# Expected utility theory (Lottery notation)

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.

## 1 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.

• 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? May 15, 2021 at 14:28
• (just to take a look at the usage of the preferences) May 15, 2021 at 15:00
• This should be a separate question. May 15, 2021 at 21:29
• @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 May 16, 2021 at 17:13
• @mathstudent23 It can be simplified, but your fractions do not add up to $1$. Instead $\left(\frac{5}{16}\cdot1000 \oplus \frac{11}{16} \cdot 0\right)$ May 17, 2021 at 0:10