Given the following pay-off matrix (in rupees) for three strategies and two states of nature. | Strategy | $E_1$ (States-of-nature) | $E_2$ (States-of-nature) | | --- | --- | --- | | $S_1$ | $40$ | $60$ | | $S_2$ | $10$ | $-20$ | | $S_3$ | $-40$ | $150$ | Select a strategy using each of the following rule : (i) Maximin (ii) Minimax

> > [!TLDR] > Maximin selects the best of the row minima ($S_1$); Minimax regret selects the strategy whose largest regret is smallest ($S_3$). > > Pay-off matrix (rupees): > > | Strategy | $E_1$ | $E_2$ | Row min | > | --- | --- | --- | --- | > | $S_1$ | 40 | 60 | 40 | > | $S_2$ | 10 | $-20$ | $-20$ | > | $S_3$ | $-40$ | 150 | $-40$ | > > **(i) Maximin criterion** (pessimistic — best of the worst pay-offs). Take each row's minimum, then the maximum of those minima: > $$\max\{40,\ -20,\ -40\} = 40 \ \Rightarrow\ \textbf{Strategy } S_1.$$ > > **(ii) Minimax (regret) criterion** (Savage). Build the regret (opportunity-loss) table: in each state, regret $=$ (best pay-off in that column) $-$ (pay-off). Column maxima are $E_1 = 40$, $E_2 = 150$. > > | Strategy | $E_1$ regret | $E_2$ regret | Max regret | > | --- | --- | --- | --- | > | $S_1$ | $40-40=0$ | $150-60=90$ | 90 | …

Given the following pay-off matrix (in rupees) for – Class 12 Business Mathematics Statistics

Question 18 of 38

Given the following pay-off matrix (in rupees) for three strategies and two states of nature.

Strategy	$E_1$ (States-of-nature)	$E_2$ (States-of-nature)
$S_1$	$40$	$60$
$S_2$	$10$	$-20$
$S_3$	$-40$	$150$

Select a strategy using each of the following rule :

Maximin
Minimax

Puducherry TnboardTamil Nadu HSC (DGE) Commerce Board 2020Subjective· 2mImportance★★★★★

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Maximin selects the best of the row minima ( $S_1$ ); Minimax regret selects the strategy whose largest regret is smallest ( $S_3$ ).

Pay-off matrix (rupees):

Strategy	$E_1$	$E_2$	Row min
$S_1$	40	60	40
$S_2$	10	$-20$	$-20$
$S_3$	$-40$	150	$-40$

Maximin criterion (pessimistic — best of the worst pay-offs). Take each row's minimum, then the maximum of those minima: $\max\{40,\ -20,\ -40\} = 40 \ \Rightarrow\ \textbf{Strategy } S_1.$
Minimax (regret) criterion (Savage). Build the regret (opportunity-loss) table: in each state, regret $=$ (best pay-off in that column) $-$ (pay-off). Column maxima are $E_1 = 40$ , $E_2 = 150$ .

Strategy	$E_1$ regret	$E_2$ regret	Max regret
$S_1$	$40-40=0$	$150-60=90$	90

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