<p>These axioms are in general not enough. If you look at the proof in Kreps' book, most of the work is done by the mixture-space theorem, and for many variants of the problem, one can get a representation <span class="math-container">$U:F\to\mathbb{R}$</span> such that for all <span class="math-container">$f,g\in F$</span> and <span class="math-container">$\alpha\in[0,1]$</span> one has <span class="math-container">$$U(\alpha f+(1-\alpha)g)=\alpha U(f)+(1-\alpha)U(g).$$</span>
This is not the same as having a representation of the form <span class="math-container">$$U(f)=\int_\Omega\int_S u~\mathrm df(\omega)~\mathrm d\mu.$$</span>
Without additional assumptions, such a representation is not even possible if <span class="math-container">$\Omega$</span> contains a single state, which reduces the problem to pure objective expected utility. What goes wrong and how one can fix the issue in this case is discussed in the later parts of chapter 5 of Kreps's book on non-simple lotteries.</p>
<p>A reference with detailed proofs of what you are looking for in general is</p>
<blockquote>
<p>Fishburn, Peter C. &quot;The foundations of expected utility.&quot; <em>Theory &amp;
Decision Library</em> (1982).
The book makes for rather dry reading, though.</p>
</blockquote>

CC BY-SA 4.0
2024-11-10T14:13:12.450Z
2024-11-10T14:13:12.450Z
Michael Greinecker
<p>I am taking a look in the book of <a href="https://www.cambridge.org/core/books/theory-of-decision-under-uncertainty/18B73C5038191683B3F7F30B72B77FCB" rel="nofollow noreferrer">Gilboa </a>. In chapter <span class="math-container">$14$</span> there is a presentetion on Anscombe–Aumann’s Theorem. Although he does not present the assumptions about the sets of outcomes and states, and argues that these are finite, the theorem is written for more general spaces.</p>
<p>The theorem in the book is presented as follows, where the sets <span class="math-container">$X$</span> and <span class="math-container">$\Omega$</span> (of outcomes and states) instead of discrete are continuous.</p>
<p>The Anscombe–Aumann axioms are the following</p>
<ol>
<li><p>Rationality: Each <span class="math-container">$\succsim^{F}$</span> is complete and transitive, where <span class="math-container">$F$</span> is set of acts.</p>
</li>
<li><p>Continuity: For every <span class="math-container">$f, g, h \in F$</span>, if <span class="math-container">$f \succ g \succ h$</span>, there exist <span class="math-container">$\alpha,\beta \in (0,1)$</span> such that <span class="math-container">$\alpha f + (1 − \alpha)h \succ g \succ \beta f + (1 − \beta)h$</span>.</p>
</li>
<li><p>Independence: For every <span class="math-container">$f, g, h \in F$</span>, and every <span class="math-container">$\alpha \in (0, 1)$</span>,</p>
</li>
</ol>
<p><span class="math-container">$$f \succsim g \text{ iff } \alpha f + (1 − \alpha)h \succsim \alpha g + (1 − \alpha)h$$</span>.</p>
<ol start="4">
<li><p>Monotonicity: For every <span class="math-container">$f, g, h \in F$</span>, <span class="math-container">$f_{\omega} \succsim g_{\omega}$</span> for all <span class="math-container">$\omega\in\Omega$</span> implies <span class="math-container">$f \succsim g$</span>.</p>
</li>
<li><p>Nontriviality: There exist <span class="math-container">$f,g\in \Delta(X)$</span> such that <span class="math-container">$f \succ g$</span>.</p>
</li>
</ol>
<p>The theorem is the followig</p>
<p><span class="math-container">$\textbf{Theorem 14.1 (Anscombe–Aumann):}$</span> <span class="math-container">$\succsim$</span> satisfies AA<span class="math-container">$1$</span>–AA<span class="math-container">$5$</span> if and only if there exist a probability measure <span class="math-container">$\mu$</span> on <span class="math-container">$\Omega$</span> and a non-constant function <span class="math-container">$u : X →\mathbb{R}$</span> such that, for every <span class="math-container">$f, g \in \mathcal{F}$</span></p>
<p><span class="math-container">$$\text{$f\succsim g$ if and only if $\int_{\Omega}\mathbb{E}_{f(\omega)}ud\mu(\omega)\geq \int_{\Omega}\mathbb{E}_{g(\omega)}ud\mu(\omega)$}\tag{1}$$</span></p>
<p>Furthermore, in this case, <span class="math-container">$\mu$</span> is unique and <span class="math-container">$u$</span> is unique up to positive linear transformations.</p>
<p>You can check the book for more. I only changed the notation of the state of the world from <span class="math-container">$S$</span> to <span class="math-container">$\Omega$</span>.</p>
<p>My questions are the following</p>
<ol>
<li><p>Does anyone know where to look for a detailed presentation of the Anscombe-Aumann model when the assumptions of the spaces are more general, namely like compact and spaces (like space <span class="math-container">$[0,1]$</span>)?</p>
</li>
<li><p>Also, where could I find a detailed prooof of the theorem (I checked also an old book of Kreps, but does not say too much, about technicalities and how to present the model introducing more general spaces.</p>
</li>
</ol>
<p>Thank you in advance!</p>

<microeconomics><mathematical-economics><game-theory><preferences>
Anscombe–Aumann model presentation for more general spaces
Oliver Queen
# Anscombe–Aumann model presentation for more general spaces

(Asked by: username_0 on 2024-10-17)

I am taking a look in the book of [Gilboa](https://www.cambridge.org/core/books/theory-of-decision-under-uncertainty/18B73C5038191683B3F7F30B72B77FCB) . In chapter $14$ there is a presentetion on Anscombe–Aumann’s Theorem. Although he does not present the assumptions about the sets of outcomes and states, and argues that these are finite, the theorem is written for more general spaces.

The theorem in the book is presented as follows, where the sets $X$ and $\Omega$ (of outcomes and states) instead of discrete are continuous.

The Anscombe–Aumann axioms are the following

1. Rationality: Each $\succsim^{F}$ is complete and transitive, where $F$ is set of acts.
2. Continuity: For every $f, g, h \in F$, if $f \succ g \succ h$, there exist $\alpha,\beta \in (0,1)$ such that $\alpha f + (1 − \alpha)h \succ g \succ \beta f + (1 − \beta)h$.
3. Independence: For every $f, g, h \in F$, and every $\alpha \in (0, 1)$,

$$f \succsim g \text{ iff } \alpha f + (1 − \alpha)h \succsim \alpha g + (1 − \alpha)h$$.

4. Monotonicity: For every $f, g, h \in F$, $f\_{\omega} \succsim g\_{\omega}$ for all $\omega\in\Omega$ implies $f \succsim g$.
5. Nontriviality: There exist $f,g\in \Delta(X)$ such that $f \succ g$.

The theorem is the followig

$\textbf{Theorem 14.1 (Anscombe–Aumann):}$ $\succsim$ satisfies AA$1$–AA$5$ if and only if there exist a probability measure $\mu$ on $\Omega$ and a non-constant function $u : X →\mathbb{R}$ such that, for every $f, g \in \mathcal{F}$

$$\text{$f\succsim g$ if and only if $\int\_{\Omega}\mathbb{E}\_{f(\omega)}ud\mu(\omega)\geq \int\_{\Omega}\mathbb{E}\_{g(\omega)}ud\mu(\omega)$}\tag{1}$$

Furthermore, in this case, $\mu$ is unique and $u$ is unique up to positive linear transformations.

You can check the book for more. I only changed the notation of the state of the world from $S$ to $\Omega$.

My questions are the following

1. Does anyone know where to look for a detailed presentation of the Anscombe-Aumann model when the assumptions of the spaces are more general, namely like compact and spaces (like space $[0,1]$)?
2. Also, where could I find a detailed prooof of the theorem (I checked also an old book of Kreps, but does not say too much, about technicalities and how to present the model introducing more general spaces.

Thank you in advance!

**Tags:** microeconomics, mathematical-economics, game-theory, preferences

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## 1 answers

### Answer by username_1 on 2024-11-10. Score: 2

These axioms are in general not enough. If you look at the proof in Kreps' book, most of the work is done by the mixture-space theorem, and for many variants of the problem, one can get a representation $U:F\to\mathbb{R}$ such that for all $f,g\in F$ and $\alpha\in[0,1]$ one has $$U(\alpha f+(1-\alpha)g)=\alpha U(f)+(1-\alpha)U(g).$$
This is not the same as having a representation of the form $$U(f)=\int\_\Omega\int\_S u~\mathrm df(\omega)~\mathrm d\mu.$$
Without additional assumptions, such a representation is not even possible if $\Omega$ contains a single state, which reduces the problem to pure objective expected utility. What goes wrong and how one can fix the issue in this case is discussed in the later parts of chapter 5 of Kreps's book on non-simple lotteries.

A reference with detailed proofs of what you are looking for in general is

> Fishburn, Peter C. "The foundations of expected utility." *Theory &
> Decision Library* (1982).
> The book makes for rather dry reading, though.


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