A previous post introduced the theory of intertheoretic utility comparison. This post will give examples of how to do that comparison, by normalising individual utility functions.
All methods presented here obey the axioms of Relevant data, Continuity, Individual normalisation, and Symmetry. Later, we'll see which ones follow Utility reflection, Cloning indifference, Weak irrelevance, and Strong irrelevance.
Max, min, mean
The maximum of a utility function is , while the minimum is . The mean of .
The max-min normalisation of is the such that the maximum of is and the minimum is .
The max-mean normalisation of is the such that the maximum of is and the mean is .
The max-mean normalisation has an interesting feature: it's precisely the amount of utility that an agent completely ignorant of its own utility, would pay to discover that utility (as a otherwise the agent would employ a random, 'mean', strategy).
For completeness, there is also:
- The mean-min normalisation of is the such that the mean of is and the minimum is .
Controlling the spread
The last two methods find ways of controlling the spread of possible utilities. For any utility , define the mean difference: . And define the variance: , where is the mean defined previously.
These lead naturally to:
The mean difference normalisation of is the such that has a mean difference of .
The variance normalisation of is the such that has a variance of .
The different normalisation methods obey the following axioms:
As can be seen, max-min normalisation, despite its crudeness, is the only one that obeys all the properties. If we have a measure on , then ignoring the cloning axiom becomes more reasonable. Strong irrelevance can in fact be seen as an anti-variance; it's because of its second order aspect that it fails this.