I want to discuss the reasoning behind PAR, and to prove some lemmas about it. (Note: I've included the latest version of the PAR rules at bottom. I apologize that these rules have now gone through many iterations, and even changed how they deal with some edge cases; but I think the basic ideas of PAR, and the results in most simple realistic scenarios, have remained consistent.) First off: there are two basic ideas in PAR: the default rule, and the tally rule. These ideas are more-or-less independent conceptually, but they both work together to give the method a good tradeoff between non-slippery-slope performance in the chicken dilemma without damaging its performance in a center squeeze situation. The default rule is intended to "nudge" reasonably-lazy voters to cooperate in a chicken dilemma. A voter in a chicken dilemma faction has at least one candidate they support, and one candidate they strongly and saliently oppose. If they explicitly vote these two "prefer" and "reject", and leave all others blank, how should that be interpreted? In PAR, that is interpreted as "accept" for candidates with significant support (defined as at least 25% top-ranks), but "reject" for relatively-unknown candidates or clear chicken-dilemma losers (under 25% top-ranks). If highly-engaged strategic voters know that a large portion of their faction will vote in the "lazy" style above, it becomes unlikely that strategy will benefit them. I believe that many voters will be aggressively strategic if and only if they expect most of the "other side" to be so. If that's true, nudging the lazy voters away from aggressive strategy will serve to nudge such "copycat" voters in the same way. The tally rule in the latest version of PAR is: tally all "prefer" ratings, and all "accept" ratings except on ballots which prefer the frontrunner, where the frontrunner is the candidate X (if any) who meets the following three criteria: - X has the highest tally using the rules above when considering X as frontrunner - X is not rejected by a majority - X has more top-ranks than any candidate Y not rejected by a majority Thus, if no candidate meets these criteria, the PAR winner is simply the least-rejected candidate. This method will always elect a voted majority Condorcet winner C, because such a candidate will always either meet the three criteria, or win by the fallback rule. - For any Z, C's tally when considering C as a frontrunner will include (more than) all ballots with C>Z, by assumption a majority; but Z's tally will include only ballots with Z>C, by assumption a minority. - C will not be rejected by a majority. - If Z has more top-ranks than C, and Z is not rejected by a majority, then Z will not have the highest tally when considering Z as a frontrunner, and C will have fewer rejections than Z. (I realize that the above "proof" is not actually solid. I have not shown it's impossible for Z to have more top-ranks than C, so that the fallback tally ends up being used, and then for some other candidate Y to have fewer rejections than C. But it's pretty hard to do that without creating a Condorcet cycle.) However, PAR does not always elect the CW if it is not a majority CW. Consider the following election: 33: A>B 22: B>A 12: C>B 33: C A is the most-preferred non-majority-rejected, so start the tally with A as the frontrunner. A tallies 55, B tallies 34, C tallies 45; A is still in the lead, so A wins. But B would beat A pairwise by (a nonmajority tally of) 34 to 33. In a sense, this is a failure of center squeeze. But if all the C voters voted C>B, B would win; it is only in cases where it's a minority of them who do so that A can still win. And this behavior makes it less necessary for the A voters to "defensively" truncate B, which I think will lead to more honestly cooperative voting overall. ... Here are the rules 1. *Voters Prefer, Accept, or Reject each candidate.* On ballots which don't explicitly use "Reject", or for candidates with less than 25% "Prefer", blanks count as "Reject"; otherwise, blanks count as "Accept". 2. *Tally 1 point for each "Prefer"* for each candidate. 3. Out of the candidates (if any) with no more than 50% "Reject", find the one with the most points. *For every ballot which doesn't "Prefer" this frontrunner, add 1 point for each "Accept".* 4. If the frontrunner still has the most points, they win. Otherwise, the winner is the candidate with fewest "Reject" ratings.

I said earlier: > This method will always elect a voted majority Condorcet winner C, because > such a candidate will always either meet the three criteria, or win by the > fallback rule. > - For any Z, C's tally when considering C as a frontrunner will include > (more than) all ballots with C>Z, by assumption a majority; but Z's tally > will include only ballots with Z>C, by assumption a minority. > - C will not be rejected by a majority. > - If Z has more top-ranks than C, and Z is not rejected by a majority, > then Z will not have the highest tally when considering Z as a frontrunner, > and C will have fewer rejections than Z. > This last statement is not true. Say there is a Z who has more top-ranks than C and is not rejected by a majority. Without loss of generality, assume Z is initially selected as the frontrunner. Now Z's tally will be their number of above-bottom votes, while C's tally will be the C>= Z votes. The latter number is, by assumption, a majority; but it's not necessarily larger than the former number. For instance, take the following scenario: 49: Z 48: C>Z 3: >C C beats Z 51-49, but Z wins PAR 97-3. There are two different semi-honest strategies available for C to win — either 2 of the >C voters can switch to C, or 47 of the C>Z voters can switch to C. But C clearly does not win the election as presented. What if you add the requirement that every voter will prefer at least one candidate? That changes the above to: 49: Z 48: C>Z 3: Y>C But wait a minute; C is no longer the majority Condorcet winner! In order to restore that property, we must go to: 46: Z 3: Z>C 48: C>Z 3: Y>C Z still wins. So my claimed property is totally shot. But I have a hard time imagining that this would really happen. Essentially, 54% of the voters are "wasting" either the top or bottom of their ballot on Y, when Z versus C is clearly the contest that matters. What if we try to make Y more relevant? 23: Z 26: Z>C 25: C>Z 26: Y>C Yes, Y is more relevant here; they actually have more first-choice support than C. But still, we have 51% of the voters "wasting" the bottom of their ballot on Y, even though Y has just 26% above any other candidate. And furthermore, the Y>C is not at all reciprocated by any C>Y; in that sense, preferences have some "cyclical tendency". So I still believe that, in real-world elections, any voted majority Condorcet winner that exists will win; even though this is definitely not true as a criterion compliance.

Basically, the TL;DR message of my previous two messages in this thread is: if there is a voted majority Condorcet winner, then that candidate's supporters should be sure to reject any rival who will get more first-choice votes, even if that means bullet voting; and supporters of a rival who will get a minority of above-bottom votes should top-rank the VMCW if the VMCW is their second choice. Both of these "strategies" are relatively obvious, and thus I believe likely on naive ballots, but they cannot be guaranteed. 2016-11-15 11:59 GMT-05:00 Jameson Quinn <jameson.quinn@gmail.com>: > I said earlier: > > >> This method will always elect a voted majority Condorcet winner C, >> because such a candidate will always either meet the three criteria, or win >> by the fallback rule. >> - For any Z, C's tally when considering C as a frontrunner will include >> (more than) all ballots with C>Z, by assumption a majority; but Z's tally >> will include only ballots with Z>C, by assumption a minority. >> - C will not be rejected by a majority. >> - If Z has more top-ranks than C, and Z is not rejected by a majority, >> then Z will not have the highest tally when considering Z as a frontrunner, >> and C will have fewer rejections than Z. >> > > This last statement is not true. > > Say there is a Z who has more top-ranks than C and is not rejected by a > majority. Without loss of generality, assume Z is initially selected as the > frontrunner. Now Z's tally will be their number of above-bottom votes, > while C's tally will be the C>= Z votes. The latter number is, by > assumption, a majority; but it's not necessarily larger than the former > number. > > For instance, take the following scenario: > 49: Z > 48: C>Z > 3: >C > > C beats Z 51-49, but Z wins PAR 97-3. There are two different semi-honest > strategies available for C to win — either 2 of the >C voters can switch to > C, or 47 of the C>Z voters can switch to C. But C clearly does not win the > election as presented. > > What if you add the requirement that every voter will prefer at least one > candidate? That changes the above to: > > 49: Z > 48: C>Z > 3: Y>C > > But wait a minute; C is no longer the majority Condorcet winner! In order > to restore that property, we must go to: > > 46: Z > 3: Z>C > 48: C>Z > 3: Y>C > > Z still wins. So my claimed property is totally shot. But I have a hard > time imagining that this would really happen. Essentially, 54% of the > voters are "wasting" either the top or bottom of their ballot on Y, when Z > versus C is clearly the contest that matters. > > What if we try to make Y more relevant? > > > 23: Z > 26: Z>C > 25: C>Z > 26: Y>C > > Yes, Y is more relevant here; they actually have more first-choice support > than C. But still, we have 51% of the voters "wasting" the bottom of their > ballot on Y, even though Y has just 26% above any other candidate. And > furthermore, the Y>C is not at all reciprocated by any C>Y; in that sense, > preferences have some "cyclical tendency". > > So I still believe that, in real-world elections, any voted majority > Condorcet winner that exists will win; even though this is definitely not > true as a criterion compliance. >