There's a result that's almost a theorem, which is that an agent that is an expected utility maximiser, is an agent that is stable under self-modification (or the creation of successor sub-agents).
Of course, this needs to be for "reasonable" utility, where no other agent cares about the internal structure of the agent (just its decisions), where the agent is not under any "social" pressure to make itself into something different, where the boundedness of the agent itself doesn't affect its motivations, and where issues of "self-trust" and acausal trade don't affect it in relevant ways, etc...
I know you aren't trying to list all caveats but I think there are others that are other important ways this can go wrong. An agent may not be able to tell that a self-modification will be successful but it may have a high expected utility even as there's some risk of changing one's preferences.
A putative new idea for AI control; index here.
There's a result that's almost a theorem, which is that an agent that is an expected utility maximiser, is an agent that is stable under self-modification (or the creation of successor sub-agents).
Of course, this needs to be for "reasonable" utility, where no other agent cares about the internal structure of the agent (just its decisions), where the agent is not under any "social" pressure to make itself into something different, where the boundedness of the agent itself doesn't affect its motivations, and where issues of "self-trust" and acausal trade don't affect it in relevant ways, etc...
So quite a lot of caveats, but the result is somewhat stronger in the opposite direction: an agent that is not an expected utility maximiser is under pressure to self-modify itself into one that is. Or, more correctly, into an agent that is isomorphic with an expected utility maximiser (an important distinction).
What is this "pressure" agent are "under"? The known result is that if an agent obeys four simple axioms, then its behaviour must be isomorphic with an expected utility maximiser. If we assume the Completeness axiom (trivial) and Continuity (subtle), then violations of Transitivity or Independence correspond to situations where the agent has been money pumped - lost resources or power for no gain at all. The more likely the agent is to face these situations, the more pressure they're under to behave as an expected utility maximiser, or simply lose out.
I have two models for how idealised agents could deal with this sort of pressure. The first, post-hoc, is the unlosing agent I described here. The agent follows whatever preferences it had, but kept track of its past decisions, and whenever it was in a position to violate transitivity or independence in a way that it would suffer from, it makes another decision instead.
Another, pre-hoc, way of dealing with this is to make an "ultra choice" and choose between not decisions, but all possible input output maps (equivalently, between all possible decision algorithms), looking to the expected consequences of each one. This reduces the choices to a single choice, where issues of transitivity or independence need not necessarily apply.
Actual agents will be bounded, unlikely to be able to store and consult their entire history when making every single decision, and unable to look at the whole future of their interactions to make a good ultra choice. So how would they behave?
This is not determined directly by their preferences, but by some sort of meta-preferences. Would they make an approximate ultra-choice? Or maybe build up a history of decisions, and then simplify it (when it gets to large to easily consult) into a compatible utility function? This is also determined by their interactions, as well - an agent that makes a single decision has no pressure to be an expected utility maximiser, one that makes trillions of related decisions has a lot of pressure.
It's also notable that different types of boundedness (storage space, computing power, time horizons, etc...) have different consequences for unstable agents, and would converge to different stable preference systems.
So what is the point of this post? It isn't presenting new results; it's more an attempt to launch a new sub-field of investigation. We know that many preferences are unstable, and that the agent is likely to make them stable over time, either through self-modification, subagents, or some other method. There are also suggestions for preferences that are known to be unstable, but have advantages (such as resistance to Pascal Muggings) that standard maximalisation does not.
Therefore, instead of saying "that agent design can never be stable", we should be saying "what kind of stable design would that agent converge to?", "does that convergent stable design still have the desirable properties we want?" and "could we get that stable design directly?".
The first two things I found in this area were that traditional satisficers could converge to vastly different types of behaviour in an essentially unconstrained way, and that a quasi-expected utility maximiser of utility u might converge to an expected utility maximiser, but it might not be u that it maximises.
In fact, we need not look only at violations of the axioms of expected utility; they are but one possible reason for decision behaviour instability. Here are some that spring to mind:
Now, some categories (such as "Adversaries and social pressure") may not possess a tidy stable solution, but it is still worth asking what setups are more stable than others, and what the convergence rules are expected to be.