KEMENY CHALLENGE ================= Here is an attempt by me to intentionally create small elections for which it is difficult to determine the Kemeny winner.
Consider this pairwise matrix: http://www.RangeVoting.org/Tourn27.html and replace all the +1s by random numbers in the interval [A, B] and all the -1s by ditto but negated, to get the pairwise margins matrix for a 27-candidate election. Here B>A>0 are two parameters chosen by the Devil to try to cause these problems to be hardest [I'd originally suggested A=9million B=10million, but maybe some other choice like A=0 and B=10billion would tend to make it harder]. Also of course randomly permute the 27 candidate-names in a way unknown to the solver, before giving the problem to the solver [equivalently permute both the rows and columns of the 27x27 matrix by one random permutation]. THE CHALLENGE: Find the Kemeny winner or order... can anybody do either reliably for 27-candidate elections of this class, or is this usually beyond humankind's abilities? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step) ---- Election-Methods mailing list - see http://electorama.com/em for list info
