Because of the tremendous interest by my students in the recent presidential election, I decided on Thursday to scrap the previously-planned lesson and instead to do a lesson on voting theory. We compared :
* plurality voting (our current system, where the person with the greatest number of votes, even if only around 27%, wins -- and that happens to be the percentage tallied by the winner in my school board district);
* runoff voting, where one takes the top two vote-getters and holds a runoff election between them;
* repeated runoff voting, where one takes the lowest vote-getter and eliminates him or her, and does a runoff between the remainder, repeating if necessary until a majority is reached;
* Borda counting, where the voters give a numerical preference to the candidates, and an algorithm of some sort is used to decide how many points to give to each voter's first choice, second choice, third choice, and so on.
(I mentioned that some countries have instant-runoff voting. Students complained about the Electoral College and all of the irregularities in this year's election, but that was not the main focus of the lesson.)
I made up an example with four candidates, and the following preferences among the following groups of voters. The candidates were Julia, Kate, Larry, and Max. From memory, I think I had the preferences as follows:
# of voters first choice second choice third choice fourth choice 6 Julia Kate Larry Max 8 Max Kate Larry Julia 5 Larry Kate Julia Max 3 Kate Julia Larry Max
(obviously there are 24 ways of arranging preferences for 4 candidates, but I was trying to keep it simple.)
This means that for 6 voters, Julia is the first choice, but they think that Max is the worst; and so on...
Max wins the plurality vote, even though 14 of the voters think that Max is the worst.
Julia wins a runoff between the 2 top vote getters by 14 to 8, a majority, because she gets the votes from those who wanted Larry or Kate as their first choice.
In a repeated runoff count, Kate is eliminated from the first round, and those votes go to Julia. In the second round, Julia has 6+3=9 votes, Max has 8, and Larry has 5, so he is eliminated, and his votes again go to Julia, who wins again in the third round, 14 to 8 against Max.
In a borda count, we award 3 points to 1st choice, 2 to 2nd choice, 3 to 3rd choice, and 0 to 4th choice. So Julia gets 6*3 + 8*0 + 5*1 + 3*2 = 29 points. And Max gets 24 + 0 + 0+0=24. And Larry gets 6+8+15+3=32. And Kate gets 12+16+15+9=52, and wins by a lot.
Notice that Kate was either the first or second choice of all of the voters. I wonder how McCain would have done if we used a Borda count?
My students generally concluded that the plurality method we use is the worst of the four methods I mentioned, and that the Borda count is the best at actually expressing mathematically the will of the people.
No, I did not go into Condorcet voting. I am not of the opinion that it is a good way of doing voting. There are too many cases where voters will prefer A to B, and B to C, and C to A. Thus we become irrational because we are intransitive.
Lani Guinier was blackballed because she proposed a slight variation on the Borda count: a voter could cast all of his or her points for 1 candidate, i.e. calling one candidate your first AND second AND third AND fourth AND fifth AND sixth choice if one wants. To me, that minor variation is not all that important.
Certainly a Borda count could be handled by voting machines. With a well-designed national or state-wide ballot, we could have instructions that made more sense than what they had to face in Palm Beach County, FL. And clearly there are voting machines that will simply not permit a voter to attempt to cast a vote that does not follow the rules, hence forcing the voter to re-do his or her ballot until it makes sense.
But it'll never happen. Too bad.
My souces for this lesson were, more or less, the COMAP book For All Practical purposes, which is an interesting source of ideas; columns I had read a long time ago in Scientific American by Martin Gardner and I suspect others on Kenneth Arrow's apparent proof that the only type of election procedure that actually follows a few apparently simple axioms is a dictatorship; and a much more recent article on voting theory by a person whose name I cannot recall right now because I forgot to bookmark it and instead simply printed out and took to school. He contradicts Arrow. I will attempt to find this citation on Monday.