You have the following payoff table for a decision analysis problem. Payoffs are in units of thousands of dollars of profit.

State of Nature

State of Nature

State of Nature

Alternatives

S$1$

S$2$

S$3$

D$1$

$3$

$3$

$9$

D$2$

$4$

$7$

$6$

D$3$

$2$

$6$

$7$

Which decision alternative would you choose using an optimistic approach?

Which decision alternative would you choose using a conservative approach?

There are $2$ steps to solve this one.

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Step $1$

Answer:

In decision analysis, the optimistic approach involves selecting the decision alternative that maximizes the best possible outcome for each state of nature. The conservative approach, on the other hand, focuses on minimizing the maximum possible regret for each decision alternative.

Optimistic Approach:

For each decision alternative, select the maximum payoff among all possible states of nature:

For D$1$: Max payoff $=9($in State of Nature S$3)$

For D$2$: Max payoff $=7($in State of Nature S$2)$

For D$3$: Max payoff $=7($in State of Nature S$3)$

Explanation:

Based on the optimistic approach, the decision alternative with the highest maximum payoff is D$1$ with a payoff of $9($in State of Nature S$3).$

Step $2$

Conservative Approach:

For each decision alternative, calculate the regret for each state of nature $($the maximum payoff in that state of nature minus the payoff for the chosen decision alternative$).$ Then, select the decision alternative that minimizes the maximum regret.

Regret for D$1$: $(9-3)=6($in State of Nature S$3)$

Regret for D$2$: $(7-4)=3($in State of Nature S$2)$

Regret for D$3$: $(7-2)=5($in State of Nature S$3)$

The maximum regret for D$1$ is $6,$ for D$2$ is $3,$ and for D$3$ is $5.$ The conservative approach would select the decision alternative with the smallest maximum regret, which is D$2$ with a maximum regret of $3($in State of Nature S$2).$

Explanation:

In summary:

Optimistic approach: Choose D$1($payoff of $9$ in State of Nature S$3).$

Conservative approach: Choose D$2($maximum regret of $3$ in State of Nature S$2).$