# Intertheoretic utility comparison: examples

1 min read17th Jul 20191 comment

# 6

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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 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 .

## 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 , 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.

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