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 ${max}_{s\in S}u\left(s\right)$, while the minimum is ${min}_{s\in S}u\left(s\right)$. The mean of $u$ ${\sum }_{s\in S}u\left(s\right)/||S||$.

• The max-min normalisation of $\left[u\right]$ is the $u\in \left[u\right]$ such that the maximum of $u$ is $1$ and the minimum is $0$.

• The max-mean normalisation of $\left[u\right]$ is the $u\in \left[u\right]$ such that the maximum of $u$ is $1$ and the mean is $0$.

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 $\left[u\right]$ is the $u\in \left[u\right]$ 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: ${\sum }_{s,s^{\prime }\in S}|u\left(s\right)-u\left(s^{\prime }\right)|$. And define the variance: ${\sum }_{s\in S}\left(u\right(s)-\mu {)}^2$, where $\mu$ is the mean defined previously.

These lead naturally to:

• The mean difference normalisation of $\left[u\right]$ is the $u\in \left[u\right]$ such that $u$ has a mean difference of $1$.

• The variance normalisation of $\left[u\right]$ is the $u\in \left[u\right]$ such that $u$ has a variance of $1$.

Properties

The different normalisation methods obey the following axioms:

Property	Max-min	Max-mean	Mean-min	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, max-min normalisation, despite its crudeness, is the only one that obeys all the properties. If we have a measure on $S$, 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.