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In my previous question, I aksed about they we care so much about expected utility, rather than e.g., the variance in utility (Why care so much about expected utility?)

From the helpful answers, I understood that it was based on a result known as the Von Neumann–Morgenstern utility theorem. From a few axioms, this theorem shows that only the expected utility is relevant. Although I accept the proof and thought the answers were great, I added my own to help close the matter (https://stats.stackexchange.com/a/388320/103407).

I do have a follwup question though. The proof hinges upon four axioms. One in particular struck me as quite strong. It was the axiom of continuity (my phrasing):

We rank three possible choices in order, say, $L \preceq M \preceq N$, where $A \preceq B $ indicates that an outcome $A$ is worse than or no better than outcome $B$. The axiom of continuity states that there must exist a probability, $p$, such that taking option $L$ with probability $p$ and option $N$ with probability $1 - p$ must be just as good as just taking option $M$, i.e., there exists a $p$ such that $$ p L + (1 - p) N \sim M $$

I'm not sure why I should accept this axiom as common sense or one that results in good decision making. I just don't think there does always exist such a $p$ for me, personally. E.g., I think I would always much rather stick with a sure outcome $M$ than gamble between two extreme outcomes $L$ and $N$ with any probability $p$. I don't like the huge variance in outcomes in the latter.

Is there any compelling reason why to accept this axiom? Am I clearly being irrational!? Are there any 'weaker' theorems like the Von Neumann–Morgenstern one that relax this axiom? Are there any alternative paradigms of decision theory that reject this axiom?

I suppose I am hinting that I want to introduce a lexicographical preference for small variance. E.g., I first consider how much I like the choices. If I like them equally, I then favor the choice with the smallest variance. In this way, there does not exist a $p$ in which they are equally preferable. Where does this lead? Do we have still have a theorem like the expected utility one?

innisfree
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  • After some more reading: I guess I am expressing a lexicographic preference for small variance :) – innisfree Jan 21 '19 at 10:22
  • Consider a dialog in which your interlocutor asks you to compare $pL+(1-p)N$ to $M$ again and again, beginning with $p=1/10$ and multiplying $p$ by $1/10$ each time. Are you asserting that you would *never* prefer $M$ in *any* of these situations? Even when $p$ becomes so tiny that for any practical purpose it should be treated as zero? It seems to me that would contradict your initial assumption that $M$ is preferable to $N.$ – whuber Jan 21 '19 at 17:10
  • Well, if I absolutely don’t like variance, I would make the switch only when $p=0$ such that there was no variance with that choice. There would never though be a $p$ when the choices were equally preferable. – innisfree Jan 22 '19 at 02:35
  • I do see that this might be considered problematic :/ I’m just trying to break out of the strong conclusion of the expected utility theorem – innisfree Jan 22 '19 at 02:37
  • Maybe indeed this is wrong-headed: rather than trying to shoe-horn variance into the procedure, I should just be assigning extremely small utilities to the ‘extreme’ outcomes that I don’t like. – innisfree Jan 22 '19 at 02:42
  • Ok, that’s also the solution on wiki https://en.m.wikipedia.org/wiki/Risk_neutral_preferences – innisfree Jan 22 '19 at 02:46
  • $p$ is allowed to be 0, no? – shimao Feb 15 '19 at 15:39

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