So I thought to make a preference chart where a person's 3rd favorite always comes halfway between their first and second favorite on a circle... First I solved these equations:

In complex numbers on the unit circle this is saying the variable in the right hand side of each equation is the point on the circle halfway between the two variables on the left hand side around the circle... And the last equation a=1 is to disallow rotational symmetry we set one variable to equal the real axis...

There were quite a few solutions, but most were trivial that all the variables equal 1 or 0 or all real number solutions, the two interesting ones are mirror images across the real axis, one of them is:

Which plotted on the unit circle is:

And see that it encodes the information perfectly for example that f is halfway around the circle between j and e, when two variables are equal the halfway point is either equal to either or on the opposite side of the circle..

So once plotted the question is can we extrapolate that someone who listed, for example, a and e as their first two favorites, which was not information given, will usually like h as a third choice... I'll have to find a real dataset to statistically determine whether that is the case or not...

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