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Emile comments on The Savage theorem and the Ellsberg paradox - Less Wrong
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Comments (55)
A style note: the beginning of your post is really long and boring, I ended up skipping most of it so I could get to the damned paradox already.
... and, on reading the problem description, my first reaction was "both bets are obviously worth the same to me", so - as your footnote notes - I don't see any paradox here, just an anecdote that most people are bad at Maths (or rather, bad at abstraction, it's a very understandable "mistake").
Ambiguity aversion makes sense as a heuristic in more realistic situations: in real life, it's stupid to bet when someone else may know more than you. So you should be cautious when there's information someone might know (like how many of each balls are in the jar). Real life betting situations are often a question of who really has the most information (on horses, or stock, or trivia).
There are many thought experiments like this that are just built to break a heuristic that works in real life (trolley problem, I'm looking at you); and since most people don't investigate the deep reasons for all their heuristics (which can be hard to figure out), they apply them incorrectly in the thought experiment. Nothing mysterious about that.
(edit: reworded a bit)
Heh :-) I'm okay with people being more interested in the Ellsberg paradox than the Savage theorem. Sections headers are there for skipping ahead. There's even colour :-)
I think it would be unfair to ask me to make the Savage theorem as readable as the Ellsberg paradox. For starters, the Ellsberg paradox can be described really quickly. The Savage theorem, even redux, can't. Second, just about everyone here agrees with the conclusion of the Savage theorem, and disagrees with the Ellsberg-paradoxical behaviour.
My goal was just to make it clearer than the previous post -- and this is not an insult against the previous author, he presented the full theorem and I presented a redux version covering only the relevant part, as I explained in the boring rationality section before the boring representation theorem. I'd be happy if some people who did not understand the previous set of axioms understood the four rules here.
As for the rest, yes, consensus here so far (only a few hours in, of course, but still impressively unanimous) seems to be that it's a bias. Of course, in that case, it's a very famous bias, and it hasn't been covered on LW before. I can still claim to have accomplished something I think, no? And if it turns out it's not so irrational after all, well!
This is not a reply to this comment. I wanted to comment on the article itself, but I can't find the comment box under the article itself.
According to Robin Pope's article, "Attractions to and Repulsions from Chance," section VII, Savage's sure-thing principle is not his P2, although it is easily confused with it. The sure-thing principle says that if you do prefer (A but not B) over (B but not A), then you ought to prefer A over B. That is, in case of a violation of P2, you should resolve it by revising the latter preference (the one between bets with overlapping outcomes), not the former. This is apparently how Savage revised his preferences on the Allais paradox to align them with EU theory.
The article:section is in the book "Game Theory, Experience, and Rationality, Foundations of Social Sciences, Economics and Ethics, in honor of J.C. Harsanyi," pp 102-103.