THREE BRIEF PROOFS OF ARROW S IMPOSSIBILITY THEOREM JOHN GEANAKOPLOS COWLES FOUNDATION PAPER NO. 1116

Size: px
Start display at page:

Download "THREE BRIEF PROOFS OF ARROW S IMPOSSIBILITY THEOREM JOHN GEANAKOPLOS COWLES FOUNDATION PAPER NO. 1116"

Transcription

1 THREE BRIEF PROOFS OF ARROW S IMPOSSIBILITY THEOREM BY JOHN GEANAKOPLOS COWLES FOUNDATION PAPER NO COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box New Haven, Connecticut

2 Economic Theory 26, (2005) DOI: /s Exposita Notes Three brief proofs of Arrow s Impossibility Theorem John Geanakoplos Cowles Foundation, Box , New Haven, CT , USA ( john.geanakoplos@yale.edu) Received: July 9, 2001; revised version: September 2, 2004 Summary. Arrow s original proof of his impossibility theorem proceeded in two steps: showing the existence of a decisive voter, and then showing that a decisive voter is a dictator. Barbera replaced the decisive voter with the weaker notion of a pivotal voter, thereby shortening the first step, but complicating the second step. I give three brief proofs, all of which turn on replacing the decisive/pivotal voter with an extremely pivotal voter (a voter who by unilaterally changing his vote can move some alternative from the bottom of the social ranking to the top), thereby simplifying both steps in Arrow s proof. My first proof is the most straightforward, and the second uses Condorcet preferences (which are transformed into each other by moving the bottom alternative to the top). The third proof proceeds by reinterpreting Step 1 of the first proof as saying that all social decisions are made the same way (neutrality). Keywords and Phrases: Arrow Impossibility Theorem, Pivotal, Neutrality. JEL Classification Numbers: D7, D70, D71. Let A = {A, B,..., C} be a finite set of at least three alternatives. A transitive preference over A is a ranking of the alternatives in A from top to bottom, with ties allowed. We consider a society with N individuals, each of whom has a (potentially I wish to thank Ken Arrow, Chris Avery, Don Brown, Ben Polak, Herb Scarf, Chris Shannon, Lin Zhou, and especially Eric Maskin for very helpful comments and advice. I was motivated to think of reproving Arrow s theorem when I undertook to teach it to George Zettler, a mathematician friend. After I presented this paper at MIT, a graduate student there named Luis Ubeda Rives told me he had worked out the same neutrality argument as I give in my third proof while he was in Spain nine years ago. He said he was anxious to publish on his own and not jointly, so I encourage the reader to consult his forthcoming working paper. The proofs appearing here appeared in my 1996 CFDP working paper. Proofs 2 and 3 originally used May s notation, which I have dropped on the advice of Chris Avery.

3 212 J. Geanakoplos different) transitive preference. A constitution is a function which associates with every N-tuple (or profile) of transitive preferences a transitive preference called the social preference. A constitution respects unanimity if society puts alternative α strictly above β whenever every individual puts α strictly above β. The constitution respects independence of irrelevant alternatives if the social relative ranking (higher, lower, or indifferent) of two alternatives α and β depends only on their relative ranking by every individual. The constitution is a dictatorship by individual n if for every pair α and β, society strictly prefers α to β whenever n strictly prefers α to β. Arrow s Theorem. Any constitution that respects transitivity, independence of irrelevant alternatives, and unanimity is a dictatorship. The strategy in all three proofs is to find a limited dictator n, and then to prove that n must be a genuine dictator. We say that n is pivotal for alternatives α,β at a profile of preferences π if by putting α strictly above β, or the reverse, n induces the social preference to do the same, holding all other individual preferences fixed as in π. Barbera showed that there is a pivotal individual, who must then be a dictator. We say that an individual n is extremely pivotal for an alternative β at a profile π if n is pivotal for all pairs α, β involving β at π. Such an individual can move β from the very bottom of the social profile to the very top. In the first proof, we show that there is an extremely pivotal individual who must be a dictator. The critical step in the proof is an extremal lemma asserting that if every individual preference ranks β at the very top or the very bottom, then so must the social preference, even if half the individuals put β at the top and the other half put β at the bottom. We say that an individual n is a local dictator at a profile π if n is pivotal for all pairs α, β at π. In our second proof, we show that there is a local dictator, who must then be a genuine dictator. The critical idea here is to look at Condorcet profiles, where each preference is a cyclical permutation of the others. Such a permutation is generated by moving the very top alternative to the very bottom. Arrow motivated his impossibility theorem by Condorcet s paradox. In our last proof we begin with a lemma proving that Arrow s axioms imply neutrality, namely that every decision must be made the same way, independent of the names of the alternatives. We use it to show that any pivotal voter is a dictator. The neutrality lemma is equivalent to the extremal lemma, but for variety we give a different proof of it. First Proof Extremal Lemma. Let alternative b be chosen arbitrarily. At any profile in which every voter puts alternative b at the very top or very bottom of his ranking of alternatives, society must as well (even if half the voters put b at the top and half put b at the bottom). Proof. Suppose to the contrary that for such a profile and for distinct a, b, c, the social preference put a b and b c. By independence of irrelevant alternatives, this would continue to hold even if every individual moved c above a, because that could be arranged without disturbing any ab or cb votes (since b occupies an

4 Three brief proofs of Arrow s Impossibility Theorem 213 extreme position in each individual s ranking, as can be seen from the diagram). By transitivity the social ranking would then continue to put a c, but by unanimity it would also put c>a, a contradiction, proving the lemma. Next we argue that there is a voter n = n(b) who is extremely pivotal in the sense that by changing his vote at some profile he can move b from the very bottom of the social ranking to the very top. To see this, let each voter put b at the very bottom of his (otherwise arbitrary) ranking of alternatives. By unanimity, society must as well. Now let the individuals from voter 1 to N successively move b from the bottom of their rankings to the very top, leaving the other relative rankings in place. Let n be the first voter whose change causes the social ranking of b to change. (By unanimity, a change must occur at the latest when n = N.) Denote by profile I the list of all voter rankings just before n moves b, and denote by profile II the list of all voter rankings just after n moves b to the top. Since in profile II b has moved off the bottom of the social ranking, we deduce from our first argument that the social preference corresponding to profile II must put b at the very top. We argue third that n = n(b) is a dictator over any pair ac not involving b. To see this, choose one element, say a, from the pair ac. Construct profile III from profile II by letting n move a above b, so that a> n >b> n c, and by letting all the agents n n arbitrarily rearrange their relative rankings of a and c while leaving b in its extreme position. By independence of irrelevant alternatives, the social preferences corresponding to profile III would necessarily put a>b(since all individual ab votes are as in profile I where n put b at the bottom), and b>c (since all individual bc votes are as in profile II where n put b at the top). By transitivity, society must put a>c. By independence of irrelevant alternatives, the social preference over ac must agree with n whenever a> n c. We conclude by arguing that n is also a dictator over every pair ab. Takea third distinct alternative c to put at the bottom in the construction of paragraph 2. From the third argument, there must be a voter n(c) who is an αβ dictator for any pair αβ not involving c, such as ab. But agent n can affect society s ab ranking, namely at profiles I and II, hence this ab dictator n(c) must actually be n.

5 214 J. Geanakoplos Second Proof In a Condorcet profile each voter n N is assigned one of the Condorcet preferences (cyclial permutations of the alphabetical order) described below: C A C B C C A B C B C A. C. A If all n N are assigned to the first preference C A, then by unanimity, C A must be the social preference. Among Condorcet profiles π such that the social preference is C A, find π A that minimizes the number of voters with preferences C A. There must be at least one voter n in π A with preferences C A, for otherwise C would be unanimously preferred to A. We shall show that n is a local dictator at π A. Suppose alternative β immediately follows α alphabetically. Suppose at π A that n unilaterally switches to C β, giving the Condorcet profile π β. By IIA, we still have A> πβ > πβ α and β> πβ > πβ C. Hence, for the social order to change, we must get α πβ β. (Furthermore, if α = πβ β, then by transitivity and the fact #A 3, A> πβ C.) Suppose instead that n switches to C A, where A<B< <C, giving the non-condorcet profile π Ā. Take two alphabetically consecutive alternatives α, β. Then C β and C A agree on αβ (β >α) and on AC (C >A). Hence by IIA, α πā β since α πβ β. Since α, β are arbitrary, this gives A πā πā C. Furthermore, if α = πā β, then by IIA, α = πβ β, and from the last paragraph, this would imply that A> πβ C, and thus by IIA, A> πā C, contradicting A πā B πā πā C. We conclude that A< πā B< πā < πā C. Thus n is indeed a local dictator at π A. Let π be any nearly strict profile (where each individual preference has at most one pair of alternatives ranked equally) at which n is a local dictator. Change π to another nearly strict profile π by letting a single voter n n raise some alternative half a step higher in his ranking in such a way that either he breaks a single tie αβ or creates a single tie αβ (but not both). We shall show that n is also a local dictator at π. By IIA, this change by n cannot change the social ranking of any pair except possibly αβ. Modify π and π by letting n rank α> n γ> n β, for some third alternative γ. By hypothesis the social ranking at the modified π has α> π γ> π β. Hence by IIA, α> π γ and γ> π β, so by transitivity α> π β. Thus by IIA, the half-step move by n cannot change the power of n to enforce his strict preference over every pair at π. Since π A has no ties, for any pair αβ, a sequence of such half-moves can always be found to achieve arbitrary preferences for every voter n n over the given pair αβ. By IIA, n is a dictator. Third Proof Strict Neutrality Lemma. All binary social rankings are made the same way. Consider two pairs of alternatives ab and αβ. Suppose that in some profile π each voter strictly prefers a to b, orb to a, and suppose that in another profile π each B.

6 Three brief proofs of Arrow s Impossibility Theorem 215 voter has the same relative ranking of αβ as he does of ab in π. Then the social preference between ab in π is identical to the social preference between αβ in π and both social preferences are strict. Proof. Suppose WLOG that socially a b in π. Take c/ {a, b} and suppose first that αβ = ac or αβ = cb or αβ = cd with d / {a, b, c}. Create a new profile π with α just above a for each voter n (if α a), and β just below b for each voter n (if β b). Since all ab preferences are strict, this can be arranged while maintaining the same ab and αβ preferences, as the diagram makes clear. α α α b b a a a β β b b b α α β β β a a By unanimity, α> π a (if α a) and b> π β (if β b). By transitivity, α> π β. Reversing the roles of ab and αβ, we conclude by IIA that socially a> π b in the original profile. All that remains is to consider the situation where αβ = ba. This is handled by considering in turn profiles where the preferences are identical between the pairs (ab, ac), then (ac, bc), and then finally (bc, ba). This concludes the proof of the lemma. Next, take two distinct alternatives a and b and start with a profile in which b> n a for all n. Beginning with n =1, let each voter successively move a above b. By unanimity and the strict neutrality lemma, there will be a voter n who moves the social preference from b>ato a>bwhen he moves a up. Clearly n is pivotal. The situation is described below. 1 n N 1 n N aa ab b bb baa a aa bb b b bb aa a b a We now show that n is a dictator. Take an arbitrary pair of alternatives α, β and let n rank α> n β. Let the αβ rankings for n n be arbitrary. Take c/ {α, β} and put c above everything for 1 n<n, c below everything for n <n N, and α> n c> n β. By IIA, neutrality, and the pivotal profile discovered above, socially c>βand α>c, and so by transitivity, α>β. 1 n N c c α β α References β α αβ c β Arrow, K.J.: Social choice and individual values. New York: Wiley 1951 Barbera, S.: Pivotal voters. A new proof of Arrow s theorem. Economics Letters 6, (1980) Geanakoplos, J.: Three brief proofs of Arrow s impossibility theorem. Cowles Foundation Discussion Paper 1123R (1996) May, K.: A set of independent, necessary, and sufficient conditions for simple majority decisions. Econometrica 20, (1952) Ubeda, Luis: Neutrality in Arrow and other impossibility theorems. Economic Theory 23(1), (2003) α c β c

Arrow s General (Im)Possibility Theorem

Division of the Humanities and ocial ciences Arrow s General (Im)Possibility Theorem KC Border Winter 2002 Let X be a nonempty set of social alternatives and let P denote the set of preference relations

Social Choice Theory. Felix Munoz-Garcia School of Economic Sciences Washington State University. EconS Advanced Microeconomics II

Social Choice Theory Felix Munoz-Garcia School of Economic Sciences Washington State University EconS 503 - Advanced Microeconomics II Social choice theory MWG, Chapter 21. JR, Chapter 6.2-6.5. Additional

Non-Manipulable Domains for the Borda Count

Non-Manipulable Domains for the Borda Count Martin Barbie, Clemens Puppe * Department of Economics, University of Karlsruhe D 76128 Karlsruhe, Germany and Attila Tasnádi ** Department of Mathematics, Budapest

Arrow s Paradox Prerna Nadathur January 1, 2010 Abstract In this paper, we examine the problem of a ranked voting system and introduce Kenneth Arrow s impossibility theorem (1951). We provide a proof sketch

arxiv: v2 [math.co] 14 Apr 2011

Complete Characterization of Functions Satisfying the Conditions of Arrow s Theorem Elchanan Mossel and Omer Tamuz arxiv:0910.2465v2 [math.co] 14 Apr 2011 April 15, 2011 Abstract Arrow s theorem implies

Cowles Foundation for Research in Economics at Yale University

Cowles Foundation for Research in Economics at Yale University Cowles Foundation Discussion Paper No. 1904 Afriat from MaxMin John D. Geanakoplos August 2013 An author index to the working papers in the

6.207/14.15: Networks Lecture 24: Decisions in Groups

6.207/14.15: Networks Lecture 24: Decisions in Groups Daron Acemoglu and Asu Ozdaglar MIT December 9, 2009 1 Introduction Outline Group and collective choices Arrow s Impossibility Theorem Gibbard-Satterthwaite

Dependence and Independence in Social Choice Theory

Dependence and Independence in Social Choice Theory Eric Pacuit Department of Philosophy University of Maryland, College Park pacuit.org epacuit@umd.edu March 4, 2014 Eric Pacuit 1 Competing desiderata

Social Choice. Jan-Michael van Linthoudt

Social Choice Jan-Michael van Linthoudt Summer term 2017 Version: March 15, 2018 CONTENTS Remarks 1 0 Introduction 2 1 The Case of 2 Alternatives 3 1.1 Examples for social choice rules............................

Social choice theory, Arrow s impossibility theorem and majority judgment

Université Paris-Dauphine - PSL Cycle Pluridisciplinaire d Etudes Supérieures Social choice theory, Arrow s impossibility theorem and majority judgment Victor Elie supervised by Miquel Oliu Barton June

Online Appendix to Strategy-proof tie-breaking in matching with priorities

Online Appendix to Strategy-proof tie-breaking in matching with priorities Lars Ehlers Alexander Westkamp December 12, 2017 Section 1 contains the omitted proofs of Lemma 5, Lemma 6 and Lemma 7 Subsection

On the Chacteristic Numbers of Voting Games

On the Chacteristic Numbers of Voting Games MATHIEU MARTIN THEMA, Departments of Economics Université de Cergy Pontoise, 33 Boulevard du Port, 95011 Cergy Pontoise cedex, France. e-mail: mathieu.martin@eco.u-cergy.fr

Strategy-Proofness on the Condorcet Domain

College of William and Mary W&M ScholarWorks Undergraduate Honors Theses Theses, Dissertations, & Master Projects 5-2008 Strategy-Proofness on the Condorcet Domain Lauren Nicole Merrill College of William

Coalitionally strategyproof functions depend only on the most-preferred alternatives.

Coalitionally strategyproof functions depend only on the most-preferred alternatives. H. Reiju Mihara reiju@ec.kagawa-u.ac.jp Economics, Kagawa University, Takamatsu, 760-8523, Japan April, 1999 [Social

The Axiomatic Method in Social Choice Theory:

The Axiomatic Method in Social Choice Theory: Preference Aggregation, Judgment Aggregation, Graph Aggregation Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss

13 Social choice B = 2 X X. is the collection of all binary relations on X. R = { X X : is complete and transitive}

13 Social choice So far, all of our models involved a single decision maker. An important, perhaps the important, question for economics is whether the desires and wants of various agents can be rationally

Arrow s Theorem as a Corollary

Arrow s Theorem as a Corollary Klaus Nehring University of California, Davis October 2002 Abstract Arrow s Impossibility Theorem is derived from a general theorem on social aggregation in property spaces.

Rationalization of Collective Choice Functions by Games with Perfect Information. Yongsheng Xu

Rationalization of Collective Choice Functions by Games with Perfect Information by Yongsheng Xu Department of Economics, Andrew Young School of Policy Studies Georgia State University, Atlanta, GA 30303

GROUP DECISION MAKING

1 GROUP DECISION MAKING How to join the preferences of individuals into the choice of the whole society Main subject of interest: elections = pillar of democracy Examples of voting methods Consider n alternatives,

DICTATORIAL DOMAINS. Navin Aswal University of Minnesota, Minneapolis, USA Shurojit Chatterji Indian Statistical Institute, New Delhi, India and

DICTATORIAL DOMAINS Navin Aswal University of Minnesota, Minneapolis, USA Shurojit Chatterji Indian Statistical Institute, New Delhi, India and Arunava Sen Indian Statistical Institute, New Delhi, India

From social choice functions to dictatorial social welfare functions Antonio Quesada Universidad de Murcia Abstract A procedure to construct a social welfare function from a social choice function is suggested

Volume 31, Issue 1. Manipulation of the Borda rule by introduction of a similar candidate. Jérôme Serais CREM UMR 6211 University of Caen

Volume 31, Issue 1 Manipulation of the Borda rule by introduction of a similar candidate Jérôme Serais CREM UMR 6211 University of Caen Abstract In an election contest, a losing candidate a can manipulate

Positively responsive collective choice rules and majority rule: a generalization of May s theorem to many alternatives

Positively responsive collective choice rules and majority rule: a generalization of May s theorem to many alternatives Sean Horan, Martin J. Osborne, and M. Remzi Sanver December 24, 2018 Abstract May

Coalitional Structure of the Muller-Satterthwaite Theorem

Coalitional Structure of the Muller-Satterthwaite Theorem Pingzhong Tang and Tuomas Sandholm Computer Science Department Carnegie Mellon University {kenshin,sandholm}@cscmuedu Abstract The Muller-Satterthwaite

The Gibbard random dictatorship theorem: a generalization and a new proof

SERIEs (2011) 2:515 527 DOI 101007/s13209-011-0041-z ORIGINAL ARTICLE The Gibbard random dictatorship theorem: a generalization and a new proof Arunava Sen Received: 24 November 2010 / Accepted: 10 January

Finite Dictatorships and Infinite Democracies

Finite Dictatorships and Infinite Democracies Iian B. Smythe October 20, 2015 Abstract Does there exist a reasonable method of voting that when presented with three or more alternatives avoids the undue

Ordered Value Restriction

Ordered Value Restriction Salvador Barberà Bernardo Moreno Univ. Autònoma de Barcelona and Univ. de Málaga and centra March 1, 2006 Abstract In this paper, we restrict the pro les of preferences to be

A topological approach to Wilson s impossibility theorem

Journal of Mathematical Economics 43 (2007) 184 191 A topological approach to Wilson s impossibility theorem Yasuhito Tanaka Faculty of Economics, Doshisha University, Kamigyo-ku, Kyoto 602-8580, Japan

Multiple Equilibria in the Citizen-Candidate Model of Representative Democracy.

Multiple Equilibria in the Citizen-Candidate Model of Representative Democracy. Amrita Dhillon and Ben Lockwood This version: March 2001 Abstract De Sinopoli and Turrini (1999) present an example to show

On the topological equivalence of the Arrow impossibility theorem and Amartya Sen s liberal paradox

Applied Mathematics and Computation 181 (2006) 1490 1498 www.elsevier.com/locate/amc On the topological equivalence of the Arrow impossibility theorem and Amartya Sen s liberal paradox Yasuhito Tanaka

An Equivalence result in School Choice

An Equivalence result in School Choice Jay Sethuraman May 2009 Abstract The main result of the paper is a proof of the equivalence of single and multiple lottery mechanisms for the problem of allocating

Lecture Notes, Lectures 22, 23, 24. Voter preferences: Majority votes A > B, B > C. Transitivity requires A > C but majority votes C > A.

Lecture Notes, Lectures 22, 23, 24 Social Choice Theory, Arrow Possibility Theorem Paradox of Voting (Condorcet) Cyclic majority: Voter preferences: 1 2 3 A B C B C A C A B Majority votes A > B, B > C.

APPLIED MECHANISM DESIGN FOR SOCIAL GOOD

APPLIED MECHANISM DESIGN FOR SOCIAL GOOD JOHN P DICKERSON Lecture #21 11/8/2016 CMSC828M Tuesdays & Thursdays 12:30pm 1:45pm IMPOSSIBILITY RESULTS IN VOTING THEORY / SOCIAL CHOICE Thanks to: Tuomas Sandholm

MAXIMAL POSSIBILITY AND MINIMAL DICTATORIAL COVERS OF DOMAINS

MAXIMAL POSSIBILITY AND MINIMAL DICTATORIAL COVERS OF DOMAINS Gopakumar Achuthankutty 1 and Souvik Roy 1 1 Economic Research Unit, Indian Statistical Institute, Kolkata Abstract In line with the works

Voting Systems. High School Circle II. June 4, 2017

Voting Systems High School Circle II June 4, 2017 Today we are going to resume what we started last week. We are going to talk more about voting systems, are we are going to being our discussion by watching

A DISTANCE-BASED EXTENSION OF THE MAJORITY JUDGEMENT VOTING SYSTEM

A DISTANCE-BASED EXTENSION OF THE MAJORITY JUDGEMENT VOTING SYSTEM EDURNE FALCÓ AND JOSÉ LUIS GARCÍA-LAPRESTA Abstract. It is common knowledge that the political voting systems suffer inconsistencies and

A Characterization of Single-Peaked Preferences via Random Social Choice Functions

A Characterization of Single-Peaked Preferences via Random Social Choice Functions Shurojit Chatterji, Arunava Sen and Huaxia Zeng September 2014 Paper No. 13-2014 ANY OPINIONS EXPRESSED ARE THOSE OF THE

Comparing impossibility theorems

Comparing impossibility theorems Randy Calvert, for Pol Sci 507 Spr 2017 All references to A-S & B are to Austen-Smith and Banks (1999). Basic notation X set of alternatives X set of all nonempty subsets

Condorcet Efficiency: A Preference for Indifference

Condorcet Efficiency: A Preference for Indifference William V. Gehrlein Department of Business Administration University of Delaware Newark, DE 976 USA Fabrice Valognes GREBE Department of Economics The

3.1 Arrow s Theorem. We study now the general case in which the society has to choose among a number of alternatives

3.- Social Choice and Welfare Economics 3.1 Arrow s Theorem We study now the general case in which the society has to choose among a number of alternatives Let R denote the set of all preference relations

Talking Freedom of Choice Seriously

Talking Freedom of Choice Seriously Susumu Cato January 17, 2006 Abstract In actual life, we face the opportunity of many choices, from that opportunity set, we have to choose from different alternatives.

Binary Aggregation with Integrity Constraints

Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence Binary Aggregation with Integrity Constraints Umberto Grandi and Ulle Endriss Institute for Logic, Language and

The Arrow Impossibility Theorem Of Social Choice Theory In An Infinite Society And Limited Principle Of Omniscience

Applied Mathematics E-Notes, 8(2008), 82-88 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ The Arrow Impossibility Theorem Of Social Choice Theory In An Infinite

Arrow s Impossibility Theorem: Preference Diversity in a Single-Profile World

Arrow s Impossibility Theorem: Preference Diversity in a Single-Profile World Brown University Department of Economics Working Paper No. 2007-12 Allan M. Feldman Department of Economics, Brown University

Arrow s Impossibility Theorem and Experimental Tests of Preference Aggregation

Arrow s Impossibility Theorem and Experimental Tests of Preference Aggregation Todd Davies Symbolic Systems Program Stanford University joint work with Raja Shah, Renee Trochet, and Katarina Ling Decision

Alternative Characterizations of Boston Mechanism

Alternative Characterizations of Boston Mechanism Mustafa Oǧuz Afacan April 15, 2013 Faculty of Arts and Social Sciences, Sabancı University, 34956, İstanbul, Turkey. Abstract Kojima and Ünver (2011) are

Partial lecture notes THE PROBABILITY OF VIOLATING ARROW S CONDITIONS

Partial lecture notes THE PROBABILITY OF VIOLATING ARROW S CONDITIONS 1 B. Preference Aggregation Rules 3. Anti-Plurality a. Assign zero points to a voter's last preference and one point to all other preferences.

Unanimity, Pareto optimality and strategy-proofness on connected domains

Unanimity, Pareto optimality and strategy-proofness on connected domains Souvik Roy and Ton Storcken y February 2016 Abstract This is a preliminary version please do not quote 1 Extended Abstract This

Condorcet Consistency and the strong no show paradoxes

Condorcet Consistency and the strong no show paradoxes Laura Kasper Hans Peters Dries Vermeulen July 10, 2018 Abstract We consider voting correspondences that are, besides Condorcet Consistent, immune

"Arrow s Theorem and the Gibbard-Satterthwaite Theorem: A Unified Approach", by Phillip Reny. Economic Letters (70) (2001),

February 25, 2015 "Arrow s Theorem and the Gibbard-Satterthwaite Theorem: A Unified Approach", by Phillip Reny. Economic Letters (70) (2001), 99-105. Also recommended: M. A. Satterthwaite, "Strategy-Proof

Fair Divsion in Theory and Practice

Fair Divsion in Theory and Practice Ron Cytron (Computer Science) Maggie Penn (Political Science) Lecture 6-b: Arrow s Theorem 1 Arrow s Theorem The general question: Given a collection of individuals

THE UNIVERSITY OF KANSAS WORKING PAPERS SERIES IN THEORETICAL AND APPLIED ECONOMICS

THE UNIVERSITY OF KANSAS WORKING PAPERS SERIES IN THEORETICAL AND APPLIED ECONOMICS AN EFFICIENCY CHARACTERIZATION OF PLURALITY SOCIAL CHOICE ON SIMPLE PREFERENCE DOMAINS Biung-Ghi Ju University of Kansas

A NEW SET THEORY FOR ANALYSIS

Article A NEW SET THEORY FOR ANALYSIS Juan Pablo Ramírez 0000-0002-4912-2952 Abstract: We present the real number system as a generalization of the natural numbers. First, we prove the co-finite topology,

Binary Aggregation with Integrity Constraints

Binary Aggregation with Integrity Constraints Umberto Grandi and Ulle Endriss Institute for Logic, Language and Computation, University of Amsterdam u.grandi@uva.nl, ulle.endriss@uva.nl Abstract We consider

Chapter 12: Social Choice Theory

Chapter 12: Social Choice Theory Felix Munoz-Garcia School of Economic Sciences Washington State University 1 1 Introduction In this chapter, we consider a society with I 2 individuals, each of them endowed

Preference, Choice and Utility

Preference, Choice and Utility Eric Pacuit January 2, 205 Relations Suppose that X is a non-empty set. The set X X is the cross-product of X with itself. That is, it is the set of all pairs of elements

Survey of Voting Procedures and Paradoxes

Survey of Voting Procedures and Paradoxes Stanford University ai.stanford.edu/ epacuit/lmh Fall, 2008 :, 1 The Voting Problem Given a (finite) set X of candidates and a (finite) set A of voters each of

The Review of Economic Studies, Ltd.

The Review of Economic Studies, Ltd. Oxford University Press http://www.jstor.org/stable/2297086. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at.

First-Order Logic Formalisation of Impossibility Theorems in Preference Aggregation

First-Order Logic Formalisation of Impossibility Theorems in Preference Aggregation Umberto Grandi and Ulle Endriss Institute for Logic, Language and Computation, University of Amsterdam, Postbus 94242,

Implementation of the Ordinal Shapley Value for a three-agent economy 1

Implementation of the Ordinal Shapley Value for a three-agent economy 1 David Pérez-Castrillo 2 Universitat Autònoma de Barcelona David Wettstein 3 Ben-Gurion University of the Negev April 2005 1 We gratefully

Recap Social Choice Fun Game Voting Paradoxes Properties. Social Choice. Lecture 11. Social Choice Lecture 11, Slide 1

Social Choice Lecture 11 Social Choice Lecture 11, Slide 1 Lecture Overview 1 Recap 2 Social Choice 3 Fun Game 4 Voting Paradoxes 5 Properties Social Choice Lecture 11, Slide 2 Formal Definition Definition

Winning probabilities in a pairwise lottery system with three alternatives

Economic Theory 26, 607 617 (2005) DOI: 10.1007/s00199-004-059-8 Winning probabilities in a pairwise lottery system with three alternatives Frederick H. Chen and Jac C. Heckelman Department of Economics,

Reference-based Preferences Aggregation Procedures in Multicriteria Decision Making

Reference-based Preferences Aggregation Procedures in Multicriteria Decision Making Antoine Rolland Laboratoire ERIC - Université Lumière Lyon 2 5 avenue Pierre Mendes-France F-69676 BRON Cedex - FRANCE

Static Decision Theory Under Certainty

Static Decision Theory Under Certainty Larry Blume September 22, 2010 1 basics A set of objects X An individual is asked to express preferences among the objects, or to make choices from subsets of X.

Approximation Algorithms and Mechanism Design for Minimax Approval Voting

Approximation Algorithms and Mechanism Design for Minimax Approval Voting Ioannis Caragiannis RACTI & Department of Computer Engineering and Informatics University of Patras, Greece caragian@ceid.upatras.gr

Logic and Social Choice Theory. A Survey. ILLC, University of Amsterdam. November 16, staff.science.uva.

Logic and Social Choice Theory A Survey Eric Pacuit ILLC, University of Amsterdam staff.science.uva.nl/ epacuit epacuit@science.uva.nl November 16, 2007 Setting the Stage: Logic and Games Game Logics Logics

Probabilistic Aspects of Voting

Probabilistic Aspects of Voting UUGANBAATAR NINJBAT DEPARTMENT OF MATHEMATICS THE NATIONAL UNIVERSITY OF MONGOLIA SAAM 2015 Outline 1. Introduction to voting theory 2. Probability and voting 2.1. Aggregating

Social Welfare Functions that Satisfy Pareto, Anonymity, and Neutrality: Countable Many Alternatives. Donald E. Campbell College of William and Mary

Social Welfare Functions that Satisfy Pareto, Anonymity, and Neutrality: Countable Many Alternatives Donald E. Campbell College of William and Mary Jerry S. Kelly Syracuse University College of William

A Note on the McKelvey Uncovered Set and Pareto Optimality

Noname manuscript No. (will be inserted by the editor) A Note on the McKelvey Uncovered Set and Pareto Optimality Felix Brandt Christian Geist Paul Harrenstein Received: date / Accepted: date Abstract

Economics Bulletin, 2012, Vol. 32 No. 1 pp Introduction. 2. The preliminaries

1. Introduction In this paper we reconsider the problem of axiomatizing scoring rules. Early results on this problem are due to Smith (1973) and Young (1975). They characterized social welfare and social

On Domains That Admit Well-behaved Strategy-proof Social Choice Functions

On Domains That Admit Well-behaved Strategy-proof Social Choice Functions Shurojit Chatterji, Remzi Sanver and Arunava Sen May 2010 Paper No. 07-2010 ANY OPINIONS EXPRESSED ARE THOSE OF THE AUTHOR(S) AND

Intro to Economic analysis

Intro to Economic analysis Alberto Bisin - NYU 1 Rational Choice The central gure of economics theory is the individual decision-maker (DM). The typical example of a DM is the consumer. We shall assume

arxiv: v1 [cs.gt] 29 Mar 2014

Testing Top Monotonicity Haris Aziz NICTA and UNSW Australia, Kensington 2033, Australia arxiv:1403.7625v1 [cs.gt] 29 Mar 2014 Abstract Top monotonicity is a relaxation of various well-known domain restrictions

Dominance and Admissibility without Priors Jörg Stoye Cornell University September 14, 2011 Abstract This note axiomatizes the incomplete preference ordering that reflects statewise dominance with respect

Boolean Inner-Product Spaces and Boolean Matrices

Boolean Inner-Product Spaces and Boolean Matrices Stan Gudder Department of Mathematics, University of Denver, Denver CO 80208 Frédéric Latrémolière Department of Mathematics, University of Denver, Denver

UNIVERSITY OF NOTTINGHAM. Discussion Papers in Economics CONSISTENT FIRM CHOICE AND THE THEORY OF SUPPLY

UNIVERSITY OF NOTTINGHAM Discussion Papers in Economics Discussion Paper No. 0/06 CONSISTENT FIRM CHOICE AND THE THEORY OF SUPPLY by Indraneel Dasgupta July 00 DP 0/06 ISSN 1360-438 UNIVERSITY OF NOTTINGHAM

Rationality and solutions to nonconvex bargaining problems: rationalizability and Nash solutions 1

Rationality and solutions to nonconvex bargaining problems: rationalizability and Nash solutions 1 Yongsheng Xu Department of Economics Andrew Young School of Policy Studies Georgia State University, Atlanta,

A Quantitative Arrow Theorem

University of Pennsylvania ScholarlyCommons Statistics Papers Wharton Faculty Research 10-2012 A Quantitative Arrow Theorem Elchanan Mossel University of Pennsylvania Follow this and additional works at:

Fairness and Redistribution: Response

Fairness and Redistribution: Response By ALBERTO ALESINA, GEORGE-MARIOS ANGELETOS, AND GUIDO COZZI This paper responds to the comment of Di Tella and Dubra (211). We first clarify that the model of Alesina

Metainduction in Operational Set Theory

Metainduction in Operational Set Theory Luis E. Sanchis Department of Electrical Engineering and Computer Science Syracuse University Syracuse, NY 13244-4100 Sanchis@top.cis.syr.edu http://www.cis.syr.edu/

INTEGER PROGRAMMING AND ARROVIAN SOCIAL WELFARE FUNCTIONS

MATHEMATICS OF OPERATIONS RESEARCH Vol. 28, No. 2, May 2003, pp. 309 326 Printed in U.S.A. INTEGER PROGRAMMING AND ARROVIAN SOCIAL WELFARE FUNCTIONS JAY SETHURAMAN, TEO CHUNG PIAW, and RAKESH V. VOHRA

On the Impossibility of Certain Ranking Functions

On the Impossibility of Certain Ranking Functions Jin-Yi Cai Abstract Suppose all the individuals in a field are linearly ordered. Groups of individuals form teams. Is there a perfect ranking function

14.770: Introduction to Political Economy Lectures 1 and 2: Collective Choice and Voting

14.770: Introduction to Political Economy Lectures 1 and 2: Collective Choice and Voting Daron Acemoglu MIT September 6 and 11, 2017. Daron Acemoglu (MIT) Political Economy Lectures 1 and 2 September 6

THE PROFILE STRUCTURE FOR LUCE S CHOICE AXIOM

THE PROFILE STRUCTURE FOR LUCE S CHOICE AXIOM DONALD G SAARI Abstract A geometric approach is developed to explain several phenomena that arise with Luce s choice axiom such as where differences occur

Economic Core, Fair Allocations, and Social Choice Theory

Chapter 9 Nathan Smooha Economic Core, Fair Allocations, and Social Choice Theory 9.1 Introduction In this chapter, we briefly discuss some topics in the framework of general equilibrium theory, namely

Political Economy of Institutions and Development. Lectures 2 and 3: Static Voting Models

14.773 Political Economy of Institutions and Development. Lectures 2 and 3: Static Voting Models Daron Acemoglu MIT February 7 and 12, 2013. Daron Acemoglu (MIT) Political Economy Lectures 2 and 3 February

Approximation Algorithms and Mechanism Design for Minimax Approval Voting 1

Approximation Algorithms and Mechanism Design for Minimax Approval Voting 1 Ioannis Caragiannis, Dimitris Kalaitzis, and Evangelos Markakis Abstract We consider approval voting elections in which each

(a i1,1 a in,n)µ(e i1,..., e in ) i 1,...,i n. (a i1,1 a in,n)w i1,...,i n

Math 395. Bases of symmetric and exterior powers Let V be a finite-dimensional nonzero vector spaces over a field F, say with dimension d. For any n, the nth symmetric and exterior powers Sym n (V ) and

Implementing the Nash Extension Bargaining Solution for Non-convex Problems. John P. Conley* and Simon Wilkie**

Implementing the Nash Extension Bargaining Solution for Non-convex Problems John P. Conley* and Simon Wilkie** Published as: John P. Conley and Simon Wilkie Implementing the Nash Extension Bargaining Solution

Online Appendix for Incentives in Landing Slot Problems

Online Appendix for Incentives in Landing Slot Problems James Schummer Azar Abizada April 14, 2017 This document contains supplementary results and proofs for Incentives in Landing Slot Problems, published

Recognizing single-peaked preferences on aggregated choice data

Recognizing single-peaked preferences on aggregated choice data Smeulders B. KBI_1427 Recognizing Single-Peaked Preferences on Aggregated Choice Data Smeulders, B. Abstract Single-Peaked preferences play

Strategic Manipulability without Resoluteness or Shared Beliefs: Gibbard-Satterthwaite Generalized

Strategic Manipulability without Resoluteness or Shared Beliefs: Gibbard-Satterthwaite Generalized Christian Geist Project: Modern Classics in Social Choice Theory Institute for Logic, Language and Computation

MATH 324 Summer 2011 Elementary Number Theory. Notes on Mathematical Induction. Recall the following axiom for the set of integers.

MATH 4 Summer 011 Elementary Number Theory Notes on Mathematical Induction Principle of Mathematical Induction Recall the following axiom for the set of integers. Well-Ordering Axiom for the Integers If

Existence Theorems of Continuous Social Aggregation for Infinite Discrete Alternatives

GRIPS Discussion Paper 17-09 Existence Theorems of Continuous Social Aggregation for Infinite Discrete Alternatives Stacey H. Chen Wu-Hsiung Huang September 2017 National Graduate Institute for Policy

Quasi-transitive and Suzumura consistent relations

Quasi-transitive and Suzumura consistent relations Walter Bossert Department of Economics and CIREQ, University of Montréal P.O. Box 6128, Station Downtown, Montréal QC H3C 3J7, Canada FAX: (+1 514) 343

a (b + c) = a b + a c

Chapter 1 Vector spaces In the Linear Algebra I module, we encountered two kinds of vector space, namely real and complex. The real numbers and the complex numbers are both examples of an algebraic structure

Introductory notes on stochastic rationality. 1 Stochastic choice and stochastic rationality

Division of the Humanities and Social Sciences Introductory notes on stochastic rationality KC Border Fall 2007 1 Stochastic choice and stochastic rationality In the standard theory of rational choice