[jason1990]

I think what you may be missing here is that my EV calculations are based on the fact that A and n are independent. If they are dependent, then that EV formula and subsequent analysis goes out the window. If you look back at the Call-Off criterion, you will see that if n is of the form CA^2, then you will just be calling off your bet with some fixed probability that does not depend on A. Your EV will then simply be $50*(prob. of not calling off).

Let me throw something else out there. Suppose you have a function f taking values between 0 and 1. You can use this function to generate a strategy for calling off. Namely, if you see A, then you call off your bet with probability 1 - f(A). Conversely, whatever strategy you might employ will correspond to some f. For example, the Never Call-Off strategy corresponds to f(A) = 1 for all A. The Group n strategy corresponds to f(A) = e^{-A^2/n}.

If your strategy corresponds to f, then your EV will be 50(2f(y) - f(2y)). In order to outperform the Never Call-Offers, you need this to be greater than 50. Let us call a strategy "uniformly dominating" if

2f(y) - f(2y) >= 1 for all y.

I claim that the only uniformly dominating strategy is the Never Call-Off strategy. I will leave it as a puzzle to prove this result.