Repeated games under epistemic uncertainty actually allows you to get an edge by extracting information and reducing that uncertainty, in comparison to a game which is guaranteed to have no edge for you.
To make clear: playing this game with an urn which has an equal amount of red and green balls, is the worst case scenario when you get to pick the colour that wins. The second urn can only be the same or better, whether or not you can figure out any information about the balls in it.
For instance, you are not better off discovering that the urn has a 50:50 ratio than you would be if you knew that 50:50 ratio ahead of time.
If you're saying that you could discover a better ratio, then the thing that gives you the edge is the better ratio itself; if you knew the better ratio ahead of time, the epistemic uncertainty would only harm you.