Elliott Thornley

I'm a Postdoctoral Research Fellow at Oxford University's Global Priorities Institute.

Previously, I was a Philosophy Fellow at the Center for AI Safety.

So far, my work has mostly been about the moral importance of future generations. Going forward, it will mostly be about AI.

You can email me at __elliott.thornley@philosophy.ox.ac.uk__.

The point is: there are no theorems which state that, unless an agent can be represented as maximizing expected utility, that agent is liable to pursue strategies that are dominated by some other available strategy. The VNM Theorem doesn't say that, nor does Savage's Theorem, nor does Bolker-Jeffrey, nor do Dutch Books, nor does Cox's Theorem, nor does the Complete Class Theorem.

But suppose we instead define 'coherence theorems' as theorems which state that

If you are not shooting yourself in the foot in sense X, we can view you as having coherence property Y.

Then you can fill in X and Y any way you like. Either it will turn out that there are no coherence theorems, or it will turn out that coherence theorems cannot play the role they're supposed to play in coherence arguments.

I think of coherence theorems loosely as things that say

ifan agent follows such and such principles,thenwe can prove it will have a certain property.

If you use this definition, then VNM (etc.) counts as a coherence theorem. But Premise 1 of the coherence argument (as I've rendered it) remains false, and so you can't use the coherence argument to get the conclusion that sufficiently-advanced artificial agents will be representable as maximizing expected utility.

I’m following previous authors in defining ‘coherence theorems’ as

theorems which state that, unless an agent can be represented as maximizing expected utility, that agent is liable to pursue strategies that are dominated by some other available strategy.

On that definition, there are no coherence theorems. VNM is not a coherence theorem, nor is Savage’s Theorem, nor is Bolker-Jeffrey, nor are Dutch Book Arguments, nor is Cox’s Theorem, nor is the Complete Class Theorem.

there are theorems that are relevant to the question of agent coherence

I'd have no problem with authors making that claim.

I would urge the author to create an actual concrete situation that doesn't seem very dumb in which a highly intelligence, powerful and economically useful system has non-complete preferences

Working on it.

Great post! Lots of cool ideas. Much to think about.

But I still think both these claims are wrong.

And that’s because you only consider one rule for decision-making with incomplete preferences: a myopic veto rule, according to which the agent turns down a trade if the offered option is ranked lower than its current option according to one or more of the agent’s utility functions.

The myopic veto rule does indeed lead agents to pursue dominated strategies in single-sweetening money-pumps like the one that you set out in the post. I made this point in my coherence theorems post:

But the myopic veto rule isn’t the only possible rule for decision-making with incomplete preferences. Here’s another. I can’t think of a better label right now, so call it ‘Caprice’ since it’s analogous to Brian Weatherson’s rule of the same name for decision-making with multiple probability functions:

The Caprice Rule implies the policy that I suggested in my coherence theorems post:

And that makes the agent immune to single-souring money-pumps (in which the agent first has the opportunity to trade in A for B and then (conditional on making that trade) has the opportunity to trade in B for A-).

The Caprice Rule also implies the following policy:

And that makes the agent immune to single-sweetening money-pumps like the one that you discuss. If the agent recognises that – conditional on trading in mushroom (analogue in my post: A) for anchovy (B) – they will be able to trade in anchovy (B) for pepperoni (A+), then they will make at least the first trade, and thereby avoid pursuing a dominated strategy. As a result, an agent abiding by the Caprice Rule can’t shift probability mass from mushroom (A) to pepperoni (A+) by probabilistically precommitting to take certain trades in a way that makes their preferences complete. The Caprice Rule already does the shift.

And an agent abiding by the Caprice Rule can’t be represented as maximising utility, because its preferences are incomplete. In cases where the available trades

aren’tarranged in some way that constitutes a money-pump, the agent can prefer (/reliably choose) A+ over A, and yet lack any preference between (/stochastically choose between) A+ and B, and lack any preference between (/stochastically choose between) A and B. Those patterns of preference/behaviour are allowed by the Caprice Rule.For a Caprice-Rule-abiding agent to avoid pursuing dominated strategies in single-sweetening money-pumps, that agent must be non-myopic: specifically, it must recognise that trading in A for B and then B for A+ is an available sequence of trades. And you might think that this is where my proposal falls down:

actualagents will sometimes be myopic, so actual agents can’t always use the Caprice Rule to avoid pursuing dominated strategies, so actual agents are incentivised to avoid pursuing dominated strategies by instead probabilistically precommitting to take certain trades in ways that make their preferences complete (as you suggest).But there’s a problem with this response. Suppose an agent is myopic. It finds itself with a choice between A and B, and it chooses A. As a matter of fact, if it had chosen B, it would have later been offered A+. Then the agent leaves with A when it could have had A+. But since the agent is myopic, it won’t be aware of this fact, and so note two things. First, it’s unclear whether the agent’s behaviour deserves the name ‘dominated strategy’. The agent pursues a dominated strategy only in the same sense that I pursue a dominated strategy when I fail to buy a lottery ticket that (unbeknownst to me) would have won. Second and more importantly, the agent’s failure to get A+ won’t lead the agent to change its preferences, since it’s myopic and so unaware that A+ was available.

And so we seem to have a dilemma for money-pumps for completeness. In money-pumps where the agent is non-myopic about the available sequences of trades, the agent can avoid pursuit of dominated strategies by acting in accordance with the Caprice Rule. In money-pumps where the agent is myopic, failing to get A+ exerts no pressure on the agent to change its preferences, since the agent is not aware that it could have had A+.

You recognise this in the post and so set things up as follows: a non-myopic optimiser decides the preferences of a myopic agent. But this means your argument doesn’t vindicate coherence arguments as traditionally conceived. Per my understanding, the conclusion of coherence arguments was supposed to be: you can’t rely on advanced agents not to act like expected-utility-maximisers, because even if these agents start off not acting like EUMs, they’ll recognise that acting like an EUM is the only way to avoid pursuing dominated strategies. I think that’s false, for the reasons that I give in my coherence theorems post and in the paragraph above. But in any case, your argument doesn’t give us that conclusion. Instead, it gives us something like: a non-myopic optimiser of a myopic agent can shift probability mass from less-preferred to more-preferred outcomes by probabilistically precommitting the agent to take certain trades in a way that makes its preferences complete. That’s a cool result in its own right, and maybe your post isn’t trying to vindicate coherence arguments as traditionally conceived, but it seems worth saying that it doesn’t.

You’re right that a non-myopic designer might set things up so that their myopic agent’s preferences are complete. And maybe SGD makes this hard to avoid. But if I’m right about the shutdown problem, we as non-myopic designers should try to set things up so that our agent’s preferences are incomplete. That’s our best shot at getting a corrigible agent. Training by SGD might present an obstacle to this (I’m still trying to figure this out), but coherence arguments don’t.

That’s how I think the argument in your post can be circumvented, and why I still think we can use incomplete preferences for shutdownability/corrigibility:

That’s the main point I want to make. Here’s a more minor point: I think that even in the case where you have a non-myopic optimiser deciding the preferences of a myopic agent, non-domination by itself doesn’t imply utility maximisation. You also need the assumption that the non-myopic optimiser takes some kinds of money-pumps to be more likely than others. Here’s an example to illustrate why I think that. Suppose that our non-myopic optimiser predicts that each of the following money-pumps are equally likely to occur, with probability 0.5. Call the first ‘the A+ money-pump’ and the second ‘the B+ money-pump’:

## A+ money-pump

## B+ money-pump

The non-myopic optimiser knows that the agent will be myopic in deployment. Currently, the agent’s preferences are incomplete: it lacks a preference between A and B. Either it abides by the veto rule and sticks with whatever it already has, or it chooses stochastically between A and B. That difference won’t matter here: we can just say that the agent chooses A with probability p and chooses B with probability 1-p. The non-myopic optimiser is considering precommitting the agent to choose either A or B with probability 1, with the consequence that the agent’s preferences would then be complete. Does precommitting dominate not precommitting?

No. The agent pursues a dominated strategy if and only if the A+ money-pump occurs and the agent chooses A or the B+ money-pump occurs and the agent chooses B. As it stands, those probabilities are 0.5, p, 0.5, and 1-p respectively, so that the agent’s probability of pursuing a dominated strategy is 0.5p+0.5(1-p)=0.5. And the non-myopic optimiser can’t change this probability by precommitting the agent to choose A or B. Doing so changes only the value of p, and 0.5p+0.5(1-p)=0.5 no matter what the value of p.

That’s why I think you also need the assumption that the non-myopic optimiser believes that the myopic agent is more likely to encounter some kinds of money-pumps than others in deployment. The non-myopic optimiser has to think, e.g., that the A+ money-pump is more likely than the B+ money-pump. Then making the agent’s preferences complete can decrease the probability that the agent pursues a dominated strategy. But note a few things:

(1) If the probabilities of the A+ money-pump and the B+ money-pump are each non-zero, then precommitting the agent to choose one of A and B doesn’t

justshift probability mass from a less-preferred outcome to a more-preferred outcome. It also shifts probability mass between A and B, and between A+ and B+. For example, precommitting to always choose A sends the probability of B and of A+ down to zero. And it’s not so clear that the new probability distribution is superior to the old one. This new probability distribution does give a smaller probability of the agent pursuing a dominated strategy, but minimising the probability of pursuing a dominated strategy isn’t always best. Consider an example with complete preferences:## First A- money-pump

## Second A- money-pump

Suppose the probability of the First A- money-pump is 0.6 and the probability of the Second A- money-pump is 0.4. Then precommitting to always choose A- minimises the probability of pursuing a dominated strategy. But if the difference in value between A- and A is much greater than the difference in value between A and A+, then it would be better to precommit to choosing A.

(2) As the point above suggests, given your set-up of a non-myopic optimiser deciding the preferences of a myopic agent, and the assumption that some kinds of decision-trees are more likely than others, it can also be that the non-myopic optimiser can decrease the probability that an agent with

completepreferences pursues a dominated strategy by precommitting the agent to take certain trades. You make something like this point in the ‘Value vs Utility’ section: if there are lots of vegetarians around, you might want to trade down to mushroom pizza. And you can see it by considering the First A- money-pump above: if that’s especially likely, the non-myopic optimiser might want to precommit the agent to trade in A for A-. This makes me think that the lesson of the post is more about the instrumental value of commitments in your non-myopic-then-myopic setting than it is about incomplete preferences.(3) Return to the A+ money-pump and the B+ money-pump from above, and suppose that their probabilities are 0.6 and 0.4 respectively. Then the non-myopic optimiser can decrease the probability of the myopic agent pursuing a dominated strategy by precommitting the agent to always choose B, but doing so will only send that probability down to 0.4. If the non-myopic optimiser wants the probability of a dominated strategy lower than that, it has to make the agent non-myopic. And in cases where an agent with incomplete preferences is non-myopic, it can avoid pursuing dominated strategies by acting in accordance with the Caprice Rule.