Arrovian juntas

This article explicitly constructs and classifies all arrovian voting systems on three or more alternatives. If we demand orderings to be complete, we have, of course, Arrow's classical dictator theorem, and a closer look reveals the classification of all such voting systems as dictatorial hierarchies. If we leave the traditional realm of complete orderings, the picture changes. Here we consider the more general setting where alternatives may be incomparable, that is, we allow orderings that are reflexive and transitive but not necessarily complete. Instead of a dictator we exhibit a junta whose internal hierarchy or coalition structure can be surprisingly rich. We give an explicit description of all such voting systems, generalizing and unifying various previous results.Comment: 22 pages, 1 figur

Arrow’s Impossibility Theorem and the distinction between Voting and Deciding

Arrow’s Impossibility Theorem in social choice finds different interpretations. Bordes-Tideman (1991) and Tideman (2006) suggest that collective rationality would be an illusion and that practical voting procedures do not tend to require completeness or transitivity. Colignatus (1990 and 2011) makes the distinction between voting and deciding. A voting field arises when pairwise comparisons are made without an overall winner, like in chess or basketball matches. Such (complete) comparisons can form cycles that need not be transitive. When transitivity is imposed then a decision is made who is the best. A cycle or deadlock may turn into indifference, that can be resolved by a tie-breaking rule. Since the objective behind a voting process is to determine a winner, then it is part of the very definition of collective rationality that there is completeness and transitivity, and then the voting field is extended with a decision.economic crisis; voting theory; democracy; economics and mathematics;

Arrovian juntas

This article explicitly constructs and classifies all arrovian voting systems on three or more alternatives. If we demand orderings to be complete, we have, of course, Arrow's classical dictator theorem, and a closer look reveals the classification of all such voting systems as dictatorial hierarchies. If we leave the traditional realm of complete orderings, the picture changes. Here we consider the more general setting where alternatives may be incomparable, that is, we allow orderings that are reflexive and transitive but not necessarily complete. Instead of a dictator we exhibit a junta whose internal hierarchy or coalition structure can be surprisingly rich. We give an explicit description of all such voting systems, generalizing and unifying various previous results.rank aggregation problem; Arrow's impossibility theorem; classification of arrovian voting systems; partial ordering; partially ordered set; poset; dictator; oligarchy; junta

Voting Rules that are Unbiased but not Transitive-Symmetric

We explore the relation between two natural symmetry properties of voting rules. The first is transitive-symmetry – the property of invariance to a transitive permutation group – while the second is the "unbiased" property of every voter having the same influence for all i.i.d. probability measures. We show that these properties are distinct by two constructions – one probabilistic, one explicit – of rules that are unbiased but not transitive-symmetric

Equitable voting rules

May's Theorem (1952), a celebrated result in social choice, provides the foundation for majority rule. May's crucial assumption of symmetry, often thought of as a procedural equity requirement, is violated by many choice procedures that grant voters identical roles. We show that a weakening of May's symmetry assumption allows for a far richer set of rules that still treat voters equally. We show that such rules can have minimal winning coalitions comprising a vanishing fraction of the population, but not less than the square root of the population size. Methodologically, we introduce techniques from group theory and illustrate their usefulness for the analysis of social choice questions.Comment: 43 pages, 5 figure

Arrow's Theorem for One-Dimensional Single-Peaked Preferences

In one-dimensional environments with single-peaked preferences we consider social welfare functions satisfying Arrow''s requirements, i.e. weak Pareto and independence of irrelevant alternatives. When the policy space is a one-dimensional continuum such a welfare function is determined by a collection of 2N strictly quasi-concave preferences and a tie-breaking rule. As a corollary we obtain that when the number of voters is odd, simple majority voting is transitive if and only if each voter�s preference is strictly quasi-concave.mathematical economics;

Using Ramsey theory to measure unavoidable spurious correlations in Big Data

Given a dataset we quantify how many patterns must always exist in the dataset. Formally this is done through the lens of Ramsey theory of graphs, and a quantitative bound known as Goodman's theorem. Combining statistical tools with Ramsey theory of graphs gives a nuanced understanding of how far away a dataset is from random, and what qualifies as a meaningful pattern. This method is applied to a dataset of repeated voters in the 1984 US congress, to quantify how homogeneous a subset of congressional voters is. We also measure how transitive a subset of voters is. Statistical Ramsey theory is also used with global economic trading data to provide evidence that global markets are quite transitive.Comment: 21 page