[EM] Preference functions

[Juho Laatu]

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