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Thursday, August 7, 2008

Moving On

Hello everyone,

After doing a lot of thinking and reading over vacation, I've decided to shut this blog down. By no means am I done with blogging, but this site really was my first attempt at the beast, and I need to make some fundamental changes. Thank you all for reading (and occasionally commenting). You can follow me to:



Friday, July 25, 2008

Technology Hiatus

Hey all,

I'm going on vacation in Ocean City, MD for the next 9 days. No computer means no posting. I'll be back on August 4th!


Sunday, July 13, 2008

Reversal of Social Preference in Plurality Voting

I continued working through the exercises in Chapter 1 of Donald Saari's Basic Geometry of Voting, and found another interesting problem in Exercise (just one exercise after my last posts' excursion, considering when a plurality winner can be a Condorcet loser).

This time, we're given some information about voters' preferences in an election. All we're told is how many voters ranked each alternative first. The example in the text has four alternatives; A, B, C, and D with 9, 8, 7, and 6 votes, respectively. The election uses the plurality method, so that the social preference is A beats B beats C beats D.

We're then told that if a candidate drops out of the election, all of those people who voted for them instead vote for their second choice preference. It turns out that such an occurrence can actually reverse the resultant social preference. For example, removing candidate D can lead to the plurality result:

C beats B beats A!

I found this quite strange, and decided to dig deeper. I determine an algorithm which "optimizes" this strange result, and find conditions to see what elections permit such a result. You can read about it all right here:

Reversal of Social Preference in Plurality Voting

Oh, and I managed to finish the rest of the exercises in the section without feeling the need to write another blog post. Good thing, otherwise I might never get past chapter 2. The introductory material does, however, give one a lot of interesting things to think about (obviously)!

Saturday, July 12, 2008

Plurality Winner, Condorcet Loser?

I recently received my copy of Donald Saari's Basic Geometry of Voting. I'm using the book both for research purposes and to prepare to TA a class on the mathematics involved in Democracy, voting systems, and the like. I'll be posting some extensions of what comes out of the book and my own research here in the future.

Today's work comes courtesy of Exercise in the book, which asks us to investigate when it is possible for a plurality winner to be a Condorcet loser. The exercise only looks at elections with three alternatives, but I was able to generalize to an arbitrary number of voters and alternatives.

For those unfamiliar, here are some preliminaries necessary to understand the work:

Each voter ranks the alternatives from first-choice to last. A strict plurality winner is an alternative which receives more first-place votes than any other alternatives. (Non-strict definitions typically allow for ties).

A Condorcet winner is one which is ranked higher than every other alternative in a majority of decisions. This is often confusing, as the same majority is not required for each pair of alternatives. For example, if

4 people prefer A to B to C
3 people prefer B to C to A
2 people prefer C to B to A

Then A is the plurality winner, because it has the most first-place votes. B is the Condorcet winner, as 5 people prefer B to A (a majority), and 7 people prefer B to C (also a majority, though not the same voters).

In the example above, A is what we call a Condorcet loser; it is ranked lower than every other alternative in a majority of decisions (5 to 4 in both cases).

This example is what we seek, when the plurality winner is simultaneously the Condorcet loser, an admittedly strange (though not necessarily uncommon) result. The complete analysis, in which we explicitly work through the case of three alternatives, then generalize to an arbitrary number of alternatives is right here:

When Can a Plurality Winner Be A Condorcet Loser?

Thursday, July 3, 2008

Abstract Bracketology 2

Here's my next step in analyzing the NCAA tournament with some mathematics. Today we look at the basics of multiround theory, setting up some computations and determining some relevant quantities. The file can be found right here:

Abstract Bracketology 2

Enjoy, and let me know if you come up with any other things that should be studied here!

Saturday, June 28, 2008

Abstract Bracketology 1

Sorry for the delay in posting readers! I've been doing some construction work, which has been taking up a great deal of my time. I'm back with a possibly poorly timed application of mathematics to the world of sports.

Year after year my NCAA March Madness Bracket falls apart, and I've had enough. It's time to mathematize (this, rather than increasing my knowledge of college basketball, is of course the easiest way to improve my results). I'll be continuing my work on this indefinitely, as I've already come up with a number of interesting questions to look at. We start small, looking at strategies for a single game tournament.

Abstract Bracketology 1

If anyone has ideas of where to go with these, or questions to ask/answer, please comment!

...and GO DUKE!

Thursday, June 19, 2008

GRE Math Quick Review: Conversions

Rounding out the subseries on fractions, decimals, and percentages, this document covers how to convert between the three forms and offers advice on how to approach problems with such conversions. Exercises, solutions, and explanations at the end, as always. You can access the document below:

GRE Math Quick Review: Conversions

For the rest of the GRE Math Quick Review Series, access the portal page.