Aggregation of Preferences in Crisp and Fuzzy Settings: Functional Equations Leading to Possibility Results

We analyze various models introduced in social choice to aggregate individual preferences. We show that on the basis of most of these models there is a system of functional equations such that, in many cases, the origin of impossibility results in a social choice model is the non-existence of a solution for the corresponding system. Among the functional equations considered, we pay a particular attention to general means and associativity, proving that the existence of an associative bivariate mean is equivalent to the existence of a semilatticial partial order. This key result allows us to explain how the knowledge of associative bivariate means can be used to solve social choice paradoxes. In our analysis we deal both with crisp and fuzzy settings.

[1]  M. J. Frank,et al.  Associative Functions: Triangular Norms And Copulas , 2006 .

[2]  E. Czogala,et al.  Associative monotonic operations in fuzzy set theory , 1984 .

[3]  Graciela Chichilnisky,et al.  On Fixed Point Theorems and Social Choice Paradoxes , 1979 .

[4]  J. Aczel,et al.  A Short Course on Functional Equations: Based Upon Recent Applications to the Social and Behavioral Sciences , 1986 .

[5]  Juan Carlos Candeal,et al.  Homothetic and weakly homothetic preferences , 1995 .

[6]  H. Nurmi.  Approaches to collective decision making with fuzzy preference relations , 1981 .

[7]  Jean-Luc Marichal,et al.  On the associativity functional equation , 2000, Fuzzy Sets Syst..

[8]  Z. Maguar,et al.  Functional equations in several variables , 1993 .

[9]  F. J. Juan.  A note on Fung-Fu's theorem , 1985 .

[10]  Bernard Monjardet.  On the use of ultrafilters in social choice theory , 1983 .

[11]  Peter C. Fishburn,et al.  Should social choice be based on binary comparisons , 1971 .

[12]  K. S. Fu,et al.  AN AXIOMATIC APPROACH TO RATIONAL DECISION MAKING IN A FUZZY ENVIRONMENT , 1975 .

[13]  Walter Trockel,et al.  An alternative proof for the linear utility representation theorem , 1992 .

[14]  E. Induráin,et al.  Existence of Additive and Continuous Utility Functions on Ordered Semigroups , 1999 .

[15]  Juan Carlos Candeal-Haro,et al.  UTILITY FUNCTIONS ON PARTIALLY ORDERED TOPOLOGICAL GROUPS , 1992 .

[16]  José Luis García-Lapresta,et al.  Individual-valued preferences and their aggregation: consistency analysis in a real case , 2005, Fuzzy Sets Syst..

[17]  Dieter Sondermann,et al.  Arrow's theorem, many agents, and invisible dictators☆ , 1972 .

[18]  Jean-Luc Marichal,et al.  On nonstrict means , 1997 .

[19]  Madan M. Gupta,et al.  Fuzzy mathematical models in engineering and management science , 1988 .

[20]  Luc Lauwers,et al.  Topological aggregation, the case of an infinite population , 1997 .

[21]  Philip Bacon.  Unicoherence in means , 1970 .

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

[23]  Graciela Chichilnisky.  Topological Aggregation of Preferences: The Case of a Continuum of Agents , 1994 .

[24]  Luc Lauwers,et al.  Ultraproducts and aggregation , 1995 .

[25]  János Aczél,et al.  The Associativity Equation Re‐Revisited , 2004 .

[26]  K. Arrow.  Social Choice and Individual Values , 1951 .

[27]  K. Arrow,et al.  Handbook of Social Choice and Welfare , 2011 .

[28]  Michel Grabisch,et al.  Fuzzy aggregation of numerical preferences , 1999 .

[29]  Graciela Chichilnisky,et al.  Necessary and Sufficient Conditions for a Resolution of the Social Choice Paradox , 1981 .

[30]  Peter C. Fishburn,et al.  LEXICOGRAPHIC ORDERS, UTILITIES AND DECISION RULES: A SURVEY , 1974 .

[31]  G. D. Birkhoff.  Landau on Number Theory , 1929 .

[32]  L. Gevers.  On Interpersonal Comparability and Social Welfare Orderings , 1979 .

[33]  G. Chichilnisky,et al.  Topological aggregation of preferences: the case of a continuum of agents , 1997 .

[34]  J. Kacprzyk,et al.  Group decision making and consensus under fuzzy preferences and fuzzy majority , 1992 .

[35]  A. Gibbard.  Manipulation of Voting Schemes: A General Result , 1973 .

[36]  Graciela Chichilnisky,et al.  Social Choice and the Topology of Spaces of Preferences , 1980 .

[37]  H. Bustince,et al.  Migrativity of aggregation functions , 2009, Fuzzy Sets Syst..

[38]  Irving Kaplansky,et al.  Infinite Abelian groups , 1954 .

[39]  A. Sen,et al.  The Impossibility of a Paretian Liberal , 1970, Journal of Political Economy.

[40]  Zsolt Páles,et al.  The associativity equation revisited , 1989 .

[41]  Graciela Chichilnisky.  Action of Symmetry Groups , 1996 .

[42]  M. Satterthwaite.  Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions , 1975 .

[43]  Georg Aumann,et al.  Über Räume mit Mittelbildungen , 1944 .

[44]  Juan Carlos Candeal,et al.  Aggregation of preferences from algebraic models on groups , 1995 .

[45]  Peter C. Fishburn,et al.  Arrow's impossibility theorem: Concise proof and infinite voters , 1970 .

[46]  Bonifacio Llamazares,et al.  Aggregation of fuzzy preferences: Some rules of the mean , 2000, Soc. Choice Welf..