If we’re using the Independence II as an axiom, you should be a little more precise, when you introduced it above, you referred to the base four axioms, including continuity.
Now, I only noticed consistency needed to convert between the two Independence formulations, which would make your statement correct. But on the face of things, it looks like you are trying to show a money pump theorem under discontinuous preferences by calling upon the continuity axiom.
Correct, by definition, if you have a dense set (which by default we treat the probability space as) and we map it into another space than either that space is also dense, in which case the converging sequences will have limits or it will not be dense (in which case continuity fails). In the former case, continuity reduces to point-wise continuity.
Note, setting the limit to “no preference” does not resolve the discontinuity. But by intermediate value, there will exist at least one such point in any continuous approximation of the discontinuous function.
Nice to see Europe catching up with, say India in this regard.
Does that answer your question?
This has been helpful. I’m much more familiar with the mathematics than the economics. Presently, I’m more worried about the mathematical chicanery involved in approximating a consistent continuous utility function out of things.
But does doesn’t the money pump result for non-independence rely on continuity? Perhaps I missed something there.
(Of note, this is what happens when I try to pull out a few details which are easy to relate and don’t send entirely the wrong intuition—can’t vouch for accuracy, but at least it seems we can talk about it.)
Sorry I left this out. It’s a huge simplification, but treat the set of p as a discrete subset set in the standard topology.
I’m very busy at the moment, but the short version is that one of my good candidates for a utility component function, c, has, c(A) < c(B) < c(pA + (1-p)B) for a subset of possible outcomes A and B, and choices of p.
This is only a piece of the puzzle, but if continuity in the von Neumann-Morgenstern sense falls out of it, I’ll be surprised. Some other bounds are possible I suspect.
Of note, you don’t explain why discontinuous preferences necessarilly cause vulnerability to money pumping.
I’m concerned about this largely because the von Neumann-Morgenstern continuity axiom is problematic for constructing a functional utility theory from “fun theory”.
Glad to see something like this.
Fair enough. Although in considering the implications of more than two options for the other conditions, I noticed something else worrisome.
The solution you present weakens a social welfare function, after all if I have two voters, and they vote (10,0,5) and (0,10,5) the result is an ambiguous ordering, not a strict ordering as required by Arrow’s theorem (which is really a property of very particular endomorphisms on permutation groups).
It seems like a classic algorithmic sacrifice of completeness for power. Was that your intent?
Note, according to the wikipedia article listed, Arrow’s theorem is valid “if the decision-making body has at least two members and at least three options to decide among”. This makes me suspect the Pareto-efficiency counter-example as this assumes we have only 2 options.
What worries me about this tact is that I’m sufficiently clever to realize that in conducting a vast and complex research program to empirically test humanity to determine a global reflectively consistent utility function, I will be changing the utility trade-offs of humanity.
So I might as well make sure that I conduct my mass studies in such a way to ensure that the outcome is both correct and easier for me to perform my second much longer (essentially infinitely longer) time phase of my functioning.
So said AI would determine and then forever follow exactly what humanity’s hidden utility function is. But there is no guarantee that this is a particularly friendly scenario.
I have a similar result, except I’ve never experienced a stimulant effects from anything other than blood sugar I’m not certain I can discount sleepiness. Also, I suffer from a migraine condition which has a much more severe affect on my mental faculties on a day-to-day basis.
And since improper sleeping is one of my triggers—“Happiness is getting enough sleep.” Not too much, not too little.
This seems like the conflict between two deep seated heuristics, hence it would be difficult at best to argue for the right one.
Instead, I suggest a synthetic approach. Stop treating the two intuitions as a false dichotomy, and consider the continuum between them (or even beyond them).
This is essentially an instance of availability bias. Of course, the most interesting case, rather than just a declarative hypothesis elevated among the other inhabitants of the hypothesis space for that particular question, models have other effects that go far beyond merely availability.
This is because our initial model won’t just form the first thing we think of when we examine the question, but some of the very structures we use when we formulate the question. Indeed, how we handle our models is easily responsible for the majority of the biases that have been discussed here and at Overcoming Bias.
In the case of the models mentioned in this post about quantum mechanics—we can look at the first quantum mechanics interpretation as having its own version of hypothesis privilege. This means we should downgrade it. Of course we should also do the same with its immediate successor, Many Worlds (although perhaps not as much). After all, it is the interpretations we haven’t thought of which are being penalized the most by the effect of privileging the hypothesis.
But to adequately apply this discounting beyond the direct route we need to understand the way which these model affect our thinking. How do these models encourage other models being developed, and how do they blind our cognitive architecture from different avenues?
This becomes even more pronounced when more is at stake than epistemic rationality. Since these two models are also sides (even fairly politely) in cultural and political conflicts, there are further biases that arise, as parts of these ideas become further tied to status, self-esteem, social behaviors, and habits.
I expect that one source of the problem is seen in equating these two situations. On one hand you have 100 copies of the same movie. On the other hand, you have 100 distinct humans you could pay to save. To draw a direct comparison you would need to treat these as 100 copies of some idealized stranger. In which case the scope insensitivity might (depending on how you aggregate the utility of a copy’s life) make more sense as a heuristic.
And this sort of simplification is likely one part of what is happening when we naively consider the questions:
How much would you pay to save a random stranger’s life? How much would you pay to save 100 strangers?
I wonder how this experiment would change if you presented lists of names? If you encouraged a different model for the 100 strangers.
It’s just that with two distinctly different conclusions from the results mentioned from two different sources: the article authors (in the abstract) and Gerald Weissmann, M.D., Editor-in-Chief (in the news article), I place a much lower confidence in later being a reasonable reading of the research paper.
But of course we could quite safely argue about readings and interpretations indefinitely. I’d point you to Derrida and Hermeneutics if you want to go that route.
In any case, I’ll update my estimates on the likelihood of the research paper having an errant conclusion based on Weismann’s quote, and I suggest you do the same based on the evidence in the abstract—and then I suspect we have little more to discuss on the subject.
So, perhaps the news article was based on press release that was based on the journal article. My point was that it was not produced solely from the abstract.
I don’t see why this is your point? In the very least it doesn’t present counter evidence to my claim that the abstract contains information not present in the news article which mitigates or negates the concerns of the original comment.
But the abstract does not make any “just right” claims, unlike the summary on science daily. Which is what you where complaining about.
The abstract reads—we did an incremental test, and even at the lowest dosage we found an effect. This suggest that low dosages could be effective. I don’t see anything wrong with that reasoning.
The science daily summary is simply misrepresenting it. So, the original commenter isn’t missing something in the science news, it is science daily who made the error.
Note, Independence II does not imply Independence, without using at least the consistency axiom.