TITLE: Why is this coin-flipping probability problem unsolved?
QUESTION [50 upvotes]: You play a game flipping a fair coin.  You may stop after any trial, at which point you are paid in dollars 
the percentage of heads flipped.  So if on the first trial you flip a head, you should stop and earn \$100
because you have 100% heads.  If you flip a tail then a head, you could either stop and earn \$50,
or continue on, hoping the ratio will exceed 1/2.  This second strategy is superior.
A paper by Medina and Zeilberger (arXiv:0907.0032v2 [math.PR]) says that it is an unsolved
problem to determine if it is better to continue or stop after you have flipped 5 heads in 8 trials: accept \$62.50 or hope for more.  It is easy to simulate this problem and it is clear
from even limited experimental data that it is better to continue (perhaps more than 70% chance you'll improve over \$62.50):



My question is basically: Why is this difficult to prove?  Presumably it is not that difficult
to write out an expression for the expectation of exceeding 5/8 in terms of the cumulative binomial distribution.
 (5 Dec 2013).
A paper on this topic was just published:
Olle Häggström, Johan Wästlund.
"Rigorous computer analysis of the Chow-Robbins game."
(pre-journal arXiv link).
The American Mathematical Monthly, Vol. 120, No. 10, December 2013.
(Jstor link).
From the Abstract:

"In particular, we confirm that with 5 heads and 3 tails, stopping is optimal."

REPLY [5 votes]: This seems to be related to Gittins Indices. Gittins Indices are a way of solving these kind of optimal stopping problems for some classes of problems, and basically give you a way of balancing how much you are expected to gain given your current knowledge and how much more you could gain by risking obtaining more information about the process (or probability of flipping heads, etc).
Bruno