回複：回複：讓人煩惱的禮金卡問題

Sorry, seems to be
0.577 * (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 * (1-t) + 0.9t ExpectReturn(1)

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

So, test at 0.5+0.45ExpectReturn give you the max, it is

(0.5 + 0.45 ExpectReturn)^2 = ExpectReturn

==>
ExpectReturn(1)=0.577
==>
ExpectReturn(m)=0.577m