Question 8
Yan and Jacob play the following game. Yan shows Jacob a weighted 4-sided die labelled $1$, $2$, $3$, $4$, with weights $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{7}$, $\frac{1}{42}$, respectively. Then, Jacob specifies $4$ positive real numbers $x_{1}, x_{2}, x_{3}, x_{4}$ such that $x_{1}+x_{2}+x_{3}+x_{4}=1$. Finally, Yan rolls the die, and Jacob earns $10+\log \left(x_{k}\right)$ dollars if the die shows $k$ (note this may be negative). Which $x_{i}$ should Jacob pick to maximize his expected payoff?
(Here log is the natural logarithm, which has base $e \approx 2.718$.)
(Here log is the natural logarithm, which has base $e \approx 2.718$.)