Pairwise Relative Comparisons

Absolute Measurement
Measurement is the process of estimating the magnitude of some attribute of an object, such as its length or weight, relative to some standard (unit of measurement), such as a meter or a kilogram. The term is also used to indicate the number that results from that process.  Most measurement is done in this 'absolute' fashion. 

Pairwise Relative Measurement
Pairwise relative measurement consists of comparing two elements, one to the other.  For example, one can compare the relative length of two sticks and estimate that one stick is twice as long as the other.  (Or if one is exactly twice that of the other, one can lay the smaller down next to the larger twice to see that it is exactly half as long).  Because pairwise relative measurement can be applied without the need for a standard, it is possible to measure what used to be thought of as qualitative.  Not only can pairwise relative measures be made objectively (meaning the results should be the same -- subject to measurement error no matter who is doing the measuring), but we can make subjective relative measures.  For example, we can measure the relative sizes of two apples or we can ask subjects how much sweeter one apple tastes than another, or how much more important safety is than esthetics when buying a car.

Pairwise Relative Comparisons to Derive Probability Distributions for Stock Prices
We will be using pairwise relative comparisons in order to derive probability distributions for stock prices.  We will ask participants to compare the relative likelihood of  different stock prices, taken two at a time, at some given point in the future. For example, if a stock is currently selling for $100, do you think that at a specified period of time (e.g. one month) that $100 is equally as likely as $105, less likely, or equally likely as $100?  But the direction is not all we will be information we will be gathering.  We will ask for estimates of the ratios as well.  If the two prices are not as likely, how many times more likely is one than the other? 

Three Modes of Pairwise Relative Comparisons
Pairwise relative comparisons can be made in either of three modes.  We might ask that this judgment be conveyed numerically (e.g. 2 times, or 3.5 times), or graphically (by adjusting the lengths of two bars or two portions of a circle such with a probability wheel), or verbally, using the AHP pairwise verbal mode.

Redundancy and Accuracy
If we had two possible outcomes, say A and B, we would need only one judgment about their relative likelihood to derive the probabilities for these two possible outcomes.  If we said that A and B were equally likely, the probability would be distributed .5 to A and .5 to B.  If we said that A were twice as likely as B, the probability would be distributed 2/3 to A and 1/3 to B.

Suppose we considered three possible outcomes, A, B and C.  We would need only two judgments to derive a probability distribution.  For example, if we said that A is twice as likely as B, and B is three times as likely as C, then using some algebra, we could compute the probability distribution:

However, suppose we add another judgment that A is 5 times more likely than C.  Two things happen.  First, we have introduced some inconsistency because if A were really twice as likely as B and B were really three times as likely as C, then A should be 6 times as likely as C, not 5 times.  Secondly, and more importantly, the resulting estimates of the likelihoods are likely to be more accurate than estimates obtained from just two judgments.  This is analogous to what is typically done using statistics.  If we wanted to measure the length of a board, we could measure it once.  There would be no variance.  If we took additional measurements and averaged them, the variance is no longer zero, but more importantly, the average is more likely to be closer to the true length than the estimate from just one judgment.  One interpretation of the eigenvector computation that we use to derive priorities from pairwise judgments is that it is equivalent to taking the average of an infinite number of paths of lengths 1, 2, 3, ….. involving the pairwise comparisons between the elements being prioritized.  For example, the ratio of two elements a to b can be computed as the average of the direct comparison of a to b, and the comparisons of a to c to b, and a to c to d to b, and a to d to c to b, and so on.

Because of the power of the eigenvector computation with redundant judgments, priorities (or likelihoods) derived from pairwise numerical or graphical judgments are likely to be more accurate than absolute measurement and  even priorities derived from 'fuzzy' pairwise relative verbal comparisons, such as relative likelihood future prices for a stock, can be surprisingly accurate.


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