Acyclic domains of linear orders: a survey

Among the many significant contributions that Fishburn made to social choice theory some have focused on what he has called "acyclic sets", i.e. the sets of linear orders where majority rule applies without the "Condorcet effect" (majority relation never has cycles). The search for large domains of this type is a fascinating topic. I review the works in this field and in particular consider a recent one that allows to show the connections between some of them that have been unrelated up to now.acyclic set;alternating scheme;distributive lattice;effet Condorcet;linear order,maximal chain,permutoèdre lattice, single-peaked domain,weak Bruhat order,value restriction.

Condorcet Domains of Degree at most Seven

In this paper we give the first explicit enumeration of all maximal Condorcet domains on $n\leq 7$ alternatives. This has been accomplished by developing a new algorithm for constructing Condorcet domains, and an implementation of that algorithm which has been run on a supercomputer. We follow this up by the first survey of the properties of all maximal Condorcet domains up to degree 7, with respect to many properties studied in the social sciences and mathematical literature. We resolve several open questions posed by other authors, both by examples from our data and theorems. We give a new set of results on the symmetry properties of Condorcet domains which unify earlier works. Finally we discuss connections to other domain types such as non-dictatorial domains and generalisations of single-peaked domains. All our data is made freely available for other researches via a new website.Comment: The paper is still undergoing some revision

Decision theory and discrete mathematics

AbstractThis paper discusses recent developments that emphasize the role of discrete mathematics in decision theory. It focuses on three topics which illustrate this role. The first is the transitive simple majorities problem of determining the maximum number of linear orders on n candidates that prevent the occurrence of cyclic majorities when voters' preferences are confined to those orders. The second involves the varieties of unique solutions to simple systems of n — 1 linearly independent homogeneous linear equations in n variables of types that arise from qualitative equivalence comparisons in the measurement of subjective probabilities and utility differences. The third topic describes a computer-efficient hierarchy of stochastic-dominance relations for comparisons of risky alternatives whose outcomes lie in a unidimensional set of evenly spaced points. Although the paper is primarily expository, new results and new conjectures are included