Condorcet's Paradox with Three Candidates

Condorcet formally developed the notion of cyclical majorities over two centuries ago (Condorcet, 1785), and Peter Fishburn introduced me to that phenomenon in 1971. When Peter first described the idea behind Condorcet's Paradox during a course in Social Choice Theory at Pennsylvania State University, my response was that the phenomenon simply could not happen. When he reproduced the classic example of its existence with three voters and three candidates, my immediate response was that this phenomenon certainly could not be very likely to ever be observed in realistic situations. Peter quickly suggested that I should work on developing some estimates of the probability that the paradox might occur, and very soon afterward that pursuit began. We completed many co-authored papers on related topics over the following Years, but it is only after more than 30 Years of effort that I feel a good answer can be given to the challenge that Peter presented in that classroom in 1971. The following essay can be viewed as a long overdue course project report, and we can finally see a theoretical model that clearly explains why observations of Condorcet's Paradox are so rare in elections on a small number of candidates.