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?
