Summary: This paper is concerned with the problem of optimizing the performance of a stochastic system over a finite set of alternatives in situations where the performance of the system cannot be evaluated analytically, but must be estimated or measured, for instance, through simulation. We present two variants of a new method for solving such discrete stochastic optimization problems. This new method uses global search to look for the optimal solution. It generates a sequence taking values in the set of feasible alternatives, where each new element of the sequence is generated by comparing the current element with another candidate alternative and letting the next element of the sequence be the one of the current and candidate alternatives that appears to yield better performance. For both versions of the proposed method, the element of the set of feasible alternatives that the generated sequence visits most often is shown to converge almost surely to a globally optimal solution of the underlying optimization problem. Thus the method spends most of the computational effort at the global optimizer. We also show how one variant of the proposed method can be used to solve discrete optimization problems in both transient and steady-state simulation, show how the other variant can be used to optimize probabilities, and present some numerical results.