Ranking: Paired Comparison Versus Range Voting Activity

Prepared by:

Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk

This activity is a follow-up to having constructed a ranking for a list of items based on paired comparisons carried out without allowing ties. Thus, you carried out the "experiment" of ranking: the following list of topics that are sometimes taught about algebra/arithmetic/geometry to 8th grade students:

1 = solving linear equations
2 = irrational numbers versus rational numbers
3 = slope of a line
4 = evaluate functions
5 = solving two linear equations with two unknowns
6 = Pythagorean theorem
7 = volume of prisms, pyramids and cones

What you did was to interpret Topic A > Topic B (in the list above) to mean that you feel it is more important for 8th grade students to master Topic A than Topic B with no allowance for being indifferent (ties). You have to decide what the word "master" means for you in this context.

Then by using paired comparisons you produced a ranking of these seven topics.

However, there are other approaches rather than paired comparisons for producing rankings. One such method sometimes called range or score voting is carried out as follows. Given a collection of alternatives to rank, you are allowed to "judge" a specific alternative by giving it a score, using from 0 to 99 points as the value you assign to the alternative. Thus, each alternative can be given a "score" from a list of 100 levels. For this particular activity you are not allowed to give the same score to two different alternatives (though in the usual version of range voting this would be allowed). For example, you could give 6 = Pythagorean theorem a 67 and 3 = slope of a line a 24. In this case presumably you "prefer" alternative 6 above alternative 3 because the number 67 is larger than 24. Using the numbers 0 to 99 and the relative sizes of the scores you assign, you can compute a ranking for the seven alternatives above. Give some thought to deciding why you might give, say, 4 = evaluate functions, 72 points rather than 73, 74, 71 or 70 points.

Question 1:

Produce a ranking of the topics 1-7 based on paired comparisons.

Question 2:

Putting aside the work you did to answer Question 1, produce a ranking of the topics 1-7 using range voting (without ties).

Question 3:

Compare the rankings you got for Question 1 and Question 2. Are you convinced you would always get identical rankings by using these two methods? Which of the two approaches is "better?" Which of the two approaches is "more reliable?"

Question 4:

Allow several hours to go by or a day to go by and repeat Questions 1 and 2 without looking at what you did previously. Did you get the same rankings in each case?

You may also want to use paired comparisons versus range voting (no ties) to determine a ranking of the 7 fruits:

Apple, Banana Cherry, Date, Grape, Orange, Raspberry