Hm, good point. We can just use the relative utilities. Or, equivalently, we just have to restrict ourselves to the class of normalizations that are only a function of the relative utilities. These may not be "any" normalization scheme, but they're pretty easy to use.

E.g. for the utility function (dollar, apple, hamburger) -> (1010,1005,1000), instead we could write it as (D-A, A-H) -> (5, 5). Then if we wanted to average it with the function (1,2,3), which could also be written (-1, -1), we'd get (2,2). So on average you'd still prefer the dollar.