In particular, which axiom should we reject, and why? It sounds like what jsteinhardt is saying is that Axiom 3, Continuity, shouldn’t hold in certain extreme cases.
The axiom I object to there is independence, not continuity. If A < B then pA+(1-p)C shouldn’t necessarily be less than pB+(1-p)C for small values of p.
I think it would be quite hard to object to the Archimedean formulation of continuity, and completeness and transitivity are obvious. But I see no reason why independence should be true. In particular, independence is basically saying that my preference over outcomes should be linear over the space of probability distributions, which will obviously lead to maximizing expected utility theory (assuming that a utility function exists, which is the part of VNM that I would agree with).
Can someone who subscribes to VNM please justify why independence is a reasonable axiom? Given that VNM implies that we should perform ridiculous actions (like pay Pascal’s mugger), I think this is quite relevant.
Given the number of downvotes here, perhaps I will in a separate post present the version of VNM where we don’t assume independence, after I work out exactly what we end up with.
The focus of this book is on normative and justificatory issues. I shall be concerned with the weak ordering and independence principles as normative for [...] preference and choice behavior.
And in the conclusion:
The conclusion to be derived from all this, however, is that CF and CIND—alternatively, WO and IND—cannot be secured by any of the versions of the pragmatic arguments I have explored. [...] And this means, in turn, that both the expected utility theorem and its correlative, the theorem concerning the existence of well-defined subjective probabilities, cannot be neatly grounded in a set of postulates that can themselves be defended by appeal to pragmatic considerations.
(Note that I just found this book today and have not yet read it. It does have more than 300 citations in Google Scholar.)
I’m impressed that you still remembered this thread 6 months after the fact.
Actually what happened was, I read a comment of yours mentioning a lab that you worked in, was curious which lab, so started scanning all your comments starting from the earliest to see if you said more about it, noticed this comment, remembered that I had read a paper about whether independence is justified, couldn’t find that paper but found the book instead.
Oops, apparently independence is also reasonable, as can be seen by flipping a coin with weight p and giving someone a choice between A and B if it comes up heads and giving them C if it comes up tails.
Transitivity? In The Lifespan Dilemma, Eliezer presents a sequence (L_n) in which we are convinced L_n { L_(n+1) throughout, but for which we’d prefer even L_0 to L_n for some large but finite n.
Transitivity? In The Lifespan Dilemma, Eliezer presents a sequence (L_n) in which we are convinced L_n { L_(n+1) throughout, but for which we’d prefer even L_0 to L_n for some large but finite n.
I’m not convinced L_n { L_(n+1), but I don’t seem to have his fixation with really big numbers.
In particular, which axiom should we reject, and why? It sounds like what jsteinhardt is saying is that Axiom 3, Continuity, shouldn’t hold in certain extreme cases.
We have discussed this somewhat before: Academian on VNM-utility theory, and my favorite comment on the post.
The axiom I object to there is independence, not continuity. If A < B then pA+(1-p)C shouldn’t necessarily be less than pB+(1-p)C for small values of p.
I think it would be quite hard to object to the Archimedean formulation of continuity, and completeness and transitivity are obvious. But I see no reason why independence should be true. In particular, independence is basically saying that my preference over outcomes should be linear over the space of probability distributions, which will obviously lead to maximizing expected utility theory (assuming that a utility function exists, which is the part of VNM that I would agree with).
Can someone who subscribes to VNM please justify why independence is a reasonable axiom? Given that VNM implies that we should perform ridiculous actions (like pay Pascal’s mugger), I think this is quite relevant.
Given the number of downvotes here, perhaps I will in a separate post present the version of VNM where we don’t assume independence, after I work out exactly what we end up with.
Apparently there is a book length treatment of this question, titled Rationality and dynamic choice: foundational explorations. The author writes in the introduction:
And in the conclusion:
(Note that I just found this book today and have not yet read it. It does have more than 300 citations in Google Scholar.)
Thanks for the reference! I’m impressed that you still remembered this thread 6 months after the fact.
Actually what happened was, I read a comment of yours mentioning a lab that you worked in, was curious which lab, so started scanning all your comments starting from the earliest to see if you said more about it, noticed this comment, remembered that I had read a paper about whether independence is justified, couldn’t find that paper but found the book instead.
BTW, what lab are you working in? ;)
You would end up with a Nobel Prize in economics.
Oops, apparently independence is also reasonable, as can be seen by flipping a coin with weight p and giving someone a choice between A and B if it comes up heads and giving them C if it comes up tails.
Transitivity? In The Lifespan Dilemma, Eliezer presents a sequence (L_n) in which we are convinced L_n { L_(n+1) throughout, but for which we’d prefer even L_0 to L_n for some large but finite n.
I’m not convinced L_n { L_(n+1), but I don’t seem to have his fixation with really big numbers.