Utility vs Probability: idea synthesis

by Stuart_Armstrong2 min read27th Mar 201515 comments


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A putative new idea for AI control; index here.

This post is a synthesis of some of the ideas from utility indifference and false miracles, in an easier-to-follow format that illustrates better what's going on.


Utility scaling

Suppose you have an AI with a utility u and a probability estimate P. There is a certain event X which the AI cannot affect. You wish to change the AI's estimate of the probability of X, by, say, doubling the odds ratio P(X):P(¬X). However, since it is dangerous to give an AI false beliefs (they may not be stable, for one), you instead want to make the AI behave as if it were a u-maximiser with doubled odds ratio.

Assume that the AI is currently deciding between two actions, α and ω. The expected utility of action α decomposes as:

u(α) = P(X)u(α|X) + P(¬X)u(α|¬X).

The utility of action ω is defined similarly, and the expected gain (or loss) of utility by choosing α over ω is:

u(α)-u(ω) = P(X)(u(α|X)-u(ω|X)) + P(¬X)(u(α|¬X)-u(ω|¬X)).

If we were to double the odds ratio, the expected utility gain becomes:

u(α)-u(ω) = (2P(X)(u(α|X)-u(ω|X)) + P(¬X)(u(α|¬X)-u(ω|¬X)))/Ω,    (1)

for some normalisation constant Ω = 2P(X)+P(¬X), independent of α and ω.

We can reproduce exactly the same effect by instead replacing u with u', such that

  • u'( |X)=2u( |X)
  • u'( |¬X)=u( |¬X)


u'(α)-u'(ω) = P(X)(u'(α|X)-u'(ω|X)) + P(¬X)(u'(α|¬X)-u'(ω|¬X)),

2P(X)(u(α|X)-u(ω|X)) + P(¬X)(u(α|¬X)-u(ω|¬X)).    (2)

This, up to an unimportant constant, is the same equation as (1). Thus we can accomplish, via utility manipulation, exactly the same effect on the AI's behaviour as a by changing its probability estimates.

Notice that we could also have defined

  • u'( |X)=u( |X)
  • u'( |¬X)=(1/2)u( |¬X)

This is just the same u', scaled.

The utility indifference and false miracles approaches were just special cases of this, where the odds ratio was sent to infinity/zero by multiplying by zero. But the general result is that one can start with an AI with utility/probability estimate pair (u,P) and map it to an AI with pair (u',P) which behaves similarly to (u,P'). Changes in probability can be replicated as changes in utility.


Utility translating

In the previous, we multiplied certain utilities by two. But by doing so, we implicitly used the zero point of u. But utility is invariant under translation, so this zero point is not actually anything significant.

It turns out that we don't need to care about this - any zero will do, what matters simply is that the spread between options is doubled in the X world but not in the ¬X one.

But that relies on the AI being unable to affect the probability of X and ¬X itself. If the AI has an action that will increase (or decrease) P(X), then it becomes very important where we set the zero before multiplying. Setting the zero in a different place is isomorphic with adding a constant to the X world and not the ¬X world (or vice versa). Obviously this will greatly affect the AI's preferences between X and ¬X.

One way of avoiding the AI affecting X is to set this constant so that u'(X)=u'(¬X), in expectation. Then the AI has no preferences between the two situations, and will not seek to boost one over the other. However, note that u(X) is an expected utility calculation. Therefore:

  1. Choosing the constant so that u'(X)=u'(¬X) requires accessing the AI's probability estimate P for various worlds; it cannot be done from outside, by multiplying the utility, as the previous approach could.
  2. Even if u'(X)=u'(¬X), this does not mean that u'(X|Y)=u'(¬X|Y) for every event Y that could happen before X does. Simple example: X is a coin flip, and Y is the bet of someone on that coin flip, someone the AI doesn't like.

This explains all the complexity of the utility indifference approach, which is essentially trying to decompose possible universes (and adding constants to particular subsets of universes) to ensure that u'(X|Y)=u'(¬X|Y) for any Y that could happen before X does.

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