Social Choice and Welfare PDF

Social Choice and Welfare 1. Introduction Daniel M. A. Barreto PPLE College - University of Amsterdam Fall 2024 Introduction...

Social Choice and Welfare 1. Introduction Daniel M. A. Barreto PPLE College - University of Amsterdam Fall 2024 Introduction Social Choices In this chapter, we will study situations with the following “ingredients”: Every human society has and will continue to face the: A group of people. Fundamental Economic Problem (Normative version) How should human societies organize themselves in order to produce (and A list of possible “public” policies (the alternatives) with a choice to be distribute) the things that they need or want? made. To start studying this question, we’ll start by asking ourselves how groups of Policies affect all members of the group. individuals can/should make collective choices. Individuals have (typically somewhat opposing) policy preferences.  Many Examples Many Examples The description sounds abstract, but it fits many common situations: Presidential elections Course enrollment Referenda Spectrum allocation Whether and how to provide a public good and many (many!) more... School choice Many Examples Many Examples Many Examples Many Examples Many Examples Institutions The main institutions to make social decisions in liberal democracies are: Market-like institutions for allocation problems Other money-free allocation methods Voting Delegation Questions: How well can we expect these institutions to perform? Can we design better (optimal) ones? In this course, we will generally take an institution free approach, start from (simple) fundamental situations and try to derive institutional solutions.  Institutions Formalizing our Problem The main institutions to make social decisions in liberal democracies are: We will describe a Social Choice Problem as follows: Market-like institutions for allocation problems (Fish market; stock Group: A group of N individuals indexed by i = 1, · · · , N. market; ad placement; spectrum allocation; game tickets) For example: citizens of a country, board of directors of a private firm, members of a jury, students applying to universities and universities Other money-free allocation methods (Kidney exchange; school choice; course allocation) Alternatives: A set of possible choices A = {a, b, · · · }. For example: presidential candidates, dividends, guilty or innocent, a Voting (presidential elections; brexit) matching of students to universities Delegation (Delta Works) Preferences: Each individual has a (strict) preference ordering over alternatives ≻i Questions: So a ≻i b means i prefers a to b. How well can we expect these institutions to perform? Can we design better Preferences of all individuals in society are regrouped in a Preference (optimal) ones? Profile:
In this course, we will generally take an institution free approach, start from ≻1 ,... , ≻N (simple) fundamental situations and try to derive institutional solutions. Examples of Social Choice Problems Our Working Example Social Choice Problem: our movie club must choose which movie to screen on our movie night. The jury is composed of 3 individuals: 1, 2 and 3 Our alternatives are the following 3 candidates Social Choice Problem Group Alternatives Presidential elections Citizens Candidates Pulp Fiction (1994) Yearly dividends Board of director 0 to all profits Criminal trial Jury Guilty or Innocent University Admissions Students and Universities All possible matchings Inside Out (2015) Parasite (2019)  Our Working Example What are we assuming about people? Individuals are assumed to be rational in the following sense: Preferences are given by the following preference profile Complete preferences: For any two alternatives a and b, individuals know and can say whether they prefer a to b, or b to a. 1 2 3 Coherent Preferences: Individuals have coherent preferences in the sense that, if an individual prefers a to b, and b to c, then she also prefers a to c. This type of coherence rules out problematic preference cycles such as: wine ≻ beer ≻ water ≻ wine Which means that 1 has the following preferences: This very basic ability of individuals to compare any two alternatives in a coherent way (no cycles) is our (basic) individual rationality assumption Pulp Fiction ≻ Inside Out ≻ Parasite for today. Question: Can we ask that social preferences be equally rational? Outline 1. Introduction 2. Aggregating Preferences 3. Voting 4. Arrow’s impossibility theorem 2. Aggregating Preferences 5. Impossibility for Choices 6. Manipulability and the Gibbard-Satterthwaite theorem 7. Sen’s Critique 8. Conclusion  Social Welfare Functions Social Welfare Functions We want to generate a complete and coherent (rational) social ranking ≻∗ , that: We can think of SWF as a set of rules, or an algorithm that takes the preferences of the population as an input and outputs a social ranking of is our principle for deciding which outcomes are “good” for society. all alternatives. naturally, should depend on the preferences of individuals. We can think of these rules as defining an institution A definition of institutions: This dependence is described abstractly by a Social Welfare Function: institutions “are the rules of the game in a society, or more formally, are the humanly devised constraints that shape human interaction” INPUT OUTPUT Douglas North (1990) Preference Profile Social Ranking
SWF
≻1 ,... , ≻ N ≻∗ = SWF ≻1 ,... , ≻N The task performed by SWF is one of aggregating preferences, that is transforming individual preferences into social preferences. Voting 3. Voting 3.1 Majority Voting Voting is a common way of aggregating preferences 3.2 Plurality But there are many ways of voting.  Examples of SWF: Majority Voting The brexit referendum: A = {Brexit, EU} Preference profile:
3.1. Majority Voting Brexit ≻ EU, · · ·, EU ≻ Brexit, · · · | {z }| {z } ×17,410,742 ×16,141,241 Majority voting outcome: Brexit ≻∗ EU Majority voting seems to work as a SWF, but can we generalize it to problems with more than 2 alternatives? A Problem with (Pairwise) Majority Voting A Problem with (Pairwise) Majority Voting Ranking our candidates for movie night: 3 candidates 1 2 3 Pulp Fiction (1994) Inside Out (2015) Parasite (2019) A Problem with (Pairwise) Majority Voting A Problem with (Pairwise) Majority Voting Majority Voting Contest: Majority Voting Contest: 1 2 3 1 2 3 vs. ⇒ A Problem with (Pairwise) Majority Voting A Problem with (Pairwise) Majority Voting Majority Voting Contest: Majority Voting Contest: 1 2 3 1 2 3 vs. ⇒ vs. ⇒ vs. ⇒ vs. ⇒ vs. ⇒ Social Preferences: ≻ ≻ ≻  A Problem with (Pairwise) Majority Voting Condorcet Paradox Majority Voting Contest: 1 2 3 vs. ⇒ vs. ⇒ This problem is known as the Condorcet paradox, after Nicolas de Condorcet. vs. ⇒ Social Preferences: ≻ ≻ ≻ Nicolas de Condorcet (1743-1794) The resulting social preferences have cycles. Majority voting fails as a SWF. Brexit and Condorcet Here is a real world example from the Brexit referendum: 3.2. Plurality Remain ≻ May Deal ≻ No Deal ≻ Remain  SWF Example: Plurality rule Plurality Rule: each member of the group gives 1 point to her favorite candidate, each candidate is ranked according to its total score. 4. Arrow’s impossibility theorem Example: First round of the 2002 French presidential elections Jacques Chirac: 19.88% Jean Saint-Josse: 4,23% 4.1 Universal Domain Jean-Marie Le Pen: 16.86% Alain Madelin: 3,91% 4.2 Pareto or Unanimity Lionel Jospin: 16.18% Robert Hue: 3.37% 4.3 Independence of Irrelevant François Bayrou: 6.84% Bruno Mégret: 2.34% Arlette Laguiller: 5.72% Christiane Taubira: 2,32% Alternatives Jean-Pierre Chevènement: 5.33% Corinne Lepage: 1,88% 4.4 Dictatorship Noël Mamère: 5.25% Christine Boutin: 1,19% 4.5 Impossibility Theorem Olivier Besancenot: 4.25% Daniel Gluckstein: 0,47% Plurality rule works as a SWF because numerical scores can always be used to produce a complete and coherent ranking Some Normative Criteria for SWF Next, we need to ask what is a “good” SWF. This is as much a philosophical question as an economic question! It is important to first have this normative reflection before examining how things work in practice. 4.1. Universal Domain We are going to proceed by defining criteria that it is reasonable to think a good SWF function should satisfy. Then we can check whether specific SWF satisfy these criteria. Or we can try to construct SWF that satisfy these criteria.... or maybe realize that this is a doomed endeavor?  Universal Domain Universal Domain Motivation: Consider the choice of a president or a governing party for example, or referenda. It would be strange if our method of choice only worked when people Definition (Universal Domain) hold certain preferences. A SWF satisfies universal domain (UD) if every possible preference profile Majority voting does not satisfy UD because of the Condorcet paradox: results in a well-defined (i.e. complete and transitive) social ranking. it is possible to find preferences of individuals such that majority voting does not generate a linear ordering of alternatives. In other contexts, relaxing UD seems reasonable. For example, in school choice it is reasonable to assume that students only care about the school they get, but not about the school assigned to others. Unanimity 4.2. Pareto or Unanimity Definition (Pareto property or Unanimity) A SWF satisfies the Pareto property or unanimity if whenever all individuals prefer an alternative a to another alternative b (for every individual i, a ≻i b), then the output social ranking ranks a above b (a ≻∗ b).  Unanimity Motivation and a matching example: Certainly desirable in elections. Consider the course allocation problem and suppose the final allocation (the one that is ranked first) is such that: 4.3. Independence of Irrelevant Ann would prefer the Sociology course she did not get to the Economics Alternatives course she got. Bob would prefer the Economics course he did not get to the Sociology course he got. Then the allocation method is not Pareto efficient: everybody would be better off if Ann and Bob could swap (assuming other students only care about the courses they get). Independence of Irrelevant Alternatives Independence of Irrelevant Alternatives First round of the 2002 French presidential elections: This criterion is more subtle. Jacques Chirac: 19.88% Jean Saint-Josse: 4,23% Jean-Marie Le Pen: 16.86% Alain Madelin: 3,91% The general idea is that society’s ranking of two alternatives should be Lionel Jospin: 16.18% independent of context. Robert Hue: 3.37% François Bayrou: 6.84% Bruno Mégret: 2.34% In our social choice problem, context is provided by other alternatives Arlette Laguiller: 5.72% Christiane Taubira: 2,32% available. Jean-Pierre Chevènement: Corinne Lepage: 1,88% 5.33% We will start with an example. Christine Boutin: 1,19% Noël Mamère: 5.25% Daniel Gluckstein: 0,47% Olivier Besancenot: 4.25%  Independence of Irrelevant Alternatives Independence of Irrelevant Alternatives We will assume that all individuals who voted for Chevènement had the following preferences: Lionel Jospin: 21.51% Alain Madelin: 3,91% Chevènement ≻ Jospin ≻ everybody else Jacques Chirac: 19.88% Robert Hue: 3.37% Jean-Marie Le Pen: 16.86% Bruno Mégret: 2.34% François Bayrou: 6.84% Christiane Taubira: 2,32% Now suppose Chèvenement was not available to run for the election and Arlette Laguiller: 5.72% Corinne Lepage: 1,88% replace him by another candidate that everybody ranks last. Noël Mamère: 5.25% Christine Boutin: 1,19% What we are really doing is changing the preference orderings without Olivier Besancenot: 4.25% Daniel Gluckstein: 0,47% affecting the ordering of other candidates. Jean Saint-Josse: 4,23% new candidate: 0% IIA requires that it should not affect the social ordering of any two other candidates, say Chirac and Jospin, yet.... Independence of Irrelevant Alternatives Practice M. 1 M. 2 M. 3 M. 1 M. 2 M. 3 Definition (Independence of Irrelevant Alternatives) A SWF satisfies IIA if the social ordering of any two alternatives a versus b depends only on the individual orderings of a versus b. Formally, if we are given two preference profiles which are identical in terms of how individuals order a and b, then the SWF should output the same ordering of a versus b ↓ SWF for the two profiles. ↓ SWF ≻ ≻  Practice Practice M. 1 M. 2 M. 3 M. 1 M. 2 M. 3 M. 1 M. 2 M. 3 M. 1 M. 2 M. 3 ↓ SWF ↓ SWF ↓ SWF ↓ SWF ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ IIA U Practice Practice M. 1 M. 2 M. 3 M. 1 M. 2 M. 3 M. 1 M. 2 M. 3 M. 1 M. 2 M. 3 ↓ SWF ↓ SWF ↓ SWF ↓ SWF ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ IIA U  Practice Practice M. 1 M. 2 M. 3 M. 1 M. 2 M. 3 M. 1 M. 2 M. 3 M. 1 M. 2 M. 3 ↓ SWF ↓ SWF ↓ SWF ↓ SWF ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ IIA U A good SWF? We have argued that a good SWF should satisfy UD, U and IIA. But these are minimal requirements. 4.4. Dictatorship Note that they say very little about equity, fairness etc. So a good SWF should satisfy at least these criteria, but probably more. To illustrate, here is a SWF that satisfies UD, U and IIA but is clearly not a good SWF.  Dictatorship Definition (Dictatorship) A SWF is a dictatorship if there is some individual i such that the social ranking is always exactly ≻i regardless of the preferences of other individuals. 4.5. Impossibility Theorem It satisfies UD because it always produces a well-defined ordering ≻∗ =≻i. It satisfies U because if everyone prefers a to b, then so does i. It satisfies IIA, because the social ordering of any pair a, b only depends on i’s ordering of a, b. Our last criterion will be that a SWF should not be a dictatorship: ND. Example: Borda Rule Example: Borda Rule 1 2 3 Candidate Score Let ri (a) be the rank of a in ≻i. a’s score is s(a) = ∑Ni=1 ri (a). Rank alternatives so that a ⪰B (L) b ⇔ s(a) ≤ s(b) Jean-Charles de Borda Jean-Charles de Borda (1733-1799) (1733-1799)  Example: Borda Rule Example: Borda Rule 1 2 3 Candidate Score 1 2 3 Candidate Score 1+3+1=5 1+3+1=5 3+2+3=8 3+2+3=8 2+1+2=5 2+1+2=5 ⇓ ⇓ Jean-Charles de Borda Jean-Charles de Borda (1733-1799) ∼ ≻ (1733-1799) ∼ ≻ UD U ND IIA (Exercise: prove this) Arrow’s Impossibility Theorem Theorem (Arrow, 1963) Suppose A contains at least three alternatives. There is no Social Welfare Function satisfying Universal Domain, Independence of Irrelevant Alternatives, Unanimity and No-Dictator. 5. Impossibility for Choices A negative result: maybe we can get a positive result if we limit ourselves to Social Choice Functions... If A has only 2 alternatives, then majority voting is a well-defined SWF and satisfies all properties.  Social Choice Function Minimal Requirements for SCF If we just wanted to make a choice rather than obtain a full ranking, the task A Social Choice Function SCF might be easier. is Pareto Efficient if whenever a is unanimously most preferred for a We want to generate a social choice a∗ that: preference profile (≻1 , · · · , ≻N ) then SCF chooses a, that is SCF (≻1 , · · · , ≻N ) = a. is our principle for deciding which outcome is “good” for society. is Monotonic if whenever SCF (≻1 , · · · , ≻N ) = a and, for every i, and naturally, should depend on the preferences of individuals. every alternative b, the profile ≻′i ranks a above b whenever ≻i does (i.e. a does not go down in any ranking), then SCF (≻′1 , · · · , ≻′N ) = a. This dependence is described abstractly by a Social Choice Function: is Dictatorial if there is some individual i such that SCF (≻1 , · · · , ≻n ) is always i’s preferred alternative. INPUT OUTPUT has Universal Domain if SCF generates a choice for every preference profile. Preference Profile Social Choice
SCF
≻1 ,... , ≻ N a∗ = SCF ≻1 ,... , ≻N in A EXAMPLE OF SCF: Pick the best alternative from any social ranking output by a SWF. Muller-Satterthwaite Theorem Comments There are some ways around these problems. Theorem (Muller and Satterthwaite, 1977) Suppose A contains at least 3 alternatives. If a Social Choice Function has In particular, restricting the domain of possible preferences is a way out Universal Domain, is Monotonic and Pareto Efficient then it is Dictatorial. of these impossibility results. For example: with single-peaked preferences on an ordered set, If A has only 2 alternatives, then majority voting is a well-defined SCF and majority voting satisfies IIA and Pareto efficiency satisfies all properties. However, these restrictions on possible preferences have to be justified.  Implementability Even if we can find a good SCF, there is another issue that we ignored: in order to get the correct output from the SCF, we need to input the true 6. Manipulability and the preferences of individuals. Gibbard-Satterthwaite theorem But it is generally the case that these preferences are only known to them (private information). Is it in their best interest to communicate their true preferences? Or would they sometimes be better off pretending that they have other preferences? Strategic Manipulations Strategy Proofness and another definition 1 2 3 4 1 2 3 4 Definition (Strategy Proofness) A SCF is strategy proof if every individual prefers to communicate her true preferences regardless of the preferences of other individuals. Formally, SCF is strategy proof if, for every i, every ≻−i (the preferences of other agents), every ≻i (i’s true preferences) and every ≻′i (a potential lie), we have: ⇓ Plurality ⇓ Plurality SCF (≻i , ≻−i ) = SCF (≻′i , ≻−i ) or SCF (≻i , ≻−i ) ≻i SCF (≻′i , ≻−i ) 1 1 2 + 2 M. 2 prefers to lie  Strategy Proof? Onto Consider applying to an undergraduate degree for example. In the UK, university applications are centralized on a platform which asks students to report their 5 preferred institutions. Strategy-proofness here would mean that students never benefit by Definition (Onto) reporting something other than their 5 preferred choices... We say that a SCF is onto if for every alternative a, there is a preference profile such that society chooses a. Regardless of what other students are voting. This is not the case. It is typically in student’s interest to factor in their This is a very weak requirement. For example, any social choice estimated chances of being accepted, and be strategic. function that satisfies unanimity and UD must be onto because any alternative a is chosen when everybody ranks it first. Other Examples: Elections? Referendum? Gibbard-Satterthwaite Theorem Theorem (Gibbard-Satterthwaite) Suppose A contains at least 3 alternatives. If a Social Choice Function f has Universal Domain, is strategy-proof and onto, then it is Dictatorial. 7. Sen’s Critique Why? Essentially, these properties imply that the SCF must satisfy Pareto efficiency and Monotonicity, so we can conclude using the previous theorem.  A problem with unanimity? Sen’s Critique Amartya Sen argued that even this seemingly Before closing this chapter, let’s discuss a last issue with SWFs. uncontroversial principle might conflict with another Out of the conditions we’ve defined for a SWF to be desirable, probably the principle he argued we ought to hold. least contentious is unanimity. Sen’s principle is a certain notion of liberalism: one’s Afterall, if society unanimously prefers alternative a to alternative b, it preferences should be decisive in matters that concern seems obvious that we should want our SWF to rank a above b. only themselves. Right? For instance, whether I sleep on my back or on my Amartya Sen belly should be entirely up to me. Sen’s Principle of Liberalism Sen’s Impossibility of a Paretian Liberal Take A to be a set of alternatives. Here we should think of each alternative as a richly specified state of society, rather than coarse-electoral options. By richly-specified we mean: each alternative describes many dimensions of the state of society. e.g. what is our monetary policy, whether we are a monarchy or a republic, Theorem (Sen, 1970) whether I sleep on my back or my belly, etc. There is no SWF that can simultaneously satisfy Universal Domain, Definition (Liberalism) Unanimity and Liberalism. For each individual i, there is at least one pair of alternatives, say a and b, such that if a ≻i b then a ≻∗ b, and if b ≻i a then b ≻∗ a. Intuition: if a and b differ only in the dimension defining how I sleep, then my preference over these alternatives should define the social ranking between them.  Sen’s illustrative example: Sen’s illustrative example: Imagine a society consisting of two individuals, Prude and Lewd. The two are faced with the decision of who (if any) should read the Sen’s liberalism principle defines that society’s ranking between a (Prude controversial book “Lady Chatterley’s Lover ” by D. H. Lawrence. reads the book) and c (no one reads the book) should depend only on ≻P. The alternatives are: Which yields c ≻∗ a a: Prude reads the book. And that society’s ranking between b (Lewd reads the book) and c (no one b: Lewd reads the book. reads the book) should depend only on ≻L. c: No one reads the book. Which yields b ≻∗ c And the preferences are: Finally, unanimity requires that a ≻∗ b Prude: c ≻P a ≻P b But then we have a cycle! b ≻∗ c ≻∗ a ≻∗ b Lewd: a ≻L b ≻L c Comments Again, we will find our way around these issues. If Prude is indifferent between b (Lewd reads the book) and c (no one 8. Conclusion reads the book) and Lewd is indifferent between a (Prude reads the book) and c (no one reads the book), then there is no issue. For the rest of the course we will restrict attention to preferences in which people “mind their own business”. This will be a reasonable restriction in the situations we’ll be studying.  Social Choice Theory Preference Aggregation Main lesson from this class: there is no ideal way of aggregating Social Choice Theory is the study of collective decision processes and preferences. procedures. This is not to say that social choice theory is useless. Pioneered in the 18th century by Nicolas de Condorcet and Different aggregation rules have different properties that can be studied. Jean-Charles de Borda, and in the 19th century by Charles Dodgson New properties of different methods can be found. In contexts where it (also known as Lewis Carroll), it took off in the 20th century with the is natural to restrict preferences, satisfying methods can be found. Etc. works of Kenneth Arrow, Amartya Sen and Duncan Black. Depending on the particular social choice problem at hand, one method Its influence extends across economics, political science, philosophy, may be more or less acceptable than another. For example, some mathematics, and recently computer science and biology. school allocation algorithms are more prone to strategic manipulation than others. In the end it may be an empirical issue. See for example Apart from contributing to our understanding of collective decision the fast-growing empirical literature on school choice algorithms. procedures, social choice theory has applications in the areas of The remainder of this course will examine various such applications of institutional design, welfare economics, and social epistemology. these basic principles in economic theory. Proof (Reny, 2000) L1 L2 · · · Ln−1 Ln Ln + 1 · · · LN A Proof of Muller-Satterthwaite a a ··· a a a ··· a · · · · · · Theorem · · · · · · Social Choice · · · · · · a · · · · · · (Bonus Material) · · · · · · · · · · · · b b ··· b b b ··· b  Proof (Reny, 2000) Proof (Reny, 2000) L1 L2 · · · Ln−1 Ln Ln + 1 · · · LN L1 L2 · · · Ln−1 Ln Ln + 1 · · · LN a a ··· a a a ··· a a a ··· a a a ··· a · · · · · · b · · · · · · · · · · · Social Choice · · · · · · Social Choice · · · · · · a · · · · · · a · · · · · · · · · · · · · · · · · · · · · · · · b · · · · · · · · · · · · b ··· b b b ··· b · b ··· b b b ··· b Proof (Reny, 2000) Proof (Reny, 2000) L1 L2 · · · Ln−1 Ln Ln + 1 · · · LN L1 L2 · · · Ln−1 Ln Ln + 1 · · · LN b a ··· a a a ··· a b a ··· a a a ··· a a · · · · · a · · · · · · · · · · · Social Choice · · · · · · Social Choice · · · · · · a · · · · · · a · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · b ··· b b b ··· b · b ··· b b b ··· b  Proof (Reny, 2000) Proof (Reny, 2000) L1 L2 · · · Ln−1 Ln Ln + 1 · · · LN L1 L2 · · · Ln−1 Ln Ln + 1 · · · LN b b ··· a a a ··· a b b ··· a a a ··· a a a · · · · a a · · · · · · · · · · Social Choice · · · · · · Social Choice · · · · · · a · · · · · · a · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ··· b b b ··· b · · ··· b b b ··· b Proof (Reny, 2000) Proof (Reny, 2000) L1 L2 · · · Ln−1 Ln Ln + 1 · · · LN L1 L2 · · · Ln−1 Ln Ln + 1 · · · LN b b ··· a a a ··· a b b ··· b b a ··· a a a · · · · a a a a · · Social Choice · · · · · · Social Choice · · · · · · b · · · · · · a · · · · · · · · · · · · · · · · · · n is pivotal. · · · · · · · · · · · · · · · · · · · · · · · · · · ··· b b b ··· b · · ··· · · b ··· b  Proof (Reny, 2000) Proof (Reny, 2000) L1 L2 · · · Ln−1 Ln Ln + 1 · · · LN L1 L2 · · · Ln−1 Ln Ln + 1 · · · LN b b ··· b b · ··· · b b ··· b b · ··· · Social Choice · · · a · · · · · a · · b · · · · · · · · · · · · Social Choice · · · · · · · · · · · · b · · · · · · Monotonicity ⇒ No · · · · · · · · · · · · change · · · · · · · · · · a a · · · · a a a a ··· a · b ··· b a a ··· a · b ··· b Proof (Reny, 2000) Proof (Reny, 2000) L1 L2 · · · Ln−1 Ln Ln + 1 · · · LN L1 L2 · · · Ln−1 Ln Ln + 1 · · · LN b b ··· b a · ··· · b b ··· b a · ··· · Social Choice Social Choice · · · b · · · · · b · · a a · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · n is pivotal for this · · · · · · Introduce a new · · · · · · configuration too. · · · · · · alternative c · · · · a a · · · · a a a a ··· a · b ··· b a a ··· a · b ··· b  Proof (Reny, 2000) Proof (Reny, 2000) Social Choice a L1 L2 · · · Ln−1 Ln Ln + 1 · · · LN L1 L2 · · · Ln−1 Ln Ln + 1 · · · LN The choice is now a or b · · ··· · a · ··· · · · ··· · a · ··· · by monotonicity. But c is · · · c · · · · · c · · Social Choice unanimously preferred to b · · · b · · · · · b · · a and if b was the social · · · · · · · · · · · · choice it would remain so · · · · · · · · · · · · in the profile where c is c c c · c c c c c · c c moved at the top of every b b b · a a b b b · b b ranking, a contradiction to a a ··· a · b ··· b a a ··· a · a ··· a Pareto Efficiency. Hence the choice can only be a. Proof (Reny, 2000) Proof (Reny, 2000) Social Choice L1 L2 · · · Ln−1 Ln Ln + 1 · · · LN L1 L2 · · · Ln−1 Ln Ln + 1 · · · LN Social Choice a · · ··· · a · ··· · · · ··· · a · ··· · a · · · c · · · · · c · · · · · b · · · · · b · · Any change on this profile · · · · · · Any change on this profile · · · · · · that keeps a at the top of · · · · · · that keeps a at the top of · · · · · · Ln cannot change the c c c · c c Ln cannot change the c c c · c c social choice. b b b · b b social choice. b b b · b b a a ··· a · a ··· a a a ··· a · a ··· a Hence n is a dictator for a.  Proof (Reny, 2000) Since a was chosen arbitrarily, the argument shows that there is a dictator for every alternative. But then, by definition, it has to be the same dictator. Hence it is a Dictatorship.