Exercise 3: Borda vs M^cBorda Voting Rules

Navigate to http://www.math.union.edu/locate/MFA-09-2008/voting-rules.html to complete this assignment.

In this demonstration, you can not add new fixed points, but you can add voters to each ranking by repeatedly clicking on the corresponding hex vertex, or remove voters from a vertex by ALT-clicking on it. As the red point moves around, the green point shows which hex vertex is closes to the movable point.

When the mean button is pushed, the label on this closest vertex gives the winner for the standard Borda count election.
When the mediancentre button is pushed, the green line similarly indicates the "M^cBorda" election winner. "M^CBorda" is a new voting rule — we have only just begun to explore its properties.

Show that Borda and M^CBorda are different as voting rules by coming up with a profile^* for which the Borda winner and M^CBorda winners are different. Describe your setup.

^*Recall that a profile describes how many ballots were case for each of the size rankings. You will generate a profile by clicking a certain number of times for each vertex (and zero times is certainly allowed). The total number of times that you clicked corresponds to the total number of voters in your profile.

Try to find an example of a profile in which there is a tie in the Borda election (you will see more than one green line) but no tie in the M^CBorda election.

Is the opposite situation also possible (a profile that yields a tie in the M^CBorda election but no tie in the Borda election?

One attractive property of the M^CBorda rule is that it is very decisive — there are ties, but very few of them. Can you come up with a general rule that states exactly which profiles yield a M^CBorda tie?

Print your results and turn them in to your instructor.