Not sure how much sense it makes to take the arithmetic mean of probabilities when the odds vary over many orders of magnitude. If the average is, say, 30%, then it hardly matters whether someone answers 1% or .000001%. Also, it hardly matters whether someone answers 99% or 99.99999%.

I guess the natural way to deal with this would be to average (i.e., take the arithmetic mean of) the order of magnitude of the odds (i.e., log[p/(1-p)], p someone's answer). Using this method, it would make a difference whether someone is "pretty certain" or "extremely certain" that a certain statement is true or false.

Does anyone know what the standard way for dealing with this issue is?

Although I don't believe it to be impossible that a gene causes you to think in specific ways, in the setting of the game such a mechanism is not required.

It is required. If Omega is making true statements, they are (leaving aside those cases where someone is made aware of the prediction before choosing) true independently of Omega making them. That means that everyone with gene A makes choice A and everyone with gene B makes choice B. This strong entanglement implies the existence of some sort of causal connection, whether or not Omega exists.

More generally, I think that every one of those problems would be made clear by exhibiting the causal relationships that are being presumed to hold. Here is my attempt.

For the School Mark problem, the causal diagram I obtain from the description is one of these:

pupil's character ----> teacher's prediction ----> final mark
|
|
V
studying ----> exam performance

or

pupil's character ----> teacher's prediction
|
|
V
studying ----> exam performance ----> final mark

For the first of these, the teacher has waived the requirement of actually sitting the exam, and the student needn't bother. In the second, the pupil will not get the marks except by studying for and taking the exam. See also the decision problem I describe at the end of this comment.

For Newcomb we have:

person's qualities --> Omega's prediction --> contents of boxes
| |
| |
V V
person's decision --------------------------> payoff

(ETA: the second down arrow should go from "contents of boxes" to "payoff". Apparently Markdown's code mode isn't as code-modey as I expected.)

The hypotheses prevent us from performing surgery on this graph to model do(person's decision). The do() operator requires deleting all in-edges to the node operated on, making it causally independent of all of its non-descendants in the graph. The hypotheses of Newcomb stipulate that this cannot be done: every consideration you could possibly employ in making a decision is assumed to be already present in the personal qualities that Omega's prediction is based on.

A-B:

Unknown factors ---> Gene ---> Lifespan
|
|
V
Choice

or:

Gene ---> Lifespan
|
|
V
Choice

or both together.

Here, it may be unfortunate to discover oneself making choice B, but by the hypotheses of this problem, you have no choice. As with Newcomb, causal surgery is excluded by the problem. To the extent that your choice is causally independent of the given arrow, to that extent you can ignore lifespan in making your choice -- indeed, it is to that extent that you have a choice.

For Solomon's Problem (which, despite the great length of the article, you didn't set out) the diagram is:

charisma ----> overthrow
|
|
V
commit adultery

This implies that while it may be unfortunate for Solomon to discover adulterous desires, he will not make himself worse off by acting on them. This differs from A-B because we are given some causal mechanisms, and know that they are not deterministic: an uncharismatic leader still has a choice to make about adultery, and to the extent that it is causally independent of the lack of charisma, it can be made, without regard to the likelihood of overthrow.

Similarly CGTA:

CGTA gene ----> throat abcesses
|
|
V
chew gum

and the variant:

CGTA gene ----> throat abcesses
| ^
| |
V |
chew gum ---------/

(ETA: the arrow from "chew gun" to "throat abcesses" didn't come out very well.)

in which chewing gum is protective against throat abscesses, and positively to be recommended.

Newcomb's Soda:

soda assignment ---> $1M
|
|
V
choice of ice cream ---> $1K

Here, your inclination to choose a flavour of ice-cream is informative about the $1M prize, but the causal mechanism is limited to experiencing a preference. If you would prefer $1K to a chocolate ice-cream then you can safely choose vanilla.

Finally, here's another decision problem I thought of. Unlike all of the above, it requires no sci-fi hypotheses, real-world examples exist everywhere, and correctly solving them is an important practical skill.

I want to catch a train in half an hour. I judge that this is enough time to get to the station, buy a ticket, and board the train. Based on a large number of similar experiences in the past, I can confidently predict that I will catch the train. Since I know I will catch the train, should I actually do anything to catch the train?

The general form of this problem can be applied to many others. I predict that I'm going to ace an upcoming exam. Should I study? I predict I'll win an upcoming tennis match. Should I train for it? I predict I'll complete a piece of contract work on time. Should I work on it? I predict that I will post this. Should I click the "Comment" button?

The results you quote are very interesting and answer questions I've been worrying about for some time. Apologies for bringing up two purely technical inquiries:

1. Could you provide a reference for the result you quote? You referred to Eq. (34) in Everett's original paper in another comment, but this doesn't seem to make the link to the VNM axioms and decision theory.

2. <<If we have a pair of entangled particles and a classical phone line, I can send you an exact quantum state - it's called quantum teleportation, and it's sweet. But if one of our particles leaks even the tiniest bit, even if we just end up with three particles entangled instead of two, our ability to transmit quantum states is gone completely.>>

That seems wrong to me. There has to be a formulation of the form if the two initially perfectly entangled particles get only slightly entangled with other particles, then quantum teleportation still works with high fidelity / a high probability of success -- otherwise quantum teleportation wouldn't be feasible in reality.