Contemporary Developments and Prospect Theory
The success of the von Neumann and Morgenstern model was not overshadowed by research results documenting that expected utility maximization is not a good predictor of real choices.
Maurice Allais (1953) provided the first evidence that people tend to violate axioms of von Neumann and Morgenstern, by placing more weight on certainty than the standard theory predicts. Daniel Ellsberg (1961) showed that people behave in a way that cannot be described by Savage’s subjective probability. In particular people tend to be “ambiguity averse”, that is people prefer situations where probabilities are known to situations where probabilities are unknown, thus violating one of the Savage’s axioms, the so called sure thing principle. (An excellent review of this literature may be found again in Schoemaker 1982).Contemporary developments try to cope with these kinds of problems by adopting different strategies. The first strategy is probably the more conservative, because it maintains the subjective notion of probability. It consists in generalizations of the expected utility model by removing linearity in the probabilities and by positing non-linear functional forms for the preference function. Several such forms have been formally proposed and axiomatized, most are capable of generating well known features, such as risk aversion and violations of independence axioms (Machina 2008). A second strategy consists in replacing probability with alternative notions of belief able to describe the way that people make decisions, or the way they can be convinced to make rational decisions when standard probabilities cannot be defined. The basic intuition may be grasped by the following example. Consider two coins. The experience of repeated flips of the first coin suggest that it is a fair one; therefore, it is reasonable to give probability 0.5 to head and 0.5 to tail.
Suppose that the other coin is completely unknown; you have no reason to prefer one side to the other (symmetric information). If we decide to give probability 0.5 to both head and tail, then this assignment is very different from the preceding one based on frequency experience. This last kind of assignment does not necessarily respect the additive rule for probability. For example: we can legitimately assign, respecting symmetric lack of information, a non-additive probability v(H) = v(T) = 0.4; nonetheless it is true that v(H < T) = 1. Building on this intuition, in the so-called Choquet expected utility the standard probability is replaced by a capacity or a non-additive probability. These models explain most of the observed paradoxes but they also offer simple but flexible representations, and allow for more diversified patterns of behaviour under uncertainty (Schmeidler 1989). A third strategy consists in considering explicitly how beliefs are constructed. In the case-based decision theory (Gilboa and Schmeidler 1995), cases are considered primitive and a system of axiom was construed permitting the choice of the best act based on its past performance in similar cases. Each act is evaluated by the sum of the utility levels that resulted from using this act in past cases, each weighted by the similarity of that past case to the problem at hand.Probably, the most innovative contributions to this stream of literature came from Daniel Kahneman and Amos Tversky. From the early 1960s they developed a systematic study of several violations of the standard assumptions of stability of preferences, and of invariance of choices with respect to the particular kind of description of risky prospects. Their laboratory experiments and those of their followers appear to be “a knockdown refutation of the claim that the von Neumann and Morgenstern theory is usually a good predictor of how ordinary people behave” (Binmore 2009: 58). In their approach, the so-called prospect theory, the von Neumann and Morgenstern utility is replaced by the psychological values of gain and loss.
This psychological value is similar to the Edgeworthian notion of utility as experienced pleasure, objectively measurable by means of a technical device called “hedonimeter” (Baccini 2011). Also “the decision weights that people assign to outcomes are not identical to the probabilities of these outcomes, contrary to the expectation principle.... The expectation principle, by which values are weighted by their probability, is poor psychology” (Kahneman 2011). In this case the line of reasoning recalls Keynes’s notion of the weight of an argument. In particular Kahneman and Tversky (1979) documented the following: (1) a psychophysics of value according to which people are risk averse in the domain of gains and risk seeking in the domain of losses; (2) a psychophysics of chance in which people overweight sure things and improbable events, relative to events of moderate probability; and (3) that decision problems can be described or framed in multiple ways that give rise to different preferences, contrary to the invariance criterion of rational choice.Prospect theory is considered as the best available description of how people evaluate risk in experimental settings. Despite this, there are relatively few broadly accepted applications to economics (for a review see Barberis 2013). A probable explanation may be that prospect theory represents a paradigm shift in the theory of choice under uncertainty and therefore it is not easily adaptable to the problems discussed in the domain of normal economics.
Alberto Baccini