Throughout chapters 2–5, we studied the aggregation problem within the framework of arbitrary finite sets of discrete alternatives. These alternatives could have been political parties or candidates representing these parties, these alternatives could also have stood for well-specified economic and (or) social programmes determining, for example, particular distributions of commodities and particular tax schemes that involve major indivisibilities.

In various economic problems, the possible choices can be envisaged to constitute a set of points in some appropriately defined multi-dimensional choice space, and the individual preferences are represented by quasi-concave, differentiable utility functions defined over this space. The points are n-dimensional vectors which specify, for example, the final consumptions of both private and public goods of all members of the society under consideration. In the present chapter, we shall examine the existence of continuous aggregation rules within such a framework. We also discuss the issue of manipulability in continuous space.

The standard exclusion conditions in continuous space

Kramer (1973) has shown that for the issue of domain restriction, the transition from finite sets of discrete alternatives to multi-dimensional choice spaces has serious consequences. It is demonstrated that in Euclidean choice space, the standard restriction conditions such as value restriction, extremal restriction and limited agreement are inconsistent with even a modest degree of heterogeneity of individual preferences.