Both amalgamation rules give the same result in the case of n=3, and perhaps that is the only case that is really needed:
Each voter rates each candidate with a three, one, or two to express definite approval, definite disapproval, or somewhere in between. In this case the two slots not containing candidate X are the slots that are amalgamated to yield the approval ballot. On Wed, 30 Jul 2003, Forest Simmons wrote: > I promised I would give the rationale, so here it is: > > If we were to go from resolution n to resolution two in one step, we would > approve (i.e. amalgamate at the top) all of the candidates rated on the > ballot above the front runner, and disapprove (i.e. amalgamate at the > bottom) all of the candidates rated below the front runner. Whether or not > we approved the front runner would depend on other considerations such as > its rating relative to the second placer. > > In this method we conservatively amalgamate only two slots at a time > instead of the sweeping amalgamation described above. > > Which two slots should be amalgamated? > > Any two strictly below or strictly above the front runner would be > consistent with the above one step strategy, but we want to do better than > that. > > In this method we amalgamate the two slots furthest from the front runner, > and if the front runner is in the exact center, we use the highest scoring > candidate X that is not in the center to break the tie, i.e. we amalgamate > the two slots furthest from X. > > A refinement is possible. > > What we really want to do is amalgamate the adjacent slots that have the > least chance of making us regret their amalgamation. We would regret > their amalgamation if at any subsequent stage the two front runners ended > up in the same slot (but wouldn't have if we had not amalgamated). > > Suppose that we amalgamate the two adjacent slots whose leading candidates > have the lowest sum of CR scores. In other words suppose that the leading > candidates in slots k and k+1 have scores of x and y, respectively, and > that x+y is the minimum in this regard. > > Since this sum is so small, at least one of the two slots is very unlikely > to produce a front runner at a later stage, so there is little chance that > this amalgamation will be regretted later. > > It may well happen that two pairs of adjacent slots will produce the same > sum x+y, especially if this sum is zero due to empty slots on the ballot. > > In that case the tie can be resolved by going with the minimal sum pair > closest to the end of the range that is furthest from candidate X, where X > is the highest scoring candidate not in the exact center. > > It's hard to imagine how one could get any benefit from voting an > insincere order without it backfiring. > > If one thinks he can get an advantage by voting only at the extremes of > the original CR ballot, let him do so. The final approval ballot will be > equivalent to the original CR ballot, with no chance of an adjustment to > take advantage of the information in the other ballots. > > Personally, I would feel very confident in rating the candidates in > proportion to my respective utilities for them, and would trust the method > to sort out my optimal approval strategy more than I would trust the > information from any earthly pre-election polls. > > Forest > > > On Tue, 29 Jul 2003, Forest Simmons wrote: > > > This method recursively converts a set of CR ballots with resolution n to > > a set of CR ballots of resolution n-1, and stops at resolution two, i.e. > > approval ballots. > > > > > > We start with a set of Cardinal Ratings (CR) ballots on which every > > candidate is rated by a whole number in the range one to n. > > > > Provisional CR scores are calculated for the candidates by summing or > > averaging their ballot ratings. > > > > Then each ballot is modified to a CR ballot with resolution n-1 as > > follows: > > > > Let X be the candidate with the highest provisional score whose rating on > > this ballot is not precisely at the midrange value. > > > > If X is above midrange, then move all of the candidates except the lowest > > level candidates down one level, i.e. decrement their ratings in such a > > way that the lowest two levels are amalgamated. > > > > Otherwise lower only the top level, i.e. amalgamate the top two levels. > > > > > > Now recursively convert these n-1 resolution CR ballots to resolution n-2, > > etc. until they are completely converted to approval ballots. > > > > The approval winner is the method winner. > > > > > > That's it. > > > > I'll explain the rationale behind this in another posting. > > > > > > I like it best with Five Slot grade ballots. Each voter gives an A to the > > candidates that she is sure that she wants to approve in any case, and an > > F to each candidate that she would approve in no case. The undecided are > > sorted into three piles ... leaning towards approval, leaning away from > > approval, and not knowing which way to lean. > > > > > > Forest > > > > ---- > > Election-methods mailing list - see http://electorama.com/em for list info > > > > ---- > Election-methods mailing list - see http://electorama.com/em for list info > ---- Election-methods mailing list - see http://electorama.com/em for list info
