Random Preferences, Acyclicity, and the Threshold of Rationality

19 Pages Posted: 24 Feb 2020 Last revised: 2 Jun 2020

See all articles by Bassel Tarbush

Bassel Tarbush

University of Oxford - Merton College

Date Written: January 25, 2020

Abstract

Draw an agent's preferences over $n$ alternatives at random as follows: independently, for each pair of distinct alternatives, with probability $1-p$ the agent is indifferent between the alternatives in the pair, and with probability $p$ the agent is equally likely to prefer one alternative over the other. The agent's preferences are rational if they are transitive or, equivalently, if there are no preference cycles. I show that rationality exhibits a threshold behavior: if $p$ is asymptotically smaller than $n^{-2}$ then the agent's preferences are rational with high probability, and if $p$ is asymptotically larger than $n^{-2}$ then the agent's preferences are not rational with high probability. The main result of this paper embeds some existing results on the probability of preference cycles and of Condorcet cycles.

Keywords: Random preferences, random graphs, transitivity, preference cycles

JEL Classification: D01, D85

Suggested Citation

Tarbush, Bassel, Random Preferences, Acyclicity, and the Threshold of Rationality (January 25, 2020). Available at SSRN: https://ssrn.com/abstract=3525382 or http://dx.doi.org/10.2139/ssrn.3525382

Bassel Tarbush (Contact Author)

University of Oxford - Merton College ( email )

Merton College
Oxford, OX1 4JD
United Kingdom

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