Arrow, Hausdorff, and Ambiguities in the Choice of Preferred States in Complex Systems
This work addresses preference aggregation issues in economics and politics, but it appears incremental as it extends existing theoretical frameworks mathematically.
The paper tackles the problem of aggregating individual preferences into collective preferences in complex systems, extending Arrow's impossibility theorem from a sociological to a mathematical setting by using Hausdorff measures on sets, with a key result being that reversibility can be expressed in terms of set configurations.
Arrow's `impossibility' theorem asserts that there are no satisfactory methods of aggregating individual preferences into collective preferences in many complex situations. This result has ramifications in economics, politics, i.e., the theory of voting, and the structure of tournaments. By identifying the objects of choice with mathematical sets, and preferences with Hausdorff measures of the distances between sets, it is possible to extend Arrow's arguments from a sociological to a mathematical setting. One consequence is that notions of reversibility can be expressed in terms of the relative configurations of patterns of sets.