The VNM independence axiom ignores the value of information

I'd rephrase it as, when there is such a period of uncertainty, you can no longer factor the problem into neat little chunks and everything gets way way more complicated to work with.

So, my point is that the independence axiom still holds. (.5A + .5B) is preferable to (.5A + .5C), where A, B, and C are world-histories where you know what's going to happen in advance. And (.5A'' + .5C'') is preferable to (.5A' + .5B'), where A', B', A'', and C'' are world-histories that involve periods of uncertainty. There is no violation of the VNM axioms.

• A', B': You're uncertain for a while whether you'll go to Ecuador or whether you'll get a laptop.
• A'', C'': You're uncertain where you will go.

A steelmanning of your position would be that a good decision-making heuristic should take into account not simply the assets that will eventually be made available to you, but the assets you'll realistically be able to take advantage of.

So a rephrasing of your point that takes Nisan et al.'s objections into account would be: Risk-aversion is a valuable heuristic because usually uncertainty may lead to suboptimal decisions (take days off when it turns out you don't need to, or the other way around). The Allais paradox is a case where such risk aversion isn't justified, but some careless phrasing of the paradox (or of other situations in decision theory) the risk aversion may be justified.

The independence principle still holds, because (.5A' + .5B') is preferable to (.5A' + .5C').

Given that people can rationally have preferences that make essential reference to history and to the way events came about, why can't risk be one of those historical factors that matter? What's so "irrational" about that?

Nothing. Whoever said there was?

If your goal is to not be a thief, then expected utility theory recommends that you do not steal.

I suspect most of us do have 'do not steal' preferences on the scale of a few hundred pounds or more.

On the other hand, once you get to, say, a few hundred human lives, or the fate of the entire species, then I stop caring about the journey as much. It still matters, but the amount that it matters is too small to ever have an appreciable effect on the decision. This preference may be unique to me, but if so then I weep for humanity.

For example, given that in outcomes X, Y, and Z, I end up with $100, I still prefer outcome X, where I earned it, to Y, where it was a gift, to Z, where I stole it. We can complicate the examples to cover any other differences that you might (wrongly) suppose explain my preference without regard to history.

I really really doubt you have preferences for history. Your preferences are fully summarised by the current world state with no reference to histories - you prefer not remembering having stolen something and prefer remembering having earned it and having others remember the same. Note that this is a description of the present, not of the past.

To really care about a history, you'd have to construct a scenario like "I start out in a simulation along the lines of Z, but then the simulation is rearranged to a worldstate with X instead. Alternatively, I can be in scenario X all along. I like being in state X (at a point in time after the rearrangement/lack thereof) less in the former case than in the latter, even if no one can tell the difference between them." And I'm not sure that scenario would even work (it's not clear that there is meaningful continuity between Z!you and X!you), but I can't think of a better one off-hand.

Gwern said pretty much everything I wanted to say to this, but there's an extra distinction I want to make

What you're doing is saying you can't use A, B, and C when there is dependency, but have to create subevents like C1="C when you are you sure you'll have either A or C".

The distinction I made was things like A2="A when you prepare" not A2="A when you are sure of getting A or C". This looks like a nitpick, but is in fact incredibly important. The difference between my A1 and A2, is important, they are fundamentally different outcomes which may have completely different utilities, they have no more in common than B1 and C2. They are events in their own right, there is no 'sub-' to it. Distinguishing between them is not 'mangling', putting them together in the first place was always an error.

you in fact have shown the axioms don't work in the general case

It is easily possible to imagine three tennis players A, B and C, such that A beats B, B beats C and C beats A (perhaps A has a rather odd technique, which C has worked hard at learning to deal with despite being otherwise mediocre). Then we have A > B and B > C but not A > C, I have just shown that the axiom of transitivity is not true in the general case!

Well, no, I haven't.

I've shown that the axiom of transitivity does not hold for tennis players. This may be an interesting fact about tennis, but it has not 'disproven' anything, nobody ever claimed that transitivity applied to tennis players.

What the VNM axioms are meant to refer to, are outcomes, meaning a complete description of what will happen to you. "Trip to Ecuador" is not an outcome, because it does not describe exactly what will happen to you, and in particular leaves open whether or not you will prepare for the trip.

This sort of thing is why I think everyone with the intellectual capacity to do so should study mathematical logic. It really helps you learn to keep things cleanly separated in your mind and avoid mistakes like this.

Well, sure, by mangling enough the events you can re-establish the axioms...Of course, by splitting the events like that, you'll reestablish independence - but by showing the need to mangle choices to make fit the axioms, you in fact have shown the axioms don't work in the general case, when the choices you're given are not independent, as it often is in real life.

'If I write out my arithmetic like "one plus two", your calculator can't handle it! Proving that arithmetic doesn't work in the general case. Sure, you can mangle these words into these things you call numbers and symbols like "1+2", but often in real life we don't use them.'

I don't see Nisan or benelliott's first comment doing any of that. (gwern could stand to be more civil.) What do you think the core problem is, and in what sense are the other comments avoiding it?

Perhaps I should elaborate on what I think the mistake is. First, let me tell you about when I made the same mistake. I once tried to point out that a rational agent may not want to do what it believes is rational if there are other agents around because it may not want to reveal to other agents information about itself. For example, if I were trying to decide between A = a trip to Ecuador and B = a trip to Iceland, I might prefer A to B but decide on B if I thought I was being watched by spies who were trying to ascertain my travel patterns and use this information against me in some way.

Someone else correctly pointed out that in this scenario I was not choosing between A and B, but between A' = "a trip to Ecuador that spies know about" and B' = "a trip to Iceland that spies know about," which is a different choice. I tried to introduce a new element into the scenario without thinking about whether that element affected the outcomes I was choosing between.

I think you're making the same kind of mistake. You start with a statement about your preferences regarding trips to Ecuador and Iceland and a new laptop, but then you introduce a new element, namely preparation time, without considering how it affects the outcomes you're choosing between. As Nisan points out, as an agent in a timeline, you're choosing between different world-histories, and those world-histories are more than just the instants where you get the trip or laptop. As benelliott points out, once you've introduced preparation time, you need to re-specify your outcomes in more detail to accommodate that, e.g. you might re-specify option E as E' = "preparation time for a trip + 50% chance the trip is to Iceland and 50% chance the trip is to Ecuador." And as gwern points out, any preparation you make for a trip when choosing option D will still pay off with probability 50%; you just need to consider whether that's worth what will happen if you prepare for a trip that doesn't happen, which is a perfectly ordinary expected value calculation.

The collision I'm seeing is that between formal, mathematical axioms, and English language usage. Its clear that Benelliot is thinking of the axiom in mathematical terms: dry, inarguable, much like the independence axioms of probability: some statements about abstract sets. This is correct-- the proper formulation of VNM is abstract, mathematical.

Kilobug is right in noting that information has value, ignorance has cost. But that doesn't subvert the axiom, as the axioms are mathematically, by definition, correct; the way they were mapped to the example was incorrect: the choices aren't truly independent.

Its also become clear that risk-aversion is essentially the same idea as "information has value": people who are risk-averse are people who value certainty. This observation alone may well be enough to 'explain' the Allais paradox: the certainty of the 'sure thing' is worth something. All that the Allais experiment does is measure the value of certainty.

It needs to be added an additional option to see where it contradicts the VNM independence hypothesis: consider grandma could either offer one of us a trip around the world (T), or offer one of us a driving license (L). I value a trip around the world $2000 (in subjective dollars), and a driving license $1850 (in subjective dollars). To make the trip around the world, as you said, I've to spend $100 in paperwork, but there is no such cost for driving lessons. So the total the gain $1900 if I'm offered a trip around the world, and $1850 if offered a driving lesson, just for me. I prefer T over L. But if we are in the situation of your dialogue, I'm offered (50% chance of T) but I need to make the paperwork and spend the $100 anyway. So in fact, the total value of that offer is $2000/2-$100 = $900. While if I'm offered (50% chance of L) then the total value of that offer is $1850/2 = $925. So I prefer T over L, but I prefer 50% chance of L to 50% chance of T. Which violates the independence principle.

Yeah, I'm going to go with benelliott on this one. You're inputting the wrong stuff which hasn't been split up right, and complaining that it doesn't seem to work.