A 31" tyre on a 16" rim is exactly the same overall diameter as a 31" tyre on a 15" rim
It used to be that you'd have more options on a 15" than any other rim, but the gap is closing as larger rims become more standard (the JK comes with 17" rims!)
A smaller rim/larger tyre however does allow more bagging of the sidewalls for better traction offroad.