Arrovian Aggregation of Convex Preferences

We consider social welfare functions that satisfy Arrow's classic axioms of independence of irrelevant alternatives and Pareto optimality when the outcome space is the convex hull of some finite set of alternatives. Individual and collective preferences are assumed to be continuous and convex, which guarantees the existence of maximal elements and the consistency of choice functions that return these elements, even without insisting on transitivity. We provide characterizations of both the domains of preferences and the social welfare functions that allow for anonymous Arrovian aggregation. The domains admit arbitrary preferences over alternatives, which completely determine an agent's preferences over all mixed outcomes. On these domains, Arrow's impossibility turns into a complete characterization of a unique social welfare function, which can be readily applied in settings involving divisible resources such as probability, time, or money.

[1]  John A. Weymark,et al.  Interpersonal Comparisons of Well-being: A reconsideration of the Harsanyi–Sen debate on utilitarianism , 1991 .

[2]  H. Houthakker.  Revealed Preference and the Utility Function , 1950 .

[3]  K. May.  Intransitivity, Utility, and the Aggregation of Preference Patterns , 1954 .

[4]  E. Kalai,et al.  An Admissible Set Occurring in Various Bargaining Situations , 1977 .

[5]  P. Fishburn.  Dominance in SSB utility theory , 1984 .

[6]  M. Breton,et al.  Arrovian theorems for economic domains. The case where there are simultaneously private and public goods , 1990 .

[7]  P. Fishburn.  Nontransitive measurable utility , 1982 .

[8]  Aanund Hylland.  Aggregation Procedure for Cardinal Preferences: A Comment , 1980 .

[9]  Nicolas de Condorcet.  Essai Sur L'Application de L'Analyse a la Probabilite Des Decisions Rendues a la Pluralite Des Voix , 2009 .

[10]  W. Gaertner.  Kenneth Arrow’s impossibility theorem stretching to other fields , 2019 .

[11]  G. Thompson,et al.  The Theory of Committees and Elections. , 1959 .

[12]  Jeffrey S. Banks.  Acyclic social choice from finite sets , 1995 .

[13]  A. Sen,et al.  Collective Choice and Social Welfare , 2017 .

[14]  Felix Brandt,et al.  On the tradeoff between efficiency and strategyproofness , 2018, Games Econ. Behav..

[15]  M. Breton,et al.  Arrovian theorems for economic domains: Assignments, matchings and pairings , 1990 .

[16]  John A. Weymark,et al.  Justice, political liberalism, and utilitarianism: Themes from Harsanyi and Rawls , 2008 .

[17]  Dongmo Zhang,et al.  The logic of collective choice , 1988, AAMAS.

[18]  R. Sugden,et al.  Regret Theory: An alternative theory of rational choice under uncertainty Review of Economic Studies , 1982 .

[19]  David Gale,et al.  Review: R. Duncan Luce and Howard Raiffa, Games and decisions: Introduction and critical survey , 1958 .

[20]  Avishai Margalit,et al.  How vicious are cycles of intransitive choice? , 1988 .

[21]  Peter C. Fishburn,et al.  The Invention of the Independence Condition for Preferences , 1995 .

[22]  C. Blyth.  Some Probability Paradoxes in Choice from among Random Alternatives , 1972 .

[23]  Alex Gershkov,et al.  Optimal Voting Rules , 2013 .

[24]  J. Weymark,et al.  Linear Aggregation of SSB Utility Functionals , 1999 .

[25]  Rajat Deb,et al.  Social Decision Functions and the Veto , 1977 .

[26]  J. Kelly.  Arrow Impossibility Theorems , 1978 .

[27]  Peter C. Fisi-Iburn.  Nontransitive preferences in decision theory , 1991 .

[28]  P. Mongin,et al.  Social Evaluation under Risk and Uncertainty , 2016 .

[29]  Felix Brandt,et al.  Welfare Maximization Entices Participation , 2015, Games Econ. Behav..

[30]  J. Harsanyi.  Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparisons of Utility , 1955 .

[31]  Amrita Dhillon,et al.  Extended Pareto rules and relative utilitarianism , 1998 .

[32]  Pavlo R. Blavatskyy,et al.  Predictably Intransitive Preferences , 2016, Judgment and Decision Making.

[33]  E. McClennen.  Sure-Thing Doubts , 1983 .

[34]  Ariel Rubinstein,et al.  On the likelihood of cyclic comparisons , 2012, J. Econ. Theory.

[35]  B. Bernheim,et al.  Beyond Revealed Preference: Choice Theoretic Foundations for Behavioral Welfare Economics , 2008 .

[36]  Tilman Borgers,et al.  Revealed Relative Utilitarianism , 2017, SSRN Electronic Journal.

[37]  R. Pollak,et al.  ACYCLIC COLLECTIVE CHOICE RULES , 1982 .

[38]  Ken-ichi Inada.  THE SIMPLE MAJORITY DECISION RULE , 1969 .

[39]  Lin Zhou,et al.  Impossibility of Strategy-Proof Mechanisms in Economies with Pure Public Goods , 1991 .

[40]  Amartya Sen,et al.  Quasi-Transitivity, Rational Choice and Collective Decisions , 1969 .

[41]  M. Pistek.  Continuous SSB representation of preferences , 2018, Journal of Mathematical Economics.

[42]  S. Afriat.  THE CONSTRUCTION OF UTILITY FUNCTIONS FROM EXPENDITURE DATA , 1967 .

[43]  Richard J. Zeckhauser,et al.  Majority Rule with Lotteries on Alternatives , 1969 .

[44]  Ton Storcken,et al.  Arrow's Possibility Theorem for one-dimensional single-peaked preferences , 2008, Games Econ. Behav..

[45]  P. Samuelson.  A Note on the Pure Theory of Consumer's Behaviour , 1938 .

[46]  Efe A. Ok,et al.  A Comprehensive Approach to Revealed Preference Theory , 2016 .

[47]  Paul Anand,et al.  The Philosophy of Intransitive Preference , 1993 .

[48]  T. Bergstrom.  When Non-Transitive Relations Take Maxima and Competitive Equilibrium Can't Be Beat , 1992 .

[49]  Kim C. Border.  Social Welfare Functions for Economic Environments with and without the Pareto Principle , 1983 .

[50]  Thomas Schwartz.  Choice functions, “rationality” conditions, and variations on the weak axiom of revealed preference , 1976 .

[51]  A. Sen,et al.  Social Choice Theory , 1980 .

[52]  Aggregation theory for SSB utility functionals , 1987 .

[53]  John A. Weymark,et al.  Arrovian Social Choice Theory on Economic Domains , 2011 .

[54]  Douglas H. Blair,et al.  Polychromatic Acyclic Tours in Colored Multigraphs , 1983, Math. Oper. Res..

[55]  David Schmeidler,et al.  Aggregation Procedure for Cardinal Preferences: A Formulation and Proof of Samuelson's Impossibility Conjecture , 1977 .

[56]  M. Breton,et al.  The Bipartisan Set of a Tournament Game , 1993 .

[57]  D. Black.  On the Rationale of Group Decision-making , 1948, Journal of Political Economy.

[58]  A. Sen,et al.  Social Choice Theory: A Re-Examination , 1977 .

[59]  L. A. Goodman,et al.  Social Choice and Individual Values , 1951 .

[60]  Pavlo R. Blavatskyy,et al.  Axiomatization of a Preference for Most Probable Winner , 2005 .

[61]  F. Brandl.  Zero-Sum Games in Social Choice and Game Theory , 2018 .

[62]  P. Schoemaker.  The Expected Utility Model: Its Variants, Purposes, Evidence and Limitations , 1982 .

[63]  Mark J. Machina,et al.  Generalized Expected Utility Analysis and the Nature of Observed Violations of the Independence Axiom , 1983 .

[64]  B. Douglas Bernheim,et al.  Behavioral Welfare Economics , 2008 .

[65]  S. Chew.  A Generalization of the Quasilinear Mean with Applications to the Measurement of Income Inequality and Decision Theory Resolving the Allais Paradox , 1983 .

[66]  Michel Le Breton,et al.  Arrovian Theorems with Private Alternatives Domains and Selfish Individuals , 1989 .

[67]  S. Vajda,et al.  GAMES AND DECISIONS; INTRODUCTION AND CRITICAL SURVEY. , 1958 .

[68]  S. Chew.  Axiomatic utility theories with the betweenness property , 1989 .

[69]  P. Fishburn.  Probabilistic Social Choice Based on Simple Voting Comparisons , 1984 .

[70]  C. Plott,et al.  Economic Theory of Choice and the Preference Reversal Phenomenon , 1979 .

[71]  François Maniquet,et al.  Approval voting and Arrow’s impossibility theorem , 2015, Soc. Choice Welf..

[72]  M. Machina.  Dynamic Consistency and Non-expected Utility Models of Choice under Uncertainty , 1989 .

[73]  Efe A. Ok,et al.  Coalitional Expected Multi‐Utility Theory , 2019, Econometrica.

[74]  Felix Brandt,et al.  On Popular Random Assignments , 2013, SAGT.

[75]  박상용.  [경제학] 현시선호이론(Revealed Preference Theory) , 1980 .

[76]  Douglas Muzzio,et al.  APPROVAL VOTING , 1983 .

[77]  Fedor Pakovich.  A counterexample to the , 2002 .

[78]  Robert A. Pollak,et al.  Rationality and Social Choice , 1986 .

[79]  Telikepalli Kavitha,et al.  Popular mixed matchings , 2009, Theor. Comput. Sci..

[80]  P. Fishburn.  Transitive measurable utility , 1983 .

[81]  Edi Karni.  Archimedean and continuity , 2007, Math. Soc. Sci..

[82]  Jean-François Laslier,et al.  Tournament Solutions And Majority Voting , 1997 .

[83]  M. Allais.  Le comportement de l'homme rationnel devant le risque : critique des postulats et axiomes de l'ecole americaine , 1953 .

[84]  Hiroki Nishimura.  The transitive core: inference of welfare from nontransitive preference relations , 2018 .

[85]  D. E. Campbell.  Arrow's theorem for economic environments and effective social preferences , 1989 .

[86]  A. Tversky,et al.  Prospect theory: an analysis of decision under risk — Source link , 2007 .

[87]  Peter C. Fishburn,et al.  Retrospective on the utility theory of von Neumann and Morgenstern , 1989 .

[88]  R. Sugden,et al.  Some implications of a more general form of regret theory , 1987 .

[89]  Walburga Rödding,et al.  On the aggregation of preferences , 1978 .

[90]  C. d'Aspremont,et al.  Social welfare functionals and interpersonal comparability , 2002 .

[91]  Semih Koray,et al.  Self‐Selective Social Choice Functions Verify Arrow and Gibbard‐Satterthwaite Theorems , 2000 .

[92]  Y. Cabannes.  Participatory budgeting: a significant contribution to participatory democracy , 2004 .

[93]  Felix Brandt,et al.  Set-rationalizable choice and self-stability , 2009, J. Econ. Theory.

[94]  Andreu Mas-Colell,et al.  General Possibility Theorems for Group Decisions , 1972 .

[95]  A. Sen,et al.  Choice Functions and Revealed Preference , 1971 .

[96]  Philippe Mongin,et al.  The Utilitarian Relevance of the Aggregation Theorem , 2015 .

[97]  Paul Anand.  Rationality and Intransitive Preference: Foundations for The Modern View , 2008 .

[98]  E. Rowland.  Theory of Games and Economic Behavior , 1946, Nature.

[99]  Peter C. Fishburn,et al.  Nonlinear preference and utility theory , 1988 .

[100]  P. Pattanaik,et al.  Necessary and Sufficient Conditions for Rational Choice under Majority Decision , 1969 .

[101]  Ehud Kalai,et al.  Social welfare functions when preferences are convex, strictly monotonic, and continuous , 1979 .

[102]  Donald E. Campbell,et al.  Impossibility theorems in the arrovian framework , 2002 .

[103]  D. Packard.  Cyclical preference logic , 1982 .

[104]  Philip J. Reny,et al.  A Characterization of Rationalizable Consumer Behavior , 2014 .

[105]  Jean-François Laslier,et al.  Aggregation of preferences with a variable set of alternatives , 2000, Soc. Choice Welf..

[106]  Felix Brandt,et al.  Universal pareto dominance and welfare for plausible utility functions , 2014, EC.

[107]  H. Young.  Condorcet's Theory of Voting , 1988, American Political Science Review.

[108]  E. Maskin,et al.  On the Robustness of Majority Rule , 2008 .

[109]  Felix Brandt,et al.  Consistent Probabilistic Social Choice , 2015, ArXiv.

[110]  Introduction to a special issue in honor of Kenneth Arrow , 2018, Public Choice.

[111]  Philippe Mongin,et al.  Harsanyi's Aggregation Theorem: multi-profile version and unsettled questions , 1994 .

[112]  P. Fishburn.  SSB Utility theory: an economic perspective , 1984 .