EXAMPLE 5 Rank Methods and a Basketball Poll

In February 2014, the Associated Press issued its weekly ranking of the top 25 teams in men’s college basketball, shown at right.

An interesting question is whether or not this is a ranking system. If it is, who are the candidates and how many are there? In fact, this can be regarded as a ranking system, but the number of candidates is not 25. That is, although 25 teams appeared on each ballot, at least one ballot included each of the teams listed at the bottom in the category “Others receiving votes.”

For this to be regarded as a ranking system, the set of candidates must include the entire set of eligible collegiate men’s basketball teams. We must also infer that each ballot lists all teams other than that voter’s top 25 below that voter’s top 25, perhaps in alphabetical order. The point assignments are then like those in the newspaper clipping, except that we also assign 0 points for a 26th place vote, 0 points for a 27th place vote, and so on. This is why our definition states that a rank method “assigns points in a nonincreasing manner” instead of “assigns points in a decreasing manner.”

We can use this poll to illustrate how total points are arrived at with a ranking method. With the top-ranked team, Syracuse, it’s quite easy. Each first-place vote is worth 25, and Syracuse received all 65 first-place votes. This accounts for its total of points. But the calculation is more interesting for the second-ranked team, Arizona, and requires some speculation on our part because we don’t actually have the ballots to examine. We know that there were 65 ballots (because there were exactly 65 first-place votes altogether), and we know that Arizona had no first-place votes. It stands to reason that Arizona’s 1517 points must have come from a vast majority of the second-place votes, together with a few lower rankings. (We know that the team didn’t receive all the second place-votes; otherwise, its point total would have been .)

One possibility is that Arizona received:

The total would then be