One simulation method is as good as another?

Fallacy 1   One simulation method is as good as another.

Counter Statement 1   The error bound on a method is based on the Star Discrepancy of the points used. The lower the Star Discrepancy the lower the error bound.

The error bound is the product of the Star Discrepancy and the Variation of the function over the relevant range of the function. Here the function is the product of some function like loss times the probability density function of the loss, if known, or of the variables that drive the loss, the more usual case.