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.
The methods
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 \(u\) is \(\max_{s\in \mathbb{S}} u(s)\), while the minimum is \(\min_{s\in \mathbb{S}} u(s)\). The mean of \(u\) \(\sum_{s\in \mathbb{S}} u(s)/\mathbb{S}\).
The maxmin normalisation of \([u]\) is the \(u\in [u]\) such that the maximum of \(u\) is \(1\) and the minimum is \(0\).
The maxmean normalisation of \([u]\) is the \(u\in [u]\) such that the maximum of \(u\) is \(1\) and the mean is \(0\).
The maxmean 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 meanmin normalisation of \([u]\) is the \(u\in [u]\) such that the mean of \(u\) is \(1\) and the minimum is \(0\).
Controlling the spread
The last two methods find ways of controlling the spread of possible utilities. For any utility \(u\), define the mean difference: \(\sum_{s,s'\in\mathbb{S}} u(s)u(s')\). And define the variance: \(\sum_{s\in\mathbb{S}} (u(s)\mu)^2\), where \(\mu\) is the mean defined previously.
These lead naturally to:
The mean difference normalisation of \([u]\) is the \(u\in [u]\) such that \(u\) has a mean difference of \(1\).
The variance normalisation of \([u]\) is the \(u\in [u]\) such that \(u\) has a variance of \(1\).
Properties
The different normalisation methods obey the following axioms:

Maxmin 
Maxmean 
Meanmin 
Mean difference 
Variance 
Utility reflection 
YES 
NO 
NO 
YES 
YES 
Cloning indifference 
YES 
NO 
NO 
NO 
NO 
Weak Irrelevance 
YES 
YES 
YES 
NO 
YES 
Strong Irrelevance 
YES 
YES 
YES 
NO 
NO 
As can be seen, maxmin normalisation, despite its crudeness, is the only one that obeys all the properties. If we have a measure on \(\mathbb{S}\), then ignoring the cloning axiom becomes more reasonable. Strong irrelevance can in fact be seen as an antivariance; it’s because of its second order aspect that it fails this.
