Those functions make sense to me. I would maybe separate the basic WV, margins and LV definitions from the additional tie breaker definitions ("W1=W2 and L2 > L1" and "L1 = L2 and W1 > W2").

The strongest argument in favour of margins must be that it is a relatively natural preference function. WV and LV are discontinuous functions and therefore can not really be called natural. Interest in using them comes mainly form strategic defence reasons, not from studying what would be a natural way of measuring strengths of preference. In addition the results that you get close to those discontinuities may appear strange (one additional vote may change the outcome radically).

Preference function between two candidates (A, B) can be given as a function of x and y, where x is the proportion of votes (0..1) ranking A over B, and y is the proportion of votes (0..1) ranking B over A. The values of the function range from 1 (100% of voters rank A over B) to -1 (100% of voters rank B over A). Value 0 refers to a tie.

Margins preference function is x-y
WV preference function is 0 if x=y, x if x>y, -y if y>x
LV preference function is 0 if x=y, 1-y if x>y, x-1 if y>x

WV and LV are not continuous around the x=y line. It is for example strange that in WV result 51-49 is considered a strong victory to A (0.51), while 50-50 gives 0 and 49-51 gives -0.51.

There are also other possible preference functions. One could measure for example the proportion of A>B preferences against all given preferences (A>B or B>A). This one is close to margins in the sense that it can be seen as one natural preference function. It is also smooth and continuous, except that at point 0-0 it is not even defined. The value of result 1-0 is 1, and the value of result 0-1 is -1, which is a quite radical approach.

Margins and proportions have different philosophy with respect to votes that have not given any preference between A and B (indifferent voters). Margins thinks that indifferent voters gave half a vote to support A>B and half a vote to support B>A. Preferences thinks that indifferent voters want to vote in favour of A vs B in same proportion as those voters did that gave a preference. Both approaches can thus be explained as natural approaches to explaining how the indifferent votes should be read.

This (comparison of different results) is a good approach to evaluating how the indifferent votes should be handled. I already wrote above on how proportions could be considered bad around the 0-0 result. In margins it is a good question if 2-0 should be considered to have equal preference strength to 51-49.

I'll define some additional preference strength functions (P). I start from the assumption that margins philosophy can be taken as the starting point for results where all voters have expressed preference (= no indifferent votes). As a result preference value for 100-0 is 1 (for all preference functions of this group of P functions), i.e. P(100,0)=1, and P(50,50)=0, and P(75,25)=0.5. The preference function is thus linear at that line (y = 1-x). Preference value should also always be 0 at line x=y. More formally, P(x,x)=0 and P(x,1-x)=2x-1. We should require also symmetry, P(x,y)=-P(y,x).

One family of preference functions that meets these criteria is (x-y)*(x+y)^k. We get margins when k=0. We get proportion when k=-1. The next interesting possibility is to use value k=1. This one is interesting because it has some WV like properties without introducing the weird discontinuities of WV. I'll study this function (k=1) a bit more.

(Note that when k=1 the function becomes quite simple, (x-y)*(x+y)^1 = x^2 - y^2.)

First few words about sincere votes. In margins the preference values of 50-0 and 75-25 are equal. Pm(50,0) = Pm(75,25). If one doesn't think this is ideal, one can adjust this relation by adjusting the value of k. When k=1 (P1) we have P1(75,25) = 0.5 (by definition). P1(50,0) = 0.25, Pm(50,0) = 0.5 and Pp(50,0) = 1 (p = proportion). The point of this paragraph is to demonstrate that if you want to have a P that gives best possible results with sincere votes, one can adjust k to reach best performance (within this family of preference functions). One parameter may be enough to address the most typical concerns (?).

What about defence against strategic voting then. In many large public elections such defences may be unnecessary. But for for people that like the WV properties P1 may be of interest also for strategy defence related reasons. It for example gives P1(10,0) = 0.01 while P1(55,45) = 0.1. I will not go further into details although it could be interesting to study the impact of parameters to strategic defence capabilities. The point here is that there are also continuous "natural" preference functions that can reach some of the targets of discontinuous functions like WV.

If one uses parameterized preference functions like the one described above, one can also adjust the parameters (k) so that the function reaches the required strategy resistance, but at the same time deviates from the preferred sincere preference function only as little as possible.

I just note that the parameterized versions may also reduce the probability of ties (e.g. if one uses value k=0.00001 instead of k=0).

Juho

At least one correction / improvement already detected.

I should have written "0 if x=y, 1-2y if x>y, 2x-1 if y>x". That does not change the results, but gives more logical values to the preference function. P_LV(x,y) is now close to 0 when x and y are close to 0.5.

Juho

---------------------------- Original Message ----------------------------

Subject: [EM] Fwd: Ordering defeats in Minimax

From: "Juho Laatu" juho.laatu@gmail.com

Date: Wed, April 26, 2017 2:09 pm

To: "Election Methods" election-methods@lists.electorama.com

natural way of measuring strengths of preference. In addition the results that you get close to those discontinuities may appear strange (one additional vote may change the outcome radically).

Another "natural" reason to favor margins over winning votes to compare one pairwise outcome over another is that the net margin, in votes, is the product of two other indicators of salience. it is the product of the margin in percent and the number of voters participating.
�Or "How loud did the voters speak in this pairwise election?" times "How many voters were there?"
I mean, which more important? �That 1025 voters prefer Tom to 975 voters who prefer Harry? �Or that 525 voters that prefer Dick to 475 prefer Tom? � Or
�2024 voters prefer Harry over 1976 voters who prefer Dick?
The second pair has the most decisive defeat from a percentage POV. �A 5% win for Dick.
The third pair has the most winning votes, Harry gets a wopping 2024 votes but Dick gets nearly as much, and despite just killing
Tom, percentwise, Dick is eliminated. �and for Harry to win when Tom defeats him by more votes than Harry defeats Dick seems odd.

r b-j � � � � � � � � �rbj@audioimagination.com
"Imagination is more important than
knowledge.

On 04/26/2017 08:09 PM, Juho Laatu wrote:

Do you consider Plurality a strategic criterion? It seems to me to be
more of a "natural behavior" criterion: if A gets more first preferences
than B gets any preferences, then B shouldn't win. This seems reasonable
from a natural behavior perspective because A dominates B in some
Approval-ish sense.

If that's a "natural behavior" criterion, then you could say that
margins is more natural from a descriptive point of view (no
discontinuities) while wv is more natural from a criterion point of
view. Though, if we're to go by the apparent popularity of IRV, it seems
that descriptive clarity weighs heavier than criterion clarity.

First of all, my thinking when it comes to practical election methods is not very criterion oriented. I tend to see criteria and criterion compatibility as important theoretical results that are mostly too far from practical election method considerations to be applied directly on them as viability criteria. The relevance of different criteria to practical election methods is almost as low as the relevance of latest mathematical inventions to practical everyday economic calculations (well, not quite, but something in that direction). I'm about at the level of accepting Condorcet criterion if we seriously want to have a neutral majority based method for consensus oriented single winner elections. One has to take also into account the fact that all election methods are bound to break some potentially useful criteria. All this means that I classify Plurality and most other criteria as an interesting discussion points but not something to be followed categorically. There are many criteria that are useful in the sense that most elections should have strong orientation in the described direction, but no need, or possibly with strong reasons to deviate from some criteria in some special (usually marginal) situations.

I think Plurality is a bit strange. Actually it is not even a criterion of of ranked methods. It is a criterion for ranked methods with implicit approval cutoff. It makes the assumption that a voter that casts a short vote has somehow approved those candidates that he marked, and not approved the others. In different elections the behaviour of voters with respect to which candidates will be marked on the ballot may vary a lot, and that may have nothing to do with how much the voters support or approve those candidates. In order to make any sense of the Plurality criterion we are thus tied to having an assumption of implicit approval in the ballots, where marking a candidate means approving that candidate at some level.

One reason why I don't like implicit approval in general (as a fact that is known by the voters) is that it encourages voters not to rank candidates that they don't like. Ranked methods work well only if most voters do rank explicitly at least all the potential winners (or all of them except one). If there is an approval cutoff, it would be better if it was an explicit one (this comment is not Plurality criterion specific but a general one).

Plurality criterion is a "heuristic" criterion in the sense that its message somehow sounds good (e.g. to people that do not regularly deal with election methods and their peculiarities). People would like also criterion "if voters would prefer A to B, then B should not win". But EM experts know that this criterion would not be a very good one, although it states something that we all would like to be true in all elections. What I'm trying to say here is only that we should be careful with cyclic group opinions. They will contain some nasty features. Instead of trying to pick a set of criteria that should be met 100%, my preferred approach is to see what kind of problems each method would be likely to face in real elections (typically but not necessarily large public elections with many different kind of voters that the strategists can not control), and evaluate them based on their performance in such real life situations.

I'm not well prepared to comment how margins can handle Plurality criterion but I'll address one basic (but theoretical and extreme, i.e. unlikely to happen in typical elections) example. 35:A, 34:B>C, 31:C. A has more first preferences than B has ballots where B is marked. B's worst defeat margin is however smallest (1), so it will win in typical margins based methods. Plurality criterion says that B should not win. B is however two votes short of being a Condorcet winner, so it can't be the worst of the worst. What if A would win? Plurality criterion pays special attention to A's high number of first preferences. But on the other hand voters would like to elect C instead of A with large majority (C>A voters would not be happy with the result). How about C then? Plurality criterion accepts C too, but using the high number of first preference votes of A as an argument that supports C does not make much sense. My conclusion is that this is a typical mess that we can get with circular preferences. Our voters were quite stupid when they didn't sufficiently rank the potential winners. There are many different possible scenarios on what the truncated opinions might have been, and different results emerging from that. In this example my recommendation would be to tell to the voters that they should rank all potential winners (except maybe the worst one). I don't see any need to start blaming (or praising) margins on what happened. Maybe you have some realistic examples in your mind, that would give better justification to the Plurality criterion.

Juho

I agree that it is important to understand how strong different pairwise preference results should be considered. In the generic preference function that I gave I to some extent tried to answer your question "How many voters were there?", and find a parameter (k) that could be adjusted to set the balance right (between high number and low number of voters that indicated their preference). In the function ( (x-y)*(x+y)^k ) the "x-y" part sets the margins approach as a starting point. The "(x+y)^k" part can be seen as an adjustment factor that takes into account the number of votes that had an opinion "x+y". Constant k tells us how much we should weaken (k>0) or strengthen (k<0) the pairwise comparison result in the case that not all voters gave their preference.

Juho

---------------------------- Original Message ----------------------------

Subject: Re: [EM] Ordering defeats in Minimax

From: "Juho Laatu" juho.laatu@gmail.com

Date: Thu, April 27, 2017 6:15 pm

To: "Election Methods" election-methods@lists.electorama.com

adjusted to set the balance right (between high number and low number of voters that indicated their preference). In the function ( (x-y)*(x+y)^k ) the "x-y" part sets the margins approach as a starting point. The "(x+y)^k" part can be seen as an adjustment factor that takes into
account the number of votes that had an opinion "x+y". Constant k tells us how much we should weaken (k>0) or strengthen (k<0) the pairwise comparison result in the case that not all voters gave their preference.

okay, i wanna restate this with the Wn and Ln symbols.
�
� (W1, L1) > (W2, L2) � means
�
� �(W1-L1)(W1+L1)^k � > � (W2-L2)(W2+L2)^k�
�
now, if k=0, this is the same as what i am coining as
"Arithmetic Margins" (for lack of a better term). �if k = -1, then it's the same as the percentage margin, where a larger race has no more weight than a smaller race if the percent margins are the same. �this is, i believe, going to have equivalent outcome as Markus's margins of
logarithms (what i coined "Geometric Margins").
If someone has better terminology for naming these different forms of margins, please correct my neologism before it takes root.

--
r b-j � � � � � � � � �rbj@audioimagination.com
"Imagination is more important than knowledge."

Hi all,
Kristofer wrote:>Do you consider Plurality a strategic criterion? It seems to me to be>more of a "natural behavior" criterion
FWIW I don't consider Plurality a strategy criterion. We can bring up strategy criteriaif we wanted to, but they are more obscure than the Woodall criteria. The best knownone worth bringing up is favorite betrayal, but WV methods don't completely satisfy it,so I would be stuck talking about simulations instead of firm guarantees.
(You may ask why does favorite betrayal come into it? Because the same incentives thatare supposed to discourage margins voters from truncating at the bottom, also discouragethem from equal-ranking a set of favorites at the top. And there isn't a mechanism ensuringthat ideal strategy is to simply rank them sincerely.)
I would say margins' largest issues are strategy-related, but its most obvious issuesaren't, so I prefer to talk about "natural behavior" when I have a choice.
To Juho:
You feel criteria are "too far from practical election method considerations to be applieddirectly on them as viability criteria"; that is actually very close to how I view yourcriteria (even if you don't use the term "criteria"). You say the "strongest argument" formargins is that it is a relatively natural and continuous preference function.
I guess that you don't say that this involves "practical considerations," and it's justan aesthetic preference. I give you credit for consistency in not even mentioning mono-add-top, which could easily be argued to be a practical consideration. On the otherhand you seem to believe WV has effective strategy incentives that result in things you don't like, so you don't seem to completely dismiss practical considerations either.
I find it more accurate when you complain that WV seems like an approval hybrid thanwhen you suggest that interest in WV stems from strategic defense concerns. When peopleinvent random new methods on EM they frequently behave like WV (or IRV); it's very hard toemulate margins' results without actually being margins. Methods that show approval-likebehavior stem from a relatively common expectation that expressed votes are a goodindication of viability/merit. There may be good historical reasons to portray WV as ahorrified reaction to margins... And I can imagine the FPP supporter who says interest inIRV "comes mainly from strategic defense reasons." But that seems like a limited way oflooking at it.
You write:>I think Plurality is a bit strange. Actually it is not even a criterion >of ranked methods. It is a criterion for ranked methods with implicit>approval cutoff. It makes the assumption that a voter that casts a short>vote has somehow approved those candidates that he marked, and not approved>the others. In different elections the behaviour of voters with respect>to which candidates will be marked on the ballot may vary a lot, and that>may have nothing to do with how much the voters support or approve those >candidates. In order to make any sense of the Plurality criterion we are >thus tied to having an assumption of implicit approval in the ballots, >where marking a candidate means approving that candidate at some level.
I think voters expect this behavior, and that's why it's important, for example, forthe long-term viability of the method's adoption. Now, you may say that we could convince voters not to expect this. You might even (though I'm still not 100% sure)manage to do this by forbidding voters from truncating at all. In that case, I woulddrop all my objections dealing with naturalness and what voters expect. My remainingcomplaints at that point would be the strategy-related ones.

Sure, I would rather criticize how margins addresses this than the fact that it triesto address it. IRV does this, for instance: You can rank the worse frontrunner in asincere way and you know that it won't hurt you. In margins we "want" you to rank theworse frontrunner, but it's not expected to be of any use to you, and it can hurt thebetter frontrunner by making "worse" look like your compromise choice.
A few comments on your Plurality example:

1. I just note that "fewest votes short of being a Condorcet winner" need notstrike one as a particularly less arbitrary standard than Plurality. I do not knowwhat stops one from dismissing yours as a criterion whose "message somehow soundsgood to people that do not regularly deal with election methods." It does "soundgood," I agree with that.
2. I'm not sure what is your basis for calling these voters "stupid." Are you(rhetorically) employing Plurality's assumption that voters didn't like theoutcome? I get the sense that you don't like the outcome yourself.
3. I do understand feeling that Plurality is strange. I don't love it. My view isthat Plurality is an easily articulated subset of a larger principle. Essentially,if the voters were instead legislators selecting an outcome, could B ever prevail?It seems quite unlikely, because at the moment that C is the status quo and the Bvoters wish to assert their win over C, they can't because they are outnumbered bythe A voters. The pairwise contest doesn't occur in a vacuum. In this case Pluralityconveys what I would want.
   A more thorough version of Plurality (in my opinion, to my taste) wouldn't talkabout first preferences or votes in total. I doubt this language is necessary toexpress the idea. (The criterion we do have is easy to apply, though.)
   Kevin

I think I basically agree with Juho on this. The plurality criterion sounds like a reasonable criterion on the surface, but think about it more and it's arguably less so. To summarise, in a pairwise method, first place on a ballot doesn't hold any special status, nor does indeed last place (or joint last place or "unranked"). And a criterion shouldn't be used to impose an approval cut-off on a method that doesn't have one in its definition.
So while it sounds like a good criterion, removing the special status of these positions means that we are left with just saying that a candidate who pairwise beats another candidate should finish ahead in the overall ranking. Which is what all Condorcet methods do - except when there's a cycle.
Toby

  From: Juho Laatu <juho.laatu@gmail.com>

To: Election Methods election-methods@lists.electorama.com
Sent: Thursday, 27 April 2017, 22:57
Subject: Re: [EM] Fwd: Ordering defeats in Minimax

First of all, my thinking when it comes to practical election methods is not very criterion oriented. I tend to see criteria and criterion compatibility as important theoretical results that are mostly too far from practical election method considerations to be applied directly on them as viability criteria. The relevance of different criteria to practical election methods is almost as low as the relevance of latest mathematical inventions to practical everyday economic calculations (well, not quite, but something in that direction). I'm about at the level of accepting Condorcet criterion if we seriously want to have a neutral majority based method for consensus oriented single winner elections. One has to take also into account the fact that all election methods are bound to break some potentially useful criteria. All this means that I classify Plurality and most other criteria as an interesting discussion points but not something to be followed categorically. There are many
  criteria that are useful in the sense that most elections should have strong orientation in the described direction, but no need, or possibly with strong reasons to deviate from some criteria in some special (usually marginal) situations.

I think Plurality is a bit strange. Actually it is not even a criterion of of ranked methods. It is a criterion for ranked methods with implicit approval cutoff. It makes the assumption that a voter that casts a short vote has somehow approved those candidates that he marked, and not approved the others. In different elections the behaviour of voters with respect to which candidates will be marked on the ballot may vary a lot, and that may have nothing to do with how much the voters support or approve those candidates. In order to make any sense of the Plurality criterion we are thus tied to having an assumption of implicit approval in the ballots, where marking a candidate means approving that candidate at some level.

One reason why I don't like implicit approval in general (as a fact that is known by the voters) is that it encourages voters not to rank candidates that they don't like. Ranked methods work well only if most voters do rank explicitly at least all the potential winners (or all of them except one). If there is an approval cutoff, it would be better if it was an explicit one (this comment is not Plurality criterion specific but a general one).

Plurality criterion is a "heuristic" criterion in the sense that its message somehow sounds good (e.g. to people that do not regularly deal with election methods and their peculiarities). People would like also criterion "if voters would prefer A to B, then B should not win". But EM experts know that this criterion would not be a very good one, although it states something that we all would like to be true in all elections. What I'm trying to say here is only that we should be careful with cyclic group opinions. They will contain some nasty features. Instead of trying to pick a set of criteria that should be met 100%, my preferred approach is to see what kind of problems each method would be likely to face in real elections (typically but not necessarily large public elections with many different kind of voters that the strategists can not control), and evaluate them based on their performance in such real life situations.

I'm not well prepared to comment how margins can handle Plurality criterion but I'll address one basic (but theoretical and extreme, i.e. unlikely to happen in typical elections) example. 35:A, 34:B>C, 31:C. A has more first preferences than B has ballots where B is marked. B's worst defeat margin is however smallest (1), so it will win in typical margins based methods. Plurality criterion says that B should not win. B is however two votes short of being a Condorcet winner, so it can't be the worst of the worst. What if A would win? Plurality criterion pays special attention to A's high number of first preferences. But on the other hand voters would like to elect C instead of A with large majority (C>A voters would not be happy with the result). How about C then? Plurality criterion accepts C too, but using the high number of first preference votes of A as an argument that supports C does not make much sense. My conclusion is that this is a typical mess that we can get with c
ircular preferences. Our voters were quite stupid when they didn't sufficiently rank the potential winners. There are many different possible scenarios on what the truncated opinions might have been, and different results emerging from that. In this example my recommendation would be to tell to the voters that they should rank all potential winners (except maybe the worst one). I don't see any need to start blaming (or praising) margins on what happened. Maybe you have some realistic examples in your mind, that would give better justification to the Plurality criterion.

Juho

Election-Methods mailing list - see http://electorama.com/em for list info

Also this mail is a late rely. This time I had some additional problems since I didn't receive this mail in my unbox at all. I found it in the electorama mail archives. The mail traffic between electorama and me doesn't always work perfectly.

I already answered Kristofer's question once, but I was maybe not as clear as I should have been. I agree that Plurality can be classified as a criterion that is intended to be a "natural behaviour" criterion. I used term "heuristic", that os close to "intended to be a natural behaviour". Plurality is however often linked with winning votes, that I consider to be a very "strategic behaviour" related comparison method.

I often talk about performance with sincere votes when I'm not talking about performance against strategies. That is practical (although also aesthetic, like many things in formal systems :-) ).

My thinking is often based on the assumption that in typical large public elections with independent voters strategic risks are not very high. I'd expect almost all Condorcet elections to be quite strategy free in practice. In most countries people would vote for the best candidates, not in some other way, trusting that giving false rankings or omitting preferences would help them in the election.

For this reason my practical considerations often tend to focus on the above mentioned performance with sincere votes. I tend to "like" margins more than winning votes not because of differences in their behaviour in the presence of strategic voting, but because of their differences in performance with sincere votes. For example with votes 51: A, 40 B>C I like more the margins thinking that C is 48 (sincere) votes away from winning the election than the winning vote thinking that C is 3 (sincere) votes away from winning the election. If we can afford to use a natural pairwise comparison method, why not use it.

Yes, I'm looking forward to such use of Condorcet methods where voters rank multiple candidates, to rank all the potential winners, in the hope of allowing the fair algorithm to pick the best winner. I hope people will understand that most likely their best strategy is to do just that.

One reason behind the popularity of IRV, when compared to Condorcet, could be its tendency to favour large incumbent parties. (Another one is its interesting dramatic elimination process that appears more natural than it is.)

Yes, they should be educated if they ever get that kind of ideas somewhere.

I don't want to forbid truncation because I want voting to be easy and pleasant, and I want to have a reasonably high number of candidates. If insincere truncation turns out to be a problem that threatens the soundness of the results in some elections, maybe one could try other tricks first, like e.g. completing the vote with random candidates or using some party preference list to complete the vote. I'm however living in the hope that in most cases none of these tricks are needed.

Yes, but what is the probability of hurting one's targets by ranking numerous candidates in a Condorcet election with a bunch of potential winners and few additional candidates that are unlikely to win? And what are the benefits of ranking all the potential winners?

Democracies typically work via majorities, and majorities are generated by voters. To me the required number of additional voters to reach majority is a very natural measure of the popularity of some idea. I'm having hard time trying to invent something that would be a more natural measure.

I referred to truncation of votes that obviously made it impossible for the election method to decide who the ideal winner is. The voters maybe failed to tell whom they would like to elect.

It is also possible that all truncations were sincere (opinions where all the remaining candidates were equally good). In that case any method with good/desired performance with sincere votes would do the job fine. It is however typical that people have clear opinions about the three potential winners, and therefore it is probable that many of the voter that truncated had some opinions that they didn't express. Giving fuller rankings would probably be in their interest (in typical elections).

Voting about things vs. persons is about the same. Condorcet is a good method if one wants to elect a good compromise winner. In a small community where people think that they know beforehand how others will vote, and where the party leaders and strategists possibly can force the voters to vote as told, strategic problems are more likely to occur than in large public elections with independent voters. Condorcet could however work well also in such small and strategy prone environments. The problems of cyclic group opinions are incurable. If opinions have a top cycle, there will be a majority that would like to change the outcome to something else. In small political circles the solution could often be to make an agreement with some other party, I will get this and you will get that, and then vote accordingly. This is more probable than leaving uncertain decisions and expected circular opinions to be solved by a rather random outcome of a vote, or even more random outcome of a vote with strategic moves by multiple parties.

Yes, Plurality's reference to first preferences is not good. If one wants to say something more generic that is not tied to first preference votes, one should reformulate the criterion. The criterion would not be as sexy that way. Also the use of implicit approval should be made clearer, if the intention is to say something about voters that assume it, or a method that makes some use of it.

Juho