User:Lucasvb/An upgrade to the spatial model of voters: Difference between revisions
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Revision as of 22:08, 28 July 2020
, 3 years ago→An operational way to define opinions on issues
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= An upgrade to the spatial model of voters =
In the following article, I'll explain
I believe this model encapsulates many important aspects that have been missing from most analyses so far. But remember, it's still just a ''model''.
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Now suppose you go around and ask people their opinion regarding many such issues, under those specific framings, and that people answered you honestly and accurately to the best of their knowledge. (We can't really expect more than that, as we can't read people's minds.)
At the very least, we would like to know whether they agree or disagree with the statement. But we could also ask how strongly they feel about that position, how certain they are of it, and how important they feel it is to hold that position.
One way to convert this into a numerical scale is by considering the following parameters:
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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.
Of course, there's the
==== Stances ====
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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.
In mathematics and engineering, this is a well-studied problem of [
The intuitive notion of how "difficult" it is to convince someone to believe something else, piece by piece, is captured by the '''''[
If you replace "dirt" with "opinion unit", you'll immediately arrive at our idea here.
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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. [
=== Why the Euclidean distance anyway? ===
We could have considered other distances (or "metrics") in the same way, but why single out the
:<math>d(a,b) = \sum_{i=1}^{N} \text{EMD}(a_i(x),b_i(x))), </math>
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== Implementation ==
In practice, it's up to us to determine how sharply-peaked the distributions can get.
The earth-mover's distance is simple and efficient to compute in a discrete case, where the distribution is defined in a number of bins. This makes it readily available in many software packages.▼
As a first approximation, it is helpful to model the distribution as a simple trapezoidal distribution, instead of a normal distribution.▼
▲The earth-mover's distance is simple and efficient to compute in a discrete 1D case, where the distribution is defined in a number of bins. This makes it readily available in many software packages.
▲
In my simulations, I've defined an integer parameter <tt>L</tt>, the resolution of one side of the belief axis. In order to make 0 a valid belief, it is best to use an odd number of bins, so the total number of bins is given by <tt>W = 2*L+1</tt>.
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The following Python code generates the trapezoid distribution for a given opinion, with belief from -1 to +1, and importance from 0 to 1.
<syntaxhighlight lang="python">
def
return wasserstein_distance(_space,_space,dist1,dist2)
def
v = (w+1)/2 - L*abs(np.linspace(-1,1,W) - belief*importance)
v /= sum(v)
for w in range(1,W+1):▼
for
b1, b2 = np.random.rand()*2-1, np.random.rand()*2-1▼
▲ o1 = opinion(b1, i1)
o2 = opinion(b2, i2)
"vs",
</syntaxhighlight>
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