Spurious counterfactuals (perhaps an easier handle than "the lobian inference issues described in Section 2.1 here, in the Embedded Agency paper ), are important to address because they lead to inference problems. Having a function (or agent) rely on proofs about itself to generate its output immediately introduces Lob's Theorem-related issues -- agents may very well *want* proofs that their behavior leads to an outcome to determine their actions, but that makes a lot of actions possible to find proofs for.

Uncertainty does not help much, but reveals a greater critique: the utility in a counterfactual is not information inferred from our making the decision -- we can't decide if a choice is good just because, in the circumstances we can imagine making that choice, well, it was the right choice, wasn't it? This seems relatively unavoidable for all agents who notice they are competent decision makers, under any level of uncertainty. A smart professor will notice, in the subjunctive case where he slashes the tires on his own car, it must have made a lot of sense, he wouldn't do that for no reason. Perhaps he needed an unimpeachable excuse for not being somewhere he desperately wanted to avoid. But seeing high reward in that case isn't relevant to the counterfactual imagining where he sees, if he slashed his tires right here and now, it would be a massive bummer.

I think explicit agent self-pointers resolve this issue. Taking the example from the paper (U being the universe and A being the agent):

```
A :=
Spend some time searching for proofs of sentences of the form
“[A() = 5 →U() = x] & [A() = 10 →U() = y]” for x,y ∈{0,5,10}.
if a proof is found with x > y:
return 5
else:
return 10
U :=
if A() = 10 :
return 10
if A() = 5 :
return 5
```

We can replace it with (I've added a parameter for "which decision is being made" because that clarifies the intent behind the counterfactual agent change, but obviously this is still a highly simplified view):

```
Situation :: Nonce
Agent :: Agent -> Situation -> Choice
Universe :: Agent -> Reward
CounterfactuallyDoing(PriorAgent, CounterfactualSituation, Choice) :=
NewAgent := \Agent, Situation ->
if Situation == CounterfactualSituation:
return Choice
else:
return PriorAgent(NewAgent, Situation)
return NewAgent
A(Self, Situation == OnlySituation) :=
Spend some time searching for proofs of sentences of the form
“[U(CounterfactuallyDoing(Self, OnlySituation, 5)) = x] & [U(CounterfactuallyDoing(Self, OnlySituation, 10)) = y]” for x,y ∈{0,5,10}.
if a proof is found with x > y:
return 5
else:
return 10
U(Actor) :=
if Actor(Actor, OnlySituation) = 10:
return 10
if Actor(Actor, OnlySituation) = 5:
return 5
```

Because the proofs are acting over simpler objects, there is no Lobian issue, and the only proofs you can find are the ones with extremely simple reasoning (they operate over constant functions, essentially). And this approach can be chained into arbitrarily deep counterfactual considerations.

Consider a simple game where you must make two choices, and if they match, you get a prize corresponding to the scenario you picked. It will have two scenarios, ChooseDesire and GetReward.

```
U(Actor) :=
precommitment := Actor(Actor, ChooseDesire)
if precommitment ∈{0,5,10} & Actor(Actor, GetReward) = precommitment:
return precommitment
else:
return 0
```

I submit that any reasonably sane way of constructing the actor in this game, whether defaulting to "can I find a proof I should choose 5" or "shortest proof" or any other proof search, will find basically sane behavior, so long as reflection is done using the CounterfactuallyDoing operator. For instance:

```
A(Self, Situation == ChooseDesire) :=
Spend some time searching for proofs of sentences of the form
“[U(CounterfactuallyDoing(Self, ChooseDesire, 5)) = x] & [U(CounterfactuallyDoing(Self, ChooseDesire, 10)) = y]” for x,y ∈{0,5,10}.
if a proof is found with x > y:
return 5
else:
return 10
A(Self, Situation == GetReward) :=
Spend some time searching for proofs of sentences of the form
“[U(CounterfactuallyDoing(Self, GetReward, 5)) = x] & [U(CounterfactuallyDoing(Self, GetReward, 10)) = y]” for x,y ∈{0,5,10}.
if a proof is found with x > y:
return 5
else:
return 10
```

I think being robust to otherwise-proof-detail-contingent changes in how the agent is constructed is a quite helpful property. You don't need to use the objects only for proofs, for instance, they could be used in simulations or any other agent-appropriate setting. I do not think the Self parameter being contained inside the quotes of the proof search goal indicates an issue, unless for some reason the proof search process was meant to be static and couldn't search for proofs of dynamic statements. If you're familiar with the notation from clojure, you can imagine it as:

```
`([U(CounterfactuallyDoing(~Self, ...
```

I also think this approach matches our intuition about how counterfactuals work. We imagine ourselves as the same except we're choosing this particular behavior. Surely, in the formal reasoning, there might also be a distinction between the initial agent and the agent within that counterfactual, considering it's present in our own imaginations?

It also fixes the smaller but more obvious issue of distinguishing between reasoning-aware counterfactuals and pure counterfactuals. A pure counterfactual shouldn't, either in proof or more sloppy reasoning, make us think a choice is wise only because a wise agent made it. By constructing counterfactual agents we can assert things about their behavior without the reflective interference caused by having that assertion describe the wise agent (except where such a description would be accurate).

This approach seems straight-forward, but I was unable to find others discussing it -- I'm not certain this is even still an open problem, or if this approach has a subtle problem I've not been able to notice (I've not taken the time to actually put this into a theorem prover, although it seems extremely efficiently computable, at least in these toy examples). If it is an open problem, and the approach is promising, hopefully this helps resolve some self-proof-related issues moving forward. Hopefully, this also avoids failures-to-reason in important general scenarios, because counterfactuals have a weaker reflective consistency to the original agent, but I think this is somewhat unavoidable, and probably desirable.

[Update/Clarification of "failure to reason": in some of the multi-agent scenarios, there will be a lack of coordination if another agent distinguishes between precommitments and "spontaneously" making a choice in mutual-access-to-source-code scenarios. For instance, the mutual-source-code Prisoner's Dilemma with one agent structured similarly to the above, maximizing reward using the CounterfactuallyDo operator, going against MIRI's "cooperate iff that cooperation is what makes the other person cooperate" will defect, because it would prove the CounterfactuallyDo'ing agent would always be giving a ~constant agent to it, and the constant agent wouldn't trigger cooperation, and so it would only see defections and therefore, maximizing the reward, defect itself.

I think it's worth explicitly interrogating if there *ought* to be a distinction between precommitments and determining a decision live.

In scenarios where the other agents have your source code, it likely *isn't live anyway*, so I should be explicit: this resolution to the Lobian issues brought up in Embedded Agency explicitly relies on the idea we should be blind to the difference between reliably making a choice and precommitting to it. In scenarios without mutual source code, this difference is inaccessible, obviously, but I think the case is still strong in mutual visibility cases.]

Just leaving a sanity check, even though I'm not sure about what the people who were more involved at the time are thinking about the 5 and 10 problem these days:

Yes, I agree this works here. But it's basically CDT - this counterfactual is basically a causal do() operator. So it might not work for other problems that proof-based UDT was intended to solve in the first place, like the absent-minded driver, the non-anthropic problem, or simulated boxing.

It seems like you could use these counterfactuals to do whatever decision theory you'd like? My goal wasn't to solve

actually hard decisions-- the 5 and 10 problem is perhaps the easiest decision I can imagine -- but merely to construct a formalism such that even extremely simple decisions involving self-proofs can be solved at all.I think the reason this seems to imply a decision theory is that it's such a simple model that there are some ways of making decisions that are impossible in the model -- a fair portion of that was inherited from the psuedocode in the Embedded Agency paper. I have an extension of the formalism in mind that allows an expression of UDT as well (I suspect. Or something very close to it. I haven't paid enough attention to the paper yet to know for sure). I would love to hear your thoughts once I get that post written up? :)

Sure, I can promise you a comment or you can message me about it directly.

I don't suppose you could clarify:

It seems strange for an agent to take another agent and a situation and return a choice.

Yeah, this is essentially my position as well. My most recent attempt at articulating this is Why 1-boxing doesn't imply backwards causation.

The Agent needs access to a self pointer, and it is parameterized so it doesn't have to be a static pointer, as it was in the original paper -- this approach in particular needs it to be dynamic in this way.

There are also use cases where a bit of code receives a pointer not to its exact self -- when it is called as a subagent, it will get the parent's pointer.