Mathematics of Voting, II

The Heisman Trophy

A month or so ago we talked about one aspect of the mathematics of voting discussed by Dr. Orrison during his visit to OU.  In particular, we saw that in an election you can take the very same votes and get different winners depending on how you count the votes.  Of course the example given there was made up and you might think “Hey, crazy things happen when you are allowed to make things up.  I bet that never happens in real life!”

Sam Bradford is a big fan of voting math!

Well, our very own Sam Bradford is the beneficiary of the mathematics of voting!  In the election to determine the winner of the Heisman Trophy the voters select candidates for first place, second place, and third place.  For first place the candidate receives 3 points, for second place 2 points, and for third place 1 point.

Here is the voting data for this year’s Heisman (taken from here):

Player	School	First	Second	Third	Total points
Sam Bradford	Oklahoma	300	315	196	1,726
Colt McCoy	Texas	266	288	230	1,604
Tim Tebow	Florida	309	207	234	1,575
Graham Harrell	Texas Tech	13	44	86	213
Michael Crabtree	Texas Tech	3	27	53	116
Shonn Greene	Iowa	5	9	32	65
Patrick White	West Virginia	3	1	8	19
Nate Davis	Ball State	0	1	8	10
Rey Maualuga	USC	2	1	1	9
Javon Ringer	Michigan State	1	0	5	8

Notice that the Heisman voting method is exactly the same as voting method 3. in our earlier post (it isn’t hard to see that if you scale the points by multiplying by a constant or shift them by adding a constant, then it doesn’t change the order you get by totalling the points.  So multiplying by 2 and adding 1 won’t change the winner in method 3. and, hence, it’s the same as the Heisman voting method).  As we all know, by this voting method Sam Bradford was declared the winner of the Heisman Trophy this year.

Sadly, Tim Tebow doesn't know that you need more than just votes for #1

However, let’s see what happens if you use voting method 1.  Then you only count the first place votes (so, just like in the presidential election, you really only vote for your number one candidate).  Under this scheme Tim Tebow would win the Heisman!

What about voting method 2?   Then you give each first place and second place vote 1 point, and 0 points for third place votes.  Under this scheme Sam Bradford again wins.

Sadly, Colt McCoy was confused and thought being #2 was best

Of course, we’re mathematicians so the first thing we wonder is if there is a scheme where Colt McCoy wins.   This is a good linear algebra problem.  Say first place votes are worth 1 point, second place votes are worth s points, and third place votes are worth t points.  If you assume 1 ≥ s ≥ t ≥ 0, then are there values for s and t so that Colt McCoy wins the Hiesman?

Note that since we can rescale by multiplying by a constant, it doesn’t harm anything to assume that first place is worth 1 point.  And of course second place should be worth less than or equal to first place, and third place should be worth less than or equal to second place.  And of course Javon Ringer is SOL no matter what.