# Overton Window Shift Points and ElasticityA Mathematical Representation of Policy Acceptance in Society

Abstract

A model to rigorously describe the Overton Window, a concept from political science which models "how ideas in society change over time and influence politics," [1] is defined mathematically along with two new concepts: the elasticity and the peak potential point. The author discusses how each point within the Overtion window carries an associated real number defined as the elasticity, $$\mu$$, which expresses the potential for expansion at the point. The peak potential point, $$\epsilon = \max(\mu)$$, is defined as a point of maximum elasticity such that the elasticity decreases monotonically as $$\mu$$ approaches $$-\infty$$ and $$+\infty$$ from $$\epsilon$$.

Potential applications are explored such as being able to predict the probability of civil war, social discord, or political unrest based on the elasticity values and the degree of which division exists within a society.

While there has always been some degree of division in American politics, the Trump presidency has brought about significant changes to the political landscape, country, and society which are likely to have significant influence for the foreseeable future. A highly visible impact to society has been how numerous conspiracy theories even when thoroughly disproved by legitimate and reputable information sources are still believed and popularized by many of his supporters. Additionally, ideologies once viewed as fringe such as neo-nazism and communism, as well as behaviors once viewed as unacceptable such as political corruption and government propaganda and disinformation have become common place and even have a degree of normalcy.

In political science this phenomena is described by the Overton Window, a concept which models "how ideas in society change over time and influence politics." [1] The current theory is that Trump's behavior has been so outrageous that people are essentially numbed – the idea being that the Overton window has shifted to the right. However, this really depends on who you ask with individuals who identify as more right-leaning more likely to believe that the Overton window has shifted to the left with a significant influence of "radical leftist ideologies." This becomes clear by the deep division seen across the populace.

The question therefore becomes, is it possible that the Overton window is fragmented? Likely not. While centrist ideas may be less common, they are certainly not less acceptable. If anything, the more central ideas are more widely accepted, even if less desired. Even as people are more commonly divided in their opinions, centrist ideas are still acceptable to everyone, even if people are commonly striving for more extreme policy issues. The only other explanation would be that the Overton window is stretching or expanding with a wider range of opinions and policies being considered acceptable by a larger population of people. One may think this would be more uniting, but as more extreme, yet opposing, views become increasingly normal, there is an increasing amount of division and ultimately tension.

We represent the political spectrum by the set, $$S \subseteq \mathbb{R}$$, such that $$S = (-\infty, \infty)$$. The Overton window is given by $$\mathcal{O} \subset S$$ with $$o_i = \inf{(\mathcal{O})}$$ and $$o_j = \sup{(\mathcal{O})}$$ such that $$\mathcal{O}$$ spans $$[o_i, o_j]$$.

Definition. For every $$o \in \mathcal{O}$$, there exists an associated number $$\mu \in \mathbb{R}$$ called the elasticity of $$o$$. Therefore, every point along the political spectrum interval is defined by a tuple, $$(o, \mu)$$.

The elasticity can be physically interpreted as how flexible $$\mathcal{O}$$ is at a given point. Conceptually, it can be interpreted as how common the acceptance of a given policy or idea is. For example, suppose there is a question regarding the societal acceptance of tax policy.

Point in SpectrumPolicy Position
-10 : Fringe leftNo sales taxes and all businesses are state-owned therefore there are no corporate taxes. The state allocates 100% of individual earnings. There is no private property, therefore state-provided allowances of food and clothing are distributed equally based on number of individuals in a household.
-7 : Far leftThe state allocates the vast majority of individual earnings (60-90%) depending on total earnings with a with goal of using tax revenue to fund national programs which benefit the general populace and provide most daily needs.
-4 : LeftTaxes range from 40-60% based on individual earnings with the goal of solving key problems such as income inequality, poverty, hunger.
0 : CenterTaxes range from 30-50% based on individual earnings. Social programs exist for citizens unable to provide for themselves.
4 : RightTaxes are roughly 30% regardless of individual earnings. Very few social programs exist with the Church and charity programs providing minimal assistance for populations unable to provide for themselves.
7 : Far-rightNo income taxes, only sales tax and corporate taxes exist. No social programs.
10 : Fringe rightNo government exists, therefore there are no taxes.

where we'll assume that people who are left-leaning subscribe to Keynesian economics while people who are right-leaning subscribe to supply-side economics. While both economic theories can have legitimacy when applied appropriately and thoughtfully, neither may be applied absolutely without resulting in negative socio-economic consequences.

Definition. There exists an $$o \in \mathcal{O}$$ which we call the peak potential point, $$\epsilon$$, which is defined as $$\epsilon = \max(\mu)$$ such that values of $$\mu$$ are monotonically decreasing as $$\mu$$ approaches $$-\infty$$ and $$+\infty$$ from $$\epsilon$$. The point is determined by summing over all discrete points in $$\mathcal{O}$$ and dividing by the total, known points. Suppose that $$\vert \mathcal{O} \vert = N$$, then

$\sum_{i=1}^{N} \mu_i$

We see the above figure as an example of a society with a mild liberal leaning wherein progressive views have a slightly higher acceptance than conservative views. We also see that the progressions to and from $$\epsilon$$ are gradual and contain few large jumps with exception of the change from index $$0$$ to index $$1$$.

By examining the plots of different societies, we may make predictions and draw conclusions. For example, containing