<p><span class="math-container">$p\circ x\oplus(1-p)\circ y$</span> is a lottery that gives you the prize <span class="math-container">$x$</span> with probability <span class="math-container">$p$</span> and the price <span class="math-container">$y$</span> with probability <span class="math-container">$(1-p)$</span>. Unless <span class="math-container">$x,y$</span> can be identified with numbers, such as amounts of money, it makes no sense to take the expectation of this lottery. What is the expected value of a lottery that gives you a cow with a probability of <span class="math-container">$0.5$</span> and a sheep with a probability of <span class="math-container">$0.5$</span>?</p>
<p>You can also take prizes to be lotteries over prizes which reduce ultimately to lotteries over prizes using the laws of probability.</p>
<p>If <span class="math-container">$U$</span> gives your the expected utility of each lottery, then you have
<span class="math-container">$$U(p_1\circ L_1\oplus p_2 L_2\oplus\cdots\oplus p_n\circ L_n)=p_1U(L_1)+p_2 U(L_2)\cdots+p_n U(L_n).$$</span>
On the right, you are actually taking an expectation, on the left you apply <span class="math-container">$U$</span> to a compound lottery. Sometimes people write down the compound lottery on the left as if it were a weighted sum and hope that the context makes the difference clear, but these are really different things that get related by expected utility theory.</p>

CC BY-SA 4.0
2021-06-08T22:00:49.817Z
Oh I see. But what if $x,y$ are numbers, such as amounts of money? Then wouldn't this equal the expected value?
j3141592653589793238
CC BY-SA 4.0
2021-06-08T22:10:33.720Z
A lottery that gives you $0$ dollars with a probability of $0.5$ and $1000$ dollars with a probability of $0.5$ is still different than a lottery that gives you $500$ dollars with probability one.
Michael Greinecker
CC BY-SA 4.0
2021-06-08T23:02:54.103Z
Oh lord. Yeah that is horribly confusing tbh. One should write $EU(L)$ or use a capital U for expected utility and lowercase u for the actual utility function or similar. So I understood that the utility of the second lottery (your latter example) should be higher than the utility of the first lottery. This would be the case for f.ex. CobbDouglas preferences where $u(x)=lnx$ except if $u(x)=x$ (perfect substitutes). Thanks (you're great, the example you gave in your comment was especially helpful and clear)!
j3141592653589793238
CC BY-SA 4.0
2021-06-10T19:33:37.053Z
@j3141592653589793238, almost everybody gets confused by this in the beginning. It might be better to consult MCWG on expected utility theory. They distinguish between the Bernoulli-utility function $u$ (defined on prizes) and the vNM-utility function $U$ defined on lotteries over prizes, where $U=Eu$. That helps.
VARulle
CC BY-SA 4.0
2021-06-10T19:35:53.120Z
@VARulle Thanks! I'm sorry MCWG -- motorcycle working group? EDIT: Ah, no, Mas-Colell, A., Whinston, M. und J. Green. I'll have a look, though I'm still an undergraduate in first year so not sure how deep I'm expected to dive in but it's certainly worth a look!
j3141592653589793238
CC BY-SA 4.0
2021-06-10T19:43:57.477Z
@j3141592653589793238, sorry, it's an acronym referring to Mas-Colell, Whinston, Green: Microeconomic Theory (Oxford University Press, 1995). A 1000+ pages tome that many view as the most authoritative PhD-level textbook on micro theory. (Some call it the "micro bible", even though many game theorists think the game theory chapter gets it all wrong.)
VARulle
CC BY-SA 4.0
2021-06-08T21:56:01.843Z
2021-06-08T21:56:01.843Z
Michael Greinecker
<p>I understand the expected <em>value</em> of a lottery is <span class="math-container">$\sum_n^N{p_nL_n}$</span> where there are <span class="math-container">$N$</span> possible outcomes, each with a probability <span class="math-container">$p_n$</span> with <span class="math-container">$n=1,...,N$</span> and <span class="math-container">$\sum_{n}p_n=1$</span> (that's rather trivial I believe).</p>
<p>Now we say <span class="math-container">$U(\sum_{n=1}^{N}{p_nL_n})=\sum_{n}^{N}{p_nU(L_n)}$</span> or in other words <span class="math-container">$U(L)=u_1p_1+...+u_np_N$</span> where <span class="math-container">$U:\mathbb{L} \mapsto \mathbb{R}$</span> with <span class="math-container">$L=(p_1,...,p_n) \in \mathbb{L}$</span>.</p>
<p>Why is that? Is that just the &quot;definition&quot; of expected utility?</p>
<p>Why is, for a two-outcomes-case,</p>
<p><span class="math-container">$U(pL_1+(1-p)L_2)=pU(L_1)+(1-p)U(L_2)$</span>?</p>
<p>Wouldn't this imply that the utility of the expected value equals the expected utility since the expression <span class="math-container">$pL_1+(1-p)L_2$</span> <em>is</em> the expected value? This should not be true though (except if U(x)=x) but it's not stated anywhere this is assumed.</p>
<p>I saw this in Varian, Microeconomic Analysis, chapter 11.3 &quot;uniqueness of the expected utility function&quot;:</p>
<p><a href="https://i.sstatic.net/dfehx.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/dfehx.png" alt="Varian Mic. Analy." /></a></p>
<p>Shouldn't this be <span class="math-container">$v(u(px+(1-p)y)) = ...$</span> on the first line?</p>
<p>What's wrong here?
I'm confused.</p>

<microeconomics><mathematical-economics><utility><risk><uncertainty>
Proof that $U(\sum_{n=1}^{N}{p_nL_n})=\sum_{n}^{N}{p_nU(L_n)}$
j3141592653589793238
# Proof that $U(\sum_{n=1}^{N}{p_nL_n})=\sum_{n}^{N}{p_nU(L_n)}$

(Asked by: username_0 on 2021-06-08)

I understand the expected *value* of a lottery is $\sum\_n^N{p\_nL\_n}$ where there are $N$ possible outcomes, each with a probability $p\_n$ with $n=1,...,N$ and $\sum\_{n}p\_n=1$ (that's rather trivial I believe).

Now we say $U(\sum\_{n=1}^{N}{p\_nL\_n})=\sum\_{n}^{N}{p\_nU(L\_n)}$ or in other words $U(L)=u\_1p\_1+...+u\_np\_N$ where $U:\mathbb{L} \mapsto \mathbb{R}$ with $L=(p\_1,...,p\_n) \in \mathbb{L}$.

Why is that? Is that just the "definition" of expected utility?

Why is, for a two-outcomes-case,

$U(pL\_1+(1-p)L\_2)=pU(L\_1)+(1-p)U(L\_2)$?

Wouldn't this imply that the utility of the expected value equals the expected utility since the expression $pL\_1+(1-p)L\_2$ *is* the expected value? This should not be true though (except if U(x)=x) but it's not stated anywhere this is assumed.

I saw this in Varian, Microeconomic Analysis, chapter 11.3 "uniqueness of the expected utility function":

[![Varian Mic. Analy.](https://i.sstatic.net/dfehx.png)](https://i.sstatic.net/dfehx.png)

Shouldn't this be $v(u(px+(1-p)y)) = ...$ on the first line?

What's wrong here?
I'm confused.

**Tags:** microeconomics, mathematical-economics, utility, risk, uncertainty


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## 1 answers

### Answer by username_1 on 2021-06-08. Score: 3 | Accepted answer

$p\circ x\oplus(1-p)\circ y$ is a lottery that gives you the prize $x$ with probability $p$ and the price $y$ with probability $(1-p)$. Unless $x,y$ can be identified with numbers, such as amounts of money, it makes no sense to take the expectation of this lottery. What is the expected value of a lottery that gives you a cow with a probability of $0.5$ and a sheep with a probability of $0.5$?

You can also take prizes to be lotteries over prizes which reduce ultimately to lotteries over prizes using the laws of probability.

If $U$ gives your the expected utility of each lottery, then you have
$$U(p\_1\circ L\_1\oplus p\_2 L\_2\oplus\cdots\oplus p\_n\circ L\_n)=p\_1U(L\_1)+p\_2 U(L\_2)\cdots+p\_n U(L\_n).$$
On the right, you are actually taking an expectation, on the left you apply $U$ to a compound lottery. Sometimes people write down the compound lottery on the left as if it were a weighted sum and hope that the context makes the difference clear, but these are really different things that get related by expected utility theory.

> **Comments:**
> **username_0**: Oh I see. But what if $x,y$ are numbers, such as amounts of money? Then wouldn't this equal the expected value?  
> **username_1**: A lottery that gives you $0$ dollars with a probability of $0.5$ and $1000$ dollars with a probability of $0.5$ is still different than a lottery that gives you $500$ dollars with probability one.  
> **username_0**: Oh lord. Yeah that is horribly confusing tbh. One should write $EU(L)$ or use a capital U for expected utility and lowercase u for the actual utility function or similar. So I understood that the utility of the second lottery (your latter example) should be higher than the utility of the first lottery. This would be the case for f.ex. CobbDouglas preferences where $u(x)=lnx$ except if $u(x)=x$ (perfect substitutes). Thanks (you're great, the example you gave in your comment was especially helpful and clear)!  
> **username_2**: @username_0, almost everybody gets confused by this in the beginning. It might be better to consult MCWG on expected utility theory. They distinguish between the Bernoulli-utility function $u$ (defined on prizes) and the vNM-utility function $U$ defined on lotteries over prizes, where $U=Eu$. That helps.  
> **username_0**: @username_2 Thanks! I'm sorry MCWG -- motorcycle working group? EDIT: Ah, no, Mas-Colell, A., Whinston, M. und J. Green. I'll have a look, though I'm still an undergraduate in first year so not sure how deep I'm expected to dive in but it's certainly worth a look!  
> **username_2**: @username_0, sorry, it's an acronym referring to Mas-Colell, Whinston, Green: Microeconomic Theory (Oxford University Press, 1995). A 1000+ pages tome that many view as the most authoritative PhD-level textbook on micro theory. (Some call it the "micro bible", even though many game theorists think the game theory chapter gets it all wrong.)  

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