<p><span class="math-container">$$
\left(\frac{2}{100} \cdot 1000 \oplus \frac{98}{100}\cdot 0\right)
$$</span>
is the lottery where you get <span class="math-container">$1000$</span> with probability <span class="math-container">$2/100$</span> and <span class="math-container">$0$</span> with probability <span class="math-container">$98/100$</span>.</p>
<p>The expression
<span class="math-container">$$
20 \sim \left(\frac{2}{100} \cdot 1000 \oplus \frac{98}{100}\cdot 0\right)
$$</span>
usually says that the decision maker is indifferent (in terms of preferences) between taking the lottery and having <span class="math-container">$20$</span> for sure.</p>

CC BY-SA 4.0
2021-05-15T14:28:45.723Z
So say that the consumer face 2 wheels. the first wheel the consumer wins $1000$ with probability $1/4$, $50$ with probability $1/4$ and $20$ for the rest. The second he wins $1000$ with probability $1/5$, $100$ with probability $1/4$ and nothing for the rest. How do I determine from the preference which he chooses from the $3$ equations?
mathstudent23
CC BY-SA 4.0
2021-05-15T15:00:43.990Z
(just to take a look at the usage of the preferences)
mathstudent23
CC BY-SA 4.0
2021-05-15T21:29:52.473Z
This should be a separate question.
VARulle
CC BY-SA 4.0
2021-05-16T00:43:28.920Z
@mathstudent23 you assume rationality (empirical evidence is against such an assumption) and substitute for $50$, $20$ and $100$ using the indifferences you have been told to turn the two wheels into equivalent $1000$ & $0$ lotteries
Henry
CC BY-SA 4.0
2021-05-16T17:02:27.867Z
@VARulle I agree. Up now.
mathstudent23
CC BY-SA 4.0
2021-05-16T17:03:03.143Z
@Henry What do you mean by substitute in this context?
mathstudent23
CC BY-SA 4.0
2021-05-16T17:13:43.490Z
@mathstudent23 $\left ( \frac{1}{5}\cdot1000 \oplus \frac{1}{4} \cdot 100 \oplus \frac{11}{20} \cdot 0\right ) \sim$ $\left ( \frac{1}{5}\cdot1000 \oplus \frac{1}{4} \cdot  \frac{45}{100}\cdot1000 \oplus \frac{1}{4}\cdot\frac{55}{100} \cdot 0 \oplus \frac{11}{20} \cdot 0\right )$ or something like that, which you can simplify
Henry
CC BY-SA 4.0
2021-05-16T17:28:52.510Z
@Henry I am not sure on why this lottery was written but wouldn't the simplified one be $\left ( \frac{1}{5}\cdot 1000\oplus \frac{70}{100}\cdot 1000 \oplus \frac{80}{100}\cdot 0\oplus \frac{11}{20}\cdot 0  \right )$
mathstudent23
CC BY-SA 4.0
2021-05-17T00:10:53.753Z
@mathstudent23 It can be simplified, but your fractions do not add up to $1$.  Instead $\left(\frac{5}{16}\cdot1000  \oplus \frac{11}{16} \cdot 0\right)$
Henry
CC BY-SA 4.0
2021-05-15T13:19:13.403Z
2021-05-15T13:19:13.403Z
tdm
<p>A wheel of fortune has outcomes <span class="math-container">$S=\left \{ 1000,100,50,20,0 \right \}$</span> as money prices. A consumer has the preferences</p>
<p><span class="math-container">$$20\sim \left ( \frac{2}{100}\cdot1000 \oplus \frac{98}{100} \cdot 0 \right )$$</span>
<span class="math-container">$$50\sim \left ( \frac{15}{100}\cdot1000 \oplus \frac{85}{100} \cdot 0 \right )$$</span>
<span class="math-container">$$100\sim \left ( \frac{45}{100}\cdot1000 \oplus \frac{55}{100} \cdot 0 \right )$$</span></p>
<p>What does this notation mean? I figure that the direct sum is just showing that the percentage for winning <span class="math-container">$1000$</span> in the first equation is <span class="math-container">$2/100$</span> and <span class="math-container">$98/100$</span> for <span class="math-container">$0$</span>. But what does the <span class="math-container">$20$</span> denote? Is this some utility and how would that be read with the indifference sign in front of it? Never seen the notation like this before.</p>

<microeconomics><utility><expected-utility>
Expected utility theory (Lottery notation)
mathstudent23
# Expected utility theory (Lottery notation)

(Asked by: username_0 on 2021-05-15)

A wheel of fortune has outcomes $S=\left \{ 1000,100,50,20,0 \right \}$ as money prices. A consumer has the preferences

$$20\sim \left ( \frac{2}{100}\cdot1000 \oplus \frac{98}{100} \cdot 0 \right )$$
$$50\sim \left ( \frac{15}{100}\cdot1000 \oplus \frac{85}{100} \cdot 0 \right )$$
$$100\sim \left ( \frac{45}{100}\cdot1000 \oplus \frac{55}{100} \cdot 0 \right )$$

What does this notation mean? I figure that the direct sum is just showing that the percentage for winning $1000$ in the first equation is $2/100$ and $98/100$ for $0$. But what does the $20$ denote? Is this some utility and how would that be read with the indifference sign in front of it? Never seen the notation like this before.

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


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## 1 answers

### Answer by username_1 on 2021-05-15. Score: 5 | Accepted answer

$$
\left(\frac{2}{100} \cdot 1000 \oplus \frac{98}{100}\cdot 0\right)
$$
is the lottery where you get $1000$ with probability $2/100$ and $0$ with probability $98/100$.

The expression
$$
20 \sim \left(\frac{2}{100} \cdot 1000 \oplus \frac{98}{100}\cdot 0\right)
$$
usually says that the decision maker is indifferent (in terms of preferences) between taking the lottery and having $20$ for sure.

> **Comments:**
> **username_0**: So say that the consumer face 2 wheels. the first wheel the consumer wins $1000$ with probability $1/4$, $50$ with probability $1/4$ and $20$ for the rest. The second he wins $1000$ with probability $1/5$, $100$ with probability $1/4$ and nothing for the rest. How do I determine from the preference which he chooses from the $3$ equations?  
> **username_0**: (just to take a look at the usage of the preferences)  
> **username_2**: This should be a separate question.  
> **username_3**: @username_0 you assume rationality (empirical evidence is against such an assumption) and substitute for $50$, $20$ and $100$ using the indifferences you have been told to turn the two wheels into equivalent $1000$ & $0$ lotteries  
> **username_0**: @username_2 I agree. Up now.  
> **username_0**: @username_3 What do you mean by substitute in this context?  
> **username_3**: @username_0 $\left ( \frac{1}{5}\cdot1000 \oplus \frac{1}{4} \cdot 100 \oplus \frac{11}{20} \cdot 0\right ) \sim$ $\left ( \frac{1}{5}\cdot1000 \oplus \frac{1}{4} \cdot  \frac{45}{100}\cdot1000 \oplus \frac{1}{4}\cdot\frac{55}{100} \cdot 0 \oplus \frac{11}{20} \cdot 0\right )$ or something like that, which you can simplify  
> **username_0**: @username_3 I am not sure on why this lottery was written but wouldn't the simplified one be $\left ( \frac{1}{5}\cdot 1000\oplus \frac{70}{100}\cdot 1000 \oplus \frac{80}{100}\cdot 0\oplus \frac{11}{20}\cdot 0  \right )$  
> **username_3**: @username_0 It can be simplified, but your fractions do not add up to $1$.  Instead $\left(\frac{5}{16}\cdot1000  \oplus \frac{11}{16} \cdot 0\right)$  

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