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= An upgrade to the spatial model of voters =
In the following article, I'll explain an alternate spatial model of voters, which is an upgrade of the typical model and has it as a special case.
I believe this model encapsulates many important aspects that have been missing from most analyses so far.
== An operational way to define opinions on issues ==
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* '''Belief''': we create a scale from "completely disagree" (-1) to "completely agree" (+1).
* '''Importance''': we create a scale from "completely indifferent" (0%) to "extremely important" (100%).
We can group both these values into an '''''opinion''''', notated by '''(belief|importance)''', for every issue in our quiz.
Everyone answering this quiz gets assigned an opinion on every one of these issues.
This is a good model because it is:
# ''Operational'': (you could in principle go around asking these questions for any number of issues).
# ''Absolute'': in principle, the questions can be as specific as required to avoid ambiguity, and there is no relative center.
# ''Bounded'': the belief has well-defined extremes.
This kind of model has been used extensively in political polls for decades. The popular website [https://isidewith.com/ I Side With] uses a very similar model.
==== Stances ====
Let's call the ''collection of opinions'' a person holds their '''''stance'''''. For example, the following would be a 5-issue stance:
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Now that we have our model for opinions, we can look into comparing them. Remember how we started with the idea of "opinion units" people can put around their belief axis? Let's build upon that idea.
Suppose Alice is trying to talk to Bob about an issue where they share different opinions. In this model, what she is trying to do is turn Bob's opinion into
This would look like Alice trying to morph Bob's opinion distribution into her own opinion distribution, piece by piece. This is illustrated below.
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In the above example, Alice is trying to convince Bob that he should somewhat agree with this particular issue, and that he should consider it less of a priority. For example, this could be something like allowing gay marriage, and Alice is trying to convince Bob that this is only about civil unions and not religious weddings, and that there are more important issues like the budget.
== Measuring distance ==
How difficult is this task?
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In mathematics and engineering, this is a well-studied problem of [https://en.wikipedia.org/wiki/Transportation_theory_(mathematics) optimal transport], and it has found uses everywhere, from artificial intelligence to traffic management.
The intuitive notion of how "difficult" it is to convince someone to believe something else, piece by piece, is captured by the '''''[https://en.wikipedia.org/wiki/Earth_mover%27s_distance earth-mover's distance]''''' ('''''EMD''''') between two distributions. It is, intuitively, the least amount of effort you would need to
With this notion of distance between distributions and this model of opinions, we can now compare two opinions in a reasonable way. To see that this is reasonable, note that:
* The earth-mover's distance between (-1|100%) and (+1|100%) is maximal for our model: you need to move all the opinion from one extreme to the other. This is very hard to achieve!
* The earth-mover's distance between two equal distributions is zero. There is no work (convincing) necessary!
* The distance between every other pair of distributions is somewhere in the middle, and the farthest the change in belief, the more distant the distributions are.
* The distance is symmetric. It takes the same amount of effort to change one distribution into the other, and vice versa.
== Comparing stances ==
So now we can compare individual opinions, we can begin to look into comparing entire stances. (Remember that a stance is a collection of opinions on multiple issues.)
In the typical [[spatial model]] of elections, voters are perfect points in an opinion space. This would be equivalent to assuming everyone's opinion distributions are infinitely sharp. In our model, however, voters are '''multiple distributions of opinion''' in this space.
In this model, the Euclidean distance between two voters <math>a = (a_1, \cdots, a_N)</math> and <math>b = (b_1, \cdots, b_N)</math> in a space with <math>N</math> issues is given by:
:<math>d(a,b) = \sqrt{\sum_{i=1}^{N} (a_i - b_i)^2} </math>
That is, we compare the separation on each issue (<math>a_i - b_i</math>). In our model of opinion distributions, we would replace this with the earth-mover's distance, as described above:
:<math>d(a,b) = \sqrt{\sum_{i=1}^{N} \text{EMD}(a_i(x),b_i(x)))^2}, </math>
where <math>\text{EMD}(a_i(x),b_i(x)))</math> is the earth-mover's distance between the ''distributions'' <math>a_i(x)</math> and <math>b_i(x)</math>.
Note that different issues are never compared with one another here. Only opinions on the same issue count towards each term.
== Benefits ==
One major benefit of this model is that we now have a direct way to embed importance into our model.
A low-importance opinion is a wider distribution, which means it has a smaller distance to other opinions than a sharp one. So a voter with a low-importance on an issue effectively sees that axis as "compressed", that is, distances are shorter along that axis. On the other hand, if a voter has a high importance to an issue, they will perceive differences more aggressively, making them see that axis as "stretched", that is, the distances are larger.
In this way, each voter has their own perception of how important each issue is and this is accounted for when computing the distance between different stances.
Note that there's still a distance between someone who is indifferent and anyone with a different opinion. This makes sense, as it takes effort to convince someone to care.
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