The Emergence of Subjective Expected Utility as the Mainstream Paradigm
As we have seen, von Neumann and Morgenstern derived utility given the notion of statistical probability. Some years earlier, Bruno de Finetti (1937) had developed a notion of subjective probability starting from the choice problem of an individual maximizing an expected monetary value.
Frank P. Ramsey (1926 [1931]), in discussing Keynes’s probability, developed a subjective view of probability and utility without assuming any of these concepts as primitive. Savage adopted also this strategy. He “virtually copied” (Savage 1954 [1972]: 97) the treatment reserved by von Neumann and Morgenstern to utility by developing the idea of a subjective probability. Savage’s model includes two primitive concepts: outcomes, as usual, and states, as the list of all scenarios that may happen. The outcome is the conjoined result of the agent’s choice of an act and of the unfolding of a state of the world. When the agent makes her choice over acts, she does not reason in terms of numbers, utility and probabilities; but on a very simple framework containing the description of possible states and the unique outcome resulting from the choice of every act in every state (that is, when an act has been chosen by an agent, and a state of the world has been unfolded, only an outcome is necessarily verified). With this simple structure and seven axioms, Savage demonstrated that both a (bounded) utility function and probability measure exist in such a way that decisions are made as if the agent is maximizing the expectation of the utility relative to the probability measure (for a presentation, see Gilboa 2009: 94-112). More precisely, the subjective expected utility hypothesis is equivalent to the joint hypothesis that the agent possesses a subjective probabilistic belief about the states of the world μ(si), and a von Neumann and Morgenstern utility function over outcomes U(x); and she evaluates acts according to a preference function (for a finite state space) of the form W = 2.~jU (xi) μ (si). Savage restricted the applicability of his theory to what he called a “small world”. In a small world it is always possible for an agent to “look... before you leap” given that it is possible to have a description so complete that the consequences of every action would be known. This idea can be expressed from another point of view: an agent in a small world can take account in advance of the impact of all future possible information on his or her subjective beliefs about the state of the world. The consistency of an agent’s choices are guaranteed if his or her personal degrees of belief are coherent in such a way that a Dutch book - a system of bets which guarantee that anyone who takes them all on will lose no matter what happens - could not be made against him or her.Savage’s results reinforced the von Neumann and Morgenstern construction, and the subjective expected utility became the major paradigm in decision making in the second half of the twentieth century. According to this view the expected utility maximization with respect to a subjective probability is the only rational way of behaviour suitable for cogently treating every kind of economic problem characterized by the presence of uncertainty. The Arrow-Pratt measure of risk aversion provided a powerful operative tool (Pratt 1964; Arrow 1965). Since then applications of the expected utility model flowered in problems of optimal savings, international trade, portfolio selection, environmental economics, and economic analysis of law.