*All of the results in this post, and most of the informal observations/interpretations, are due to Sam Eisenstat.*

Troll Bridge is a decision problem which has been floating around for a while, but which has lacked a good introductory post. The original post gives the essential example, but it lacks the "troll bridge" story, which (1) makes it hard to understand, since it is just stated in mathematical abstraction, and (2) makes it difficult to find if you search for "troll bridge".

The basic idea is that you want to cross a bridge. However, there is a troll who will blow up the bridge with you on it, * if* (and only if) you cross it "for a dumb reason" — for example, due to unsound logic. You can get to where you want to go by a worse path (through the stream). This path is better than being blown up, though.

We apply a Löbian proof to show not only that you choose not to cross, but furthermore, that your counterfactual reasoning is confident that the bridge would have blown up if you had crossed. This is supposed to be a counterexample to various proposed notions of counterfactual, and for various proposed decision theories.

The pseudocode for the environment (more specifically, the utility gained from the environment) is as follows:

IE, if the agent crosses the bridge and is inconsistent, then U=-10. (□⊥ means "PA proves an inconsistency".) Otherwise, if the agent crosses the bridge, U=+10. If neither of these (IE, the agent does not cross the bridge), U=0.

The pseudocode for the agent could be as follows:

This is a little more complicated, but the idea is supposed to be that you search for every "action implies utility" pair, and take the action for which you can prove the highest utility (with some tie-breaking procedure). Importantly, this is the kind of proof-based decision theory which eliminates spurious counterfactuals in 5-and-10 type problems. It isn't that easy to trip up with Löbian proofs. *(Historical/terminological note: This decision theory was initially called MUDT, and is still sometimes referred to in that way. However, I now often call it proof-based decision theory, because it isn't centrally a UDT. "Modal DT" (MDT) would be reasonable, but the modal operator involved is the "provability" operator, so "proof-based DT" seems more direct.)*

Now, the proof:

- Reasoning within PA (ie, the logic of the agent):
- Suppose the agent crosses.
- Further suppose that the agent proves that crossing implies U=-10.
- Examining the source code of the agent, because we're assuming the agent crosses, either PA proved that crossing implies U=+10, or it proved that crossing implies U=0.
- So, either way, PA is inconsistent -- by way of 0=-10 or +10=-10.
- So the troll actually blows up the bridge, and really, U=-10.

- Therefore (popping out of the second assumption), if the agent proves that crossing implies U=-10, then in fact crossing implies U=-10.
- By Löb's theorem, crossing really implies U=-10.
- So (since we're still under the assumption that the agent crosses), U=-10.

- Further suppose that the agent proves that crossing implies U=-10.
- So (popping out of the assumption that the agent crosses), the agent crossing implies U=-10.

- Suppose the agent crosses.
- Since we proved all of this in PA, the agent proves it, and proves no better utility in addition (unless PA is truly inconsistent). On the other hand, it will prove that not crossing gives it a safe U=0. So it will in fact not cross.

The paradoxical aspect of this example is not that the agent doesn't cross -- it makes sense that a proof-based agent can't cross a bridge whose safety is dependent on the agent's own logic being consistent, since proof-based agents can't know whether their logic is consistent. Rather, the point is that the agent's "counterfactual" reasoning looks crazy. *(However, keep reading for a version of the argument where it does make the agent take the wrong action.)* Arguably, the agent should be uncertain of what happens if it crosses the bridge, rather than certain that the bridge would blow up. Furthermore, the agent is reasoning as if it can control whether PA is consistent, which is arguably wrong.

In a comment, Stuart points out that this reasoning seems highly dependent on the code of the agent; the "else" clause could be different, and the argument falls apart. I think the argument keeps its force:

- On the one hand, it's still very concerning if the sensibility of the agent depends greatly on which action it performs in the "else" case.
- On the other hand, we can modify the troll's behavior to match the modified agent. The general rule is that the troll blows up the bridge if the agent would cross for a "dumb reason" -- the agent then concludes that the bridge would be blown up if it crossed. I can no longer complain that the agent reasons as if it were controlling the consistency of PA, but I can still complain that the agent thinks an action is bad because that action indicates its own insanity, due to a troublingly circular argument.

## Analogy to Smoking Lesion

One interpretation of this thought-experiment is that it shows proof-based decision theory to be essentially a version of EDT, in that it has EDT-like behavior for Smoking Lesion. The analogy to Smoking Lesion is relatively strong:

- An agent is at risk of having a significant internal issue. (In Smoking Lesion, it’s a medical issue. In Troll Bridge, it is logical inconsistency.)
- The internal issue would bias the agent toward a particular action. (In Smoking Lesion, the agent smokes. In Troll Bridge, an inconsistent agent crosses the bridge.)
- The internal issue also causes some imagined practical problem for the agent. (In Smoking Lesion, the lesion makes one more likely to get cancer. In Troll Bridge, the inconsistency would make the troll blow up the bridge.)
- There is a chain of reasoning which combines these facts to stop the agent from taking the action. (In smoking lesion, EDT refuses to smoke due to the correlation with cancer. In Troll Bridge, the proof-based agent refuses to cross the bridge because of a Löbian proof that crossing the bridge leads to disaster.)
- We intuitively find the conclusion nonsensical. (It seems the EDT agent should smoke; it seems the proof-based agent should not expect the bridge to explode.)

Indeed, the analogy to smoking lesion seems to strengthen the final point -- that the counterfactual reasoning is wrong.

## Probabilistic Version

For the purely logical version, I said that we can't fault the conclusion (because the agent can't prove that it is safe to cross the bridge) – only the *reasoning* is being critiqued. We might hope that an agent who uses some uncertainty, rather than reasoning with logic alone, would decide to cross the bridge.

More specifically, suppose that we make the risk of crossing extremely small: rather than -10 payoff for the case where the bridge is blown up, we get a very small negative payoff, -epsilon, which is barely worse than refusing to cross. This should make crossing the bridge worth the risk; very little to lose, and everything to gain.

Unfortunately, this is not how things work out.

Suppose that the agent uses a probability distribution which "respects logic" in the sense that it assigns probability zero to anything which is logically refutable. (This assumes logical omniscience. I'll drop that assumption later.)

The agent is as follows: *if P(cross)=0, cross; if P(¬cross)=0, don't cross; else, take the action with highest expected utility, breaking ties by not crossing.* The idea is that we are trying to take the highest-expected-utility action; however, we must handle cases where this isn't well-defined. Choosing to take an action when the probability of the action is zero guarantees that the agent can never prove its own action; it is a version of the chicken rule. Breaking ties by not crossing is arbitrary, but gives the simplest version of the example, because this choice reduces the number of cases we need to check.

The troll is as follows: *if the agent crosses the bridge due to the P(cross)=0 clause, blow up the bridge.*

Reasoning in the logic of the agent:

- Suppose
- Suppose
- Examining the agent, either crossing had higher expected utility, or P(cross)=0. But we assumed , so it must be the latter. So the bridge gets blown up.

- Therefore, .

- Suppose
- Therefore,

Thus, by Löb's theorem, we see that .

Therefore, since the agent is certain of things which are provable, the agent assigns expected value -10 to crossing. Since the expected value of the other action will always be higher than this, and since we know the agent doesn't ever really enter into the P=0 conditions, the agent will choose not to cross.

Notice that this reasoning did not depend much on the values 10, 0, and -10. The utility of the bridge getting blown up could be -0.000001, and the agent still won't cross. *It isn't weighing the risk; it's decided that the worst outcome is inevitable.* In the case of proof-based agents, I said that the overall decision not to cross was understandable, since proof-based agents are unable to weigh the risks. A probabilistic agent, however, should intuitively be able to say "I don't know whether the bridge will get blown up, because it involves reasoning about properties of myself which I'm fundamentally uncertain about; but, the odds look decent." But that's not what happens: instead, it is sure that crossing is unfavorable, *no matter what overall probability it assigns to P(A=cross)=0*.

So, in this case we conclude that the Troll Bridge example results in a chicken-rule-based agent taking the wrong action overall. The agent shouldn't be sure that it would cross "for the right reason" (it should assign some probability to P(A=cross)=0, since it can't know that its own logic is consistent). However, intuitively, it should be able to assign some probability to this, and balance the risks. If the downside risk is U=-0.000001, and the probability it assigns to its logic being consistent is not similarly small, it should cross -- and in doing so, it would get +10.

As mentioned for the proof-based agent, the agent's code is a bit arbitrary, and it is worth asking how important the details were. In particular, the default in the case of a tie was to not cross. What if the default in case of a tie were to cross?

We then modify the troll's algorithm to blow up the bridge if and only if P(A=cross)=0 **or** there is a tie. The proof then goes through in the same way.

Perhaps you think that the problem with the above version is that I assumed logical omniscience. It is unrealistic to suppose that agents have beliefs which perfectly respect logic. (Un)Fortunately, the argument doesn't really depend on this; it only requires that the agent respects proofs which it can see, and eventually sees the Löbian proof referenced. We can analyze this using logical inductors, with a version of LIDT which has a chicken rule. (You can use LIDT which plays epsilon-chicken, taking any action with probability less than epsilon; or, you can consider a version which takes actions which the deductive state directly disproves. Either case works.) We consider LIDT on a sequence of troll-bridge problems, and show that it eventually notices the Löbian proof and starts defecting. This is even more frustrating than the previous example, because the agent can cross for a long time, apparently learning that crossing is safe and reliably gets +10 payoff. Then, one day, it suddenly sees the Löbian proof and stops crossing the bridge!

I leave that analysis as an exercise for the reader.

## Conclusions

All of the examples have depended on a version of the chicken rule. This leaves us with a fascinating catch-22:

- We need the chicken rule to avoid spurious proofs. As a reminder: spurious proofs are cases where an agent would reject an action if it could prove that it would not take that action. These actions can then be rejected by an application of Löb's theorem. The chicken rule avoids this problem by ensuring that agents cannot know their own actions, since if they did then they'd take a different action from the one they know they'll take (and they know this, conditional on their logic being consistent).
- However, Troll Bridge shows that the chicken rule can lead to another kind of problematic Löbian proof.

So, we might take Troll Bridge to show that the chicken rule does not achieve its goal, and therefore reject the chicken rule. However, this conclusion is very severe. We cannot simply drop the chicken rule and open the gates to the (much more common!) spurious proofs. We would need an altogether different way of rejecting the spurious proofs; perhaps a full account of logical counterfactuals.

Furthermore, it is possible to come up with variants of Troll Bridge which counter some such proposals. In particular, Troll Bridge was originally invented to counter proof-length counterfactuals, which essentially generalize chicken rules, and therefore lead to the same Troll Bridge problems).

Another possible conclusion could be that Troll Bridge is simply too hard, and we need to accept that agents will be vulnerable to this kind of reasoning.