This is a continuation of my last post.
Risk is probability times consequence of a given scenario. The Wikipedia gives an example of a risk matrix as:
Risk has been granularized to reflect uncertainty, lack of data, and subjectivity. The vertical axis is probability. The horizontal axis is consequence. As can be seen from the figures in my previous post, this matrix is logically inconsistent with respect to iso-risk curves (lines of constant risk) whether or not the axis scales are linear or log-log. That is, for example, it is mathematically impossible to have any value other than "Low" in the first row or first column if the scales are linear. Or different values for any diagonal if the axes are log-log.
However, IMHO, the above example does not illustrate what is actually wrong with risk matrices. It is that these table values are being interpreted as risk values in the first place. Instead, risk matrix table values should be interpreted as risk categories. They measure our level of concern about a particular category of risk, not its actual value. The axes measure the "worst case scenario" values of probability and consequence for a given risk.
A risk value for any expected scenario at, say, a nuclear power plant could never be anything other than "Low". The very idea of an acceptable "Extreme" risk of "Certain" probability and "Catastrophic" consequences is ridiculous. No stakeholder would sign-off on that. To be of practical utility, the table values cannot be interpreted as risk values.
A risk category is defined as the management process used to manage risks that are at a certain level of concern. There should be stakeholder consensus on risk categories. Risk categories can be ordered by the amount of process grading and tailoring they allow for ordered levels of concern.
For example, for the most "Extreme" risk category, it may be required that each and every risk, regardless of its probability or consequence in a worst case scenario, be formally documented, analyzed to an extreme level of detail and rigor, prioritized, mitigated to an extremely low level, tracked throughout its lifetime, and signed-off on periodically by all stakeholders. Risk categories of lower concern (High, Moderate, and Low) will allow grading and tailoring of these Extreme requirements to lower levels.