Handbook of Social Choice and Welfare

CONTENTS	3
Preface	4
List of Contributors	6
Part Five Foundations	8
	13 Kenneth Arrow on Social Choice Theory _Kenneth Arrow, Amartya Sen, and Kotaro Suzumura	8
		Part I: An Editorial Note	8
		Part II: An Interview with Kenneth J. Arrow	9
		References	28
		Part III: The Classification of Social Choice Propositions	29
	14 The Informational Basis of Social Choice_ Amartya Sen	34
		1. Introduction	34
		2. Democratic Priorities and Informational Inclusiveness	37
		3. Informational Exclusions and Social Choice Framework	39
		4. Axiomatic Exclusions and Arrow's Impossibility Theorem	42
			4.1. Arrow's Theorem	42
			4.2. Proof of Arrow's Theorem	43
			4.3. Interpretation of the Proof	44
		5. Enriching Information for the Possibility of Social Choice	45
		6. On Combining Utility and Nonutility Information	46
		References	48
	15 Competitive Market Mechanisms as Social Choice Procedures _ Peter J. Hammond	51
		1. Introduction and Outline	54
			1.1. Markets and Social Choke	54
			1.2. Finite Economies	54
			1.3. Continuum Economies	55
			1.4. Public Goods and Externalities	56
		2. Agent Types	56
		2.1.Commodity Space	56
			2.2. Notation	56
			2.3. Consumption and Production Sets	57
			2.4. Preferences for Consumption	57
			2.5. Regular Smooth Preferences	59
			2.6. Preferences for Net Trades	60
			2.7. A Compact Polish Space of Agents' Types	61
			2.8. Smooth Type Domains	62
		3. Walrasian Equilibrium and Pareto Efficiency	63
			3.1. Finite Set of Agents	63
			3.2. Feasible and Pareto Efficient Allocations	64
			3.3 Wealth Distribution and Walrasian Equilibrium	64
			3.4. Compensated Walrasian Equilibrium	65
			3.5. Properties of Compensated and Walrasian Equilibria	67
			3.6. Walrasian Equilibrium with Equal Budgets	68
			3.7. The Cheaper Point Lemma	70
			3.8 The Unique Cheapest Point Case	71
			3.9 Aggregate Interiority	71
		4. Characterizing Pareto Efficient Allocations	72
			4.1. First Efficiency Theorem	72
			4.2. Second Efficiency Theorem	72
			4.3. Identical Agents	73
			4.4. Nonoligarchic Allocations	74
			4.5. Exact Characterization of Nonoligarchic Allocations	76
			4.6. Oligarchic Allocations and Hierarchical Prices	76
		5. Walrasian Equilibrium in Finite Economies	78
			5.1. Autarky	79
			5.2. Gains from Trade	79
			5.3. Existence of Compensated Walrasian Equilibrium	80
			5.4. Directional Irreducibility and Existence of Walrasian Equilibrium	83
			5.5. Extended Irreducibility	86
		6. Characterizations of WE with a Fixed Profile of Agents' Types	88
			6.1. Walrasian Acceptability	88
			6.2. Equal Rights to Multiple Proportional Trade	89
			6.3. Interactive Opportunity Sets	91
		7. Characterizations of WE with a Variable Profile of Agents' Types	93
			7.1. Minimal Message Spaces	93
			7.2. Gevers' Theorem	95
			7.3. A Maximally Selective Social Choice Rule	97
			7.4. Local Independence in Smooth Economies	98
			7.5. Strategyproofness with Exogenous Prices	99
		8. Characterizations of WE with a Varying Number of Agents	101
			8.1. The Core and Edgeworth Equilibrium	101
			8.2. Another Limit Theorem	103
			8.3. Stability	103
			8.4. Consistency and Converse Consistency	107
			8.5. Consistency and Converse Consistency with Interactive Opportunity Sets	108
			8.6. Minimal Message Spaces	109
		9. Statistical Continuum Economies	109
			9.1. Continuum Economies	109
			9.2. Statistical Economies	110
			9.3. Statistical Continuum Economies	110
			9.4. Allocation Mechanisms	111
			9.5. Pareto Efficiency	112
			9.6. Walrasian and Compensated Equilibria	112
			9.7. Lump-Sum Wealth Redistribution	113
			9.8. Walrasian Equilibrium with Equal Budgets	113
		10. Efficiency Theorems in Continuum Economies	113
			10.1. First Efficiency Theorem	113
			10.2. Second Efficiency Theorem	114
			10.3. Nonoligarchic Allocations	115
			10.4. Individual Nonconvexities	116
		11. Statistical Continuum Economies: Existence Theorems	118
			11.1. Integrably Bounded Gains from Trade Sets	118
			11.2. Continuity of the Budget and Demand Correspondences	119
			11.3. Integrably Bounded Restricted Budget and Demand Correspondences	120
			11.4. Existence of Compensated Equilibrium with Free Disposal	121
			11.5. Monotone Preferences and Existence of Compensated Equilibrium	124
			11.6. Existence of Walrasian Equilibrium	124
			11.7. Indivisible Goods and Constrained Monotone Preferences	125
			11.8. Dispersion and Existence of Walrasian Equilibrium	126
		12. Equivalence Theorems for the Core and f-Core	127
			12.1. The Core and f-Core	127
			12.2. Walrasian Equilibria Belong to the Core	128
			12.3. .f-Core Allocations are Compensated Equilibria	128
			12.4. Coalitional Fairness	130
			12.5. A Bargaining Set	131
		13. Envy-Free Mechanisms	133
			13.1. Full f-Pareto Efficiency	133
			13.2. Self-Selective Allocations	133
			13.3. Counterexample	136
			13.4. Fully Fair Allocations	136
		14. Strategyproof Mechanisms in a Continuum Economy	137
			14.1. Individual St rategyproofness	137
			14.2. Decentralization Theorem	138
			14.3. Limits to Redistribution	140
			14.4. Anonymity	140
			14.5. Weak Gains from Trade with an Inclusive Type Domain	141
			14.6. Strategyproof Mechanisms of Maximal Dimension	142
		15. Manipulation by Finite Coalitions	143
			15.1. Multilateral Strategyproofness	143
			15.2. Arbitrage-Free Allocations	144
		16. Other Environments	146
			16.1. Public Goods	146
		14 Externalities	145
		Acknowledgments	148
		References	148
	16 Functionings and Capabilities _Kaushik Basu and Luis F. Lopez-calva	156
		1. The Capabilities Approach: An Introduction	157
		2. The Ideas in History	159
		3. Sen's Critique and Formulation	162
		4. Further Forma lint ions	167
		5. Capability. Preference, arid Choice	172
		6. Applications of the Capabilities Approach to Poverty Measurement and Gender issues	178
		7. Capability and Exclusion	180
		8. The Human Development index	182
		9. Concluding Remarks	185
		Acknowledgments	185
		References	186
Part Six Developments of the Basic Arrovian Schemes	191
	17 Arrovian Social Choice Theory on Economic Domains  _Michel Le Breton and John A. Weymark	191
		1. Introduction	192
		Part I: Social Welfare Functions	195
		2. Arrow's Theorem	195
		3. Single Peaked Preferences	198
		4. Saturating Preference Domains	203
			4.1. Preliminaries	203
			4.2. The Kalai -Muller satterthwaite Theorem	206
			4.3. Further Properties of Saturating Preference Domains	208
		5. Examples of Saturating Preference Domains	209
		6. Topological Domain Restrictions for Public Goods	224
		7. Supersaturating Preference Domains	233
		8. Hypersaturatlng Preference Domains	239
		9. Topological Domain Restrictions for Private Goods	243
		10. NonCartesian Sets of Alternatives	245
			10.1. Assignments of Indivisible Objects	245
			10.2. Exchange Economies	250
		11. Effective Social Welfare Functions	255
		Part II: Social Choice Correspondences	257
		12. A Choice-Theoretic Version of Arrow's Theorem	257
			12.1. The Arrow Choke Axioms	257
			12.2. Arrow's Theorem	262
		13. Unrestricted Preference Domains	262
			13.1. The Grether-Plott Theorem	263
			13.2. Strategic Candidacy	265
			13.3. Fixed Feasible Alternatives	267
		14. Single-Peaked Preferences and Interval Agenda Domains	271
		15. Analytic Preference Domains	274
			15.1. Euclidean Spatial Preferences	275
			15.2. Monotone Analytic Preferences	276
		16. Classical Domains of Spatial and Economic Preferences	277
			16.1. Exchange Economies	278
			16.2 Production Economies	281
			16.3. General Spatial Preferences	283
		17. Independence of Pareto Irrelevant Alternatives	286
			17.1. Nonconvex Agendas	286
			17.2. Convex Agendas	288
		18. Concluding Remarks	292
		Acknowledgments	296
		References	296
	18 Topological Theories of Social Choice _Nicholas Baigent	300
		1. Introduction	301
		2. Chichilnisky's Theorem: An Elementary Introduction	301
			2.1. Linear Preferences	301
			2.2. Chichilnisky's Impossibility Theorem	300
			2.3. Loops	305
			2.4. Fundamental Equation of Topological Social Choice Theory	309
			2.5. Proof of Theorem 1	312
		3. Domain Restriction	314
			3.1. Domain Restriction	315
			3.2. Contractibility	315
		4. Weak Pareto and No Veto	316
			4.1. The Cone Property and the No Veto Property	316
		5. Strategic Manipulators and Homotopic Dictators	315
			5.1. Dropping the No Veto Property	319
		6. Continuity	322
			6.1. Justification	322
			6.2. Limitations of Continuity	324
		7. Topological Proof of Arrow's Theorem	325
			7.1. Nerves	325
			7.2. Loops in Nerves	328
			7.3. Arrow's Theorem	330
		8. ConcIusion	331
		Acknowledgments	332
		References	332
	19 Nonbinary Social Choice _Rajas Deb	334
		1. Introduction	334
		2. Choice Functions and Domain Restrictions	336
		3. Consistency and Rationalizability	338
		4. Nonbinary Social Choice: A Definition	341
		5. Social Choice: Some Standard Concepts	342
		6. Dictatorship and Oligarchy; The Fixed Agenda Case	344
			6.1. Social Choice with a Fixed Agenda: Hansson's Independence Condition	345
			6.2. Social Choice with a Fixed Agenda: Sen's Approach to the Independence Condition											348
		7. Dictatorship and 'Oligarchy: With Multiple Agendas	354
		8. Conclusion	363
		Acknowledgments	364
		References	364
		20 Social Choice with Fuzzy Preferences _Richard Barrett and Maurice Salles	366
		1. Introduction	366
		1. Fuzzy Preferences	368
			2.1. Fuzzy Preferences: Numerical Values	369
			2.2. Fuzzy Preferences: Qualitative Values	371
		3. Aggregation of Fuzzy Preferences: Arrovian Theorems	373
			3.1. The Case of Numerical Values	373
			3.2. The Case of Qualitative Values	377
		4. Other Aspects	379
			4.1. Aggregation of Fuzzy Preferences and Sen's Impossibility Theorem	380
			4.2. Aggregation of Fuzzy Preferences and Economic Environments	381
		5. Concluding Remarks	385
		Acknowledgments	386
		References	386
Part Seven Fairness and Rights	389
	21. Fair Allocation Rules _William Thomson	389
		1. Introduction	391
		2. A Note on the Approach Followed	393
		3. The Classical Problem of Fair Division	395
		4. Equitable Allocations	397
			4.1. Comparisons to Equal Division	397
			4.2. No-Domination	398
			4.3. No.Envy	398
			4.4. Concepts Related to No-Envy	403
			4.5. Selections and Rankings	405
			4.6. Economies with a Large Number of Agents	407
			4.7. Equity Criteria for Groups	409
			4.8. Egalitarian-Equivalence	411
			4.9. Equitable Trades	413
			4.10. Towards a Complete Theory of Equity	415
		5. Economies with Production	419
			5.1. Adapting the Basic Concepts	419
			5.2. Agent-by-Agent Lower and Upper Bounds	422
		6. Equal Opportunities as Equal. or Equivalent, Choice Sets	424
			6.1. Equal Opportunities	425
			6.2. Equal-Opportunity-Equivalence	426
			6.3. No Envy of Opportunities	427
		7. Monotonicity	428
			7.1. Resource-Monotonicity	429
			7.2. Welfare-Domination under Preference-Replacement	432
			7.3. Technology-Monotonicity	434
			7.4. Population-Monotonicity	437
			7.5. Monotonicity in Economies with Individual Endowments	439
		8. Consistency and Related Properties	440
			8.1. Consistency and converse consistency	437
			8.2. Minimal Consistent Enlargements	445
			8.3. Consistency in Economies with Individual Endowments	446
		9. Public Goods	447
			9.1. Basic Solutions	447
			9.2. Notions of Equal, or Equivalent, Opportunities	451
			9.3. Social Endowment Monotonicity	451
			9.4. Technorogy-MonOtonicily	451
			9.5. Welfare-Domination under Preference-Replacement	452
			9.6. Monotonicity in Economies with individual Endowments	452
			9.7. Population Monotonicity	452
			9.8. Consistency	454
		10. Indivisible Goods	454
			10.1. The Model	454
			10.2. Basic Solutions	456
			10.3. Resource Monoionieity	460
			10.4. Welfare Domination under Preference Replacement	462
			10.5. Population Morvotonicity	462
			10.6. Consistency	464
			10.7. Related Models	466
		11. Single-Peaked Preferences	472
			11.1. The Model	473
			11.2. Basic Solutions	473
			11.3. Resource-Monotonicity	475
			11.4. Welfare-Domination under Preference-Replacement	476
			11.5. Separability	477
			11.6. Population-Monotonicity	477
			11.7. consistency	478
			11.8. Extensions and Related Models	479
		12. Nonhomogeneous Continuum	481
		13. Other Domains and Issues	485
		14. Conclusion	485
		Acknowledgments	488
		References	488
	22 Compensation and Responsibility _Marc Fleurbaey and Fran?ois Maniquet	503
		1. Introduction	504
		2. Fair Monetary Compensation	513
			2.1. The Model	513
			2.2. Fairness as No-Envyr and Related Solutions	515
			2.3. Axioms and Ethical Principles	522
			2.4. The Quasi-Linear Case	532
		3. Unequal Productive Skills	542
			3.1. The Model	543
			3.2. Fairness in Compensation and Reward	545
			3.3. Bundle Equality and Welfare Lower Bounds	557
			3.4. Limited Self-Ownership	562
		4. The Utilitarian Approach to Responsibility	564
			4.1. Framework	564
			4.2. Social Welfare Functions	567
			4.3. Min.of-Means or Mean-of-Mins	569
		5. Related Literature	578
			5.1. Ranking Distributions of Opportunity Sets	578
			5.2. Extended Insurance	580
			5.3. Extended Sympathy and Interpersonal Comparisons of Utility	581
			5.4. Bargaining	583
		References	594
	23 Welfarism, Individual Rights, and Procedural Fairness _Kotaro Suzumura	601
		1. Introduction	602
		2. Informational Bases of Normative Welfare Judgments	603
		3. Social Welfare, Individual Rights, and Liberal Paradox	607
		4. Proposed Resolution Schemes for Paretian Liberal Paradox	614
			4.1. Voluntary Alienation of Conferred Libertarian Rights	616
			4.2. Voluntary Exchange of Conferred Libertarian Rights	621
			4.4. Ethical Constraints on the Exercise of Conferred Libertarian Rights	629
		5. Critiques on Sen's Formulation of Individual Rights	638
		6. Formulation, Realization, and Initial Conferment of Rights	612
		7. Liberal Paradoxes with Game Form Rights	649
			7.1. Power Structure Generated by Game Form Rights	649
			7.2. Gibbard Paradox with Game Form Rights	654
			7.3. Pareto Libertarian Paradox with Game Form Rights	656
		8. Initial Conferment of Game Form Rights	662
		9. Some Further Remarks on Paretianism	666
			9.1. The Paretian Epidemic and Its Implications	666
			9.2. Nonwelfaristic Policy Assessment and the Pareto Principle	670
		10. Concluding Remarks	671
		Acknowledgments	673
		References	673
	24 Freedom, Opportunity, and Well-Being _James E. Foster	682
		1. Introduction	683
		2. Notation	688
		3. Freedom counts	688
			3.1. Rankings and Axioms	689
			3.2. Additive Representations and Cardinality Freedom	694
			3.3. Reflections on Counting as a Measure of Freedom	696
		4. Preference and Indirect Utility	697
			4.1. Rankings and Axioms	697
			4.2. Representations of Indirect Utility Freedom	700
			4.3. Combining Cardinality and Indirect Utility	701
			4.4. Other Viev?s of Quality	704
			4.5. Reflections on Indirect Utility as a Measure of Freedom	706
		5. Plural Preferences and effective Freedom	706
			5.1. Rankings and Axioms	707
			5.2. Representations and Measures	711
			5.3. Sen's Subrelation and Arrow's Extension	713
			5.4. Reflections on Plural Preferences and Effective Freedom	718
		6. Conclusions	719
		Acknowledgments	720
		References	720
Part Eight Voting and Manipulation	724
	25 Strategyproof Social Choice _Salvador Barber	724
		1. The Purpose and Limitations of This Survey	725
		2. Introduction: A Few Historical Notes	726
		3. Strategyproof Social Choice Functions for Unrestricted Domains: 
		The Gibbard-Satterthwalte Theorem	731
			3.1. Statement	731
			3.2. The Impact of the Gibbard-Satterthwaite Theorem	734
			3.3.  Proofs of the Theorem	736
		4. Game Forms and the Question of Implementation	741
			4.1. Strategyproofness in a Larger Picture	741
			4.2. Strengthenings, Weakenings, and Related Definitions	743
			4.3. Other Forms of Strategic Behavior in Voting	745
			4.4. The Analysis of Specific Social Choice Rules	746
		5. The Search for Strategyproofness_ An Outline	747
		6. Common Domains: Strategyproof Rules for the Choice of Multi-attribute Alternatives	751
			6.1. Two Alternatives	752
			6.2. Linearly Ordered Sets of Alternatives	752
			6.3. n-Dimensional Grids	758
			6.4. Constraints: A First Approach	765
			6.5. The Structure of Strategyproof Rules	767
			6.6. Constraints Revisited	772
		7. Common Domains: Probabilistic Voting Schemes	773
		8. Common Domains: Social Choice Correspondences	780
		9. Strategyproofness in Personalized Dornains	787
			9.1. Strategyproof Rationing	787
			9.2. Strategyproof Exchange	791
			9.3. Strategyproof Matching and Assignment	800
			9.4. Strategyproof Cost Sharing	804
		10. Further Comments on Preference Domains	808
	26 Probabilistic and Spatial Models of Voting _Peter Coughlin	825
		1. Black's Analysis of Committees and Related Results about Majority Rule	826
			1.1. Voters, Alternatives, and Preferences	827
			1.2. Simple Majority Rule	827
			1.3. Single.Peakedness and Median Alternatives	828
			1.4. Results for Committees and for the General Model	829
		2. Hotelling, Downs, and Electoral Competition	832
			2.1. Hotelling's Model for Duopolists	833
			2.2. Interpretation as a Model of Electoral Competition	834
		3. A Framework for Models of Electoral Competition	836
			3.1. Basic Assumptions and Notation	836
			3.2. Possible Objectives	840
		4. Deterministic Voting and Simple Majority Rule	844
			4.1. Definitions	844
			4.2. Pure-Strategy Equilibria and Condorcet Winners	845
		5. Unidimensional Models with Deterministic Voting	851
		6. Finite-Dimensional Models with Deterministic Voting	850
			6.1. Plott's Analysis	851
			6.2. Sufficient Conditions	852
			6.3. McKelvey's Theorem	857
			6.4. Summing Up	859
		7. Probabilistic Voting Models	859
			7.1. An Overview	859
			7.2. Reasons for Analyzing Probabilistic Voting Models	859
		8 Unidimensional Models with Probabilistic Voting	861
		9. Finite-Dimensional Models with Probabilistic Voting	863
			9.1. Hinich's Model	863
			9.2. Expectations Rased on a Binary Luce Model	866
	27 Geometry of Voting _Donald G. Saari	889
		1. Introduction	890
		2. From Simple Geometry to Profile Coordinates	891
			2.1. Using Geometry to Represent Profiles	891
			2.2 Different Information, Different Voting Outcomes	893
			2.3. Profile Coordinates	895
			2.4. Resolving Classical Problems and Paradoxes	903
		3. Geometry of the Likelihoods of Voting Problems	905
			3.1. Central limit Theorem	905
			3.2. Geometry of Paired Comparisons	906
			3.3. Geometric Likelihood Estimates	909
			3.4. Explaining and Interpreting Pairwise Voting Outcomes	910
			3.5. Geometry of All Three Candidate Pairwise and Positional Outcomes	912
		4. Other Geometric Voting Results	916
			4.1. Paired Comparisons: Extending McGare s Theorem	917
			4.2. Geometry of Strategic Behavior	919
			4.3. Could My Candidate Have Won with a Different Voting Rule	922
		5. Exporting Lessons Learned From Social Choice	928
			5.1. From Voting to Non parametric Statistics	929
			5.2. "Divide and Conquer'; A Generalized Arrow's Theorem	930
			5.3. Dysfunctional Societies; Lessons from Sen's Theorem	932
			5.4. Applying Chichilnisky's Topological Dictator to Psychology	933
		6. Summary	934
		Acknowiodgmcnts	935
		References	935
		Index	938