Other commenters are wrong in saying that the payout is different for an adversarial choice.
The crux of the payout derivation is: we can only cover 1 number in step 1, 2 in step 2, 4 in step 3, 8 in step 4, and so on. You can choose your initial number in binary search randomly, and as long as you meet the above condition is met (# of possible numbers covered in each step), payout should be same as 0.2