Hicks explained that, in this way, we can totally separate the consumer's indifference map from any quantitative concept of utility, thereby eliminating any backdoor to

cardinal utility.

Much work in economics distinguishes between ordinal and cardinal utility.

However, some plausible uses of SWB data do require cardinal utility.

For example, Harsanyi applies cardinal utility to welfare economics (1955, p.

While very few scholars would consider that the champions of cardinal utility are subjectivist, things are different when it comes to the advocates of ordinal utility.

Indeed, cardinal scales can be compared in an ordinal ranking after Arrow (1974): "Obviously, a cardinal utility implies an ordinal preference but not vice versa".

Robustness of cardinal numbers is based first on the saying of Arrow (1974: 256): "Obviously, a cardinal utility implies an ordinal preference but not vice versa" and second on the fact that the four essential operations of arithmetic: adding, subtracting, multiplication and division are only reserved for cardinal numbers (see Brauers 2007 and also: Brauers and Ginevicius 2009: 137-138).

To derive a Neural Network algorithm we make the assumption, that there is an unknown

cardinal utility U(x) an object x provides to the customer.

In 'Sunto di alcuni capitoli di un nuovo trattato di economia pura' (1900), Pareto applied an ordinal transformation function to a

cardinal utility function, expressed pure economic equilibrium as a system of equations where utility is based on preference ordering, and highlighted the equivalence of the equilibrium result for both cardinal and ordinal specifications of utility.

Early in the twentieth century, behaviorists in economics worried about the meaningfulness of ascribing

cardinal utility functions to agents (that is, functions measuring the strength of preference, not just reflecting preference rankings).

Van Praag and Van der Sar (1988) attempted to deal with the

cardinal utility objection.

Bruni challenges the received view that 'Pareto was confused', and rejects the existing minority views, such as the propositions that Pareto continued his use of

cardinal utility for reasons of pedagogy, political expedience or laziness.