<p>Let's write
\begin{align}
  x&amp;=ap+(1-a)r\\
  y&amp;=aq+(1-a)r
\end{align}
You want to show that $x\sim y$. But
\begin{equation}
x\sim y\quad\Leftrightarrow\quad x\not\succ y\;\text{ and }\;y\not\succ x.
\end{equation}
So the given proof tries to establish that $x\not\succ y$ (and in a similar way, $y\not\succ x$) is true. </p>

<p>To show this, use proof by contradiction. Suppose the contrary is true: $x\succ y$, and try to derive a contradiction. This is what the given proof is doing. </p>

CC BY-SA 3.0
2017-10-31T18:32:19.910Z
2017-10-31T18:32:19.910Z
Herr K.
<p>I have been asked by my professor for Decisions Theory to refer to David M. Kreps' Notes on Theory of Choice for the proof of the vNM theorem of expected utility. Before the proof of the theorem actully starts, there are two Lemmas stated which are used in the proof of the theorem. I am unable to understand the proof of one part of one of the Lemmas. Specifically, the proof for part (c) of the below Lemma. I understand the intuition behind it. Its basically the indpendence axiom but with equivalence relation, however I do not get the proof at all. Any help is much apprciated.</p>

<p><a href="https://i.sstatic.net/PHk7Y.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/PHk7Y.png" alt="Lemma"></a></p>

<p>And this is the proof for part (c). When the author talks about applying (5.2), he is referring to the independence axiom and when he talks about applying (5.3) he is referring to the Archemedean or the continuity axiom.</p>

<p><a href="https://i.sstatic.net/fYXX7.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/fYXX7.png" alt="Lemma Proof"></a></p>

CC BY-SA 3.0
2018-02-01T18:23:50.360Z
Can you tell me the book where there are all his demonstrations?
KIALA
<decision-theory><expected-utility>
vNM Expected Utility Theorem Proof
user15038
# vNM Expected Utility Theorem Proof

(Asked by: username_0 on 2017-10-31)

I have been asked by my professor for Decisions Theory to refer to David M. Kreps' Notes on Theory of Choice for the proof of the vNM theorem of expected utility. Before the proof of the theorem actully starts, there are two Lemmas stated which are used in the proof of the theorem. I am unable to understand the proof of one part of one of the Lemmas. Specifically, the proof for part (c) of the below Lemma. I understand the intuition behind it. Its basically the indpendence axiom but with equivalence relation, however I do not get the proof at all. Any help is much apprciated.

[![Lemma](https://i.sstatic.net/PHk7Y.png)](https://i.sstatic.net/PHk7Y.png)

And this is the proof for part (c). When the author talks about applying (5.2), he is referring to the independence axiom and when he talks about applying (5.3) he is referring to the Archemedean or the continuity axiom.

[![Lemma Proof](https://i.sstatic.net/fYXX7.png)](https://i.sstatic.net/fYXX7.png)

**Tags:** decision-theory, expected-utility

> **Comments:**
> **username_1**: Can you tell me the book where there are all his demonstrations?  

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## 1 answers

### Answer by username_2 on 2017-10-31. Score: 3

Let's write
\begin{align}
x&=ap+(1-a)r\\
y&=aq+(1-a)r
\end{align}
You want to show that $x\sim y$. But
\begin{equation}
x\sim y\quad\Leftrightarrow\quad x\not\succ y\;\text{ and }\;y\not\succ x.
\end{equation}
So the given proof tries to establish that $x\not\succ y$ (and in a similar way, $y\not\succ x$) is true.

To show this, use proof by contradiction. Suppose the contrary is true: $x\succ y$, and try to derive a contradiction. This is what the given proof is doing.


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