I think there is a version of VNM utility that survives and captures the core of what we wanted: i.e. a way of representing consistent ways of ordering prospects via cardinal values of individual outcomes — it it is just that these value of outcomes can be hyperreals. I really do think the ‘continuity’ axiom (which is really an Archimedean axiom saying that nothing is infinitely valuable compared to something else) is obviously false in these settings, so has to go (or to be replaced by a version that allows infinitesimal probabilities). I know that this axiom was important in deriving the real-valued representation of the utility functions (which I also need to be false), but am not sure what its role is in justifying valuing prospects by their means. I assume there are other ways to get there and don’t think dropping/modifying the axiom will lead to circularity, but it may require some care to check.
One possibility is this: I don’t value prospects by their classical expected utilities, but by the version calculated with the hyperreal sum or integral (which agrees to within an infinitesimal when the answer is finite, but can disagree when it is divergent). So I don’t actually want the classical expected utility to be the measure of a prospect. It is possible that the continuity axiom gets you there and my modified or dropped version can allow caring about the hyperreal version of expectation.
Interesting question.
I think there is a version of VNM utility that survives and captures the core of what we wanted: i.e. a way of representing consistent ways of ordering prospects via cardinal values of individual outcomes — it it is just that these value of outcomes can be hyperreals. I really do think the ‘continuity’ axiom (which is really an Archimedean axiom saying that nothing is infinitely valuable compared to something else) is obviously false in these settings, so has to go (or to be replaced by a version that allows infinitesimal probabilities). I know that this axiom was important in deriving the real-valued representation of the utility functions (which I also need to be false), but am not sure what its role is in justifying valuing prospects by their means. I assume there are other ways to get there and don’t think dropping/modifying the axiom will lead to circularity, but it may require some care to check.
Without continuity (but maybe some weaker assumptions required), I think you get a representation theorem giving lexicographically ordered ordinal sequences of real utilities, i.e. a sequence of expected values, which you compare lexicographically. With an infinitary extension of independence or the sure-thing principle, you get lexicographically ordered ordinal sequences of bounded real utilities, ruling out St Pesterburg-like prospects, and so also ruling out risk neutral expectational utilitarianism.
One possibility is this: I don’t value prospects by their classical expected utilities, but by the version calculated with the hyperreal sum or integral (which agrees to within an infinitesimal when the answer is finite, but can disagree when it is divergent). So I don’t actually want the classical expected utility to be the measure of a prospect. It is possible that the continuity axiom gets you there and my modified or dropped version can allow caring about the hyperreal version of expectation.