Aggregation of utility and social choice: A topological characterization

Abstract

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 defined on an interval scale. Since any such utility vector is specified only up to arbitrary affine transformations, the space of utility vectors R n may be partitioned into equivalence classes of the form {a u +b 1 | a∈ R 0 +, b∈ R} . The quotient space, denoted T, is shown to be the union of the ( n−2)-dimensional sphere denoted S with the singleton {0}, 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) 68–87.). 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, Q∈{ T, S}. We show that there exist continuous, anonymous, unanimous aggregation maps iff the outcome space includes the null point (i.e., 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 influence to each voter (in a specific 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.