Paired Comparison Activity (2016)

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

Suppose your feelings about fruit are being explored. Which fruits do you like more or less than other fruits? This is your "task." Given a pair of fruits, you have to decide which of the two fruits you prefer to the other.

Example:

Banana and Apple

If you prefer a banana to an apple, you write B > A and if you prefer an apple to a banana, you write A > B. You are not allowed to throw up your hands and say you like them equally. You are not allowed to say I have never heard of these fruits and I cannot compare them." You cannot say, "I prefer a granny smith apple to a green grape but I prefer a red grape to a granny smith apple." The rules of this "game" are that you must decide which you like better for each pair, hopefully, based on your true and honest feelings.

Here are the fruits to consider:

Apple
Banana
Cherry
Fig
Grape
Orange
Raspberry

Question 1:

To compare the fruits above in pairs, how many comparisons will you have to carry out?

Question 2:

Carry out the task of making all the required comparisons!

Question 3:

Based on the data you produced, which fruit is your most favorite?

Question 4:

Based on the data you produced, which fruit is your least favorite?

Question 5:

Use a directed graph (digraph) to represent the data you produced.

Question 6:

Is there a way to use the digraph you produced in Question 5 to determine your most favorite and least favorite fruits?

Question 7:

Based on the data you produced, can you produce a "ranking" of the seven fruits? That is, are you able to say which fruit you liked best, which you liked second best, ...., which fruit you liked least?

Question 8:

Give examples of "applied" situations where in essence the same task is being required but the ranking and paired comparisons involve something other than fruit.

Question 9:

The questions above are framed around the paired comparison of fruits. There are many other choices of things which might change your perceptions of paired comparison as an "effective" tool for obtaining preference information. For example, instead of fruits consider the following list of topics that are sometimes taught as topics in algebra/arithmetic/geometry to 8th grade students:

solving linear equations
irrational numbers versus rational numbers
slope of a line
evaluate functions
solving two linear equations with two unknowns
Pythagorean theorem
volume of pyramids and cones

Repeat what you did in Questions 1 - 8 where this time you interpret Topic A > Topic B 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 and where you are working with topics rather than fruits. You have to decide what the word "master" means for you in this context.

Question 10:

Compare obtaining a ranking of the 7 fruit above by using a paired comparison approach and obtaining a ranking based on assigning for each of the 7 fruits listed above, an integer number from 0 to 100 and using these numbers to rank the fruits. Higher numbers are for more preferred fruits. Also, if after you assign each fruit a number you have two (or more) fruits with the same number, go back and adjust the numbers you gave to the fruits to avoid ties.