An Equation to Save Democracy

Imagine a world of direct democracy, without political parties, without legislatures or parliaments, without laws ruined by the fallacy of the compound question, without gerrymandering or spoilers, without a class of professional rule-makers whose warped incentives are the enemies of the common good. Here is an equation to achieve that world. Think I'm wrong? Contact me and explain to me why this won't work.

Consider a set of options: 7 possible candidates for an executive office, 15 different variants of a bill, in fact any set of any number of possibilities. Now imagine a voter considering those options. The voter is free to consider any or all of those options. To “consider” an option is to cast a vote for it in the unit interval. The option with the highest final value ( $f$ ) wins (if you need 5 out of 42, the top 5 win). Here is how we calculate that final value (the dot is left in the middle intentionally to show that the equation consists of two simple components):

$\begin{aligned} f_{option}&=\frac{\left (\displaystyle\sum_{i=1}^{c} v_i \right )}{c}\cdot \sqrt{1-\left ( \frac{c-n}{n}\right )^{2}} \end{aligned}$

where:

$\begin{aligned} v_i \in [0,1]&=\text{a vote for the option}\\ c&=\text{# of voters who have considered the option}\\ n&=\text{# of voters who have considered any option} \end{aligned}$

Notice that this is simply the mean ( $\mu$ ) of the option's votes multiplied by a unit-circle function, $y=\sqrt{1-x^2}$ , which is in quadrant II because, since $0\le c\le n$ :

$\begin{aligned} x&\in [-1,0]\text{, so}\\ y&\in [0,1] \end{aligned}$

Also, since $\mu \in [0,1]$ :

$\begin{aligned} f&\in [0,1] \end{aligned}$

The purpose of the unit-circle function is to attenuate $f$ as $c$ approaches 0. Note that because a linear function in the same position attenuates more severely, a vote of 0 is (almost 🙂) always stronger disapproval than unconsidered: chart