This article compares and contrasts traditional mathematical models with computer simulations. The strengths and flexibility of algorithmic computational simulations are shown by "walking through" a program designed to investigate and extend understanding in one of the most enduring questions in social choice research—concerns over the frequency of cycling collective decisions even when individuals hold clear transitive preferences. After discussing solutions to this problem from each approach—analytic and algorithmic—the authors modify the simulation to investigate questions dealing with weak preference orderings, even-numbered electorates, and the probabilistic uncertainty inherent in iterated approximations more generally. Although traditional formal solutions have a clear advantage for estimating the limiting values of the probability of cycles (i.e., as the electorate or number of alternatives goes to infinity), computer models are far more tractable for finite values and extensions.
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