Axioms for defeat in democratic elections

We propose six axioms concerning when one candidate should defeat another in a democratic election involving two or more candidates. Five of the axioms are widely satisfied by known voting procedures. The sixth axiom is a weakening of Kenneth Arrow’s famous condition of the Independence of Irrelevant Alternatives (IIA). We call this weakening Coherent IIA. We prove that the five axioms plus Coherent IIA single out a method of determining defeats studied in our recent work: Split Cycle. In particular, Split Cycle provides the most resolute definition of defeat among any satisfying the six axioms for democratic defeat. In addition, we analyze how Split Cycle escapes Arrow’s impossibility theorem and related impossibility results.

Schrödinger’s Ballot: Quantum Information and the Violation of Arrow’s Impossibility Theorem

We study Arrow’s Impossibility Theorem in the quantum setting. Our work is based on the work of Bao and Halpern, in which it is proved that the quantum analogue of Arrow’s Impossibility Theorem is not valid. However, we feel unsatisfied about the proof presented in Bao and Halpern’s work. Moreover, the definition of Quantum Independence of Irrelevant Alternatives (QIIA) in Bao and Halpern’s work seems not appropriate to us. We give a better definition of QIIA, which properly captures the idea of the independence of irrelevant alternatives, and a detailed proof of the violation of Arrow’s Impossibility Theorem in the quantum setting with the modified definition.

Assessment criteria for aggregate social welfare

Subject. The article addresses the history of development and provides the criticism of existing criteria for aggregate social welfare (on the simple exchange economy (the Edgeworth box) case). Objectives. The purpose is to develop a unique classification of criteria to assess the aggregate social welfare. Methods. The study draws on methods of logical and mathematical analysis. Results. The paper considers strong, strict and weak versions of the Pareto, Kaldor, Hicks, Scitovsky, and Samuelson criteria, introduces the notion of equivalence and constructs orderings by Pareto, Kaldor, Hicks, Scitovsky, and Samuelson. The Pareto and Samuelson's criteria are transitive, however, not complete. The Kaldor, Hicks, Scitovsky citeria are not transitive in the general case. Conclusions. The lack of an ideal social welfare criterion is the consequence of the Arrow’s Impossibility Theorem, and of the group of impossibility theorems in economics. It is necessary to develop new approaches to the assessment of aggregate welfare.

Aggregating multiple ordinal rankings in engineering design: the best model according to the Kendall’s coefficient of concordance

Aggregating the preferences of a group of experts is a recurring problem in several fields, including engineering design; in a nutshell, each expert formulates an ordinal ranking of a set of alternatives and the resulting rankings should be aggregated into a collective one. Many aggregation models have been proposed in the literature, showing strengths and weaknesses, in line with the implications of Arrow's impossibility theorem. Furthermore, the coherence of the collective ranking with respect to the expert rankings may change depending on: (i) the expert rankings themselves and (ii) the aggregation model adopted. This paper assesses this coherence for a variety of aggregation models, through a recent test based on the Kendall's coefficient of concordance (W), and studies the characteristics of those models that are most likely to achieve higher coherence. Interestingly, the so-called Borda count model often provides best coherence, with some exceptions in the case of collective rankings with ties. The description is supported by practical examples.

Define success for yourself, given Arrow’s Impossibility Theorem

“Define success for yourself, given Arrow’s Impossibility Theorem” explains different voting methods, including plurality voting, run-off voting, sequential run-off, Borda count, and dictatorship, and explains how Kenneth Arrow’s Impossibility Theorem proves that, in an election with three or more candidates, the only fair voting system is a dictatorship. That is, every voting method other than a dictatorship has a known problem with fairness. Mathematics students and enthusiasts are encouraged to consider the many “candidates” for defining “success” in mathematical and life pursuits before letting their vote be the only vote in electing a personal definition of success. This way, Arrow’s Impossibility Theorem assures a fair outcome. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Arrow's and Gibbard-Satterthwaite's Impossibility Theorems Revisited

A voting system is demonstrated which meets Arrow's 5 conditions and also satisfies the Gibbard-Satterthwaite demand for strategyproofness. The strategy is contained in the voting procedure itself so voters are incentivized to vote sincerely. This procedure allows for the circumventing of Arrow's Impossibility Theorem.

Arrow's and Gibbard-Satterthwaite's Impossibility Theorems Revisited

A voting system is demonstrated which meets Arrow's 5 conditions and also satisfies the Gibbard-Satterthwaite demand for strategyproofness. The strategy is contained in the voting procedure itself so voters are incentivized to vote sincerely. This procedure allows for the circumventing of Arrow's Impossibility Theorem.