3B2v7 51c GML4 3 1 Prod Type FTP pp 1212 col fig NIL YJMPS 1497 ED GracyVarghese PAGN Hashmath SCAN Shubha ARTICLE IN PRESS 1 3 Journal of Mathematical Psychology 5 7 Aggregation of utility and social choice a topological characterization 9 11 Matt Jones a Jun Zhang a and Gilberto Simpsonb a 13 b Department of Psychology University of Michigan 525 E University Ave Ann Arbor MI 48109 1109 USA Department of Mathematics University of Michigan 525 E University Ave Ann Arbor MI 48109 1109 USA 15 Received 11 October 2002 revised 9 August 2003 17 Abstract 19 29 31 33 F O O 27 PR 25 D 23 Keywords Social choice Topological social choice Utility Cardinal preference Aggregation Indifference Impossibility theorem TE 21 We study the topological properties of aggregation maps combining individuals preferences over n alternatives with preference expressed by a real valued n dimensional utility vector u de ned on an interval scale Since any such utility vector is speci ed only up to arbitrary af ne transformations the space of utility vectors Rn may be partitioned into equivalence classes of the form n 2 fau b1 j aAR with the 0 bARg The quotient space denoted T is shown to be the union of the n 2 dimensional sphere S S singleton f0g which corresponds to indifference or null preference The topology of T is non Hausdorff placing it outside the scope of most existing theory e g J Econom Theory 31 1983 68 We then investigate the existence and nature of continuous aggregation maps under the four scenarios of allowing or disallowing null preference both in individual and in social choice i e maps f P P Q with P QAfT Sg We show that there exist continuous anonymous unanimous aggregation maps iff the outcome space includes the null point Q T and provide a simple well behaved example for the case f S S T Similar examples exist for f T T T but these and all other maps have a property of always either over or under allocating in uence to each voter in a speci c manner We conclude that there exist acceptable aggregation rules if and only if null preference is allowed for the society but not for the individual r 2003 Published by Elsevier Inc 35 45 47 49 51 53 55 EC R R O 43 C 41 The eld of social choice theory is concerned with methods for aggregating the preferences of individuals in a population into a social preference or outcome One common example of such a problem is popular elections in which voters preferences over candidates or political parties given as favorites approved subsets rankings or scores are used to determine the winner or the relative power of the contenders The eld has been largely motivated by impossibility results showing that certain normatively desirable characteristics of the procedure for preference aggregation turn out to be incompatible For instance Arrow 1963 showed that whenever there are at least three alternatives candi N 39 1 Introduction U 37 Corresponding author Department of Psychology The University of Texas at Austin 1 University Station A8000 Austin TX 787120187 USA Fax 313 763 7480 E mail addresses mattj psy utexas edu M Jones junz umich edu J Zhang tico umich edu G Simpson 0022 2496 see front matter r 2003 Published by Elsevier Inc doi 10 1016 j jmp 2003 08 003 dates the only aggregation rules simultaneously satisfying independence of irrelevant alternatives IR and the Pareto principle are simple dictatorships The mathematical characterization of such impossibilities has since been evolved from Arrow s combinatorial approach to a topological one where the space of preferences is endowed with a topological structure see Lauwers 2000 for a thorough review In topological choice theory the additional structural information associated with the set of preferences enriches the mathematical content of the problem and allows for de nition of further desirable properties of aggregation rules For example one often considers the continuity of aggregation maps which corresponds to graceful dependence of the aggregated outcome on the preferences of individuals which are often assumed to be noisy or imperfectly measured This continuity property plays a similar role to that of the IR axiom as a consistency requirement among the outcomes associated with different preference pro les Lauwers 57 59 61 63 65 67 69 71 73 75 YJMPS 1497 ARTICLE IN PRESS M Jones et al Journal of Mathematical Psychology 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 F 23 O 21 relation renders meaningless such axioms as Coulhon and Mongin s 1989 version of IR which states that the societal utility of a particular outcome is dependent only on the individuals utilities for that outcome Proper treatment of cardinal preference aggregation must take the equivalence relation inherent in utilities into account either by requiring the aggregation function to be invariant under positive linear transformations of the inputs D Aspremont Gevers 1977 or by de ning preferences to be equivalence classes rather than individual elements of Rn Kalai Schmeidler 1977 Chichilnisky 1985 Just as it is imperative to properly consider the measurement scale of utility functions it is also crucial to carefully consider the role of null preference i e total indifference among the alternatives In their foundational work Chichilnisky 1980 and Chichilnisky and Heal 1983 explicitly ruled out the null preference from consideration Chichilnisky 1982a treats the case of vanishing ordinal preferences but only as an isolated component of the preference space This topology which we will argue is an inadequate model of preferences in effect reduces the problem to the case where the null point is excluded In her analysis of cardinal preference aggregation Chichilnisky 1985 follows the same approach taken here and considers utilities to be de ned as equivalence classes of realvalued functions under positive linear translations with null preference corresponding to the class of constant functions As in her treatment of ordinal utility Chichilnisky 1982a the topology Chichilnisky 1985 derives for these equivalence classes is disconnected with the null point an isolated component Under this topology the resolution theorem of Chichilnisky and Heal 1983 implies that there do not exist continuous anonymous unanimous aggregation functions for the space of cardinal preferences However as we argue below Chichilnisky s 1985 topology for utilities is arbitrary and poorly motivated We believe that the correspondence between these two separate attempts to
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