Pascal's wager-like situations come up occasionally with expected utility, making some decisions very tricky. It means that events of the tiniest of probability could dominate the whole decision - intuitively unobvious, and a big negative for a bounded agent - and that expected utility calculations may fail to converge.

There are various principled approaches to resolving the problem, but how about an unprincipled approach? We could try and bound utility functions, but the heart of the problem is not high utility, but hight utility combined with low probability. Moreover, this has to behave sensibly with respect to updating.

The agent design

Consider a UDT-ish agent A looking at input-output maps {M} (ie algorithms that could determine every single possible decision of the agent in the future). We allow probabilistic/mixed output maps as well (hence A has access to a source of randomness). Let u be a utility function, and set 0 < ε << 1 to be the precision. Roughly, we'll be discarding the highest (and lowest) utilities that are below probability ε. There is no fundamental reason that the same ε should be used for highest and lowest utilities, but we'll keep it that way for the moment.

The agent is going to make an "ultra-choice" among the various maps M (ie fixing its future decision policy), using u and ε to do so. For any M, designate by A(M) the decision of the agent to use M for its decisions.

Then, for any map M, set max(M) to be the lowest number s.t P(u ≥ max(M)|A(M)) ≤ ε. In other words, if the agent decides to use M as its decision policy, this is the maximum utility that can be achieved if we ignore the highest valued ε of the probability distribution. Similarly, set min(M) to be the highest number s.t. P(u ≤ min(M)|M) ≤ ε.

Then define the utility function uMε, which is simply u, bounded between max(M) and min(M). Now calculate the expected value of uMε given A(M), call this Eε(u|A(M)).

The agent then chooses the M that maximises Eε(u|A(M)). Call this the ε-precision u-maximising algorithm.

Stability of the design

The above decision process is stable, in that there is a single ultra-choice to be made, and clear criteria for making that ultra-choice. Realistic and bounded agents, however, cannot calculate all the M in sufficient detail to get a reasonable outcome. So we can ask whether the design is stable for a bounded agent.

Note that this question is underdefined, as there are many ways of being bounded, and many ways of cashing out ε-precision u-maximising into bounded form. Most likely, this will not be a direct expected utility maximalisation, so the algorithm will be unstable (prone to change under self-modification). But how exactly it's unstable is an interesting question.

I'll look at one particular situation: one where A was tasked with creating subagents that would go out and interact with the world. These agents are short-sighted: they apply ε-precision u-maximising not to the ultra-choice, but to each individual expected utility calculation (we'll assume the utility gains and losses for each decision is independent).

A has a single choice: what to set ε to for the subagents. Intuitively, it would seem that A would set ε lower than its own value; this could correspond roughly to an agent self-modifying to remove the ε-precision restriction from itself, converging on becoming a u-maximiser. However:

Theorem: There are (stochastic) worlds in which A will set the subagent precision to be higher, lower or equal to its own precision ε.

The proof will be by way of illustration of the interesting things that can happen in this setup. Let B be the subagent whose precision A sets.

Let C(p) be a coupon that pays out 1 with probability p. xC(p) simply means the coupon pays out x instead of 1. Each coupon costs ε2 utility. This is negligible, and only serves to break ties. Then consider the following worlds:

#. In W1, B will be offered the possibility of buying C(0.75ε).
#. In W2, B will be offered the possibility of buying C(1.5ε).
#. In W3, B will be offered the possibility of buying C(0.75ε), and the offer will be made twice.
#. In W4, B will be offered, with 50% probability, the possibility of buying C(1.5ε).
#. In W5, B will be offered, with 50% probability, the possibility of buying C(1.5ε), and otherwise the possibility buying 2C(1.5ε).
#. In W6, B will be offered, with 50% probability, the possibility of buying C(0.75ε), and otherwise the possibility buying 2C(1.5ε).
#. In W7, B will be offered, with 50% probability, the possibility of buying C(0.75ε), and otherwise the possibility buying 2C(1.05ε).

From A’s perspective, the best input-output maps are: in W1, don’t buy, in W2, buy, in W3, buy both, in W4, don’t buy (because the probability of getting above 0 utility by buying, is, from A's initial perspective, 1.5ε/2 = 0.75ε).

W5 is more subtle, and interesting – essentially A will treat 2C(1.5ε) as if it were C(1.5ε) (since the probability of getting above 1 utility by buying is 1.5ε/2 = 0.75ε, while the probability of getting above zero by buying is (1.5ε+1.5ε)/2=1.5ε). Thus A would buy everything offered.

Similarly, in W6, the agent would buy everything, and in W7, the agent would buy nothing (since the probability of getting above zero by buying is now (1.05ε + 0.75ε)/2 = 0.9ε).

So in W1 and W2, the agent can leave the sub-agent precision at ε. In W2, it needs to lower it below 0.75ε. In W4, it needs to raise it above 1.5ε. In W5 it can leave it alone, while in W6 it must lower it below 0.75ε, and in W7 it must raise it above 1.05ε.

Irrelevant information

One nice feature about this approach is that it ignores irrelevant information. Specifically:

Theorem: Assume X is a random variable that is irrelevant to the utility function u. If A (before knowing X) has to design successor agents that will exist after X is revealed, then (modulo a few usual assumptions about only decisions mattering, not internal thought processes) it will make these successor agents isomorphic to copies of itself, i.e. ε-precision u-maximising algorithms (potentially with a different way of breaking ties).

These successor agents are not the short-sighted agents of the previous model, but full ultra-choice agents. Their ultra-choice is over all decisions to come, while A's ultra-choice (which is simply a choice) is over all agent designs.

For the proof, I'll assume X is boolean valued (the general proof is similar). Let M be the input-output map A would choose for itself, if it were to make all the decisions itself rather than just designing a subagent. Now, it's possible that M(X) will be different from M(¬X) (here M(X) and M(¬X) are contractions of the input-output map by adding in one of the inputs).

Define the new input-ouput map M' by defining a new internal variable Y in A (recall that A has access to a source of randomness). Since this variable is new, M is independent of the value of Y. Then M' is defined as M with X and Y permuted. Since both Y and X are equally irrelevant to u, Eε(u|A(M))=Eε(u|A(M')), so M' is an input output map that fulfils the ε-precision u-maximising. And M'(X)=M'(¬X), so M' is independent of X.

Now consider the subagent that runs the same algorithm as A, and has seen X. Because of the irrelevance of X, M'(X) will still fulfil ε-precision u-maximising (we can express any fact relevant to u in the form of Zs, with P(Z)=P(Z|X), and then the algorithm is the same).

Similarly, a subagent that has seen ¬X will run M'(¬X). Putting these together, the subagent will expect to run M'(X) with probability P(X) and M'(¬X) with probability P(¬X)=1-P(X).

Since M'(X)=M'(¬X), this whole thing is just M'. So if A creates a copy of itself (possibly tweaking the tie-breaking so that M' is selected), then it will achieve its maximum according to ε-precision u-maximising.

A putative new idea for AI control; index here.Pascal's wager-like situations come up occasionally with expected utility, making some decisions very tricky. It means that events of the tiniest of probability could dominate the whole decision - intuitively unobvious, and a big negative for a bounded agent - and that expected utility calculations may fail to converge.

There are various principled approaches to resolving the problem, but how about an unprincipled approach? We could try and bound utility functions, but the heart of the problem is not high utility, but hight utility combined with low probability. Moreover, this has to behave sensibly with respect to updating.

## The agent design

Consider a UDT-ish agent A looking at input-output maps {M} (ie algorithms that could determine every single possible decision of the agent in the future). We allow probabilistic/mixed output maps as well (hence A has access to a source of randomness). Let u be a utility function, and set 0 < ε << 1 to be the precision. Roughly, we'll be discarding the highest (and lowest) utilities that are below probability ε. There is no fundamental reason that the same ε should be used for highest and lowest utilities, but we'll keep it that way for the moment.

The agent is going to make an "ultra-choice" among the various maps M (ie fixing its future decision policy), using u and ε to do so. For any M, designate by A(M) the decision of the agent to use M for its decisions.

Then, for any map M, set max(M) to be the lowest number s.t P(u ≥ max(M)|A(M)) ≤ ε. In other words, if the agent decides to use M as its decision policy, this is the maximum utility that can be achieved if we ignore the highest valued ε of the probability distribution. Similarly, set min(M) to be the highest number s.t. P(u ≤ min(M)|M) ≤ ε.

Then define the utility function uMε, which is simply u, bounded between max(M) and min(M). Now calculate the expected value of uMε given A(M), call this Eε(u|A(M)).

The agent then chooses the M that maximises Eε(u|A(M)). Call this the ε-precision u-maximising algorithm.

## Stability of the design

The above decision process is stable, in that there is a single ultra-choice to be made, and clear criteria for making that ultra-choice. Realistic and bounded agents, however, cannot calculate all the M in sufficient detail to get a reasonable outcome. So we can ask whether the design is stable for a bounded agent.

Note that this question is underdefined, as there are many ways of being bounded, and many ways of cashing out ε-precision u-maximising into bounded form. Most likely, this will not be a direct expected utility maximalisation, so the algorithm will be unstable (prone to change under self-modification). But how exactly it's unstable is an interesting question.

I'll look at one particular situation: one where A was tasked with creating subagents that would go out and interact with the world. These agents are short-sighted: they apply ε-precision u-maximising not to the ultra-choice, but to each individual expected utility calculation (we'll assume the utility gains and losses for each decision is independent).

A has a single choice: what to set ε to for the subagents. Intuitively, it would seem that A would set ε lower than its own value; this could correspond roughly to an agent self-modifying to remove the ε-precision restriction from itself, converging on becoming a u-maximiser. However:

The proof will be by way of illustration of the interesting things that can happen in this setup. Let B be the subagent whose precision A sets.

Let C(p) be a coupon that pays out 1 with probability p. xC(p) simply means the coupon pays out x instead of 1. Each coupon costs ε2 utility. This is negligible, and only serves to break ties. Then consider the following worlds:

#. In W1, B will be offered the possibility of buying C(0.75ε). #. In W2, B will be offered the possibility of buying C(1.5ε). #. In W3, B will be offered the possibility of buying C(0.75ε), and the offer will be made twice. #. In W4, B will be offered, with 50% probability, the possibility of buying C(1.5ε). #. In W5, B will be offered, with 50% probability, the possibility of buying C(1.5ε), and otherwise the possibility buying 2C(1.5ε). #. In W6, B will be offered, with 50% probability, the possibility of buying C(0.75ε), and otherwise the possibility buying 2C(1.5ε). #. In W7, B will be offered, with 50% probability, the possibility of buying C(0.75ε), and otherwise the possibility buying 2C(1.05ε).

From A’s perspective, the best input-output maps are: in W1, don’t buy, in W2, buy, in W3, buy both, in W4, don’t buy (because the probability of getting above 0 utility by buying, is, from A's initial perspective, 1.5ε/2 = 0.75ε).

W5 is more subtle, and interesting – essentially A will treat 2C(1.5ε) as if it were C(1.5ε) (since the probability of getting above 1 utility by buying is 1.5ε/2 = 0.75ε, while the probability of getting above zero by buying is (1.5ε+1.5ε)/2=1.5ε). Thus A would buy everything offered.

Similarly, in W6, the agent would buy everything, and in W7, the agent would buy nothing (since the probability of getting above zero by buying is now (1.05ε + 0.75ε)/2 = 0.9ε).

So in W1 and W2, the agent can leave the sub-agent precision at ε. In W2, it needs to lower it below 0.75ε. In W4, it needs to raise it above 1.5ε. In W5 it can leave it alone, while in W6 it must lower it below 0.75ε, and in W7 it must raise it above 1.05ε.

## Irrelevant information

One nice feature about this approach is that it ignores irrelevant information. Specifically:

These successor agents are not the short-sighted agents of the previous model, but full ultra-choice agents. Their ultra-choice is over all decisions to come, while A's ultra-choice (which is simply a choice) is over all agent designs.

For the proof, I'll assume X is boolean valued (the general proof is similar). Let M be the input-output map A would choose for itself, if it were to make all the decisions itself rather than just designing a subagent. Now, it's possible that M(X) will be different from M(¬X) (here M(X) and M(¬X) are contractions of the input-output map by adding in one of the inputs).

Define the new input-ouput map M' by defining a new internal variable Y in A (recall that A has access to a source of randomness). Since this variable is new, M is independent of the value of Y. Then M' is defined as M with X and Y permuted. Since both Y and X are equally irrelevant to u, Eε(u|A(M))=Eε(u|A(M')), so M' is an input output map that fulfils the ε-precision u-maximising. And M'(X)=M'(¬X), so M' is independent of X.

Now consider the subagent that runs the same algorithm as A, and has seen X. Because of the irrelevance of X, M'(X) will still fulfil ε-precision u-maximising (we can express any fact relevant to u in the form of Zs, with P(Z)=P(Z|X), and then the algorithm is the same).

Similarly, a subagent that has seen ¬X will run M'(¬X). Putting these together, the subagent will expect to run M'(X) with probability P(X) and M'(¬X) with probability P(¬X)=1-P(X).

Since M'(X)=M'(¬X), this whole thing is just M'. So if A creates a copy of itself (possibly tweaking the tie-breaking so that M' is selected), then it will achieve its maximum according to ε-precision u-maximising.