A router can use 4 venues paying 8, 4, 2 and 2 ticks. Policy A always uses the first. Policy B routes 1/2, 1/4, 1/8 and 1/8 of its flow to them, in order. A policy is scored by its expected ticks plus β times its entropy in bits. At what β do the policies tie? Give a fraction.

Learn / Library / Expected value

When a router should spread

Expected valueMedium

A router can use $4$ venues paying $8$ , $4$ , $2$ and $2$ ticks. Policy A always uses the first. Policy B routes $\frac{1}{2}$ , $\frac{1}{4}$ , $\frac{1}{8}$ and $\frac{1}{8}$ of its flow to them, in order. A policy is scored by its expected ticks plus $β$ times its entropy in bits. At what $β$ do the policies tie? Give a fraction.

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

Hint

Policy A has zero entropy.

Solution

Policy B expects $\frac{1}{2} \times 8 + \frac{1}{4} \times 4 + \frac{1}{8} \times 2 + \frac{1}{8} \times 2 = \frac{11}{2}$ ticks.
Its entropy is $\frac{1}{2} \times 1 + \frac{1}{4} \times 2 + \frac{1}{8} \times 3 + \frac{1}{8} \times 3 = \frac{7}{4}$ .
The tie needs $\frac{11}{2} + \frac{7}{4} \times β = 8$ , so $β = \frac{8 - \frac{11}{2}}{\frac{7}{4}} = \frac{10}{7}$ .

Answer: $\frac{10}{7}$

Next question Back to the library