$7.($a$)$ Jim is considering investing in a venture for which the returns are as follows. There are three exogenous states of nature that mean that the investment will make either a good return $($s$1),$ no return $($s$2)$ or a large financial loss $($s$3)$:

s$1$ s$2$ s$3$

Invest $($I$)6030-940$

Do not invest $($D$)303030$

$($i$)$ Determine the optimal action under each of the following decision rules: the extreme pessimist; the extreme optimist; and the Hurwicz Criterion with a pessimism$-$optimism index of $=1/20.$

$($ii$)$ At what value for $$ would your decision change in part $($i$)?$ Sketch this in a diagram and explain why the choice changes.

$($iii$)$ Determine Jim$$s best action under minimax$-$regret.

$($b$)$ By referring to Jim$$s problem above, consider whether the decision rules provide a good guide to decision$-$making under state$-$contingent uncertainty.

$($c$)$ Now, suppose Jim has subjective beliefs about the probabilities with which the states of nature occur. What improvements can be made to form a better theory of choice under uncertainty using either the expected value or expected utility?

$[$Parts $($a$),($b$)$ and $($c$)$ have equal weight$]$