(1)
Consider the strategy: pick a cut off value b, if the last ticket value a>b, stop, otherwise continune.
Let y (as a function of b) is the future expected return if the last ticket value a <= b (hence the game continunes). We have
y = \int_b^1 x dx + y b -c
so y = 1/2(1+b)-c/(1-b)
y is maximized when b = 1-sqrt(2c), and y_max = 1-sqrt(2c)
Hence the strategy is: don't play the game when c>1/2; if c<=1/2, stop playing the game the first time the ticket value exceeds 1-sqrt(2c)
(2)
Still working on this, my first thought is that we should use the same strategy as in (1)...
Consider the strategy: pick a cut off value b, if the last ticket value a>b, stop, otherwise continune.
Let y (as a function of b) is the future expected return if the last ticket value a <= b (hence the game continunes). We have
y = \int_b^1 x dx + y b -c
so y = 1/2(1+b)-c/(1-b)
y is maximized when b = 1-sqrt(2c), and y_max = 1-sqrt(2c)
Hence the strategy is: don't play the game when c>1/2; if c<=1/2, stop playing the game the first time the ticket value exceeds 1-sqrt(2c)
(2)
Still working on this, my first thought is that we should use the same strategy as in (1)...
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