Many studies have considered the probability that a pairwise majority rule (PMR) winner exists for three candidate elections. The absence of a PMR winner indicates an occurrence of Condorcet's Paradox for three candidate elections. This paper summarizes work that has been done in this area with the assumptions of: Impartial Culture, Impartial Anonymous Culture, Maximal Culture, Dual Culture and Uniform Culture. Results are included for the likelihood that there is a strong winner by PMR, a weak winner by PMR, and (...) the probability that a specific candidate is among the winners by PMR. Closed form representations are developed for some of these probabilities for Impartial Anonymous Culture and for Maximal Culture. Consistent results are obtained for all cultures. In particular, very different behaviors are observed for odd and even numbers of voters. The limiting probabilities as the number of voters increases are reached very quickly for odd numbers of voters, and quite slowly for even numbers of voters. The greatest likelihood of observing Condorcet's Paradox typically occurs for small numbers of voters. Results suggest that while examples of Condorcet's Paradox are observed, one should not expect to observe them with great frequency in three candidate elections. (shrink)

Arrow's Independence of Irrelevant Alternatives Condition is examined. It is shown why the standard rationale for (or against) the condition tends to be inconclusive as it fails to consider the basic ‘game’ issue in social choice. Specifically it is explained how some recent results (Gibbard-Satterthwaite) on the general non-existence of strategy-proof voting procedures provide the strongest rationale for the independence condition. Also, it is shown that this rationale was exactly the one used by Condorcet in his work on decision rules (...) for juries and elections. (shrink)