<p>I've been working on it and I believe this is the way to approach it:</p>

<p>Write the lotteries as follows:</p>

<p>$L_{1}=[(0.89,\$1),(0.11,\$1)]$</p>

<p>$L_{2}=[(0.89,\$1),(0.11,L')]$ where we compound $L'$ inside $L_{2}$ with $L'=[(\frac{1}{11},\$0),(\frac{10}{11},\$5)]$</p>

<p>$L_{3}=[(0.89,\$0),(0.11,L')]$ using the same $L'$</p>

<p>$L_{4}=[(0.89,\$0),(0.11,\$1)]$</p>

<p>We have $L_{1}$ is preferred to $L_{2}$.</p>

<p>So, by the assumption of independence or substitutability, changing the lottery value from $(0.89,\$1)$ to $(0.89,\$0)$ with all else remaining constant, should not change our preferences. So, if the assumption is followed, we should have $L_{4}$ is preferred to $L_{3}$.</p>

<p>However, $L_{3}$ is preferred to $L_{4}$, so it is not the case.</p>

<p>What do you think of this approach? Is it valid?</p>

CC BY-SA 3.0
2018-04-15T17:25:09.603Z
Check out my answer. Despite the downvotes I'm not sure whats wrong with its validity
Bensstats
CC BY-SA 3.0
2018-04-15T17:18:59.800Z
2018-04-15T17:18:59.800Z
Omrane
<p>The expected utility of $L_1$ and $L_2$ are:</p>

<p>$E(L_1)=\sum_{i=1}^{n=1}{p_iu_i}=1*1= 1$ <strong>$</strong></p>

<p>$E(L_2)=\sum_{i=1}^{n=3}p_iu_i=0.01*0+0.89*1+0.1*5= 1.39$  <strong>$</strong> </p>

<p>Therefore if the agent says that he prefers $L_1$ to $L_2$ then he is not following the expected utility assumption of independence as $E(L_2) &gt; E(L_1)$.</p>

<p>Similarly with $L_3$ and $L_4$</p>

<p>$E(L_3)=\sum_{i=1}^{n=2}p_iu_i=0.05$ <strong>$</strong></p>

<p>$E(L_4)=\sum_{i=1}^{n=2}p_iu_i= 0.11$ <strong>$</strong></p>

<p>Because $E(L_4)&gt;E(L_3)$, if the agent says he prefers $L_3$ to $L_4$ then he is also not following the expected utility assumption on independence. </p>

<p>Hope this is helpful!</p>

CC BY-SA 3.0
2018-04-15T17:18:53.613Z
@WaitakereCity what do you mean? i don't know why this is wrong
Bensstats
CC BY-SA 3.0
2018-04-15T17:26:39.783Z
I think you are confusing expected utility and expected value. Even though the expected value of a lottery is bigger than another, that does not mean that its expected utility must also be bigger.

Also, it has nothing to do with the independence assumption.
Omrane
CC BY-SA 3.0
2018-04-15T18:29:22.590Z
@Sadem huh? The expected value of the utility is the expected utility!
Bensstats
CC BY-SA 3.0
2018-04-15T18:39:20.907Z
@FreakconFrank, the expected value of the lottery is the summation of probabilities times the values. The expected utility of the lottery is the summation of probabilities times the expected utility of the values.

Take for example the lottery $[(0.5,0),(0.5,100)]$ of expected value $50$ and the lottery $[(1,36)]$ of expected value $36$, although the first one has higher expected value then the second one, if you take the utility function $u(x)=\sqrt{x}$, then the agent will prefer the second lottery over the first.
Omrane
CC BY-SA 3.0
2018-04-15T20:15:03.743Z
The value of the utility need not be the same as the value of the monetary outcome. That is, in general, $u(\$1)\ne 1$.
Herr K.
CC BY-SA 3.0
2018-04-15T17:02:17.320Z
2018-04-15T17:02:17.320Z
Bensstats
<p>Your example is the classic <a href="https://en.wikipedia.org/wiki/Allais_paradox" rel="nofollow noreferrer">Allais paradox</a>. </p>

<p>I think the best way to see how the preference pattern $L_1\succ L_2$ and $L_3\succ L_4$ violates independence is to visualize it geometrically. Consider the following figure: </p>

<p><a href="https://i.sstatic.net/UKE6q.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/UKE6q.png" alt="enter image description here"></a></p>

<p>Observe that 
\begin{align}
L_2&amp;=L_1+(0.01,-0.11,0.1)\\
L_3&amp;=L_4+(0.01,-0.11,0.1).
\end{align}
Therefore, the indifference curves passing through $L_1$ and $L_4$ should be parallel. Since $L_1\succ L_2$, the latter lottery should lie below the IC through $L_1$. By implication of the parallelism of the ICs, $L_3$ should also lie below the IC through $L_4$. But we're given that $L_3\succ L_4$. This means that $L_4$ lies below the IC through $L_3$ (the blue dashed line). Since parallelism of ICs is a necessary condition of independence, the agent's preference is inconsistent with independence, because it can only be justified by a set of non-parallel ICs.</p>

<p>Alternatively, you can think of the inconsistency between the agent's preference and expected utility theory in the following way.
Suppose expected utility theory holds. Then $L_1\succ L_2$ is equivalent to the inequality
\begin{equation}
u(\$1)&gt;0.01u(\$0)+0.89u(\$1)+0.1u(\$5).
\end{equation}
Adding $0.89u(\$0)-0.89u(\$1)$ on both sides, we get 
\begin{equation}
0.89u(\$0)+0.11u(\$1)&gt;0.9u(\$0)+0.1u(\$5). 
\end{equation}
But this second inequality is simply saying $L_4\succ L_3$, which is contradictory to the agent's preference.</p>

CC BY-SA 3.0
2018-04-15T20:07:09.553Z
2018-04-15T20:07:09.553Z
Herr K.
<p>Suppose we have the following four lotteries:</p>

<p>$L_{1}=[(1,\$1)]$</p>

<p>$L_{2}=[(0.01,\$0),(0.89,\$1),(0.1,\$5)]$</p>

<p>$L_{3}=[(0.9,\$0),(0.1,\$5)]$</p>

<p>$L_{4}=[(0.89,\$0),(0.11,\$1)]$</p>

<p>If our agent says that he prefers $L_{1}$ to $L_{2}$ and prefers $L_{3}$ to $L_{4}$, then he is not following the expected utility assumption of independence (also known as substitutability).</p>

<p>Anyone know how to explain this?</p>

CC BY-SA 3.0
2018-04-16T22:05:40.683Z
Your question is unclear. Your phrasing makes it sound like you're asking for an explanation for why someone would have these preferences, rather than why these preferences would violate the assumption of independence.
Acccumulation
<mathematical-economics><expected-utility>
Lotteries and expected utility
Omrane
# Lotteries and expected utility

(Asked by: username_0 on 2018-04-15)

Suppose we have the following four lotteries:

$L\_{1}=[(1,\$1)]$

$L\_{2}=[(0.01,\$0),(0.89,\$1),(0.1,\$5)]$

$L\_{3}=[(0.9,\$0),(0.1,\$5)]$

$L\_{4}=[(0.89,\$0),(0.11,\$1)]$

If our agent says that he prefers $L\_{1}$ to $L\_{2}$ and prefers $L\_{3}$ to $L\_{4}$, then he is not following the expected utility assumption of independence (also known as substitutability).

Anyone know how to explain this?

**Tags:** mathematical-economics, expected-utility
> **Comments:**
> **username_1**: Your question is unclear. Your phrasing makes it sound like you're asking for an explanation for why someone would have these preferences, rather than why these preferences would violate the assumption of independence.  

---

## 3 answers

### Answer by username_3 on 2018-04-15. Score: 4 | Accepted answer

Your example is the classic [Allais paradox](https://en.wikipedia.org/wiki/Allais_paradox).

I think the best way to see how the preference pattern $L\_1\succ L\_2$ and $L\_3\succ L\_4$ violates independence is to visualize it geometrically. Consider the following figure:

[![enter image description here](https://i.sstatic.net/UKE6q.png)](https://i.sstatic.net/UKE6q.png)

Observe that
\begin{align}
L\_2&=L\_1+(0.01,-0.11,0.1)\\
L\_3&=L\_4+(0.01,-0.11,0.1).
\end{align}
Therefore, the indifference curves passing through $L\_1$ and $L\_4$ should be parallel. Since $L\_1\succ L\_2$, the latter lottery should lie below the IC through $L\_1$. By implication of the parallelism of the ICs, $L\_3$ should also lie below the IC through $L\_4$. But we're given that $L\_3\succ L\_4$. This means that $L\_4$ lies below the IC through $L\_3$ (the blue dashed line). Since parallelism of ICs is a necessary condition of independence, the agent's preference is inconsistent with independence, because it can only be justified by a set of non-parallel ICs.

Alternatively, you can think of the inconsistency between the agent's preference and expected utility theory in the following way.
Suppose expected utility theory holds. Then $L\_1\succ L\_2$ is equivalent to the inequality
\begin{equation}
u(\$1)>0.01u(\$0)+0.89u(\$1)+0.1u(\$5).
\end{equation}
Adding $0.89u(\$0)-0.89u(\$1)$ on both sides, we get
\begin{equation}
0.89u(\$0)+0.11u(\$1)>0.9u(\$0)+0.1u(\$5).
\end{equation}
But this second inequality is simply saying $L\_4\succ L\_3$, which is contradictory to the agent's preference.


---

### Answer by username_0 on 2018-04-15. Score: 3

I've been working on it and I believe this is the way to approach it:

Write the lotteries as follows:

$L\_{1}=[(0.89,\$1),(0.11,\$1)]$

$L\_{2}=[(0.89,\$1),(0.11,L')]$ where we compound $L'$ inside $L\_{2}$ with $L'=[(\frac{1}{11},\$0),(\frac{10}{11},\$5)]$

$L\_{3}=[(0.89,\$0),(0.11,L')]$ using the same $L'$

$L\_{4}=[(0.89,\$0),(0.11,\$1)]$

We have $L\_{1}$ is preferred to $L\_{2}$.

So, by the assumption of independence or substitutability, changing the lottery value from $(0.89,\$1)$ to $(0.89,\$0)$ with all else remaining constant, should not change our preferences. So, if the assumption is followed, we should have $L\_{4}$ is preferred to $L\_{3}$.

However, $L\_{3}$ is preferred to $L\_{4}$, so it is not the case.

What do you think of this approach? Is it valid?

> **Comments:**
> **username_2**: Check out my answer. Despite the downvotes I'm not sure whats wrong with its validity  

---

### Answer by username_2 on 2018-04-15. Score: -1

The expected utility of $L\_1$ and $L\_2$ are:

$E(L\_1)=\sum\_{i=1}^{n=1}{p\_iu\_i}=1\*1= 1$ **$**

$E(L\_2)=\sum\_{i=1}^{n=3}p\_iu\_i=0.01\*0+0.89\*1+0.1\*5= 1.39$ **$**

Therefore if the agent says that he prefers $L\_1$ to $L\_2$ then he is not following the expected utility assumption of independence as $E(L\_2) > E(L\_1)$.

Similarly with $L\_3$ and $L\_4$

$E(L\_3)=\sum\_{i=1}^{n=2}p\_iu\_i=0.05$ **$**

$E(L\_4)=\sum\_{i=1}^{n=2}p\_iu\_i= 0.11$ **$**

Because $E(L\_4)>E(L\_3)$, if the agent says he prefers $L\_3$ to $L\_4$ then he is also not following the expected utility assumption on independence.

Hope this is helpful!

> **Comments:**
> **username_2**: @WaitakereCity what do you mean? i don't know why this is wrong  
> **username_0**: I think you are confusing expected utility and expected value. Even though the expected value of a lottery is bigger than another, that does not mean that its expected utility must also be bigger.  Also, it has nothing to do with the independence assumption.  
> **username_2**: @Sadem huh? The expected value of the utility is the expected utility!  
> **username_0**: @FreakconFrank, the expected value of the lottery is the summation of probabilities times the values. The expected utility of the lottery is the summation of probabilities times the expected utility of the values.  Take for example the lottery $[(0.5,0),(0.5,100)]$ of expected value $50$ and the lottery $[(1,36)]$ of expected value $36$, although the first one has higher expected value then the second one, if you take the utility function $u(x)=\sqrt{x}$, then the agent will prefer the second lottery over the first.  
> **username_3**: The value of the utility need not be the same as the value of the monetary outcome. That is, in general, $u(\$1)\ne 1$.  

---

