1.1.1 Cardinalism and subjectivity
In economics, thanks to the concept of utility, we can compare not only between apples and between apples and pears but also between apples and clothes and apples and poetry or a good sleep. On the other hand, it is a controversial issue as to whether we can compare between such several comparisons. To do so, we would need a precise scale and measure of subjective utility that would in fact be objective enough to apply across individuals. What permits such inter-individual comparisons of utility isâthe general saying goesâthat the utility function is not subjective after all, that it does not capture real subjective aspects of the individual utility but nevertheless yields an objective indication of what this subjective utility is. Freed of essentially individual aspects in its construction, the standard (i.e., not significant in subjective terms) utility functions that represent individual preferences can be exported to social choice-theory and concerns.
Many issues to be disentangled will constitute the successive topics of this chapter and determine its progressive inclination not only toward cardinalism but toward a significantly subjective admission of this doctrine. The first issue is to remember a clear definition of what counts as a cardinal utility function and what its relationship with respect to the subjective experience of utility is. Because the two notions are not equivalent (Section 1.1.1). The second issue is whether we can adopt a subjective conception of a cardinal utility function and yet pursue the significant goals that an apparently more parsimonious ordinal function achieves in decision-theory and economics (the rest of Section 1.1). Finally, we can wonder how sound the theoretical foundations of decision-theory can be when we adopt such a cardinalist and subjectivist point of view (Sections 1.2 and 1.3 will dwell on these issues). In particular, can we obtain a representation theoremâthat is, the main conceptual link between preference orderings and utilityâwhen we accept, in our informational basis a cardinal (and subjective) notion of preference intensities (Section 1.3)? Or should we and can we dispense in decision-theory with representation theorems as foundational moves of the discipline (Section 1.3.3 in particular)? Finally, in our âTo-the-labâ experimental Section 1.4, we suggest protocols that tackle psychological issues motivating the resort to a cardinal subjective utility function and the admissibility of introspective data in relation to a choice-based decision-theoretical paradigm.
The first ambiguity to dispel is that the standard von Neumann-Morgenstern (vNM) utility function is cardinal, not ordinal. To recall, the expected utility hypothesis that these authors have developed means that an individual having some preferences and exerting some choices on a space of risky objects (a set of lotteries) and satisfying a series of sufficiently intuitive axioms (see
Fig. 1.1) can be described in terms of the maximization of the expected value of a utility function that is cardinal. It is cardinal in the sense that it is unique up to a positive affine transformation. It means that the properties we assigned to the preference relation represented in this way can be preserved through changes of interval scales (the âup to positive affine transformationsâ of an initial function). By contrast, an ordinal utility function is unique up to an increasing monotone transformation. A cardinal utility function is then by default ordinal, but the
reverse is not true. A cardinal utility function is more demanding in terms of its informational basis, since it supposes that there are possible valuations of options on a cardinal space and not simply an ordering of these options. It more exactly supposes that the ordering of the options preserves the cardinal valuation of options and that we can continue to perform comparisons of preferences themselves in the form of âI prefer
x to
y more than I prefer
w to
zâ [(
xy)>(
wz)], implying that the intensity of preferences is preserved under the cardinal utility function (which it is not under a strict ordinal function). Cardinalism, in this sense, means that we do not simply order options by a preference relation (which, with no more informational basis, yields an ordinal utility function) but that we can rank the instances of that preference relation themselves.
The main difficulty is not to distinguish cardinality and ordinality but to agree on what we call a âcardinal utility function.â The problem lies in the extent to which we want the cardinal utility function that represents the preferences of an individual to reflect her subjective experience. A classical statement in economics (see Baumol, 1951, 1958) holds that the vNM utility function has nothing to do with the neoclassical utilitarian idea that utility is the measure of the subjective satisfaction (let alone welfare or happiness) of the individual when she expresses her preferences through her choices. Suppose that an hedonimeter could measure degrees of satisfaction and that we obtain an objective measure of that subjective experience, it still would have nothing to do with what vNM utility measures or indicates. If this position is to be accepted, it would amount to saying that vNM utility is an indirect index of subjective utility, not a measure of it and that it therefore conveys richer information than ordinal utility functions. Many applied economists understand and use vNM utility in that way.
Yet this position leaves an impression of ambiguity and dissatisfaction, and we may want a vNM utility function to have more subjective significance and correlate more directly with the experience of the decision-maker. Ng (1984) identifies a psychological postulate that prevents a more subjective interpretation. It is worth analyzing Ngâs contribution in some details, as it forms a paradigmatic example of how axioms and the provability of a utility representation encapsulate implicit âpsychologies,â that is, underlying views of the mind. Ng points out that the assumption of infinite sensitivityâthe perfect ability to discriminate between the finest levels of subjective experienceâis implicitly contained in vNM axioms. If we add to that unmodified list of axioms a psychologically more realistic postulate of imperfect experiential discrimination, we can grant the vNM utility function the subjective significance that classical utilitarianism associates with the very concept of utility.
Classical utilitarianism suggests that the human perceptual apparatus is endowed with limited discriminative capacities but that it is possible to calibrate minimal perception thresholds and construct units of utility on this minimum. This is, of course, a postulate subject to objection and experimentation, but we postpone this discussion for now. Minimal sensitivity to differences i...