Michael asked what motivated XA. When I came up with XA, I believe I was thinking about how voters might manipulate a score voting election if they had a target score for each candidate and were trying to make each candidate's mean score match their target. I was thinking of an ongoing election where the voters can see the mean scores and change their votes at any time. Whenever the candidate's mean fell below their target, they would change their vote to MAX and whenever it rose above their target they would change their vote to MIN. Score voting is not even required. Approval voting would be good enough! If they could do this continuously (or hire a vote butler or write a computer program to do it for them), would there be an equilibrium point for a given candidate? Yes, it would be the chiastic median (XM), the largest number, x, where at least x% of the people wanted a score of x or higher. (For simplicity, we assume MIN=0 and MAX=100.) It's like a declared-strategy variant of score voting. Once we know that we can just ask voters for their 0-100 score of each candidate and then calculate each candidate's XM. After that, I figured out that any voter who gave a candidate a higher score than their XM, if given the opportunity to change their vote, could do nothing to raise the XM. Likewise, a voter who gave a candidate a score lower than the XM could have done nothing to lower their XM. So voters should be honest, there's no reason to be dishonest when submitting your scores! Whether the candidate ends up with an XM higher or lower than the score you submitted, your vote is, as much as possible, pulling their XM towards your score. In essence, it turns your vote into an approval vote, approving everyone who society gives a lower score than you would like, and disapproving everyone who society gives a higher score than you would like. So in this sense, the method can be called Chiastic Approval (XA). Of course, this all assumes that voters only care about making the candidate's final scores match their personal evaluations. It ignores that they would strategize to affect the winner. You can construct profiles where voters give honest scores and later discover that there were two frontrunners and they gave scores to both that were higher than their XMs. Those votes are pulling up on the XMs of both frontrunners, thus their votes are not really helping distinguish the frontrunners, which they might want. (This is related to the chicken dilemma.) You can construct profiles where the voters would change their votes and that would change the winner. But whatever it's faults here, XA must be better than plain approval, no? In any case, I note that the more strongly a voter feels about the difference between two candidates, the bigger difference he will put between their scores, and it becomes much less likely that his scores will be on the same side of both their XMs. I also note that this method (XA) is appropriate for a society trying to decide one number, say an approval rating for the president. Lastly, I note that Rob LeGrand discovered this same system and studied it in his thesis. Later, I discovered it independently and studied it in mine. I have more to say about it if you want, but these are the basics. ~ Andy

Thanks for the explanation--That completely answers my question about the reason for the XA rule. With Forest's examples of how XA does very well in chicken-dilemma situations, XA could be one of the very few methods that meets FBC & doesn't have a problem with the chicken-dilemma. ...without MMPO's Plurality Criterion failure, and the resulting criticism-causing bad-example shown by Kevin Venzke. I'd use it like Approval, top-rating my entire top-set, and bottom-rating everyone else. ...except that if I expected likely chicken-dilemma defection from the faction of a member of my top-set, I'd use Forest's strategy of giving them a 1/3 rating. But I've only read these postings once, and this is a complicated topic that calls for more than one reading, and the study of examples. I'm sure that I wouldn't, today, be qualified to vote in an XA election. Michael Ossipoff On Fri, Oct 28, 2016 at 1:17 AM, Andy Jennings <elections@jenningsstory.com> wrote: > Michael asked what motivated XA. > > When I came up with XA, I believe I was thinking about how voters might > manipulate a score voting election if they had a target score for each > candidate and were trying to make each candidate's mean score match their > target. I was thinking of an ongoing election where the voters can see the > mean scores and change their votes at any time. Whenever the candidate's > mean fell below their target, they would change their vote to MAX and > whenever it rose above their target they would change their vote to MIN. > Score voting is not even required. Approval voting would be good enough! > > If they could do this continuously (or hire a vote butler or write a > computer program to do it for them), would there be an equilibrium point > for a given candidate? Yes, it would be the chiastic median (XM), the > largest number, x, where at least x% of the people wanted a score of x or > higher. (For simplicity, we assume MIN=0 and MAX=100.) > > It's like a declared-strategy variant of score voting. Once we know that > we can just ask voters for their 0-100 score of each candidate and then > calculate each candidate's XM. > > After that, I figured out that any voter who gave a candidate a higher > score than their XM, if given the opportunity to change their vote, could > do nothing to raise the XM. Likewise, a voter who gave a candidate a score > lower than the XM could have done nothing to lower their XM. So voters > should be honest, there's no reason to be dishonest when submitting your > scores! Whether the candidate ends up with an XM higher or lower than the > score you submitted, your vote is, as much as possible, pulling their XM > towards your score. In essence, it turns your vote into an approval vote, > approving everyone who society gives a lower score than you would like, and > disapproving everyone who society gives a higher score than you would > like. So in this sense, the method can be called Chiastic Approval (XA). > > Of course, this all assumes that voters only care about making the > candidate's final scores match their personal evaluations. It ignores that > they would strategize to affect the winner. You can construct profiles > where voters give honest scores and later discover that there were two > frontrunners and they gave scores to both that were higher than their XMs. > Those votes are pulling up on the XMs of both frontrunners, thus their > votes are not really helping distinguish the frontrunners, which they might > want. (This is related to the chicken dilemma.) You can construct > profiles where the voters would change their votes and that would change > the winner. But whatever it's faults here, XA must be better than plain > approval, no? > > In any case, I note that the more strongly a voter feels about the > difference between two candidates, the bigger difference he will put > between their scores, and it becomes much less likely that his scores will > be on the same side of both their XMs. > > I also note that this method (XA) is appropriate for a society trying to > decide one number, say an approval rating for the president. > > Lastly, I note that Rob LeGrand discovered this same system and studied it > in his thesis. Later, I discovered it independently and studied it in > mine. I have more to say about it if you want, but these are the basics. > > ~ Andy > > ---- > Election-Methods mailing list - see http://electorama.com/em for list info > >