<p>Let <span class="math-container">$X$</span> be the outcome space, and <span class="math-container">$S$</span> be the statespace. I am not sure what do you mean by using &quot;<strong>mixture space</strong>&quot;. There are two interpretations:</p>
<ol>
<li><p>Consider the special case of <span class="math-container">$X$</span> as a mixture space. This is the approach taken by AA. But, this approach is not appreciated in Savage's setting as <span class="math-container">$X$</span> needs to be arbitrary space in Savage's setting.</p>
</li>
<li><p>First recover a probability measure <span class="math-container">$\mu$</span> on the sigma algebra of <span class="math-container">$S$</span> and then consider the mixture space <span class="math-container">$\Delta (X)$</span>.</p>
</li>
</ol>
<p>The second approach makes more sense, see a reference from Arrow's (1970) PhD dissertation &quot;Theory of Risk Bearing&quot;. However, Arrow's axioms are stronger then Savage's, because the <span class="math-container">$\mu$</span> in Arrow's work must be countability additive (Arrow's axiom also implies Savage's P1 to P6). In Savage's work, such measure is finitely additive, and defining a mixture space with finitely additive measure is something new and have not be done in my limited knowledge.</p>

CC BY-SA 4.0
2023-10-21T14:16:58.520Z
Mixture spaces as originally introduced by [Herstein and Milnor](https://doi.org/10.2307/1905540) are abstract algebraic ([strict](https://people.hec.edu/mongin/wp-content/uploads/sites/36/2018/08/A37.MonginDEF01.pdf)) generalizations of convex sets in vector spaces. They don't refer to any probability conception.
Michael Greinecker
CC BY-SA 4.0
2023-10-22T14:36:21.620Z
@MichaelGreinecker Yes exactly. So the OP needs to precisely define the "mixture space" he wants.
High GPA
CC BY-SA 4.0
2023-10-21T08:25:24.247Z
2023-10-21T12:04:29.647Z
2023-10-21T12:04:29.647Z
High GPA
<p>In Kreps' (1988) book <a href="http://rads.stackoverflow.com/amzn/click/0813375533" rel="noreferrer"><strong>"Notes on the Theory of Choice"</strong></a>, the issue is dealt with in <strong>chapter 9 "Savage's Theory of Choice Under Uncertainty"</strong>, after discussing subjective probability in chapter 8. As usual, Kreps' style helps: he has the ability to seamlessly inject his -always formal- approach with very down-to-earth comments and examples that are strong in intuition (and he does it better than Savage, I might add). But also, here <em>"formal" does not translate into "complete exposition"</em>: he explicitly refrains from formally <em>proving</em> parts of the whole apparatus, mentioning that "this is a two-page proof", and "this is another two-page proof", and "if you want to prove this, good luck". For these parts he falls back on Fishburn's (1970) <a href="http://books.google.gr/books/about/Utility_theory_for_decision_making.html?id=lyUoAQAAMAAJ&amp;redir_esc=y" rel="noreferrer"><strong>"Utility Theory for Decision Making"</strong></a> book, <strong>chapter 14</strong> <strong>"Savage's Expected Utility Theory"</strong>. And Fishburn is <em>formal</em> alright (more symbols than words in a page).  </p>

<p>My impression is that combining these two sources can be beneficial.</p>

CC BY-SA 3.0
2014-12-07T18:04:16.727Z
The problem is that Kreps is not really proving anything, he is sketching a proof. Fishburn's proof I will check thanks. But is it any simpler than Savage's
user157623
CC BY-SA 3.0
2014-12-07T18:35:32.033Z
I would suggest to check it out and then post your answer to your own question, commenting on the issue.
Alecos Papadopoulos
CC BY-SA 3.0
2014-12-11T15:36:41.340Z
I will give you whole bounty, I will try to post an attempt for the solution as soon as I have time. Feel free to modify.
user157623
CC BY-SA 3.0
2014-12-11T15:43:57.310Z
@user157623 Thanks. Really looking forward to your answer.
Alecos Papadopoulos
CC BY-SA 3.0
2014-12-07T17:42:04.023Z
2014-12-07T17:42:04.023Z
Alecos Papadopoulos
<p>I have tried reading and understanding Savage's proof of the subjective utility representation, it is too complicated. Is anyone aware of a shorter/more elegant proof of this?
It is not a problem if we assume a finite prices set. </p>

<p>The original is in Savage, L.J. 1954. <em>The Foundations of Statistics</em>. New York: John
   Wiley and Sons.</p>

<p>A good summary can be found at
<a href="http://www.econ2.jhu.edu/people/Karni/savageseu.pdf" rel="noreferrer">http://www.econ2.jhu.edu/people/Karni/savageseu.pdf</a>.</p>

<p>The Savage proof is known to be very elaborate, and long. It uses the sure thing principle as its main axiom. I was wondering if there is a more "modern" proof, that is both elegant and shorter. Or a nice challenge would be to try to prove collaboratively using some modern mathematics, like mixture spaces, (I am aware of <a href="http://www.econ.ucsb.edu/~tedb/Courses/GraduateTheoryUCSB/anscombeaumann.pdf" rel="noreferrer">Anscombe-Aumann</a>).</p>

CC BY-SA 3.0
2014-12-03T14:56:15.000Z
Hi! Could you maybe provide a link or reference to the paper in which the original proof is found?
jmbejara
CC BY-SA 3.0
2014-12-03T22:39:44.520Z
@jmbejara: It's an old book, I doubt that you'll an online version. Google books has limited preview: http://books.google.ca/books?id=zSv6dBWneMEC&printsec=frontcover#v=onepage&q&f=false
Herr K.
CC BY-SA 3.0
2014-12-04T11:15:29.933Z
1) What is the "Almost Sure principle". Did you mean "Sure thing" principle ? 2) The title points to a specific segment of Savage's theory, while in the question you ask of an exposition of the whole. Please clarify.
Alecos Papadopoulos
CC BY-SA 3.0
2014-12-05T20:25:35.973Z
Yeah. Are you referring to a proof of the "Savage's Theorem" that is mentioned in the paper ("Savages’ Subjective Expected Utility Model," by Edi Karni) in the link?  http://www.econ2.jhu.edu/people/Karni/savageseu.pdf
jmbejara
CC BY-SA 3.0
2014-12-05T20:27:20.593Z
I already corrected this, my mistake, it is the sure thing principle and this provides the main axiom for the subjective utility representation, however its proof is very difficult and not very intuitive.
user157623
CC BY-SA 3.0
2014-12-05T23:09:59.617Z
(+1) for the first bounty in Economics.SE (and related to a worthy subject, too).
Alecos Papadopoulos
CC BY-SA 3.0
2014-12-06T03:03:30.783Z
I don't have access to it, but supposedly there's a brief (read: two chapters) sketch of the proof in Kreps' "Notes on the Theory of Choice".
jayk
CC BY-SA 3.0
2014-12-08T00:09:29.213Z
Have a look at Itzhak Gilboa's <a href="http://web.econ.unito.it/gma/didattica/tdg/gilboalecturenotes2007.pdf">Theory of decisions under Uncertainty</a>. He provides intuition for the way the proof works, not the actual proof.
Jan
<microeconomics><decision-theory>
The Savage sure thing principle and Subjective utility representation
user157623
# The Savage sure thing principle and Subjective utility representation

(Asked by: username_0 on 2014-12-03)

I have tried reading and understanding Savage's proof of the subjective utility representation, it is too complicated. Is anyone aware of a shorter/more elegant proof of this?
It is not a problem if we assume a finite prices set.

The original is in Savage, L.J. 1954. *The Foundations of Statistics*. New York: John
Wiley and Sons.

A good summary can be found at
<http://www.econ2.jhu.edu/people/Karni/savageseu.pdf>.

The Savage proof is known to be very elaborate, and long. It uses the sure thing principle as its main axiom. I was wondering if there is a more "modern" proof, that is both elegant and shorter. Or a nice challenge would be to try to prove collaboratively using some modern mathematics, like mixture spaces, (I am aware of [Anscombe-Aumann](http://www.econ.ucsb.edu/~tedb/Courses/GraduateTheoryUCSB/anscombeaumann.pdf)).

**Tags:** microeconomics, decision-theory

> **Comments:**
> **username_1**: Hi! Could you maybe provide a link or reference to the paper in which the original proof is found?  
> **username_2**: @username_1: It's an old book, I doubt that you'll an online version. Google books has limited preview: http://books.google.ca/books?id=zSv6dBWneMEC&printsec=frontcover#v=onepage&q&f=false  
> **username_3**: 1) What is the "Almost Sure principle". Did you mean "Sure thing" principle ? 2) The title points to a specific segment of Savage's theory, while in the question you ask of an exposition of the whole. Please clarify.  
> **username_1**: Yeah. Are you referring to a proof of the "Savage's Theorem" that is mentioned in the paper ("Savages’ Subjective Expected Utility Model," by Edi Karni) in the link?  http://www.econ2.jhu.edu/people/Karni/savageseu.pdf  
> **username_0**: I already corrected this, my mistake, it is the sure thing principle and this provides the main axiom for the subjective utility representation, however its proof is very difficult and not very intuitive.  
> **username_3**: (+1) for the first bounty in Economics.SE (and related to a worthy subject, too).  
> **username_4**: I don't have access to it, but supposedly there's a brief (read: two chapters) sketch of the proof in Kreps' "Notes on the Theory of Choice".  
> **username_5**: Have a look at Itzhak Gilboa's <a href="http://web.econ.unito.it/gma/didattica/tdg/gilboalecturenotes2007.pdf">Theory of decisions under Uncertainty</a>. He provides intuition for the way the proof works, not the actual proof.  

---

## 2 answers

### Answer by username_3 on 2014-12-07. Score: 6 | Accepted answer

In Kreps' (1988) book [**"Notes on the Theory of Choice"**](http://rads.stackoverflow.com/amzn/click/0813375533), the issue is dealt with in **chapter 9 "Savage's Theory of Choice Under Uncertainty"**, after discussing subjective probability in chapter 8. As usual, Kreps' style helps: he has the ability to seamlessly inject his -always formal- approach with very down-to-earth comments and examples that are strong in intuition (and he does it better than Savage, I might add). But also, here *"formal" does not translate into "complete exposition"*: he explicitly refrains from formally *proving* parts of the whole apparatus, mentioning that "this is a two-page proof", and "this is another two-page proof", and "if you want to prove this, good luck". For these parts he falls back on Fishburn's (1970) [**"Utility Theory for Decision Making"**](http://books.google.gr/books/about/Utility_theory_for_decision_making.html?id=lyUoAQAAMAAJ&redir_esc=y) book, **chapter 14** **"Savage's Expected Utility Theory"**. And Fishburn is *formal* alright (more symbols than words in a page).

My impression is that combining these two sources can be beneficial.

> **Comments:**
> **username_0**: The problem is that Kreps is not really proving anything, he is sketching a proof. Fishburn's proof I will check thanks. But is it any simpler than Savage's  
> **username_3**: I would suggest to check it out and then post your answer to your own question, commenting on the issue.  
> **username_0**: I will give you whole bounty, I will try to post an attempt for the solution as soon as I have time. Feel free to modify.  
> **username_3**: @username_0 Thanks. Really looking forward to your answer.  

---

### Answer by username_6 on 2023-10-21. Score: 1

Let $X$ be the outcome space, and $S$ be the statespace. I am not sure what do you mean by using "**mixture space**". There are two interpretations:

1. Consider the special case of $X$ as a mixture space. This is the approach taken by AA. But, this approach is not appreciated in Savage's setting as $X$ needs to be arbitrary space in Savage's setting.
2. First recover a probability measure $\mu$ on the sigma algebra of $S$ and then consider the mixture space $\Delta (X)$.

The second approach makes more sense, see a reference from Arrow's (1970) PhD dissertation "Theory of Risk Bearing". However, Arrow's axioms are stronger then Savage's, because the $\mu$ in Arrow's work must be countability additive (Arrow's axiom also implies Savage's P1 to P6). In Savage's work, such measure is finitely additive, and defining a mixture space with finitely additive measure is something new and have not be done in my limited knowledge.

> **Comments:**
> **username_7**: Mixture spaces as originally introduced by [Herstein and Milnor](https://doi.org/10.2307/1905540) are abstract algebraic ([strict](https://people.hec.edu/mongin/wp-content/uploads/sites/36/2018/08/A37.MonginDEF01.pdf)) generalizations of convex sets in vector spaces. They don't refer to any probability conception.  
> **username_6**: @MichaelGreinecker Yes exactly. So the OP needs to precisely define the "mixture space" he wants.  

---

