Timeline for Why utility should be bounded (or unbounded)?
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| Sep 3 at 21:42 | comment | added | martinkunev | @1muflon1 I still think this doesn't show "unbounded utility" implies there is an event with ∞ utility. Suppose for all z in R, there is an event with that utility. No matter which z you take, you can find a gamble to satisfy continuity. | |
| Oct 11, 2020 at 23:22 | comment | added | 1muflon1♦ | @HighGPA in mathematics bounded function is a function for which there is some number A such that $|f(x)| \leq A$. Hence, if utility cannot be $\infty$ there will be some number A for which the above holds and function will be bounded. Furthermore, the continuity axiom would be violated even if there is a limit because limit of $\lim_{x\rightarrow \infty} x = \infty$ | |
| Oct 11, 2020 at 22:59 | comment | added | High GPA | Many thanks for your teaching! I guess in the last paragraph, you proved that $z\neq +\infty$ and $z\neq -\infty$. However, "unbounded" usually mean that $z\to +\infty$, which is different from $z= +\infty$, to my limited knowledge. | |
| Oct 11, 2020 at 22:36 | comment | added | 1muflon1♦ | when you want to construct a rigorous model it would be unsatisfactory to just 'handwave' and say that agents ignore some arbitrary small probabilities. Also, just because this particular paradox would be resolved that does not mean other logical inconsistencies would not pop-up somewhere else. It again comes down back to instrumental value of a model. | |
| Oct 11, 2020 at 22:35 | comment | added | 1muflon1♦ | @HighGPA 1. Paradox is any logical inconsistency most aren't named. 2. I could have used the same example with just making one of the options $-\infty$ I was not attempting to make a rigorous proof but rather just an example, but it works both ways - any infinity at either end just messes up with the continuity axiom. 3. You are right there are other ways of resolving St. Petersburg paradox - but in a sense they are often unsatisfactory. For example, one solution to St. Petersburg paradox is that we can just assume agents ignore small probabilities which might even be true but ... | |
| Oct 11, 2020 at 22:31 | vote | accept | High GPA | ||
| Oct 11, 2020 at 22:29 | comment | added | High GPA | I think even if the utility is unbounded, the St. Petersburg paradox might not be a problem, as realizing the final payoffs in St. Petersburg paradox take infinite amount of time, and agents can have a time discount factor. The experimenter can never convince the agents that they can pay arbitrarily high amount of money, or they can compute the reward instantly. | |
| Oct 11, 2020 at 22:27 | comment | added | High GPA | Many thanks for the literature! Two questions. First, in your paragraph 2, what are the "paradoxes" besides St Petersburg paradox? Second, in you last paragraph you proved that $z\neq\infty$. But by "unbounded" I guess people mean that $z\in (-\infty,+\infty)$ and $u\in(-\infty,+\infty)$? That is, the utility cannot be infinity but it can arbitrarily approaches infinity. | |
| Oct 11, 2020 at 14:46 | history | edited | 1muflon1♦ | CC BY-SA 4.0 |
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| Oct 11, 2020 at 14:39 | history | edited | 1muflon1♦ | CC BY-SA 4.0 |
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| Oct 11, 2020 at 14:19 | history | answered | 1muflon1♦ | CC BY-SA 4.0 |