<p>In order to do that, you need to define <span class="math-container">$u(·)$</span> as a utility function on "sure things" rather than on lotteries. In your example, you need to think in terms of the set of possible prizes to the lotteries. Say the set of possible prizes is given by <span class="math-container">$R$</span> and asume that is finite. For any <span class="math-container">$r\in R$</span>, define <span class="math-container">$w_r$</span> as a lottery that pays <span class="math-container">$r$</span> in every state of nature. Then, it should be clear that </p>

<p><span class="math-container">$$x \sim \sum_i p_i w_i,$$</span></p>

<p>(and similarly for <span class="math-container">$y$</span> and <span class="math-container">$z$</span>) where <span class="math-container">$p_i$</span> denotes the probability of state <span class="math-container">$i.$</span> Given that, as you mention, it must be the case that two equally desirable lotteries receive the same expected utility, the utility assigned to <span class="math-container">$x$</span> needs to be</p>

<p><span class="math-container">$$H(x)=H\left(\sum_i p_i w_i\right).$$</span></p>

<p>The first chapter of Huang, C. and Litzenberger, R.H. <em>Foundations for Financial Economics,</em> North Holland, 1988 explains in detail why the function representing the preferences <span class="math-container">$(H(·))$</span> has to be linear, and, with linearity stablished, you get the expected utility representation because</p>

<p><span class="math-container">$$H\left(\sum_i p_i w_i\right)=\sum_i p_i H(w_i).$$</span></p>

<p>The only thing left is to define <span class="math-container">$u(r) \equiv H(w_r),$</span> i.e, <span class="math-container">$u(r)$</span> is defined on outcomes, while <span class="math-container">$H(w_r)$</span> is defined on lotteries. This is why we say that <span class="math-container">$u(r)$</span> is a utility function on sure things, because <span class="math-container">$r$</span> receives the same utility level that a lottery paying precisely <span class="math-container">$r$</span> in every state of nature.</p>

<p>Of course, expected utility is most useful in settings with large outcomes spaces and/or large lotteries spaces. In a restricted setting such as yours, you're probably best served by assigning values ("utilities") to lotteries themselves (as you did in your question). Again, a more rigorous proof can be found in Huang and Litzenberger, op.cit.</p>

CC BY-SA 4.0
2018-11-26T18:34:36.667Z
Why do you say OP's example has 3 lotteries but just 2possible outcomes? Also what's $z_3$?
Herr K.
CC BY-SA 4.0
2018-11-26T19:48:27.617Z
@Patricio I have made an edit.
OGC
CC BY-SA 4.0
2018-11-27T10:13:53.723Z
@OGC, As I mention in my answer, in order to obtain an expected utility representation, you need to define $u(·)$ on "sure things." I'm edditing my answer to hopefully make myself clearer.
Patricio
CC BY-SA 4.0
2018-11-27T10:29:38.227Z
@HerrK., my mistake, I'll edit the answer
Patricio
CC BY-SA 4.0
2018-11-26T12:56:30.953Z
2018-11-27T10:39:27.753Z
2018-11-27T10:39:27.753Z
Patricio
<p>I am trying to prove the expected utility theorem with three outcomes. The expected utility with <span class="math-container">$n$</span> outcomes is rather cumbersome and long in the economics textbook Mas-Colell. But I was hoping that the proof with three outcomes is shorter, however, I am having some difficulty proving it. </p>

<p>Suppose I have three lotteries <span class="math-container">$x \succeq   y \succeq   z$</span>. We can think of <span class="math-container">$x$</span> as the "best" lottery and <span class="math-container">$z$</span> as the "worst". The can set  <span class="math-container">$u(x)=1$</span> and <span class="math-container">$u(z)=0$</span> and then <span class="math-container">$u(y)=p$</span> where <span class="math-container">$p$</span> is the probability at which a gamble with a <span class="math-container">$p$</span> chance of <span class="math-container">$x$</span> and a <span class="math-container">$1-p$</span> chance of <span class="math-container">$z$</span> is indifferent to <span class="math-container">$y$</span>.</p>

<p>How do I proceed from here to prove the Expected utility theorem? </p>

<p>For <span class="math-container">$n$</span> outcomes, the EU states that given <span class="math-container">$\succeq $</span> satisfies independence and continuity axioms, there is a utility function <span class="math-container">$ u:Z\rightarrow \mathbb{R}$</span> such that if <span class="math-container">$$p\succeq q\Leftrightarrow \sum_{i=1}^{n}p_{i}u(z_{i})\geq \sum_{i=1}^{n}q_{i}u(z_{i}).$$</span></p>

<p><strong>Edit:</strong> Letting <span class="math-container">$u(x)= 1$</span> and <span class="math-container">$u(z) = 0$</span> linearly scales the utility function. 
So we want to show <span class="math-container">$$u(y) = u\left ( px + (1-p)z \right ) = pu(x) + (1-p)u(z) = p \text{ by the independence axiom}.$$</span></p>

<p>Since we have assumed <span class="math-container">$x \succeq  y \succeq  z$</span> then it follows that <span class="math-container">$u(x) \geq u(y) \geq u(z)$</span>, where <span class="math-container">$p \in [0,1]$</span>. Does this make sense? </p>

<p>I am not sure about the part: <span class="math-container">$u\left ( px + (1-p)z \right ) = pu(x) + (1-p)u(z)$</span>. Is this a legitimate way of writing the proof for the case of three outcomes?</p>

<microeconomics><expected-utility><proof>
Proof of Expected utility theorem with three outcomes
OGC
# Proof of Expected utility theorem with three outcomes

(Asked by: username_0 on 2018-11-26)

I am trying to prove the expected utility theorem with three outcomes. The expected utility with $n$ outcomes is rather cumbersome and long in the economics textbook Mas-Colell. But I was hoping that the proof with three outcomes is shorter, however, I am having some difficulty proving it.

Suppose I have three lotteries $x \succeq y \succeq z$. We can think of $x$ as the "best" lottery and $z$ as the "worst". The can set $u(x)=1$ and $u(z)=0$ and then $u(y)=p$ where $p$ is the probability at which a gamble with a $p$ chance of $x$ and a $1-p$ chance of $z$ is indifferent to $y$.

How do I proceed from here to prove the Expected utility theorem?

For $n$ outcomes, the EU states that given $\succeq $ satisfies independence and continuity axioms, there is a utility function $ u:Z\rightarrow \mathbb{R}$ such that if $$p\succeq q\Leftrightarrow \sum\_{i=1}^{n}p\_{i}u(z\_{i})\geq \sum\_{i=1}^{n}q\_{i}u(z\_{i}).$$

**Edit:** Letting $u(x)= 1$ and $u(z) = 0$ linearly scales the utility function.
So we want to show $$u(y) = u\left ( px + (1-p)z \right ) = pu(x) + (1-p)u(z) = p \text{ by the independence axiom}.$$

Since we have assumed $x \succeq y \succeq z$ then it follows that $u(x) \geq u(y) \geq u(z)$, where $p \in [0,1]$. Does this make sense?

I am not sure about the part: $u\left ( px + (1-p)z \right ) = pu(x) + (1-p)u(z)$. Is this a legitimate way of writing the proof for the case of three outcomes?

**Tags:** microeconomics, expected-utility, proof


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## 1 answers

### Answer by username_1 on 2018-11-26. Score: 3 | Accepted answer

In order to do that, you need to define $u(·)$ as a utility function on "sure things" rather than on lotteries. In your example, you need to think in terms of the set of possible prizes to the lotteries. Say the set of possible prizes is given by $R$ and asume that is finite. For any $r\in R$, define $w\_r$ as a lottery that pays $r$ in every state of nature. Then, it should be clear that

$$x \sim \sum\_i p\_i w\_i,$$

(and similarly for $y$ and $z$) where $p\_i$ denotes the probability of state $i.$ Given that, as you mention, it must be the case that two equally desirable lotteries receive the same expected utility, the utility assigned to $x$ needs to be

$$H(x)=H\left(\sum\_i p\_i w\_i\right).$$

The first chapter of Huang, C. and Litzenberger, R.H. *Foundations for Financial Economics,* North Holland, 1988 explains in detail why the function representing the preferences $(H(·))$ has to be linear, and, with linearity stablished, you get the expected utility representation because

$$H\left(\sum\_i p\_i w\_i\right)=\sum\_i p\_i H(w\_i).$$

The only thing left is to define $u(r) \equiv H(w\_r),$ i.e, $u(r)$ is defined on outcomes, while $H(w\_r)$ is defined on lotteries. This is why we say that $u(r)$ is a utility function on sure things, because $r$ receives the same utility level that a lottery paying precisely $r$ in every state of nature.

Of course, expected utility is most useful in settings with large outcomes spaces and/or large lotteries spaces. In a restricted setting such as yours, you're probably best served by assigning values ("utilities") to lotteries themselves (as you did in your question). Again, a more rigorous proof can be found in Huang and Litzenberger, op.cit.

> **Comments:**
> **username_2**: Why do you say OP's example has 3 lotteries but just 2possible outcomes? Also what's $z_3$?  
> **username_0**: @username_1 I have made an edit.  
> **username_1**: @username_0, As I mention in my answer, in order to obtain an expected utility representation, you need to define $u(·)$ on "sure things." I'm edditing my answer to hopefully make myself clearer.  
> **username_1**: @HerrK., my mistake, I'll edit the answer  

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