<p>It's complicated.</p>
<p>The original formulation of the representation theorem of von Neumann and Morgenstern in their book made no reference to countable additivity. It was actually defined for an abstract set <span class="math-container">$L$</span> of lotteries endowed with a mixture operation that takes two lotteries <span class="math-container">$p,q$</span> and a number <span class="math-container">$\lambda$</span> in the unit interval and produces a mixture <span class="math-container">$\lambda p~\oplus~(1-\lambda)q$</span>. The result produces then a function <span class="math-container">$v:L\to\mathbb{R}$</span>, unique up to positive affine transformations, with the property that <span class="math-container">$$v\big(\lambda p~\oplus~(1-\lambda)q\big)=\lambda v(p)+(1-\lambda)v(q).$$</span>
This abstract approach was <a href="https://doi.org/10.2307/1905540" rel="nofollow noreferrer">later refined</a> by Hernstein and Milnor.</p>
<p>Now, if every lottery is a probability distribution over finitely many outcomes (and, therefore, trivially countably additive) with the mixture operation being the natural one, the result allows us to reduce the function <span class="math-container">$v$</span> to its restriction to &quot;certain&quot; lotteries, which correspond to outcomes. One can then view <span class="math-container">$v$</span> as being defined on outcomes only. This is the familiar expected utility representation.</p>
<p>Now, let <span class="math-container">$X$</span> be a set of outcomes and let <span class="math-container">$L$</span> consist of all finitely additive probability measures defined on all subsets of <span class="math-container">$X$</span>. If the axioms of von Neumann and Morgenstern hold, we must get a representation <span class="math-container">$v$</span> as above. Since <span class="math-container">$L$</span> includes all probability distributions over finitely many outcomes, we also get a special function <span class="math-container">$u:X\to\mathbb{R}$</span>, the restriction of <span class="math-container">$v$</span> to certain lotteries. The question is whether we get a representation of the form
<span class="math-container">$$v(p)=\int u~\mathrm dp.$$</span>
The answer might, in general, be no. To have such a representation, one can impose a suitable topology on <span class="math-container">$L$</span> and require the preferences to be represented to be continuous. Then, <span class="math-container">$v$</span> will be continuous with respect to this topology too. A topology on <span class="math-container">$L$</span> that works is the <em>topology of setwise convergence</em>, the smallest topology that makes the function <span class="math-container">$p\mapsto p(A)$</span> continuous for every subset <span class="math-container">$A$</span> of <span class="math-container">$X$</span>. Under this topology, the probability distributions over finite subsets are dense, and the integral is a continuous function. I don't know any particularly accessible source for the relevant math; I learned it from Torgersen's &quot;Comparison of Statistical Experiments.&quot;</p>

CC BY-SA 4.0
2024-08-28T18:40:47.487Z
2024-08-28T18:40:47.487Z
Michael Greinecker
<p>The main difference between vNM EU and Savage SEU is that one is objective one is subjective.</p>
<p>However, there is another difference: vNM EU uses countably additive prob while Savage use finitely additive prob.</p>
<p>I have a question: does the original vNM axioms and theorems work for finitely additive EU?</p>

<expected-utility><probability>
vNM theorem for finitely additive measure?
dodo
# vNM theorem for finitely additive measure?

(Asked by: username_0 on 2024-08-28)

The main difference between vNM EU and Savage SEU is that one is objective one is subjective.

However, there is another difference: vNM EU uses countably additive prob while Savage use finitely additive prob.

I have a question: does the original vNM axioms and theorems work for finitely additive EU?

**Tags:** expected-utility, probability

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## 1 answers

### Answer by username_1 on 2024-08-28. Score: 3 | Accepted answer

It's complicated.

The original formulation of the representation theorem of von Neumann and Morgenstern in their book made no reference to countable additivity. It was actually defined for an abstract set $L$ of lotteries endowed with a mixture operation that takes two lotteries $p,q$ and a number $\lambda$ in the unit interval and produces a mixture $\lambda p~\oplus~(1-\lambda)q$. The result produces then a function $v:L\to\mathbb{R}$, unique up to positive affine transformations, with the property that $$v\big(\lambda p~\oplus~(1-\lambda)q\big)=\lambda v(p)+(1-\lambda)v(q).$$
This abstract approach was [later refined](https://doi.org/10.2307/1905540) by Hernstein and Milnor.

Now, if every lottery is a probability distribution over finitely many outcomes (and, therefore, trivially countably additive) with the mixture operation being the natural one, the result allows us to reduce the function $v$ to its restriction to "certain" lotteries, which correspond to outcomes. One can then view $v$ as being defined on outcomes only. This is the familiar expected utility representation.

Now, let $X$ be a set of outcomes and let $L$ consist of all finitely additive probability measures defined on all subsets of $X$. If the axioms of von Neumann and Morgenstern hold, we must get a representation $v$ as above. Since $L$ includes all probability distributions over finitely many outcomes, we also get a special function $u:X\to\mathbb{R}$, the restriction of $v$ to certain lotteries. The question is whether we get a representation of the form
$$v(p)=\int u~\mathrm dp.$$
The answer might, in general, be no. To have such a representation, one can impose a suitable topology on $L$ and require the preferences to be represented to be continuous. Then, $v$ will be continuous with respect to this topology too. A topology on $L$ that works is the *topology of setwise convergence*, the smallest topology that makes the function $p\mapsto p(A)$ continuous for every subset $A$ of $X$. Under this topology, the probability distributions over finite subsets are dense, and the integral is a continuous function. I don't know any particularly accessible source for the relevant math; I learned it from Torgersen's "Comparison of Statistical Experiments."


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