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.)

Question

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&#039;s career lasts   n days.)

Yair Boyle · Accepted Answer

I&#039;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&#039;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  &amp;gt;  1,   f  (  n  )  =  0.75  +                    f        (        n        −        1        )        +        g        (        n        −        1        )            2       (no need for flowers, she&#039;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&#039;m not sure there is a closed-form solution that can be guessed using this way.Hope it helps!

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

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