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Is it possible to turn subjective claims in objective statements? And to what extent can be objective in our opinions of the world? These are the sort of questions that I enjoy trying to answer; mostly on my own rather than as a debate, the latter often turns in futile bickering with no fruits of the labour. Trying to create an objective statement from a subjective statement is contentious and full of caveats; often suggesting that such a process is possible raises alarm and offends people to their core. I've tried my best to promote logic and objectivity within discussions that rely heavily upon subjectivity and ad hominem style arguments. In order to take subjectivity as an input and output objectivity there needs to be a process, a transformation, for achieving this. I believe that the political compass is an example of this process. In this article I won't consider what the exact process is (that is examined across a few articles) but rather how do we measure the 'objectiveness' of the output statement.

Restated: I try to identify possible metrics for quantifying opinion. This article comes from my AGO system of thought and is under-pinned by my interest in the political compass.

Introduction

I am going to assume that the reader is already familiar with the Political Compass and may be familiar with my articles on the subject. As noted in previous articles I believe that a system of graphical representation of opinion is a good way to try and quantify opinion. Such a system should have clear rules from which concise representations can be created. I have spent a considerable amount of time trying to figure out a consistent set of rules and notation that allows one to create a generic "political compass" (graphical representation of opinion), but for this article it is not important that the reader has read anything that I've previously written on this topic.

Given that such a representation may exist it is pertinent that people agree upon the rules of construction of any political compass. Therefore, two people who may not agree politically can, at least, agree that the representation is fair and true. This is true of mathematics: if you accept the axioms of mathematics to be true then you should agree with all the conclusions that one finds given that the axioms are consistent. Admittedly, mathematics is not quite that simple but close enough for argument's sake. Seen another way, we could say that two agree upon an conclusion if they both agree upon the premise and the argument.

As the website for the political compass gives almost no clues as to how the final graph is constructed one is left wondering if the internal logical rules are consistent and hence the final graph is consistent from person to person. It would be naive to assume that no inconsistencies exist and only transparency of its inner workings would quell such a debate.

For this article I will assume the political compass (or any similar construction) is consistent and that people in general agree with the final representation. The key topic that I will explore here are the possible metrics for measuring agreement. These metrics are only meaningful if the basic axioms of the construction of the system are agreed upon. If two people don't even agree upon the construction of the representation then the metrics are entirely spurious. If two people didn't agree upon the statement that 1+1=2 then there is little hope for them agreeing that 1+2=3.

In the context of such generic graphical constructions of opinion (as I outline in my AGO articles), I present what I believe are the key metrics for measuring agreement: sector, line (including mirror symmetry), proximity and path.

Sector Agreement

This is probably the weakest form of agreement and hence the metric is the roughest measurement of opinion. This type of comparison is one of the first that can be drawn: "oh you are in the bottom-left quadrant, me too." In the case of the political compass the bottom left quadrant is socially progressive and leaning towards socialism economically. Each person that completes the test can be assigned a quadrant 1-4 and the comparison can be binary: yes - same quadrant, or no - not in the same quadrant.

A related comparison is to see which side of the axes two people (points) fall on: i.e. a person in quadrant one may agree with a person in quadrant two about how the role of government intervention in the lives of its citizens. Remember that this idea can be generalized beyond the political compass to any such construction. All you are doing here by identifying the quadrants of people to provide one method of categorizing people by similarity.

Note: I chose the word sector as I believe it is more general than the word quadrant. A graph doesn't need to be a square as it is in the political compass.

Line Agreement

A stronger type of agreement than quadrant agreement is when two points share one coordinate, such as two points that are both at x=3 yet they are in different quadrants. The commonality between two viewpoints is expressed by the fact that they lie upon the same line of constant x (or y if that was the case). From an argument of symmetry there has to be some type of agreement: perhaps the two points represent two people who mostly agree on economic policies but other disagree upon the amount of state-intervention in the people's lives. The fact that they agree economically would be shown by the fact that they occupy the same line (same x coordinate).

Proximity Agreement

The next strongest type of agreement is what I call proximity agreement. That is when two people (points on the graph) are close to each other, and in the strongest case they are on top of each other at the same point. The distance between the two points is essentially the measure of proximity, this is open to debate but I'd limit the use of the word to intra-sector. Each of these metrics may not be so meaningful on their own but together will make a lot more sense. In the case of proximity the points may have different coordinates but the closeness of each denotes that the two should be similar.

When I first thought about these agreement metrics I considered proximity agreement to be the strongest expression of agreement, that is the most accurate / objective. Clearly, if two people occupy the same point then they have to agree. Moreover, they would be indistinguishable within the framework provided... or at least this was what I initially thought. One of the problems with the political compass, which is a deeply subtle point, is that the boundaries grow with each question ("proposition" if you want). That is to say that the size of the graph increases with each new question. Consequently this means that there are fewer paths to the edge of the graph, while there are many paths near the centre. This lead to the obvious eureka moment when I realised that the path taken is the strongest type of agreement. In hindsight it was obvious and even seems tautological.

Path Agreement

... and so we arrive at the final type of agreement: path agreement. There may be other types but this short list should be quite sufficient for most comparisons. The greater the number of shared answers between two people then the more similar their paths will be, and therefore their opinions will also be very similar. This would mean that two people who occupy different points could have a stronger agreement score than two people who share the same point at the end. The path taken is more important than the final point that people occupy. This is perhaps the most profound point that can be made about the political compass graphs. Showing the final output of points is only a small part of the picture; it condenses all the information into an easy to read format but it hides all the details.

As an interesting special case the origin should have the most number of paths that lead to it. I'd be inclined to call this the highest number of path degeneracies. Well, the next immediate consequence is that it is possible for two people to give exactly opposite answers and still end up at the origin. Consider an example which only has two questions, person 1 provides answers which takes them +1 in the x-direction and then -1 in the x-direction. The net result is that their point lies upon the origin. Person two can answer the questions with the exact opposite response (-1 then +1) and arrive back at the same point. Here the metric of vicinity is weak, it has no way to distinguish "opposites" and neither would the other previous metrics but the path agreement metric would.

You can add more questions to the system but it is still possible to find two people with opposing points of view that land back at the origin. It is tempting to say that it doesn't matter as they wouldn't have particularly strong views about economics (or whatever you are trying to measure) but this could be fallacious. It may indicate a failure within the nature of questions to appropriately probe the solution space; that is to say that the questions may not be probing what you think they are, or that the questions are good but the way that the answers (transforms) relate to the questions is somehow skewed. However, it might not be possible to remove this problem which would strongly suggest something profound about opposites: opposites can agree?