Comparing two systems: Beyond common random numbers

Suppose one wishes to compare two closely related systems via stochastic simulation. Common random numbers (CRN) involves using the same streams of uniform random variates as inputs for both systems to sharpen the comparison. One can view CRN as a particular choice of copula that gives the joint distribution of the inputs of both systems. We discuss the possibility of using more general copulae, including simple examples that show how this can outperform CRN.