What Is The Impact Of Inconsistency?

Usually we cannot be so certain of our judgments that we would insist on forcing consistency in the pairwise comparison matrix. Rather, we guess our feelings or judgments in all the positions except the diagonal ones (which are always 1), force the reciprocals in the transpose positions, and look for an answer. We may not be perfectly consistent, but that is the way we tend to work. (It is also the way we grow. When we integrate new experiences into our consciousness, previous relationships may change and some consistency is lost. As long as there is enough consistency to maintain coherence among the objects of our experience, the consistency need not be perfect.) It is useful to remember that most new ideas that affect our lives tend to cause us to rearrange some of our preferences, thus making us inconsistent with our previous commitments. If we were to program ourselves never to change our minds, we would be afraid to accept new ideas. All knowledge has to be admitted into our narrow corridor between tolerable inconsistency and perfect consistency.

As a rule of thumb, we do not recommend proceeding if the consistency ratio is more than about 0.10 for n > 4. For n = 3 and 4 we recommend that the C.R. be less than 0.05 and 0.09 respectively. Thus in general when asked, we require that C.R. not exceed 0.10 by much. How do you explain this outcome in general?

The notion of order of magnitude is essential in any mathematical consideration of changes in measurement. When one has a numerical value say between 1 and 10 for some measurement and one wants to determine whether change in this value is significant or not, one reasons as follows: A change of a whole integer value is critical because it changes the magnitude and identity of the original number significantly. If the change or perturbation in value is of the order of a percent or less, it would be so small (by two orders of magnitude) and would be considered negligible. However if this perturbation is a decimal (one order of magnitude smaller) we are likely to pay attention to modify the original value by this decimal without losing the significance and identity of the original number as we first understood it to be. Thus in synthesizing near consistent judgment values, changes that are too large can cause dramatic change in our understanding, and values that are too small cause no change in our understanding. We are left with only values of one order of magnitude smaller that we can deal with incrementally to change our understanding. It follows that our allowable consistency ratio should be not more than about .10. The requirement of 10% cannot be made smaller such as 1% or 0.1% without trivializing the impact of inconsistency. But inconsistency itself is important because without it, new knowledge that changes preference cannot be admitted.