Three Brief Proofs Of Arrows Impossibility Theorem

Arrow Impossibility Theorems is a 10-chapter text that describes existing impossibility theorems. This book explores a number of formalizations of ethical constraints of the theorems. After an introduction to the framework and notation for Arrow impossibility theorems, this book goes on discussing some concepts and an apparatus of relations among those concepts which are important for the theorems. Other chapters present some impossibility results that serve to point out serious difficulties in some plausible escape routes from the theorems of earlier chapters. The final chapter describes important areas of research that have arisen in the collective choice field in the transition away from studying the conditions of Arrow's theorem alone to the totality of all impossibility theorems. This book is intended primarily for economists.

In this note I consider a simple proof of Arrow's Impossibility Theorem (Arrow 1963). I start with the case of three individuals who have preferences on three alternatives. In this special case there are 133 possible combinations of the three individuals' rational preferences. However, by considering the subset of linear preferences, and employing the full strength of the IIA axiom, I reduce the number of cases necessary to completely describe the SWF to a small number, allowing an elementary proof suitable for most undergraduate students. This special case conveys the nature of Arrow's result. It is well known that the restriction to three options is not really limiting (any larger set of alternatives can be broken down into triplets, and any inconsistency within a triplet implies an inconsistency on the larger set). However, the general case of individuals can be easily considered in this framework, by building on the proof of the simpler case. I hope that a motivated student, having mastered the simple case of three individuals, will find this extension approachable and rewarding.

Kenneth Arrow's pathbreaking Òimpossibility theoremÓ was a watershed in the history of welfare economics, voting theory, and collective choice, demonstrating that there is no voting rule that satisfies the four desirable axioms of decisiveness, consensus, nondictatorship, and independence. In this book, Amartya Sen and Eric Maskin explore the implications of ArrowÕs theorem. Sen considers its ongoing utility, exploring the theoremÕs value and limitations in relation to recent research on social reasoning, while Maskin discusses how to design a voting rule that gets us closer to the idealÑgiven that achieving the ideal is impossible. The volume also contains a contextual introduction by social choice scholar Prasanta K. Pattanaik and commentaries from Joseph E. Stiglitz and Kenneth Arrow himself, as well as essays by Sen and Maskin outlining the mathematical proof and framework behind their assertions.

In mathematics, it simply is not true that 'you can't prove a negative'. Many revolutionary impossibility theorems reveal profound properties of logic, computation, fairness and the universe, and form the mathematical background of new technologies and Nobel prizes. But to fully appreciate these theorems and their impact on mathematics and beyond, you must understand their proofs. This book is the first to present these proofs for a broad, lay audience. It fully develops the simplest rigorous proofs found in the literature, reworked to contain less jargon and notation, and more background, intuition, examples, explanations, and exercises. Amazingly, all of the proofs in this book involve only arithmetic and basic logic – and are elementary, starting only from first principles and definitions. Very little background knowledge is required, and no specialized mathematical training – all you need is the discipline to follow logical arguments and a pen in your hand.

Edited in collaboration with FoLLI, the Association of Logic, Language and Information, this book constitutes the refereed proceedings of the Second International Workshop on Logic, Rationality, and Interaction, LORI 2009, held in Chongqing, China, in October 2009. The 24 revised full papers presented together with 8 posters were carefully reviewed and selected from a flood of submissions. The workshops topics include but are not limited to semantic models for knowledge, for belief, and for uncertainty, dynamic logics of knowledge, information flow, and action, logical analysis of the structure of games, belief revision, belief merging, logics for preferences and utilities, logics of intentions, plans, and goals, logics of probability and uncertainty, argument systems and their role in interaction, as well as norms, normative interaction, and normative multiagent systems.

This book constitutes the refereed proceedings of the 8th International Conference on Logic and the Foundations of the Theory of Game and Decision Theory, LOFT8 2008, held in Amsterdam, The Netherlands, July 2008. This volume is based on a selection of the presented papers and invited talks. They survived a thorough and lengthy reviewing process. The LOFT conferences are interdisciplinary events that bring together researchers from a variety of fields: computer science, economics, game theory, linguistics, logic, multi-agent systems, psychology, philosophy, social choice and statistics. Its focus is on the general issue of rationality and agency. The papers collected in this volume reflect the contemporary interests and interdisciplinary scope of the LOFT conferences.