We present a Monte Carlo method for approximating the minimax expected size (MES) confidence set for a parameter known to belong to a compact set. Size refers to the measure of the confidence set; the measure can be indexed by the true parameter value, which allows the confidence procedure to be tailored for specific scientific goals. As the number of iterations increases, the Monte Carlo estimator converges to the \(\Gamma\)-minimax procedure, where \(\Gamma\) is a polytope of priors. The algorithm exploits Bayes/minimax duality by searching for the \(\Gamma\)-least favorable prior. A Fortran-90 implementation of the algorithm for both serial and parallel computers is available. We apply the method to estimate parameters of the primordial universe from observations of the cosmic microwave background radiation.