QUESTION $1$

You are given the following payoff table $($in units of thousands of Rands$)$ for a decision Analysis Problem. Show all formulas and calculations.

$\backslash$table$[[$Alternatives$,$State of Nature$],[$S$1,$S$2,$S$3],[$A$1,25000,20000,14000],[$A$2,23000,21000,19000],[$Prior Probability,$0\mathrm{.}6,0\mathrm{.}3,0\mathrm{.}1]]$

$($a$)$ Which alternative should be chosen under the maximin payoff criterion?

$($b$)$ Which alternative should be chosen under the maximum likelihood criterion?

$(1)$

$($c$)$ Which alternative should be chosen under Bayes decision rule?

$(2)$

$($d$)$ Using Bayes decision rule, do sensitivity analysis graphically with respect to the prior probabilities of states $S1$ and $S2($without changing the prior probability of state $S3)$ to determine the crossover point where the decision shifts from one alternative to the other. Then use algebra to calculate this crossover point.

$(6)$

$($e$)$ Repeat part $($d$)$ for the prior probabilities of states $S1$ and $S3.$

$(6)$

$($f$)$ Repeat part $($d$)$ for the prior probabilities of states $S2$ and $S3.$

$(6)$

$($g$)$ If you feel that the true probabilities of the states of nature are within $10$ percent of the given prior probabilities, which alternative would you choose?

$(2)$

$[24]$

QUESTION $2$

You have to make certain decisions according to Bayes' decision rule. For your current problem, you have constructed the following payoff table $($in units of Rands$)$: Show all formulas and calculations.

PRDEAPS$/2024$

$\backslash$table$[[$Alternatives$,$State of Nature$],[$S$1,$S$2,$S$3],[$A$1,8000,13000,-13000],[$A$2,0,4000,-4000],[$A$3,5000,7000,-7000],[$Prior Probability,$0\mathrm{.}5,0\mathrm{.}3,0\mathrm{.}2]]$

$($a$)$ Which alternative should you choose? $(4)$

$($b$)$ Find EVPI. $(2)$

$($c$)$ What is the most that you should consider paying to obtain more information which state of nature will occur? $(1)$