Mathematically speaking, it’s been a slow week and a slow day. My alter ego the kid has been playing with the very beginning of differential geometry – namely, curves in 3-D space – and with the very beginning of “Excursions in Modern Mathematics” by Tannenbaum and Arnold – namely, different ways of deciding an election.
It’s a wonderful book in general and I highly recommend it.
In fact, the kid has been having so much fun with voting methods that I had trouble stopping him so I could write this.
In addition, I spent some time searching for Arrow’s impossibility theorem – which says that there is no voting method satisfying a particular set of desirable properties, so we can stop looking for a voting scheme that is “fair” in that specific sense. (Arrow himself shared the 1972 Nobel Prize in economics.)
Here, let me offer you some links.
Oh, first let me say just one thing about the impossibility theorem: it applies to voting when the voters order all of the alternatives, not to votes where they each choose exactly one alternative. So it doesn’t apply to the U.S. presidential election, but rather to elections where first place wins the whatever and second place becomes the vice-whatever. It also applies to votes like the Hugo awards, the Academy Awards, and the choice of host city for the Olympic Games.
Perhaps for the first time, I am not going to include a Wikipedia link – there is an article, but it’s more advanced than introductory, and just not that readable, I think.
Here’s a short introduction from MIT.
And here are three proofs of it.
Finally, voting methods and the impossibility theorem is one of the mathematical fun facts on this site… which I think I had never seen before. It’s got 203 separate math fun facts. You might even think that a few of these are suitable things for math geeks to know – and, in fact, at least two are on my list of things a math geek should know.
I’m inserting a picture because it took me a while to figure out how to get a list of them all: look at the bottom of the picture, where it says “List All”. That will get you the entire list; it’s kind of hard to search when you have no idea what’s there.
Let me close by saying that this coming Monday’s post is almost complete; the draft is already out here, but it needs some more detail and a summary as well as the usual final edits. It will eliminate the multicollinearity from the Toyota data – by thoroughly scrambling the dummy variables.