Social choice with approximate interpersonal comparison of welfare gains

Suppose it is possible to make approximate interpersonal comparisons of welfare gains and losses. Thus, if w, x, y, and z are personal psychophysical states (each encoding all ethically relevant information about the physical and mental state of a person), then it sometimes possible to say, ``The welfare gain of the state change w --> x is greater than the welfare gain of the state change y --> z.'' We can represent this by the formula ``(w --> x)> (y --> z)'', where `>' is a `difference preorder': an incomplete preorder on the space of all possible personal state changes. A `social state change' is a bundle of personal state changes. A `social difference preorder' (SDP) is an incomplete preorder on the space of social state changes, which satisfies Pareto and Anonymity axioms. The `minimal' SDP is the natural extension of the Suppes-Sen preorder to this setting; we show that it is a subrelation of every other SDP. The `approximate utilitarian' SDP ranks social state changes by comparing the sum total utility gain they induce, with respect to all `utility functions' compatible with `>'. The `net gain' preorder ranks social state changes by comparing the aggregate welfare gain they induce upon various subpopulations. We show that, under certain conditions, all three of these preorders coincide.