Our broken Electoral System

When Americans elect a president every 4 years, the method we use is actually pretty strange when you stop to think about it:

1) Every state gets a number of votes equal to their number of representatives plus two. These are called "electoral votes".

2) 48 of the 50 states use a winner-takes-all system, where whichever presidential candidate gets the most votes in that state gets ALL the electoral votes of that state. The other two states use an adaptation of that method, where each candidate gets an electoral vote for each congressional district they win, plus two more for winning the overall state popular vote.

Electoral College for the year 2000

A notable side-effect of this policy is that someone can become President of the United States while losing the popular vote. This has happened 4 times out of 55 US presidential elections, or 7% of the time. Maybe that seems like an acceptably small fraction to you, but consider that there are also cases where it was very close to happening, like in 2004: Bush II had about 3,500,000 more nationwide votes than Kerry, but if 60,000 Bush voters had changed their minds and voted for Kerry in just one state (Ohio), he would have become the president. In the last 60 years, a "close" election like this, where fewer than 60,000 voters could've made the wrong man President, has come close to happening 6 times, meaning that 6/15 or 40% of recent elections were problematic.

For fun, I've taken the liberty of running some simulations. Each state is given its share of electoral votes as of the 2000 census, I specify the national popular vote totals and give each state its own vote total, normally distributed about the national mean, with a standard deviation taken from the last three presidential elections (about 11% each time). Then I check to see if the national popular vote winner is also the electoral college winner.

For an example election that's 48/52 (i.e. a 4% margin for one candidate), I ran this simulation 1,000,000 times, and here are the EV results:

We see that in more than 10% of the runs, the national popular vote winner does not become the president. Repeating this process for a collection of margins, I find the probability of the "wrong president" vs. national popular vote margin:

I also show the last eight elections as vertical lines on the bottom, highlighting in red the one that gave us the "wrong" person. Statistically speaking, we should have seen on average 1.3 "wrong" presidents in the past 8 elections. Reality, however, is constrained to integers in this case, so it's really no huge anomaly that we got 1 error out of 8. What's surprising to me is how astonishingly poor this system is at electing the popular vote winner to the presidency. With a national popular vote margin of 4% we get an error of 10%. With a margin of 1% we get an error of 37%. For margins smaller than 1% we may as well flip a coin, even though 1% represents more than 3,000,000 Americans.

Raw data is tabulated below. For reference, the margins of the last 8 elections ranged between 0.5% and 10%. The real miracle here is that we have had only four wrongly-elected presidents out of 55!

Popular Vote Margin	Probability of Wrong President
10%	0.06%
8%	0.52%
6%	2.7%
4%	10.4%
2%	26%
1%	37%
0.4%	44%
0.2%	47%

How big is the Universe?

Hello everyone, and sorry for the long time since my last post. Today I'll be giving an in-depth and unqualified analysis of current astronomy and cosmology. What I find interesting is the pace at which our knowledge of the universe's structure has developed.

1)Starting with ancient cultures, we have known or speculated that we live on a sphere ("Earth") nearby some other objects (the sun and planets), which are all surrounded by little twinkly lights (other stars). Assigning a date and person to this discovery is tricky, but a good guess is an African named Eratosthenes in Egypt, who measured the earth's diameter in ~200 BCE. He did this by noting that on the summer solstice, the sun shone exactly straight down in one city, but cast a shadow about 1/50th of a circle in another city. He knew the difference in the distance between the two cities, and using "math" he calculated the diameter of the earth. So the size of the known universe, in our understanding, was the earth's diameter: 12,800 km.

2)It wasn't until Johann Kepler in the 1600s that we were relatively confident that our Earth was not fixed, but was moving around the sun (and not the other way around). This had been hypothesized before, but only at that time had it been proven and accepted. This increased the size of the known universe to the diameter of the major axis of Saturn's orbit (then the furthest known planet): 1.4 billion km.

3)Even then, it wasn't until Frederich Bessel in 1838 that we confirmed that the other points of light in our sky are stars similar to our sun, and found the distances between them. He did this by measuring the angle of a "nearby" star against the background, further away stars as the earth moved in its orbit. Just like the two images from our eyes allow us to tell the distance to something, this parallax allowed us to calculate the distance to the stars. Until then the working theory was that they were unknown lights, possibly infinitely far away. Now the size of the known universe became the size of our galaxy: 950 million billion km.
4)Only in the twentieth century did we realize that our galaxy was not the entire universe, but merely one tiny, tiny piece of it. Edwin Hubble helped us realize this when he observed objects receding away from us much more quickly than the escape velocity of our galaxy, and this was later confirmed by measuring the brightness of certain standard objects (e.g. Cepheid Variables or Type 1a Supernovas) within those galaxies. This inflated our known universe to the distance between these "nearby" galaxies, about 60 billion billion km.

5) At about the same time, we discovered that the universe is expanding. Our perception of the size of the universe expanded with it, until in the 1960s we found the observational limit: a distance so far that the light has travelled for about 13-14 billion years. Wikipedia explains this barrier nicely:

In practice, we can see light only from as far back as the time of photon decoupling in the recombination epoch, which is when particles were first able to emit photons that were not quickly re-absorbed by other particles, before which the Universe was filled with a plasma opaque to photons. The collection of points in space at just the right distance so that photons emitted at the time of photon decoupling would be reaching us today form the surface of last scattering, and the photons emitted at the surface of last scattering are the ones we detect today as the cosmic microwave background radiation (CMBR).

Now, you might think that that means the limit of our observable universe is a sphere of diameter 13 billion light-years. However, because space itself has expanded during that time, the objects we are observing from 13 billion years ago are now much farther away, meaning the diameter of the observable universe is almost one million billion billion km.

5) And only very recently (1998) have we realized that the expansion of the universe is accelerating, a prediction which completely alters our view of the universe's fate. Current cosmological thinking has no estimate for the boundary of the universe. Our observable universe might be a tiny bubble in an infinite volume; the new estimate is literally "infinity until further notice".

Each of these five discoveries monumentally changed our understanding of the universe. Each time, the universe becomes monstrously larger than we had previously thought, and all of our assumptions become completely challenged. I have taken the liberty of tabulating the size of the "known universe" along with the date, for demonstration purposes:

date	size of known universe (km)	note
200 BCE	12,800	diameter of the earth
1600	1,400,000,000	size of our solar system
1838	950,000,000,000,000,000	diameter of our galaxy
1924	60,000,000,000,000,000,000	size of our "local group" of galaxies
1960s	1,000,000,000,000,000,000,000,000	size of the observable universe
1998	"literally infinity"	astronomers give up

I observe that the rate of this growth as a function of time is not governed by any reasonable function, but might be considered to be logarithmic starting in 1600, with our known universe expanding by a factor of 10 every 24 years until recently, when astronomers simply gave up. This rate of growth (in understanding) is, I'm sorry, astronomical.

There are still so many unresolved questions about the nature of our universe. So-called "dark matter" is required to explain why galaxies spin the way they do, and it has been detected indirectly, but we still have no idea about its nature. If it exists, there is postulated to be five times as much of it, whatever it is, than "regular" matter. Things become stranger: in order to explain the accelerating expansion of the universe, astronomers and cosmologists posit a "dark energy", an unknown force that would have to have 20 times as much "mass-energy" as observable mass in the universe. Mass-energy is the unit used because, as Einstein showed, energy can be converted into matter, or vice versa. If we converted all the dark energy into regular matter, there would be 20 times as much of it.

To butcher a quote by Socrates, "A wise man understands that he knows nothing." The more we study, the less we seem to understand. Maybe that's just the nature of the universe.