In the random election model, with 3 candidates and rank-order votes from V→∞ voters,
(a) D.G.Saari & M.M.Tataru: The probability of dubious election winners, Economic Theory 13 (1999) 345-363, claimed the answer was ≈69%.
But I don't understand their paper and my computer disagrees. My computer found 47% disagreement between the plurality and antiplurality winners in 3-candidate random elections, in which case the answer should instead be 53%. (Saari and Tataru claimed these two probabilities should be the same.) Who is right?
William V. Gehrlein & Dominique Lepelley: The probability that all weighted scoring rules elect the same winner, Economic Lett. 66,2 (2000) 191-197 did the same computer simulation I did and found the same answer, so it appears I was right.
(b) Vincent Merlin, Maria Tataru, Fabrice Valognes: On the likelihood of Condorcet's profiles, Social Choice & Welfare 19,1 (2002) 193-206, said the probability of a situation such as the example below in which every weighted-positional scoring voting system simultaneously elects some unique winner different than the CW (and that CW exists) is about 1.808% in 3-candidate random elections with a large number of voters.
(c) The possible votes are ABC, CBA, BCA, and BAC for three candidates A,B,C located on a line in that order. Suppose the numbers of these votes are respectively x,y,z and t. The proof is then to consider the linear program
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