Each concert a singer makes profit of 0.75, but after every concert a singert can fall into bad mood

sebadillab0 2022-06-29 Answered
Each concert a singer makes profit of 0.75, but after every concert a singert can fall into bad mood with probability = 0.5. To get a singer out of these mood producer can send her flowers. If flowers cost x money, then singer can get out of bad mood with probability x . Which x should producer choose in order to maximize his expected profit. (All the profit from concert goes to producer. One can make assumprion, that singer's career lasts n days.)
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Answers (1)

Yair Boyle
Answered 2022-06-30 Author has 10 answers
I'm not sure that this is the correct path, but one can solve it by brute force using backward induction.

Let f ( n ) be the expected profit when there are n days left and the singer is happy today and g ( n ) be the expected profit when there are n days left and the singer is sad today. I assume that if she's sad, she stays sad until flowers are sent and make her happy and there is one concert a day.

Clearly, f ( 1 ) = 0.75 and g ( 1 ) = max 0 x 1 { x 0.75 x } = 9 64 .

For every n > 1, f ( n ) = 0.75 + f ( n 1 ) + g ( n 1 ) 2 (no need for flowers, she's happy!) and g ( n ) = max 0 x 1 { x [ 0.75 + f ( n 1 ) + g ( n 1 ) 2 ] + ( 1 x ) g ( n 1 ) x }

I tried to calculate it for n = 2 , 3 but it got ugly pretty quickly so I'm not sure there is a closed-form solution that can be guessed using this way.

Hope it helps!

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