Thread: a geometric interpretation of quadratic voting.
Imagine representing a set of decisions on N issues as a point in N-dimensional space.
Imagine representing a set of decisions on N issues as a point in N-dimensional space.
If each person has k voting tokens, we can represent their vote as a right to move the decision from the center of a radius sqrt(k) circle to a point of their choice along the edge. You can see this by pythagorean theorem: (dx)^2 + (dy)^2 = r^2 = k 
Now, how would they vote? Zooming in enough, we can pretend preferences are linear, that is someone is trying to maximize ax + by + ..., where a,b are constants specific to them.
So they would choose a point on the line ax + by + ... = k for the highest possible k that still intersects with the circle. This line is tangent to the circle. From high school math, we know that the perpendicular vector to ax + by + ... = k is the vector (a, b .... )
How much you care about a parameter is proportional to the corresponding variable in your utility function: if you are maximizing ax + by + ..., then `a` determines how much one increment of `x` makes you happier. And the amount by which you end up increasing `x` is... `a`
Hence, QV satisfies the desired rule "how much influence you exert on an issue is proportional to how strongly you care about it".
