Collective choice functions on non-convex problems

Collective choice problems on sets in <InlineEquation ID="Equ1"> <EquationSource Format="TEX"> ${\frak R}_+^n$</EquationSource> </InlineEquation> arise naturally in economics. Such problems have been extensively studied both in the theory of revealed preferences (Peters and Wakker, 1991) and in axiomatic bargaining theory under the assumption of convexity. However, our knowledge of collective choice functions on non-convex problems is still patchy. In this paper I study the existence and characterisation of continuous choice functions on the domain <InlineEquation ID="Equ2"> <EquationSource Format="TEX"> $\Gamma$</EquationSource> </InlineEquation> of comprehensive problems. The main result completely characterises rational choice functions that are continuous and satisfy Weak Pareto Optimality: they form the class of Monotone Path Choice Functions on <InlineEquation ID="Equ3"> <EquationSource Format="TEX"> $\Gamma$</EquationSource> </InlineEquation>. I also show that any discontinuous rational and weakly Pareto optimal choice function must be non- anonymous. Copyright Springer-Verlag Berlin Heidelberg 2000