It can make sense to say that a utility function is bounded, but that implies certain other restrictions. For example, bounded utility functions cannot be decomposed into independent (additive) subcomponents if the number of subcomponents is unknown. Any utility function that is summed over an unknown number of independent (e.g.) societies must be unbounded. Does that mean you believe that utility functions can’t be aggregated over independent societies or that no two societies can contribute independently to the utility function? That latter implies that a utility function cannot be determined without knowing about all societies, which would make the concept useless. Do you believe that utility functions can be aggregated at all beyond the individual level?
You mentioned that a utility function should be seen as a proxy to decision making. If decisions can be independent, then their contributions to the definition of a utility function must be independent*. If the utility function is bounded, then the number of independent decisions something can decide between must also be bounded. Maybe that makes sense for individuals since you distinguished a utility function as a summary of “current” decision-making, and any individual is presumably limited in their ability to decide between independent outcomes at any given point in time. Again, though, this causes problems for aggregate utility functions.
Consider the functor F that takes any set of decisions (with inclusion maps between them) to the least-assuming utility function consistent with them. There exists a functor G that takes any utility function to the maximal set of decisions derivable from it. F,G together form a contravariant adjunction between set of decisions and utility functions. F is then left-adjoint to G. Therefore G preserves finite coproducts as finite products. Therefore for any independent sets of decisions A,B and their union A+B, the least-assuming utility function defined over them exists and is F(A+B)=F(A)*F(B).
It seems like nonsense to say that utility functions can’t be aggregated. A model of arbitrary decision making shouldn’t suddenly become impossible just because you’re trying to model, say, three individuals rather than one. The aggregate has preferential decision making just like the individual.
It can make sense to say that a utility function is bounded, but that implies certain other restrictions. For example, bounded utility functions cannot be decomposed into independent (additive) subcomponents if the number of subcomponents is unknown. Any utility function that is summed over an unknown number of independent (e.g.) societies must be unbounded. Does that mean you believe that utility functions can’t be aggregated over independent societies or that no two societies can contribute independently to the utility function? That latter implies that a utility function cannot be determined without knowing about all societies, which would make the concept useless. Do you believe that utility functions can be aggregated at all beyond the individual level?
You mentioned that a utility function should be seen as a proxy to decision making. If decisions can be independent, then their contributions to the definition of a utility function must be independent*. If the utility function is bounded, then the number of independent decisions something can decide between must also be bounded. Maybe that makes sense for individuals since you distinguished a utility function as a summary of “current” decision-making, and any individual is presumably limited in their ability to decide between independent outcomes at any given point in time. Again, though, this causes problems for aggregate utility functions.
Consider the functor F that takes any set of decisions (with inclusion maps between them) to the least-assuming utility function consistent with them. There exists a functor G that takes any utility function to the maximal set of decisions derivable from it. F,G together form a contravariant adjunction between set of decisions and utility functions. F is then left-adjoint to G. Therefore G preserves finite coproducts as finite products. Therefore for any independent sets of decisions A,B and their union A+B, the least-assuming utility function defined over them exists and is F(A+B)=F(A)*F(B).
It seems like nonsense to say that utility functions can’t be aggregated. A model of arbitrary decision making shouldn’t suddenly become impossible just because you’re trying to model, say, three individuals rather than one. The aggregate has preferential decision making just like the individual.