Learn / Library / Market making

What one fill tells a market maker

Market makingMedium

A market maker holds $11$ hypotheses about who is trading. Groups of $8$ , $2$ and $1$ hypotheses have total probabilities $\frac{1}{2}$ , $\frac{1}{4}$ and $\frac{1}{4}$ , and inside a group all hypotheses are equally likely. The next fill reveals the group. How many bits of entropy remain about the hypothesis on average after the fill? Give a fraction.

Checking an answer needs JavaScript. The hint and the solution open without it.

Hint

Average the entropy left inside each group.

Solution

Inside the groups the entropy is $log_{2} 8 = 3$ , then $log_{2} 2 = 1$ , then $0$ .
On average $\frac{1}{2} \times 3 + \frac{1}{4} \times 1 + \frac{1}{4} \times 0 = \frac{7}{4}$ bits remain.

Answer: $\frac{7}{4}$

Next question Back to the library