Abstract
Economics bases the choice theory on the mental experiment that introduces the choice correspondence, which associates to every set of possible actions the subset of preferred actions. If some conditions are satisfied, then the choice correspondence implies a binary preference ordering on actions and an ordinal utility function. This approach applies both to decisions under certainty and decisions under uncertainty. The preference ordering depends on the consequence of actions. Under certainty, there is only one consequence to every action, while, under uncertainty, many consequences are possible, associated with the states of the world. These consequences are represented by the action itself, the states of the world, and the corresponding outcomes. Current theories consider only outcomes, but some theories include state dependent preference. Preference for the action itself is not considered, but it might be relevant. The rationality of the theory is a different question from the rationality of the decision-maker. Moreover, the rationality of the theory may imply the rationality of a preference ordering, but this does not require the rationality of the decision-maker. It is only assumed that he/she behaves according to the calculation made by the theorist. The rationality of the preference ordering requires the rationality of the preference on outcomes, of the expectations on the events, and of their connection with the preference ordering on actions. The normative relevance of rational preferences is removed by the introduction of many alternative rational theories, which justify contrasting behaviors in identical situations.
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Notes
It should be remembered that utility was introduced into economics by considering a type of behavioral psychological determination. Gossen, Jevons, Menger, Walras, Marshall, Pantaleoni, and many other marginalist neoclassical economists explained the choice through the connection of the kind represented by the dependence between stimulus, sensation, emotion, need, good and utility. They used a theory that not only represents the choice but also explains its nature. This approach has been rejected by Pareto and by current economic theory, which limits itself to assuming the choice correspondence to be a "given" entity, without theorizing its formation. However, there are branches of economics, such as that of decision under uncertainty, where an attempt is made to explain choices, not just to represent them. This happened for von Neumann-Morgenstern's theory of expected utility, where the independence axiom is justified by the rationality of the decision-maker and so on, up to the behavioral approach, where psychological factors are often introduced. For example, in Mas-Colell et al., 1995, p. 42, "rationality", determined by completeness and transitivity, concerns the relation of preference (and not the consumer), and, p. 167, the choice under uncertainty is examined taking into account that “uncertain alternatives have a structure that we can use to restrict the preferences that ‘rational’ individuals may hold”.
The literature on the subject is quite extensive, with several specifications and framing. Good references, though partly dated, can be found in Chipman et al., 1971.
The separability condition was defined by Cantor and is indicated by Gilboa 2009, pp. 51–53.
Debreu 1959, pp. 55–59.
For example, preferences that generate the preference reversal (Tversky and Thaler 1990), are not representable with a utility function, which, however, is used also by the prospect theory of the behavioral approach (Kahneman and Tversky 1979; Tversky and Kahneman 1992). Obviously, the behavioral approach does not need the use of prospect theory and the associated utility function, just as the approach followed in this paper does not require an ordinal utility function to represent preferences, although this is handy when possible. See Starmer 2000 for the illustration of many theories of choices under risk, which includes theories that do not require transitive preferences.
Debreu 1959, p. 98: "This new definition of a commodity allows one to obtain a theory of uncertainty free from any probability concept and formally identical with the theory of certainty".
On insurance decision with state-dependent preferences see, for instance, Kremslehner and Muermann 2009.
The expected utility theory implies a type of uniformity in the indifference curves of the Hirshleifer-Yaari diagram that is not generally required by indifference curves resulting from a continuous utility function \(u(a) = u(x_{1} ,x_{2} )\), with \(x_{i} = x|e_{i}\) for \(i = 1,2\). Expected utility theory requires that the slope of the indifference curves in the certainty points (those on the bisector) be always the same, whatever the amount of money. This slope is equal to the ratio of the two probabilities, which does not depend on the amount of money. However, for a generic utility function, which does not necessarily require the introduction of probabilities, the slope may differ. In this regard, and on the introduction in this case of probability-price dependent on the initial amount of money, see Montesano 2019. Moreover, even the expected utility theory with state dependence (for which \(u(a) = u(x_{1} ,e_{1} )\Pr (e_{1} ) + u(x_{2} ,e_{2} )\Pr (e_{2} )\)) can have uneven slopes at the certainty points.
Pareto describes the consumer's "given" preferences (of which no decision-maker's rationality is required) by means of a field of marginal substitution rates, not by the choice function that was introduced later, and considers the possibility that they were not describable with an (ordinal) function of utility considering the case of "open cycles". In his book, however, he always considers preferences that can be described with an (ordinal) utility function, i.e., assuming, for convenience, that the cycles are "closed". In this regard, Pareto (1909) 2014 (pp. 309–20 and 621–30) and Montesano 2006.
See, for instance, Andersen et al., 2018.
On the advantages provided by axiomatization, Gilboa et al., 2019.
Regarding the economic insurance sector, considering the behavior of the decision-makers who make risky decisions, see, for example, Harrison 2019.
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Montesano, A. On the economic foundations of decision theory. Theory Decis 93, 563–583 (2022). https://doi.org/10.1007/s11238-021-09853-w
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DOI: https://doi.org/10.1007/s11238-021-09853-w