The Impartial, Anonymous, and Neutral Culture Model: A Probability Model for Sampling Public Preference Structures

Abstract

We introduce a new probability model, namely the Impartial, Anonymous, and Neutral Culture (IANC) Model, for sampling public preferences concerning a given set of alternatives. The IANC Model treats public preferences through a class of preference profiles named roots, where both names of the voters and the alternatives are immaterial. The general framework along with the theoretical formulation through group actions, an exact formula for the number of roots, and the description of a symbolic algebra package that allows for the generation of roots uniformly are presented. In order to be able to obtain uniform distribution of roots for large electorate size and high number of alternatives which lead to combinatorial explosions, the machinery we developed involves elements of symmetric functions and an application of the Dixon-Wilf algorithm. Using Monte Carlo methods, the model we develop allows for a testbed that can be used to answer various questions about the properties and behaviors of anonymous and neutral social choice rules for large parameters. As applications of the method, the results of two Monte Carlo experiments are presented: the likelihood of the existence of Condorcet winners and the probability of Condorcet and plurality rules to choose the same winner.

Keywords:

• algorithms
• consensus
• discrete choice

Acknowledgments

The authors would like to thank the anonymous referee whose insightful comments and suggestions greatly improved the presentation of this article.

Notes

Note. Each trial is the generation of a root from R(2, 4) by using GenerateRoot[2,4]. The three roots θ₁, θ₂, θ₃ area as given in (12).

Note. IC = Impartial Culture; IAC = Impartial Anonymous Culture; IANC = Impartial, Anonymous, and Neutral Culture. The formula for R(m, n) is as given in Theorem 3.

Note. IANC = Impartial, Anonymous, and Neutral Culture.

Note. Impartial Culture (IC) and Impartial Anonymous Culture (IAC) probabilities are from Gehrlein (1998). N = 10,000 samples were generated for the Impartial, Anonymous, and Neutral Culture (IANC) computations.