In another recent mail I said that I am not a criterion oriented person, and that often there is no need to require 100% compliance to all the numerous criteria. I however gave in an earlier mail (see below) some criteria for (pairwise) preference functions, and I'm about to present some more such criteria here. :-) I already listed two criteria that typical "natural" preference functions could be requited to have. P(x,x) = 0 P(x,1-x) = 2x-1 I'll add two monotonicity related requirements. if a > b then P(a,y) > P(b,y) if a > b then P(x,a) < P(x,b) Alternatively one could satisfy with less, and allow also equal results. if a > b then P(a,y) >= P(b,y) if a > b then P(x,a) <= P(x,b) Symmetry is also one natural requirement. P(x,y) = -P(y,x) One more requirement could be that the preference functions must be continuous. I'm sure that you'll forgive me that some of the functions in the preference function family that I gave are not defined at (0,0) although the first requirement says that the value of the preference function must be 0 there :-). Juho > On 26 Apr 2017, at 20:44, Juho Laatu <juho4880@yahoo.com> wrote: > >> On 25 Apr 2017, at 06:36, Andrew Myers <andru@cs.cornell.edu> wrote: > >> 1. WV: (W1, L1) > (W2, L2) if W1 > W2 or (W1=W2 and L2 > L1) [currently implemented] >> 2. Margins: (W1, L1) > (W2, L2) if W1 - L1 > W2 - L2 >> 3. LV: (W1, L1) > (W2, L2) if L1 < L2 or (L1 = L2 and W1 > W2) > > 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"). > > >> On 25 Apr 2017, at 11:06, Kristofer Munsterhjelm <km_elmet@t-online.de> wrote: > >> I'm usually a wv person, but I think Minmax is more classically >> associated with margins. Or perhaps I think that because Juho is here >> and he prefers margins :-) > > 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. > > >> On 04/25/2017 03:34 PM, Kevin Venzke wrote: > >> The desire to say that a 35-0 win is stronger than a 51-49 win could >> make sense if a strong win in itself was of some value. > > 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. > > 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. > > >> On 25 Apr 2017, at 11:06, Kristofer Munsterhjelm <km_elmet@t-online.de> wrote: > >> On a side note, Minmax can produce a lot of ties if there are few voters >> involved, so sometimes I prefer to break ties by second strongest defeat >> (and then third strongest, fourth strongest, etc). That isn't >> *classical* Minmax, but it shouldn't break any of Minmax's criteria. > > 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 > >

Here are some more thoughts on preference functions, and especially on the those preference functions that can be considered natural measures of strength of preference (of two candidates, by a group of voters, using ranked ballots). The value of a preference function P(x,y) may range from -1 to 1. Parameters x and y get values from 0 to 1. Parameter x refers to the proportion of votes that rank candidate X over candidate Y. Parameter y refers to the proportion of votes that rank candidate Y over candidate X. The proportion of votes that do not rank either candidate above the other is 1-x-y. Function P gets value 1 when all votes rank candidate X over candidate Y. Function P gets value -1 when all votes rank candidate Y over candidate X. Value 0 refers to a tie. The strengths of preferences can be compared by comparing the results of the PFs. I'll define first some criteria for all "sensible" preference functions (PF). P(x,y) = -P(y,x) - symmetric handling of x and y P(x,x) = 0 - equal amount of preferences in both directions shall always be a tie P(1,0) = 1 - unanimous support to X - the target of this requirement is just to normalise the presentation (and preference strength values) of different PFs P(0,1) = -1 - unanimous support to Y P shall be monotonic - grows or stays equal when x grows - diminishes or stays equal when y grows The result of a PF is a tie or a victory (with strength) to either candidate. Note that there are comparison methods like Pairwise Opposition that don't give or care about such results (ties, pairwise victories). PFs are thus just a subset of all comparison methods. Pairwise Opposition can't meet the first two requirements above. Pairwise Opposition could be called a strength function (SF). SFs can measure e.g. opposing votes or votes in favour. Their values could be from 0 to 1 (instead of -1 to 1). Symmetric sensible SFs can be defined as functions that can be presented as sensible PFs. Then some possible criteria for "natural" PFs. P(x,1-x) = 2x-1 - this one may be valid for all natural PFs - this criterion normalises the presentation of different PFs (to a scale that feels natural and obvious) - note that most election methods care only about which preference is stronger, not about how much the values differ (i.e. this criterion has no impact on their operation) - note that winning votes can not meet this criterion (= can not be normalised) since there is a big value gap between "strong preference" 51-49 and "strong preference" 49-51 P(x+a,y-a) = P(x,y)+k*a - most natural PFs might respect this criterion - the idea is that in addition to having linear values when x+y=1 (P(x,1-x) = 2x-1), the values should be linear also when the indifferent voters are excluded (when x+y is constant) - the value of k may be different for different "lines" (= for different values of x+y) All natural PFs shall be continuous - PFs with some kind of thresholds might violate this rule (they could be said to be "natural threshold PFs", but maybe not "natural PFs") - exceptions to this rule are allowed at point (0,0) since we want to allow functions like proportions / ratio to be called natural PFs Next I'll present a family of preference functions that meet all of the criteria above. Function f is the modifying parameter that generates the family. P1(0,0) = 0 P1(x,y) = (x-y)/(x+y)*f(x+y) when x>0 or y>0 Function f can be any function, but it shall meet some requirements. f is defined in range 0<x≤1 f is monotonic (rising) in range 0<x≤1 f is continuous in range 0<x≤1 (you could skip this one if you really want, and the system still works, maybe just losing the "natural" status) f(1) = 1 f(x)≥0 when 0<x≤1 Any continuous monotonic function will thus do if it is ≥0 and ends at 1. You'll get margins when f(x)=x. You'll get proportions / ratio when f(x)=1. Note that when y=0 P1(x,y) becomes (x-0)/(x+0)*f(x+0), and that is f(x). This helps the 3D visualisation of different functions of this preference function family. The 3D function can be thought to consist of a series of straight lines that are parallel to the x+y=1 line. All of them are at height 0 at the point where x=y. Only the tilting angle and length of the lines changes. Their end points of the lines form function f at plane y=0. Function f can make whatever monotonic curves within the rectangle (defined by points (0,0) and (1,1)). Plane x=0 has a corresponding pattern (rotated 180°, with non-positive values, z ≤ 0). If you have a program than can display these functions in 3D, use it. That helped me to understand these functions better. The smaller family of preference functions (with one parameter) that I gave earlier can be defined in this framework as f(x) = x^n, where n≥0. Within each f you can have also other modifiers (than n). The first additional one that I found interesting is one that changes the angle of the first "lines" next to point (0,0) and line at x+y=1 (see parameters up and down). The next one was parameter turn, that can flip a function e.g. from the bottom right triangle (below line f(x)=x) to the top left triangle (or somewhere in-between). This makes it easier to generate functions with wanted shape also at "the other side" of the f(x))x line. All these modifications will maintain the "natural" properties of the function. I'll give the definition (one version) of the modifier function below. It modifies the results of another function (f) that is expected to meet the same requirements that were given to f above. f1(x) = up*x + (1 - up*x - down*(1-x)) * ( (1-turn)*f(x)^bend + turn*(1 - f(1-a)^bend + f(0)^bend) ) 0 ≤ bend neutral value = 1 (bend=n) 0 ≤ up ≤ 1 neutral value = 0 0 ≤ down ≤ 1 neutral value = 0 0 ≤ turn ≤ 1 neutral value = 0 f is any function that meets the criteria (for f) above That's enough for now. Any opinions on the value of this kind of natural preference functions? Or on those preference functions that I classified as non-natural? Note that some of the natural preference functions may be quite similar to other non-natural functions. For example proportions / ratio is quite similar to losing votes. I wonder how much also their strategic properties correlate. And I wonder how much the flexibility of these natural preference functions can be used to defend against whatever strategic threats one might assume to exist in the elections. Parameterized functions are good because one can adjust them in a balanced way to protect against each of the identified treats only at suitable level (leaving space for other modifications too, thereby allowing better total defence). The number one use for the parameters may still be to define the fairest possible preference function for sincere votes. Juho