How should we make interpersonal comparisons of well-being levels and differences? One branch of welfare economics eschews such comparisons, which are seen as impossible or unknowable; normative evaluation is based upon criteria such as Pareto or Kaldor-Hicks efficiency that require no interpersonal comparability. A different branch of welfare economics, for example optimal tax theory, uses “social welfare functions” to compare social states and governmental policies. Interpersonally comparable utility numbers provide the input for SWFs. But this scholarly tradition has never adequately explained the basis for these numbers. John Harsanyi, in his work on so-called “extended preferences,” advanced a fruitful idea. While an ordinary preference is a ranking of outcomes, an “extended preference” is a ranking of individual histories. To say that individual k has an extended preference for history over means something like the following: k prefers to have the attributes of individual i in outcome x, as opposed to having the attributes of individual j in outcome y. Harsanyi’s proposal was to endow individuals with “extended preferences”; to represent such preferences using “extended” utility functions; and to employ such functions, in turn, as the basis for interpersonal comparisons of well-being levels and differences. Harsanyi’s analysis, however, had various gaps and flaws. In this Article, I both diagnose these difficulties, and show how they can be remedied - yielding a viable account of interpersonal comparisons, one sufficient for the needs of the SWF approach.