appears first in the order–, x is better than y. In our previous example, e is the Condorcet winner, as can be inferred from Table 2. Table 2. Pairwise comparisons to choose the Condorcet winner on the case in Table 1. However, there might not be a Condorcet winner. Consider the setting in Table 3, where 21 voters must choose among candidates {a, b, c}. Table 3. A case in which 21 voters must choose among candidates {a, b, c} and there is no Condorcet winner. In this case a beats b by 14-7, b beats c by 15-6, c beats a by 13-8. We commonly find ourselves in a situation in which there is no Condorcet winner. This is called a Condorcet Paradox.2,4 In the simulations we describe in Section 4 it can be appreciated that as the number n of voters increases, the probability of there being no Condorcet winner tends to a particular value. Combinatorial and probabilistic methods have been previously used to address Condorcet paradoxes;14 the frequencies of undesirable events in electoral processes, particularly preference cycles have also been studied from a computational perspective.15 Our aim is to examine the performance of the methods themselves to address a specific choice problem; to achieve this aim we will use the criteria outlined in Section 3. Setting aside procedural assessments of axiomatization, such as the property of fairness, we claim that computational simulations can provide empirical data on several questions concerning how likely is it that a method yields a successful outcome, that is, how likely is it that a method chooses the best candidate. There has been previous work on simulation to compare plurality, Borda, and Condorcet methods, using frequential data to compare their performance.16,17 We contribute to this discussion by analyzing additional voting methods and providing a rationale for their general assessment, which is illustrated with a collective decision-making problem that has not received much theoretical attention. 2.1 A collective decision problem and notation Consider the problem of 