User:Lucasvb/An upgrade to the spatial model of voters: Difference between revisions

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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 "extremelyvery important" (100%).
 
We can group both these values into an '''''opinion''''', notated by '''(belief|importance)''', for every issue in our quiz.
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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 because of the framing.
 
 
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.
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:[ (+0.5|80%), (-1.0|100%), (0.0|50%), (-0.5|50%), (+0.8|70%) ]
 
So a person who completely disagrees on an issue, and deems it extremelyvery important, would get a (-1|100%) opinion for that issue. A person who is uncertain either way, but still thinks it is an important issue that needs to be discussed would be a (0|100%) opinion. This would be a true "centrist" on that issue.
 
On the other hand, a person who says they are completely indifferent about an issue doesn't really care either way, so their opinion would be a (N/A,0%), that is, we should completely disregard the belief (agree or disagree) value of their opinion.
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On the second issue, suppose both Alice and Bob hold the same opinion, say, (-0.5|50%). Both belief and importance values are the same. We would expect this to count as "maximum agreement".
Any other type of opinion difference should fall somewhere in the middle. How can we quantify this?
 
How can we quantify this?
 
Here's one way we can consider this situation mathematically which, at least to me, makes a lot of sense.
 
Imagine everyone has a fixed amount of "''opinion units''" on every issue, and they must use it to form the ''shape'' of an opinion on an issue.
 
We'll visualize each opinion unit as a little block, which we may place it along a ''belief axis''. On one side of the axis we have the "-1" (completely disagree) and the other "+1" (completely agree) beliefs.
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A person's opinion will then be represented by a single "pile" of these opinion units. An opinion deemed important will be a tall and narrow pile, whereas a less important opinion will be a wide and short pile.
In effect, we will turn each opinion into a ''distribution''. This is shown in the examples below.
 
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[[File:Dist model undecided.png|left|frame|400px|A (0.0&#124;100%) opinion distribution, symbolizing undecidedness. This is what a true "centrist" belief looks like.]]
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This is very different from an indifferent person, who doesn't care either way. In that case, the opinion is an uniform pile all across the axis. The person's opinion isn't "anywhere".
 
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Note: In principle the opinion does not have to be symmetric or single-peaked, but those opinions would be more complicated to understand and our model doesn't need to be too complex.
 
<div style="clear:both"></div>
[[File:Dist model skew.png|left|frame|400px|A more complex opinion distribution could be represented by a skewed distribution. This is a person who strongly believes in something, but still has many caveats to their belief.]]
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So we will assume all opinions are simple, single-peaked, symmetric distributions, for our own sake.
 
=== Comparing opinions ===
 
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.
 
Remember how we started with the idea of "opinion units" people can place 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 her own.
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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 the government allowing gay marriage, and Alice is trying to convince Bob that this is only about civil unions and not religious weddings ("it's mostly a tax thing!"), and that there are more important issues to worry about, like the budget.
 
== Measuring distance ==
 
How difficult is this task for Alice?
 
We would expect that if both opinions are already similar, not a lot of convincing is required. We would also expect that the further apart the "opinion units" need to be relocated, the more difficult it is to change someone's opinion.
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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 rearrange one pile of dirt into another pile of dirt.
 
If you replace "dirt" with "opinion unit", you'll immediately arrive at our idea here.
 
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 is always positive, and is only zero if both distributions are exactly the same.
 
* 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!
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* 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 and unbiased. It takes the same amount of effort to change one distribution into the other, and vice versa.
 
== Comparing stances ==
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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. ThisThe wouldEuclidean bedistance equivalentbetween totwo assumingvoters everyone's<math>a opinion= distributions(a_1, are\cdots, infinitelya_N)</math> sharp.and In<math>b our= model(b_1, however\cdots, votersb_N)</math> arein '''multiplea distributionsspace ofwith opinion'''<math>N</math> inissues thisis space.given by:
 
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:
 
In our model, however, voters are '''multiple distributions of opinion''' in this space. We would replace the above with the earth-mover's distance:
 
:<math>d(a,b) = \sqrt{\sum_{i=1}^{N} \text{EMD}(a_i(x),b_i(x)))^2}, </math>
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Note that different issues are never compared with one another here. Only opinions on the same issue count towards each term.
 
It should be clear that that if the distributions are infinitely sharp (i.e. [https://en.wikipedia.org/wiki/Dirac_delta_function Dirac deltas]), the earth-mover's distance is simply the distance between those two sharp peaks. In this way, we recover the old traditional spatial model from our model.
 
=== Why the Euclidean distance anyway? ===
 
We could have considered other distances (or "metrics") in the same way, but why single out the Eucliean one? Why not use
 
:<math>d(a,b) = \sum_{i=1}^{N} \text{EMD}(a_i(x),b_i(x))), </math>
 
Well, if you think about it, in the space of political opinions we naturally correlate a few issues. Terms like "fiscally conservative" or "socially liberal" are common, and can be thought of as an "ideological direction" in the space of opinions that point towards multiple issues at the same time. (I like to call them "ideals".)
 
This notion of "direction" is well-modeled mathematically by the notion of an "angle" in this space. This, in turn, mathematically requires a notion of inner products, which requires the Euclidean metric.
 
== Benefits ==
 
One major benefit of this modelapproach is that we now have a direct way to embed importance into our model of voters.
 
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 toon an issue, they will perceive differences more aggressively, making them see that axis as "stretched", that is, the distances are perceived as 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. This model of distance also naturally captures the correlations between multiple issues due to this scaling, and the effect of voters and candidates giving different importance to issues.
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.
 
With the Euclidean distance, and how we embedded the different priorities voters have on multiple issues in our model, we now have a unified model which can naturally deal with voters having strong ideals, degrees of compromising, etc. We could even model the dynamics of voters by using the notion of "effort to move around opinion units".
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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