Intelligent Agent Foundations Forumsign up / log in
Intertheoretic utility comparison: examples
discussion post by Stuart Armstrong 31 days ago | discuss

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 max-min normalisation of \([u]\) is the \(u\in [u]\) such that the maximum of \(u\) is \(1\) and the minimum is \(0\).

  • The max-mean normalisation of \([u]\) is the \(u\in [u]\) 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 \([u]\) 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:

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 \(\mathbb{S}\), 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.



NEW LINKS

NEW POSTS

NEW DISCUSSION POSTS

RECENT COMMENTS

The "benign induction
by David Krueger on Why I am not currently working on the AAMLS agenda | 0 likes

This comment is to explain
by Alex Mennen on Formal Open Problem in Decision Theory | 0 likes

Thanks for writing this -- I
by Daniel Dewey on AI safety: three human problems and one AI issue | 1 like

I think it does do the double
by Stuart Armstrong on Acausal trade: double decrease | 0 likes

>but the agent incorrectly
by Stuart Armstrong on CIRL Wireheading | 0 likes

I think the double decrease
by Owen Cotton-Barratt on Acausal trade: double decrease | 0 likes

The problem is that our
by Scott Garrabrant on Cooperative Oracles: Nonexploited Bargaining | 1 like

Yeah. The original generator
by Scott Garrabrant on Cooperative Oracles: Nonexploited Bargaining | 0 likes

I don't see how it would. The
by Scott Garrabrant on Cooperative Oracles: Nonexploited Bargaining | 1 like

Does this generalise to
by Stuart Armstrong on Cooperative Oracles: Nonexploited Bargaining | 0 likes

>Every point in this set is a
by Stuart Armstrong on Cooperative Oracles: Nonexploited Bargaining | 0 likes

This seems a proper version
by Stuart Armstrong on Cooperative Oracles: Nonexploited Bargaining | 0 likes

This doesn't seem to me to
by Stuart Armstrong on Change utility, reduce extortion | 0 likes

[_Regret Theory with General
by Abram Demski on Generalizing Foundations of Decision Theory II | 0 likes

It's not clear whether we
by Paul Christiano on Infinite ethics comparisons | 1 like

RSS

Privacy & Terms