I think the most natural fix within the VNM theory is to just say S’ and D’ are the events “car is awarded so son/daughter based on a coin toss”, which are slightly better than S and D themselves, and that F is really 0.5S’ + 0.5D’. Unfortunately, such modifications undermine the applicability of the VNM theorem, which implicitly assumes that the source of probabilities itself is insignificant to the outcomes for the agent. Luckily, Bolker4 has divised an axiomatic theory whose theorems will apply without such assumptions, at the expense of some uniqueness results. I’ll have another occasion to post on this later.
I don’t know if author has made further comment on this. I don’t think this undermines the applicability of VNM. If the agent cares whether the car was assigned via a coin toss, then the relevant consequences aren’t just S and D, but richer outcomes like S′ = “son gets car via coin toss” and D′ = “daughter gets car via coin toss.” In that case, the original model just used too coarse a consequence space; VNM can still be applied to lotteries over the refined outcomes. What would challenge VNM is insisting that two lotteries over the same fully specified outcomes can still differ in value purely because of how the probabilities are generated. However, if we assume a deterministic universe, we are allowed to expand the outcome space indefinitely until there is no probability involved, so I’m having a hard time imagining such a scenario.
I don’t know if author has made further comment on this. I don’t think this undermines the applicability of VNM. If the agent cares whether the car was assigned via a coin toss, then the relevant consequences aren’t just S and D, but richer outcomes like S′ = “son gets car via coin toss” and D′ = “daughter gets car via coin toss.” In that case, the original model just used too coarse a consequence space; VNM can still be applied to lotteries over the refined outcomes. What would challenge VNM is insisting that two lotteries over the same fully specified outcomes can still differ in value purely because of how the probabilities are generated. However, if we assume a deterministic universe, we are allowed to expand the outcome space indefinitely until there is no probability involved, so I’m having a hard time imagining such a scenario.