DeepAI AI Chat
Log In Sign Up

A Phase Transition in Arrow's Theorem

by   Frederic Koehler, et al.

Arrow's Theorem concerns a fundamental problem in social choice theory: given the individual preferences of members of a group, how can they be aggregated to form rational group preferences? Arrow showed that in an election between three or more candidates, there are situations where any voting rule satisfying a small list of natural "fairness" axioms must produce an apparently irrational intransitive outcome. Furthermore, quantitative versions of Arrow's Theorem in the literature show that when voters choose rankings in an i.i.d. fashion, the outcome is intransitive with non-negligible probability. It is natural to ask if such a quantitative version of Arrow's Theorem holds for non-i.i.d. models. To answer this question, we study Arrow's Theorem under a natural non-i.i.d. model of voters inspired by canonical models in statistical physics; indeed, a version of this model was previously introduced by Raffaelli and Marsili in the physics literature. This model has a parameter, temperature, that prescribes the correlation between different voters. We show that the behavior of Arrow's Theorem in this model undergoes a striking phase transition: in the entire high temperature regime of the model, a Quantitative Arrow's Theorem holds showing that the probability of paradox for any voting rule satisfying the axioms is non-negligible; this is tight because the probability of paradox under pairwise majority goes to zero when approaching the critical temperature, and becomes exponentially small in the number of voters beyond it. We prove this occurs in another natural model of correlated voters and conjecture this phenomena is quite general.


page 1

page 2

page 3

page 4


A Quantitative Version of the Gibbard-Satterthwaite Theorem for Three Alternatives

The Gibbard-Satterthwaite theorem states that every non-dictatorial elec...

A Smoothed Impossibility Theorem on Condorcet Criterion and Participation

In 1988, Moulin proved an insightful and surprising impossibility theore...

Tradeoffs in Hierarchical Voting Systems

Condorcet's jury theorem states that the correct outcome is reached in d...

Cognitive Hierarchy and Voting Manipulation

By the Gibbard--Satterthwaite theorem, every reasonable voting rule for ...

How Many Vote Operations Are Needed to Manipulate A Voting System?

In this paper, we propose a framework to study a general class of strate...

A quantitative analysis of the 2017 Honduran election and the argument used to defend its outcome

The Honduran incumbent president and his administration recently declare...

New Results for Domineering from Combinatorial Game Theory Endgame Databases

We have constructed endgame databases for all single-component positions...