I helped judge a science fair this weekend, and in the process came across an instance of over-mathematization of the world.
The situation was this: I and a group of 5 other people were charged with the task of selecting the top three projects from about a dozen. After going through each project individually and then again as a group while talking with the students who worked on the project, we came up with our own impressions of each project. Now to the decision stage.
We first went down the list of projects and eliminated those that we felt were not 'top three' material. We did this by general consensus: if anyone felt that a project was worth talking about some more then it was added to the list of potential winners. At the end of this we had six projects to consider.
So how do we decide which project is top ranked, which is second ranked, and which is third? This can be viewed as a voting problem, and a method to solve it is called a voting system.
Now, there are lots of voting systems. The most common one is simple majority rule---vote for the choice you most prefer and count how many votes each choice (candidate) receives; the winner is the one with the most vote. Unfortunately, this obviously may not work for more than two choices.
For three or more choices, we could just determine the winner by plurality: the winner is the one with the most votes, though not necessarily the one with the majority of votes. But this method has some problems, most obviously that we need to do more than select a single winner; we also need second and third place. Plurality voting also has the problem that it may encourage voters to not vote for their top-ranked choice, especially if it appears that that their third-ranked choice will win a plurality of the votes. Perhaps by voting strategically (and not true to their preference), a voter can swing the election to their second-ranked choice.
Another method we could use is a Borda count: each person assigns `n-1` points to the top-ranked choice, `n-2` points to the second-ranked choice, and so on to 0 points for the last-ranked choice. Then the points are totaled for each candidate and the winner is the one with the most points (and the second and third place can also be determined). This will find our rankings in one vote.
However, the Borda count also has its problems. In some situations it is possible for a group of voters (or even one voter) to change their rankings and make a winner become a loser while still ranking the original winner first! For example, suppose there are 22 voters with these rankings:
| # voters: | 5 | 4 | 2 | 4 | 1 | 6 |
| 1st | A | A | B | B | C | C |
| 2nd | B | C | A | C | A | B |
| 3rd | C | B | C | A | B | A |
| Points: | ||||||
| A | 10 | 8 | 2 | 0 | 1 | 0 |
| B | 5 | 0 | 4 | 8 | 0 | 6 |
| C | 0 | 4 | 0 | 4 | 2 | 12 |
So A gets 21 points, B gets 23 points and C gets 22 points; the Borda ranking is B-C-A. But now suppose that the 4 voters who prefer B-C-A were to change their preferences to B-A-C. They still prefer B (the previous winner), but now the Borda totals come out to 25 points for A, 23 points for B and 18 points for C. Candidate A has gone from third place to first, and the voters who put him there still prefer B to win!
Other voting systems have other problems. In fact, given a set of quite reasonable criteria that a voting system should satisfy (such as not allowing the situation above), there is no voting system that satisfies all of the criteria. This is upshot of Arrow's Impossibility Theorem.
So which voting system should we have used to rank the projects? In fact we didn't use any of them, for the simple reason that we didn't need to. After ranking the projects individually, we put our rankings on the board and saw what the consensus was. In our case, it was clear which project was ranked first, and that the second-third ranks were between two others. After a brief discussion, we all agreed on an ordering and that was that.
And in our case, that was the best option. There was no need to 'quantify' the results since there was no need for a final, end of discussion vote. Unlike most elections, the group was able to continue discussion about the projects until there was a consensus. Perhaps if we had not been able to reach a consensus we would have had to vote, but luckily that wasn't necessary.
However, one member of the group insisted on getting 'objective' results that would 'quantify' the situation, using what (I think) amounted to the Borda count system. With these results, he argued, we could see that the 'right' result was, and so eliminate possible 'force of personality' effects that may arise during discussion.
But why should the Borda count give the 'right' result? Why should any one quantification be preferred over any other? I could understand why it would be nice to protect the wilting lilies of the world who cannot stand up for themselves, but when our rankings are arrived at independently, then displayed for all to see, I'm not sure that personality problem is very significant. (And lest I be accused of being the forcing personality, I should point out that I said nothing at all during the discussion of our second-third choices.)
I think he places too much emphasis on 'quantified' results, especially in this case when there was no need for them. It was especially curious given that there is no objective way to determine which system we should have been using. After all, what criteria we want our system to satisfy is a matter of preference itself; should we have a vote to determine what voting system we should use?
Mathematics is good, and quantification can be useful, but sometimes it is not necessary for reaching a good decision.