From the following pay-off matrix, find the optimal decision under each of the following rule (i) maxmin (ii) minimax. | Act | $S_1$ | $S_2$ | $S_3$ | $S_4$ | | --- | --- | --- | --- | --- | | $A_1$ | 14 | 9 | 10 | 5 | | $A_2$ | 11 | 10 | 8 | 7 | | $A_3$ | 9 | 10 | 10 | 11 | | $A_4$ | 8 | 10 | 11 | 13 | (States of nature: $S_1, S_2, S_3, S_4$)

> > [!TLDR] > Maximin $\to A_3$ (row-min $9$); Minimax-regret $\to A_3$ (max regret $5$). > > Pay-off matrix: > > | Act | $S_1$ | $S_2$ | $S_3$ | $S_4$ | > | --- | --- | --- | --- | --- | > | $A_1$ | 14 | 9 | 10 | 5 | > | $A_2$ | 11 | 10 | 8 | 7 | > | $A_3$ | 9 | 10 | 10 | 11 | > | $A_4$ | 8 | 10 | 11 | 13 | > > **(i) Maximin criterion.** Take the minimum pay-off of each act, then choose the maximum of those minima. > - $A_1:\min=5,\quad A_2:\min=7,\quad A_3:\min=9,\quad A_4:\min=8.$ > - Largest minimum $=9\Rightarrow$ optimal act $A_3$. > > **(ii) Minimax criterion (minimax regret / Savage).** Since the entries are pay-offs (gains), the minimax rule is applied to the *regret* (opportunity-loss) table. Regret $=$ (column maximum) $-$ (pay-off). > Column maxima: $S_1=14,\ S_2=10,\ S_3=11,\ S_4=13.$ > > | Act | $S_1$ | $S_2$ | $S_3$ | $S_4$ | Max regret | …

From the following pay-off matrix, find the optimal decision under each of the following rule (i) maxmin (ii)

Question 26 of 38

From the following pay-off matrix, find the optimal decision under each of the following rule (i) maxmin (ii) minimax.

Act	$S_1$	$S_2$	$S_3$	$S_4$
$A_1$	14	9	10	5
$A_2$	11	10	8	7
$A_3$	9	10	10	11
$A_4$	8	10	11	13

(States of nature: $S_1, S_2, S_3, S_4$ )

Puducherry TnboardTamil Nadu HSC (DGE) Commerce Board 2023Subjective· 3mImportance★★★★★

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Maximin $\to A_3$ (row-min $9$ ); Minimax-regret $\to A_3$ (max regret $5$ ).

Pay-off matrix:

Act	$S_1$	$S_2$	$S_3$	$S_4$
$A_1$	14	9	10	5
$A_2$	11	10	8	7
$A_3$	9	10	10	11
$A_4$	8	10	11	13

(i) Maximin criterion. Take the minimum pay-off of each act, then choose the maximum of those minima.

$A_1:\min=5,\quad A_2:\min=7,\quad A_3:\min=9,\quad A_4:\min=8.$
Largest minimum $=9\Rightarrow$ optimal act $A_3$ .

(ii) Minimax criterion (minimax regret / Savage). Since the entries are pay-offs (gains), the minimax rule is applied to the regret (opportunity-loss) table. Regret $=$ (column maximum) $-$ (pay-off).

Column maxima: $S_1=14,\ S_2=10,\ S_3=11,\ S_4=13.$

| Act | $S_1$ | $S_2$ | $S_3$ | $S_4$ | Max regret | …

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