My understanding is that in the VNM system, utility is defined over lotteries. Is this the point you're contesting or are you happy with that but unhappy with the use of the word "acts" to describe these lotteries. In other words, do you think the portrayal of the VNM system as involving preferences over lotteries is wrong or do you think that this is right but the way we describe it conflates two notions that should remain distinct.

I think I'm missing the point of what you're saying here so I was hoping that if I explained why I don't understand, perhaps you could clarify.

VNM-utility is unique up to a positive linear transformation. When a utility function is unique up to a positive linear transformation, it is an interval (/cardinal scale). So VNM-utility is an interval scale.

This is the standard story about VNM-utility (which is to say, I'm not claiming this because it seems right to me but rather because this is the accepted mainstream view of VNM-utility). Given that this is a simple mathematical property, I presume the mainstream view will be correct.

So if your comment is correct in terms of the presentation in the FAQ then either we've failed to correctly define VNM-utility or we've failed to correctly define interval scales in accordance with the mainstream way of doing so (or, I've missed something). Are you able to pinpoint which of these you think has happened?

One final comment. I don't see why ratios (a-b)/|c-d| aren't meaningful. For these to be meaningful, it seems to me that it would need to be that [(La+k)-(Lb+k)]/[(Lc+k)-(Ld+k)] = (a-b)/(c-d) for all L and K (as VNM-utilities are unique up to a positive linear transformation) and it seems clear enough that this will be the case:

[(La+k)-(Lb+k)]/[(Lc+k)-(Ld+k)] = [L(a-b)]/[L(c-d)] = (a-b)/(c-d)

Again, could you clarify what I'm missing (I'm weaker on axiomatizations of decision theory than I am on other aspects of decision theory and you're a mathematician so I'm perfectly willing to accept that I'm missing something but it'd be great if you could explain what it is)?

Does the horizontal axis of the decision tree in section 3 represent time?

Yes and no. Yes, because presumably the agent's end result re: house and money occurs after the fire and the fire will happen after the decision to take out insurance (otherwise, there's not much point taking out insurance). No, because the diagram isn't really about time, even if there is an accidental temporal component to it. Instead, the levels of the diagram correspond to different factors of the decision scenario: the first level is about the agent's choice, the second level about the states of natures and the third about the final outcome.

Given that this is how the diagram works, smearing out the triangles would mix up the levels and damage the clarity of the diagram. To model an agent as caring about whether they were insured or not, we would simply modify the text next to the triangles to something like "Insurance, no house and $99,900", "Insurance, house and - $100" and so on (and then we would assign different utilities to the agent based partly on whether they were insured or not as well as whether they had a house and how much money they had).

I think that forgetting this point sometimes leads to misapplications of decision theory.

I agree, though I think that talking of utility rather than money solves many of these problems. After all, utility should already take into account an agents desire to be insured etc and so talk about utility should be less likely to fall into these traps (which isn't to say there are never any problems)

Will be fixed in the next update. Thanks for pointing it out.

Thanks. Will be fixed in next update. Thanks also for the positive comment.

Thanks, as you note, the linked comment is right.

Thanks, will be fixed in next update.

Fixed for next update. Thanks.

Thanks, fixed for the next update.