> > As Jameson notes, CTA is the same as the chiastic median, but applied to > the thresholds instead of the scores. However, the thresholds are 100 > minus the scores, so it definitely changes the meaning of things.
I should correct myself. CTA is not exactly the same as the chiastic median of the thresholds. CTA calculates the largest number, x, between 0 and 100 such that x percent of people gave a threshold of x or below. Chiastic median applied to the thresholds would calculate the largest number, x, between 0 and 100 such that x percent of people gave a threshold of x or above. But the three concerns I mentioned are still valid. ~ Andy On Mon, Apr 8, 2013 at 10:12 AM, Andy Jennings <[email protected]>wrote: > Forest, > > This is an interesting method. It gives another good objective meaning > for numerical scores on a 0-100 scale. > > The consensus threshold would be very useful in situations where > compromise is paramount. For instance, I can pledge to support a new > taxation scheme only if 90% of the citizens support it. (This is even more > important in situations where participation is voluntary, such as > donations.) If 90% of the people say the same thing, it is probably a > pretty good compromise. > > On the other hand, I see some situations where it could be problematic. > If 90% of people give a candidate a score of 11 and the other 10% of > people give a score of 0, that candidate will have a consensus threshold > approval of 90%. I can imagine those voters being surprised that such a > large coalition was built from people who liked the candidate so little. > > As Jameson notes, CTA is the same as the chiastic median, but applied to > the thresholds instead of the scores. However, the thresholds are 100 > minus the scores, so it definitely changes the meaning of things. The CTA > value probably has more natural real world meaning than the chiastic > median, but it brings the following concerns: > > 1. In terms of the visual representation, I consider the area to be the > best measure of support if everyone is honest. Approval, median, and > chiastic approval, as you describe below, all measure something related to > the area under the curve. Examining the other diagonal, like CTA, doesn't > seem to measure anything related to the area. So it might not be a good > indicator of the total level of support. > > 2. CTA does not meet the unanimity criterion for aggregation functions, > which says that if all voters give the same score, then the societal score > should match. Approval doesn't either, but median, score, and chiastic > approval do. > > 3. CTA also can be extremely sensitive to one voter's input. One voter, > by lowering his score (raising his threshold) by one point, can scuttle the > whole coalition and cause the societal score to go from 100 to 0. > > ~ Andy > > > > On Thu, Apr 4, 2013 at 4:44 PM, Forest Simmons <[email protected]> wrote: > >> For purposes of clarification, I would like to show how Approval, >> Bucklin, Range, Chiastic Approval, and Consensus Threshold Approval >> manifest themselves relative to each other visually. >> >> >> >> I assume versions of these methods that make use of range style ballots >> on a scale of zero to 100. These methods also have in common that once >> the ballots are counted each candidate ends up with a score of some kind, >> and the candidate with the largest score is elected. >> >> >> >> So let’s concentrate on how each of these methods would assign a score to >> the same fixed candidate. >> >> >> >> All of these methods can be explained in terms of the graph of the >> function F given by >> >> >> >> p=F(r) is the percentage of the ballots that rate our candidate strictly >> greater than r. >> >> >> >> Each point (r, p) of this graph will lie somewhere in the 100 by 100 >> square with corners at (0,0), (0,100), (100, 0) and (100, 100). >> >> >> >> Furthermore, the graph will descend from left to right in steps whose >> widths are whole numbers. >> >> >> >> The left endpoint of each step will be included but the right end point >> will not be included. >> >> >> >> Color this graph blue. Now join the steps with vertical segments. The >> interior points of the vertical segments are colored red, while the top end >> point of each red segment will be colored red, and the bottom point will be >> colored blue. >> >> >> >> Now the union of the red and blue separates the lower left corner from >> the upper right corner of the square. Therefore the diagonal from (0, >> 0) to (100, 100) must cross the colored graph in either a red or blue point. >> Since the red and blue are non-increasing while the diagonal is strictly >> increasing, there can be only one point of intersection. The common >> value of the coordinates of this intersection point is the Chiastic >> Approval score. >> >> >> >> Now calculate the area of the region of the square that lies to the >> lower left of the red/blue diagonal. This area is the average rating of >> our candidate. So the candidate whose lower left area is greatest is >> the Range winner. >> >> >> >> Now bisect our square horizontally with a straight line segment from (0, >> 50) to (100, 50). The first coordinate (r) of the point of intersection >> (r, 50) of this line with the red determines the basic Bucklin score. Ties >> are broken by various methods. >> >> >> >> Now bisect the square with a vertical segment from (50, 0) to (50, 100). >> Assuming an approval cutoff of fifty, the second coordinate (p) of the >> intersection (50, p) of this segment with the blue is the approval score. >> >> >> >> Now consider the diagonal from the upper left corner (0, 100) to the >> lower right corner (100, 0). If this diagonal does not intersect the >> blue, then the candidate’s Consensus Threshold Approval score is zero. >> Otherwise >> it is the second coordinate of the highest (and therefore leftmost) blue >> point of intersection. >> >> >> >> In summary, we have bisected the 100 by 100 square vertically, >> horizontally, and diagonally. The diagonal with positive slope leads us >> to the chiastic approval winner. The other diagonal leads us to the >> consensus threshold approval winner. The horizontal bisector leads us >> to the Bucklin winner. The vertical bisector leads us to the Approval >> winner. The area cut off by the colored graph determines the Range >> winner. >> >> >> On Wed, Apr 3, 2013 at 6:18 PM, Jameson Quinn <[email protected]>wrote: >> >>> >>> >>> >>> 2013/4/3 Forest Simmons <[email protected]> >>> >>>> >>>> >>>> On Wed, Apr 3, 2013 at 12:07 AM, Kristofer Munsterhjelm < >>>> [email protected]> wrote: >>>> >>>>> On 04/03/2013 12:01 AM, Forest Simmons wrote: >>>>> >>>>>> Jobst has suggested that ballots be used to elicit voter's "consensus >>>>>> thresholds" for the various candidates. >>>>>> >>>>>> If your consensus threshold for candidate X is 80 percent, that means >>>>>> that you would be willing to support candidate X if more than 80 >>>>>> percent >>>>>> of the other voters were also willing to support candidate X, but >>>>>> would >>>>>> forbid your vote from counting towards the election of X if the total >>>>>> support for X would end up short of 80 percent. >>>>>> >>>>>> The higher the threshold that you give to X the more reluctant you are >>>>>> to join in a consensus, but as long as your threshold t for X is less >>>>>> than than 100 percent, a sufficiently large consensus (i.e. larger >>>>>> than >>>>>> t percent) would garner your support, as long as it it is the largest >>>>>> consensus that qualifies for your support. >>>>>> >>>>>> A threshold of zero signifies that you are willing to support X no >>>>>> matter how small the consensus, as long as no larger consensus >>>>>> qualifies >>>>>> for your support. >>>>>> >>>>>> I suggest that we use score ballots on a scale of 0 to 100 with the >>>>>> convention that the score and the threshold for a candidate are >>>>>> related >>>>>> by s+t=100. >>>>>> >>>>>> So given the score ballots, here's how the method is counted: >>>>>> >>>>>> For each candidate X let p(X) be the largest number p between 0 and >>>>>> 100 >>>>>> such that p(X) ballots award a score strictly greater than 100-p to >>>>>> candidate X. >>>>>> >>>>>> The candidate X with the largest value of p(X) wins the election. >>>>>> >>>>> >>>>> I think a similar method has been suggested before. I don't remember >>>>> what it was called, but it had a very distinct name. >>>>> >>>>> It went: for each candidate x, let f(x) be the highest number so that >>>>> at least f(x)% rate the candidate above f(x). >>>>> >>>>> I *think* it went like that, at least. Sorry that I don't remember the >>>>> details! >>>> >>>> >>>> Good memory, that was Andy Jennings' Chiastic method. Graphically >>>> these two methods are based on different diagonals of the same rectangle. >>>> >>> >>> Different, how? It seems to me they're just the same, but with the >>> numbers reversed. >>> >>>> >>>>> >>>>> If there are two or more candidates that share this maximum value of >>>>>> p, >>>>>> then choose from the tied set the candidate ranked the highest in the >>>>>> following order: >>>>>> >>>>>> Candidate X precedes candidate Y if X is scored above zero on more >>>>>> ballots than Y. If this doesn't break the tie, then X precedes Y if X >>>>>> is scored above one on more ballots than Y. If that still doesn't >>>>>> break >>>>>> the tie, then X precedes Y if X is scored above two on more ballots >>>>>> than >>>>>> Y, etc. >>>>>> >>>>>> In the unlikely event that the tie isn't broken before you get to 100, >>>>>> choose the winner from the remaining tied candidates by random ballot. >>>>>> >>>>> >>>>> I imagine Random Pair would also work. >>>>> >>>>> >>>>> The psychological value of this method is that it appeals to our >>>>>> natural >>>>>> community spirit which includes a willingness to go along with the >>>>>> group >>>>>> consensus when the consensus is strong enough, as long as there is no >>>>>> hope for a better consensus, and as long as it isn't a candidate that >>>>>> we >>>>>> would rate at zero. >>>>>> >>>>> >>>>> That's an interesting point. I don't think that factor has been >>>>> considered much in mechanism design in general. Condorcet, say, is usually >>>>> advocated on the basis that it provides good results and resists enough >>>>> strategy, and then one adds the reasoning "it looks like a tournament, so >>>>> should be familiar" afterwards. >>>>> >>>>> Perhaps there's some value in making methods that appeal to the right >>>>> sentiment, even if one has to trade off "objective" qualities (like BR, >>>>> strategy resistance or criterion compliance) to get there. The trouble is >>>>> that we can't quantify this, nor how much of sentiment-appeal makes up for >>>>> deficiencies elsewhere, at least not without performing costly >>>>> experiments. >>>> >>>> >>> I'm currently doing such "costly experiments" on Amazon MTurk (with >>> money from Harvard). I'm evaluating Approval, Borda, Condorcet >>> (3-candidate, so the differences between the most common varieties doesn't >>> matter), GMJ, IRV, Plurality, Score, and SODA (with honest-declaring and >>> mutually-rational-assigning AI candidates), with an 18-voter, 3-candidate >>> scenario in factions of 8, 4, and 6 (with utilities for each voter of 0-3, >>> summing to 12, 16, and 11). I'll let the list know as results are available. >>> >>> Jameson >>> >>> >> >> ---- >> Election-Methods mailing list - see http://electorama.com/em for list >> info >> >> >
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