Oscar Math for Best Picture

A course I am teaching – Mathematics for General Education – at a college in Northern California deals with various voting methods. The students are as excited as any movie buff about the 82nd Annual Academy Awards this year (3/7/2010) but they have an additional reason: they are curious to see how the new voting system for Best Picture plays out. They have been learning about the Instant Runoff Voting (IRV) method in my class.

The number of pictures that could nominated for the Best Picture award, from 1946 until last year, was 5. This year the number has been increased to 10 (Avatar, The Blind Side, District 9, An Education, The Hurt Locker, Inglourious Basterds, Precious, A Serious Man, Up and Up in the Air.)

Suppose there are 100 voters to pick a winner. If Up in the Air were to win 51 votes, the contest would be over because it had won a majority of the votes. This is as straightforward as it can get. However, it is rare that in a field of 10, one picture will get a majority of votes in the first round.

Let’s consider a more complicated (some would say, pathological) case. Suppose Up in the Air won 11 votes, eight other pictures won 10 votes each and the remaining picture won 9 votes. According to the old system, Up in the Air would still win because it had the most first-place votes (plurality, as opposed to majority), even though it did not reflect the choice of a whopping 89 voters!

How can a picture rejected by 89% of the voters still win the Oscar? It is precisely to remedy such situations that the IRV method is being used. It is fairer than the Plurality method it is replacing and isn’t that complicated either.

In IRV, voters identify not only their first choice, but second, third and fourth choices, all the way down to tenth in this year’s Best Picture category. In other words, a voter whose first choice is Avatar may identify The Hurt Locker as her second choice, Up in the Air the third, An Education the fourth, and so on. (There are around fifty-eight hundred Academy members who will cast their votes for Best Picture among the nominated 10.)

For simplicity, consider 4 pictures and only two choices to illustrate how IRV method works. Suppose this is how the 100 votes are cast:

Number of voters	33	26	23	18
1st choice	Avatar	Hurt Locker	The Blind Side	Up
2nd choice	Up in the Air	District 9	Avatar	Hurt Locker

None of the pictures won a majority of votes. Since Up has the fewest first-place votes in this example, it gets bumped from the list. The second-place Hurt Locker moves up and takes over its 18 votes. The table now looks like this:

Number of voters	33	26+18 = 44	23
1st choice	Avatar	Hurt Locker	The Blind Side
2nd choice	Up in the Air	District 9	Avatar

But Hurt Locker isn’t the winner yet because still there is no majority (at least 51 in this case). In the second round, eliminate The Blind Side because now it has the fewest first-place votes. Avatar moves up and grabs its 23 votes. The table now looks like this:

Number of voters	33 +23 = 56	26+18 = 44
1st choice	Avatar	Hurt Locker
2nd choice	Up in the Air	District 9

Now there’s a majority winner (56 out of 100) and it’s Avatar!

So how did the students cast their “votes” for The Best Picture? Of the 40 students, one chose District 9 as his first choice, another chose The Hurt Locker and the rest, all 38 of them, chose Avatar!

If Community College math students had their say, it would appear that Avatar would win by an overwhelming majority in the very first round. If only life were that simple!