There is very little in the literature on the frequency with which voting paradoxes could be expected in the real world. The point of this note is to add a little, unfortunately very little, to this scant information and to suggest a method of getting more. One of the authors was on a search committee set up to select a new chairman for the Political Science Department at V.P.I. The Political Science Department established an elaborate voting procedure under which each of the candidates was to be evaluated by each of the six committee members on six specific dimensions, such as scholarly ability, administrative skill, etc. and also given an overall evaluation. The evaluation was on a scale of 1 to 10, but this could easily be converted into a simple ordinal scale with, of course, some ties. There were 37 candidates to be graded.
Unfortunately, a number of the members of the committee shirked and did not fill out all the schedules, with the result that only the overall rating and the rating on scholarly competence were suitable for a test of cycles. Further, on the overall evaluation, one member of the committee rated almost all candidates in the group of 10 finalists identically and another member of the committee did not rate any of the candidates. The result is that effectively we have a committee of four rating on overall ability and a committee of six on scholarly ability. The result was fairly clearcut, but unfortunately of no great importance, because of the extreme small size of the sample. There was a Condorcet winner (No. 6) on scholarly ability, and on the general rating, there was one candidate (No. 32) who could not be beaten by anyone else, but who tied with candidates 7, 9 and 13. The tie was only possible, of course, because of expressions of indifference by one member of the committee in each case. 7, 9 and 13 all participate in cycles in which they can be beaten by other candidates, who were beaten by 32, so this is an example of tie intransitivity. The table below may help to explain the result. This shows the paired voting outcomes for the ten candidates who we judged stood at the top of the 37.
We also looked for the lack of independence of irrelevant alternatives, using the Borda method for the candidates and then inquiring as to the