Exercise $17\mathrm{.}9-$ Promoting Beliefs with Law of Small Numbers. A manager has

hired a new investment analyst. The manager accurately knows that $20\%$ of analysts are

high ability and $80\%$ are low ability. High$-$ability analysts make investment decisions that

yield a positive quarterly return $60\%$ of the time, whereas low$-$ability analysts make

investment decisions that yield a positive quarterly return $40\%$ of the time.

At the end of the analyst's first year, the manager reviews the analyst's performance

over the previous $4$ quarters and promotes the analyst if she believes that there is at

least a $75\%$ chance that the analyst is high ability. Suppose the analyst has a successful

first year, with positive quarterly returns in each of the previous $4$ quarters.

a Given the above information, if the manager has Bayesian beliefs, what is her belief

that the analyst is high ability at the end of the first year? Will she promote the

analyst?

b Now suppose that the manager is a believer in the law of small numbers. That is$,$ she

believes $($erroneously$)$ that across $1$ consecutive quarters, a high $($low$)$ ability analyst

will earn positive quarterly returns exactly $6(4)$ times. What is her belief in this case

that the analyst is high ability at the end of the first year? Will the manager promote

the analyst?

c Provide intuition for your answers in parts $($a$)$ and $($b$).$ In addition, explain

whether the manager's thinking in part $($b$)$ illustrates the gambler's fallacy or

extrapolation.