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Geometry of Chaotic and Stable Discussions

  • Author(s): Saari, Donald G.
  • et al.
Abstract

It always seems to be the case. No matter how hard you might work on a proposal, no matter how polished and complete the final product may be, when it is presented to a group for approval, there always seems to be some majority who wants to "improve It." Is this just an annoyance or is there a reason? The mathematical modeling provides an immediate explanation in terms of some interesting and unexpected mathematics. Even more; the mathematics describing this behavior underscores the reality that it can be surprising easy even for a group sincerely striving for excellence to make inferior decisions. Indeed, these difficulties are so pervasive and can arise in such unexpected ways that it is realistic to worry whether groups you belong to have been inadvertently victimized by these mathematical subtleties based on the orbits of symmetry groups. These problems can occur even if all decisions are reached by consensus during discus sions, such as a committee discussing the selection of a new calculus book. This paper addresses deliberations by discussing a branch of voting theory where Euclidean geometry models an "issue space." Then describing how it is possible to un intentionally make inferior choices, we will encounter mathematical behaviors remark- ably similar to "attractors" and "chaotic dynamics" from dynamical systems. Since the coexistence of chaotic and stable behavior is common in the Newtonian N -body problem and dynamical systems, it is interesting that this combination also coexists in the dynamics of discussions. Another connection arises when configurations central to the N -body problem play a suggestive role in the analysis; at another step we use singularity theory. What adds to the delight of this topic is that while the mathematics can be intricate, the issues can be described at a classroom level where some even lead to student level research projects. Keywords: spatial voting, paradoxes, singularity theory

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