fraud in the states where it is most likely to occur: the states which are completely dominated by one party only.

Isabel Lugo said...

Carl,

thanks for pointing out my error! I did indeed mean for small ε.

Anonymous said...

What do you want from me? I'm just a stupid kid.