回複：回複：回複：讓人煩惱的禮金卡問題

Sorry, seems to be
0.37987 * (upper-bound currently known)

ExpectReturn(upper_bound) must be linear
ExpectReturn(upper_bound) = upper_bound * ExpectReturn(1)

Denote,
E(1, t) = ExpectReturn for test spend t for known upper 1
So
ExpectReturn(1) = max E(1,t), for all t in [0,1]

E(1,t) = t * (1-t) + ExpectReturn(0.9t) * t
= t * (1-t) + 0.9t*t ExpectReturn(1)

So
ExpectReturn(1) = max (t*(1-t) + t*t*0.9ExpectReturn(1))

So, test at 0.5/(1-0.9ExpectReturn) give you the max, it is

0.25/(1-0.9ExpectReturn) = ExpectReturn

==>
ExpectReturn(1)=0.37987
==>
ExpectReturn(m)=0.37987m

we use the root < 0.5