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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).

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    $\begingroup$ Hi! Could you maybe provide a link or reference to the paper in which the original proof is found? $\endgroup$
    – jmbejara
    Commented Dec 3, 2014 at 14:56
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    $\begingroup$ 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. $\endgroup$
    – Alecos Papadopoulos
    Commented Dec 4, 2014 at 11:15
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    $\begingroup$ 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? econ2.jhu.edu/people/Karni/savageseu.pdf $\endgroup$
    – jmbejara
    Commented Dec 5, 2014 at 20:25
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    $\begingroup$ (+1) for the first bounty in Economics.SE (and related to a worthy subject, too). $\endgroup$
    – Alecos Papadopoulos
    Commented Dec 5, 2014 at 23:09
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    $\begingroup$ 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". $\endgroup$
    – jayk
    Commented Dec 6, 2014 at 3:03

2 Answers 2

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In Kreps' (1988) book "Notes on the Theory of Choice", 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" 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.

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  • $\begingroup$ 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 $\endgroup$
    – user157623
    Commented Dec 7, 2014 at 18:04
  • $\begingroup$ I would suggest to check it out and then post your answer to your own question, commenting on the issue. $\endgroup$
    – Alecos Papadopoulos
    Commented Dec 7, 2014 at 18:35
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    $\begingroup$ 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. $\endgroup$
    – user157623
    Commented Dec 11, 2014 at 15:36
  • $\begingroup$ @user157623 Thanks. Really looking forward to your answer. $\endgroup$
    – Alecos Papadopoulos
    Commented Dec 11, 2014 at 15:43
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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.

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    $\begingroup$ Mixture spaces as originally introduced by Herstein and Milnor are abstract algebraic (strict) generalizations of convex sets in vector spaces. They don't refer to any probability conception. $\endgroup$
    – Michael Greinecker
    Commented Oct 21, 2023 at 14:16
  • $\begingroup$ @MichaelGreinecker Yes exactly. So the OP needs to precisely define the "mixture space" he wants. $\endgroup$
    – High GPA
    Commented Oct 22, 2023 at 14:36

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