<p>We know, that given a set of possible outcomes <span class="math-container">$X$</span>, a set of states of nature <span class="math-container">$\Omega$</span>, and the set of all acts from <span class="math-container">$\Omega$</span> to <span class="math-container">$X$</span>, if a DM has rational preferences over the acts and if the Savage axioms are true, then there exists a unique subjective probability distribution <span class="math-container">$P$</span> over the states of nature and utility function unique up to an at fine transformation, <span class="math-container">$u$</span> over the outcomes such that the preferences over acts can be represented by a utility function <span class="math-container">$U$</span> which is the expected value of the utility <span class="math-container">$u$</span> of the outcomes of the acts, with respect to the probability distribution found.</p>
<p>But this is an existence theorem. Can the probabilities and the preferences over outcomes induced by the utility function be deduced if we have all the choice data available? (I guess this would be a choice under uncertainty version of the problem of integrability?)</p>
<p>Any direct answer or if inconvenient, reference to any short text to read would help.</p>

CC BY-SA 4.0
2021-08-03T03:18:21.360Z
Are you asking about the Savage framework or the Anscombe-Aumann framework? I ask since you mention lotteries.
201p
CC BY-SA 4.0
2021-08-03T06:35:07.673Z
Sorry! I should have written acts, corrected now!
Ishan Kashyap Hazarika
<expected-utility><uncertainty>
Deducing beliefs from choices when the Savage Axioms are true
Ishan Kashyap Hazarika
# Deducing beliefs from choices when the Savage Axioms are true

(Asked by: username_0 on 2021-08-02)

We know, that given a set of possible outcomes $X$, a set of states of nature $\Omega$, and the set of all acts from $\Omega$ to $X$, if a DM has rational preferences over the acts and if the Savage axioms are true, then there exists a unique subjective probability distribution $P$ over the states of nature and utility function unique up to an at fine transformation, $u$ over the outcomes such that the preferences over acts can be represented by a utility function $U$ which is the expected value of the utility $u$ of the outcomes of the acts, with respect to the probability distribution found.

But this is an existence theorem. Can the probabilities and the preferences over outcomes induced by the utility function be deduced if we have all the choice data available? (I guess this would be a choice under uncertainty version of the problem of integrability?)

Any direct answer or if inconvenient, reference to any short text to read would help.

**Tags:** expected-utility, uncertainty
> **Comments:**
> **username_1**: Are you asking about the Savage framework or the Anscombe-Aumann framework? I ask since you mention lotteries.  
> **username_0**: Sorry! I should have written acts, corrected now!  

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