# McKelvey-Schofield Chaos Theorem Without Agenda Setter

The McKelvey-Schofield Chaos Theorem states that in a multidimensional preference space, it is almost always possible to reverse engineer the implementation of your desired policy by constructing an appropriate interim sequence of policies, each of which would pass a majority rule vote. This can result in diverging from everyone else's desired policies over time. The "agenda setter" has almost complete control over the final destination, although everyone else's desired policies affect the set of viable interim sequences which can reach it. I am curious how unusual it would be to observe this divergence without the presence of an agenda setter.

Here Policy 1 is the current policy. The green area represents exactly the set of policies which all of the three circled voters would vote for. If we imagine repeating this for all $$\geq 3$$ cardinality subsets of the voters, then the union of all of those green areas would be the viable policies for the next timestep. Suppose we uniformly randomly pick a point (or a microscopically small region, if random points are philosophically problematic) from this union to constitute Policy 2. Then we reiterate this exercise for many timesteps.

How probable is it that Policy $$\lim_{N -> \infty} N$$ remains near the voters' preferences? Does this change if some motivated distribution other than the uniform random distribution is chosen (i.e., weighing area higher if more of the voters would vote for it), or in more than $$2$$ dimensions? If we found such a policy evolution in the wild diverging from voters' preferences, to what degree would this constitute evidence of intelligent design by an agenda setter?

• What does "diverging from voters' preferences" mean in this context? Jul 27, 2023 at 22:26
• The geometric distance between the current policy and its nearest voter keeps getting larger. Jul 28, 2023 at 2:03