I think you’re right that there is no epsilon such that you can guarantee 50% + epsilon performance. You can however guarantee that your performance is greater than 50%. (Since ROB isn’t allowed to use the full limit strategy of giving two identical reals, there will always be some nonzero delta.)
Monotonicity is insufficient. Your function does need to be strictly increasing. Otherwise ROB can always pick numbers in a non-increasing interval.
I think you’re right that there is no epsilon such that you can guarantee 50% + epsilon performance. You can however guarantee that your performance is greater than 50%. (Since ROB isn’t allowed to use the full limit strategy of giving two identical reals, there will always be some nonzero delta.)
Monotonicity is insufficient. Your function does need to be strictly increasing. Otherwise ROB can always pick numbers in a non-increasing interval.