I don't think you should 'guess' that the minimum error is proportional 1/N, since its relationship with N is exactly what we need to know.

I mistakenly thought I had a qualitative argument that was too simple to bother spelling out. Unfortunately, it was also too simple to be correct. :-) However, see constant's comment; my intuition appeared to be probably right after all.

... But do you realize that 1/N decreases faster than 1/sqrt(N), so that your guess would indicate that the average strategy would rarely win?

Yes. If my analysis is correct, the method would be very unlikely to win, but it would be much better than using the same method as the other competitors, precisely because 1/N decreases faster than 1/sqrt(N).