I have a feeling the half-average "should" be in the interval between 0 and 1.
I will be very surprised if the half-average is zero.
I'm guessing there is both literature and mathematical models for this interesting problem, but I'm not going to go ahead and Google it. Not knowing is way more fun.
NPR's Planet Money is running an experiment which could be an interesting way to test your other-people-modeling skills.
This is a guessing game. To play, pick a number between 0 and 100. The goal is to pick the number that's closest to half the average of all guesses.
http://www.npr.org/blogs/money/2011/10/03/133654225/please-help-us-pick-a-number?sc=fb&cc=fp
The other people guessing are self-selected, I would assume primarily NPR listeners.