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!

-III

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 1.1.3.3 (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)!

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 1.1.3.2 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?

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!