Which comes first, the nominal or real record high?

A quick puzzle: say that gas prices hit a record high in nominal dollars. And say they also, around the same time, hit a record high in real (inflation-adjusted) dollars. Which comes first?

Say the nominal price is f(t) and the real price is h(t) = f(t)g(t), where g is monotone decreasing. Then h'(t) = f'(t) g(t) + f(t) g'(t). So if the nominal price is at a maximum at time T, then f'(T) = 0 and so h'(T) = f(T) g'(T). f(T) is positive because it's a price and g'(T) is negative by assumption. So h'(T) is negative and at the time of the nominal high, the real price is already decreasing. The real high comes first, and there's a short period in which the real price is decreasing but the nominal price is still increasing. This makes sense - a nominally constant price is decreasing in real dollars.

This is brought to you by an example from Geoff Nunberg's book Talking Right, on how the Right has distorted political language in the United States in an effort to marginalize the Left. At the particular point I'm commenting on, argues that the right likes to proclaim "record highs" in gas prices which are basically always in nominal dollars and therefore make the gas price rise look worse than it is. My argument, however, says nothing about that - taking derivatives makes this a local problem, and "record highs" (nominal or real) are global maxima.

How not to visualize the electoral college

I went to the National Constitution Center in Philadelphia today.

As you may know, the Constitution provides that, in elections for the President, each state receives a number of electors equal to its total number of senators and representatives. Each state has two senators, and the number of representatives is proportional to the population. The number of representatives is adjusted after the census, which happens in years divisible by ten.

Why am I telling you this? Because at one point on the wall there was an animated map, which displayed how apportionment had changed between censuses. Each state was represented as a "cylinder", with base the state itself and height proportional to its number of electors. (Or representatives; it honestly would be impossible to tell the difference by eye, as in this scheme that would just push everything up by two units.) There was one such display in the animation for each census, with smooth transitions between them.

Since the eye wants to interpret the "volume" of a state as its number of electors, this has the effect of making geographically-large states look like they have better representation than they do. I noticed this by looking at New Jersey and Pennsylvania, which have areas of 7417 and 44817 square miles, and 15 and 21 electors respectively. The solid corresponding to Pennsylvania has about eight times the volume as that corresponding to New Jersey. New Jersey's an easy one to look at because it happens to be the most densely populated state at the present time, and in this visualization it is not the tallest.

The volume of the solid corresponding to each state is proportional to the product of its number of its electors and its area. The states for which this product is largest are, in order, Texas, California, Alaska, New York, Florida, Illinois, Arizona, Michigan, Pennsylvania, and Colorado. The first two of these, between them, have 41% of the total volume in this visualization.

I'd suggest replacing this with a model where volume is proportional to the number of electoral votes. Or, since that might have its own problems, a cartogram which evolves in time. The West would just grow out of nowhere.

How did Obama do among Prius owners?

Here's a question: Did Obama do better among African-Americans or Prius owners?

The consensus is that he did better among African-Americans. (96% of African-Americans who voted voted for him, which is a pretty high bar.)

But how would one go about estimating how he did among Prius owners?

The Iranian election

The Devil Is in the Digits, an op-ed by Bernd Beber and Alexandra Scacco in Saturday's Washington Post.

This piece claims that the distribution of insignificant digits in vote totals in the recent Iranian election look funny, and that there's a good chance this is because the numbers were made up.

I haven't looked at the numbers myself, but this seems like an avenue worth pursuing.

The third derivative of the employment rate is positive

The third derivative of the number of people employed in the United States is positive. (From 538.)

Nate Silver puts it as "the second derivative has improved", but let's face it, this is really a statement about the third derivative. Compare Nixon's 1972 statement that the rate of increase of inflation was decreasing, which Hugo Rossi pointed out in the Notices was a statement about the third derivative. (I seem to recall John Allen Paulos pointing this out in one of his books, but I don't recall which book and therefore can't date it relative to Rossi's letter in the Notices.)

Jean Chretien is secretly a mathematician?

"I don't know. A proof is a proof. What kind of a proof? It's a proof. A proof is a proof, and when you have a good proof, it's because it's proven." -- Jean Chretien, former prime minister of Canada. The context appears to be something having to do with Canada's involvement in the Iraq war, but I'm having trouble finding details. It seems that this was a Big Thing in Canada when it happened, so perhaps I have Canadian readers who can explain?

Pairing up the states

A Ballot Buddy System, by Randall Lane, an op-ed in today's New York Times.

As you may remember, there was a presidential election six weeks ago in the United States. But Barack Obama isn't officially elected president until today; today is the day that the electors cast their votes. This is the first time since 1892 that a state will have electors voting for more than one candidate. Maine and Nebraska both have laws in which two electors go to the winner of the popular vote in the state and one goes to the winner of each congressional district. Nebraska went for McCain, but the 2nd congressional district (Omaha and some of its inner suburbs) went for Obama.

It's been suggested that all states should apportion their electoral votes in this way, on the assumption that less people live in "safe districts" than "safe states". (I'm not sure if this is the case, especially with the way some districts are gerrymandered these days.) But the problem with this is that the majority of people (and legislators) in any state would see their party hurt by the passage of such a law in their state.

Lane's suggestion is that Republican-leaning states and Democratic-leaning states with approximately the same number of electoral votes (say, Texas and New York) could agree to pass these laws together. The problem is that in each pairing, it seems that you'd want two states that are roughly of equal size and are equally far from the political center; it seems that it might not be possible to construct such a pairing. The obvious problem is what to do with California? It's easy to state a few plausible pairs, as Lane does, but I'm not sure that all the states could be paired off in this way. Furthermore, things probably get weird, in terms of how much "power" each state holds in presidential elections, if some substantial number of states have enacted such laws.

A quirk of electoral apportionment

I was curious: how will the electoral vote apportionment change between now and 2012? (Reapportionment is done after each census, and censuses take place in years divisible by 10; the apportionment takes effect the year after the census. Thus the 2004 and 2008 presidential elections were done under one apportionment, and the 2012, 2016, and 2020 elections will be done under another one.)

I don't know (my first attempt at programming the apportionment gave some really strange-looking results) but I wanted to share an amusing fact.

Each of the 50 states receives a number of electoral votes equal to its number of Representatives, plus two. So the question is really one of determining the number of Representatives that each state gets. The way this works is as follows. First, each state receives one seat. Then, let the populations of the states be P₁, P₂, ..., P₅₀; let

Q_i,j = P_i / (j(j-1))^1/2

for 1 ≤ i ≤ 50 and all positive integers j. Sort these numbers, and take the 385 largest of these numbers. Now state i (the state with population P_i) gets r = r_i representatives, where r is the unique integer such that Q_i,r is one of the 385 largest Q's, and Q_i,r+1 is not. (385 is 435-50; 435 is the number of seats in the House of Representatives, and 50 seats were already assigned in the previous step, one for each state.) Essentially, this assigns the seats in the House "in sequence", so we can speak of the 51st seat, 52nd seat, ..., 435th seat.

So what if there's a tie for 385th place in that ordering? This can occur, of course, if two states have the same population, and I bet some tiebreaker is written into the law. But what if two states have different populations, but after dividing by the square root factor, two of the Q_i,j are the same? Surprisingly, this can happen. Let P₁ = 6P₂. Then it's not hard to see Q_1,9 = Q_2,2; that is, state 1 gets its ninth seat "simultaneously with" state 2 getting its second seat. More generally, if

P₁ / (m(m-1))^1/2 = P₂ / (n(n-1))^1/2

then state 1 gets its mth seat simultaneously with state 2 getting its nth seat. Note that P₁/P₂ is rational. So a tie can only occur when (m(m-1)/n(n-1))^1/2 is rational; when does this happen?

When n = 2, this amounts to asking when (m(m-1)/2) is a square; this happens for m = 2, 9, 50, ... (the indices of the square-triangular numbers in the sequence of triangular numbers) So one state can receive its second seat at the same time another one gets its 9th seat, its 50th seat, ... if the larger state has 6, 35, ... times the population of the smaller one.

Somehow I doubt the law covering apportionment has a provision for this. I suspect the provision taken would be similar to what happens if there's a tie in an election; I know there are some jurisdictions that just flip a coin in that case.

Edit, 10:53 pm: Boris points out in the comments that somebody's done the projection. Texas gains 4, Florida and Arizona each gain 2; the Carolinas, Georgia, Utah, Nevada, and Oregon each gain 1. New York and Ohio each lose 2; Massachusetts, New Jersey, Pennsylvania, Michigan, Illinois, Minnesota, Iowa, Missouri, Louisiana, and California each lose 1. At first glance this shift seems like it would favor the Republicans in the presidential race; nine of the seats created are in states that voted for McCain in '08, and only two of the seats destroyed are. But I'm not sure about this analysis; states are made of people, so as a state's population grows or shrinks its political makeup changes as well. Maybe Nate Silver will have something to say about this?

The notion of "typical" doesn't behave nicely

Matt Yglesias makes an interesting point. The "typical" American is white, in that more than half of all Americans are white. The "typical" American is Christian, in that more than half of all Americans are Christian. But does this mean that the "typical" American is a white Christian, in that more than half of all Americans are white Christians? Not necessarily; I don't have the numbers.

Moreover, the "typical" white Christian votes Republican. Thus typical people vote Republican, so the Republicans should have won last night. But they didn't.

The point is that most people are "typical" in some ways, but few people are "typical" in all ways. And a party that is based around just people who are "typical" in all ways (note that I'm not saying this describes the Republican party) is doomed to fail, because most people are unusual along some dimension. I don't think this deserves the name of "paradox", but it's just something worth keeping in mind about How Statistics Work.

Vote!

Scott Aaronson points out that the probability of your vote changing the results of the election scales like N^-1/2, where N is the number of people. But the amount of change your vote creates, if it tips the election, scales like N. So the expected amount of change you will cause, by voting, scales like N^1/2. That's a big number, so you should vote. If you live in a big country, you should vote more, although that's irrelevant; if any other country is voting today, the US media has ensured that I don't know that.

Of course, N^1/2 seems a bit high, and it comes from modeling people as flips of a fair coin; Aaronson points out that under a more realistic prior (due to Andrew Gelman), the expected probability that your vote flips the election is N^-1, so the expected amount of change your vote causes doesn't depend on the size of the country.

I won't "officially endorse" anybody. But one of the candidates in the present election trained as a lawyer, taught in a law school for a while, and likes to compare himself to a certain Senator from Illinois. That senator, in order to equip himself to understand the law better, studied Euclid. Being a mathematician, I think this is pretty cool. Who I'm voting for is left as an exercise for the reader.

A thought on expectation

Political campaigns should not campaign in such a way as to maximize their expected number of votes. Rather, they should campaign in such a way as to maximize their probability of winning.

Early in a campaign, these are pretty much the same thing, because one doesn't know how things are going to play out; late in a campaign they diverge. The candidate that's behind in the polls should, perhaps, start doing things that are, in expectation, Bad Ideas. If there were something that McCain could do that had a 1% chance of swinging 10% of the vote to him in the next 24 hours, and a 99% chance of scaring all the voters away, he should do it. (For example, let's say McCain turns out to secretly be an extraterrestrial; we probably don't want to be ruled by extraterrestrials, but who knows, we might change our mind? Of course, I'm being deliberately silly here.)

This is somewhat analogous to what happens in sports; strategies change late in a game. In the early innings of a baseball game you play, basically, in such a way as to maximize the difference between the expectation of your number of runs and your opponents' number of runs. In the late innings, when you have a better idea of how many runs you need, you change your strategies. See, for example, the bottom of the ninth inning of Game 3 of the World Series; tie game, bases loaded, nobody out. The Rays bring in one of their outfielders to create a fifth infielder; the idea is essentially that they want to maximize the probability of the Phillies scoring zero runs, and if the ball gets into the outfield a run is scoring anyway. (As it turns out, the Phillies won that game -- on a hit in the infield.) Of course, nobody saw that because it happened at two in the morning. Things are different when you're playing for one run.

Basically, what's happening as I write this is that Obama is running out the clock. (Yes, baseball doesn't have a clock. But Obama's a basketball player, so I think he'd like this metaphor.)

What politics and sports have in common, of course, is that there's a huge difference between second place and first place. If you're a company of some sort, does it matter if you sell 1% more or 1% less than your competitor? Not really, although it might have meaning for your pride. But if you're a politician, 1% of the vote makes a big difference.

Conditional probability is subtle, part 2

Nate Silver returns to the idea of conditional probability: he's saying Pennsylvania is "in play" in this election, not because the polling in Pennsylvania is close, but because conditioned on the election being close, Pennsylvania is close. In general, I've heard quite a few people argue that the candidates should focus on the states that would be close if the election were roughly tied, not the ones that actually are close, because the details of which state a candidate deliberately tries to sway things in only matter in a close election anyway.

Unfortunately, subtleties like this seem to be lost on some of Silver's commenters.

In case you're wondering, my last post titled "Conditional probability is subtle" had nothing to do with politics.

People are not balls

The margin of error is only the beginning of political polling: "If one or more of the above statements [about certain red and blue balls] are true, then the formula for margin of error simplifies to Margin of Error = Who the hell knows?"

Electoral sensitivity

By combing through my logs I found The electoral college and second terms, which is related to my post on translating popular votes to electoral votes. (Roughly speaking, in a close election, the electoral-vote margin is about five times the popular-vote margin.)

Two sorts of predictions

Lately, there are two things that a lot of people searching for this blog seem to want to know: predictions about the election, and predictions about the World Series.

Baseball Prospectus says the Phillies have a 51.7% chance of winning, and the Rays 48.3%. That seems a bit surprising, though; I would have thought (although I don't like to admit it) that the Rays would be a slight favorite. I suspect there's some difficulty in incorporating the fact that the American League (in which the Rays play) has a higher quality of play that the National League (in which the Phillies play).

And Fivethirtyeight.com says Barack Obama has a 93.8% chance of winning. (The election, not the World Series.) The prediction there is based on a Monte Carlo method which simulates the election results; there's a plot of the number of electoral votes received in the simulations, and it gets spikier and spikier as more and more states become settled one way or the other. (The site's model assumes that states may move between now and Election Day, and as "now" gets closer to Election Day, that effect diminishes.)

What do these two predictions have in common? Fivethirtyeight.com is Nate Silver's web site; he works for Baseball Prospectus.

By the way, various people have said that Fivethirtyeight.com doesn't take into account correlations between states. The most common misconception seems to be that the site assumes the results in each state are independent. This is not true; if you start from the state win probabilities displayed there and combine them under the assumption of independence, you get a distribution much different than the one currently displayed on the site.

Tao on polling

Terry Tao on the mathematics of electoral polling.

McCain's life span?

Here and here, people attempt to answer the question: what are the chances that John McCain will die in the next four or eight years?

A quick look at mortality tables says roughly 15% in four years, 30% in eight years, which are roughly ten times the corresponding figures for Obama --- although it gets more complicated than that pretty quickly. Obama smoked for a while, McCain had cancer. Obama's parents died relatively young, which seems bad for him-- but his father from an automobile accident and his mother from ovarian cancer, which Obama himself is obviously not at risk for. McCain's mother, on the other hand, is still alive at 96. The presidency is a very stressful job -- but it comes with great health care! (I don't actually know what sort of health care the president has, but somehow I don't see doctors turning away the president for inability to pay.) And so on.

And if we're talking about the probability that a president will survive his term, we also have to think about assassination. Four out of 44 US presidents have been assassinated; what are the probabilities that either of the nominees would be assassinated? I don't even know how one would begin to assess that.

Finally, as meep points out in the first post linked above, "The central limit theorem doesn't kick in at one person". We don't get to elect a president, branch off a large number parallel worlds, and see in what proportion of those worlds he survives four or eight years. (Unless you subscribe to the many-worlds interpretation, that is.) We get one shot.

(Well, we Americans get one shot. My statistics have historically shown that about half my readers are reading from outside the US.)

Weather and political polls

From a Philadelphia TV weather man, Glenn "Hurricane" Schwartz, upon observing that the low and high temperature for Philadelphia today (70 and 85, respectively) were the "normal" temperatures for the day:

"Exactly normal! It's not normal to be exactly normal!"

which, of course, is true. That's how distributions work.

The weather people on television try a lot less to explain why the weather did what it did than the political people; John and Zeno have talked about this "roller-coaster polling". I suspect this is because once the weather has happened we don't care why it happened that way, while the whole point of polls is to use them to forecast the upcoming election.

Mathematicians in politics?

Quite some time ago, the folks at 360 asked if there have been heads of state who were by training mathematicians. This is really two questions in one: people who were trained as mathematicians, and people who had a mathematical career before going into politics.

The first question doesn't seem that interesting, because it seems to include cases in which Politician X majored in math as an undergrad, then went to law school, became a lawyer, and then entered politics from the law, as so many do. That's not the question I want to answer.

For the second question, a bit of clicking around turns up this list, which inclues Alberto Fujimori (president of Peru), Paul Painlevé (prime minister of France), and Eamon de Valera (president of Ireland). Painlevé in particular made a name for himself as a mathematician; the other two appear to have at least taught it in some capacity at some point.

I had thought that Henri Poincaré had been in politics, but it appears that I was confusing him with his cousin Raymond. Borel served in the French National Assembly. I haven't done any sort of systematic sampling, but it seems like mathematician-politicians are particularly prevalent in France, that wonderful country where they name streets after mathematicians. (Here in the United States, for example in my native city of Philadelphia, we name streets after mathematical objects, namely the positive integers.)

One interesting close call is Einstein. The story has it that he was offered the presidency of Israel in 1952. Of course Einstein was a physicist, but given the title of this blog I feel I can mention him.

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 V_D and V_R be the number of popular votes obtained by the Democratic and Republican candidates, respectively; let E_D and E_R be their numbers of electoral votes. Then E_D/E_R is approximately (V_D/V_R)_n. When V_D/V_R = 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.)