Automatic consideration of risk aversion It is often the case that a person, faced with real-world
gambles with money, does not act to maximize the
expected value of their
dollar assets. For example, a person who only possesses $1000 in savings may be reluctant to risk it all for a 20% chance odds to win $10,000, even though :20\%(\$10\,000)+80\%(\$0) = \$2000 > 100\%(\$1000) However,
if the person is VNM-rational, such facts are automatically accounted for in their utility function
u. In this example, we could conclude that :20\%u(\$10\,000)+80\%u(\$0) where the dollar amounts here really represent
outcomes (cf. "
value"), the three possible situations the individual could face. In particular,
u can exhibit properties like
u($1)+
u($1) ≠
u($2) without contradicting VNM-rationality at all. This leads to a quantitative theory of
monetary risk aversion. Implications for the expected utility hypothesis In 1738,
Daniel Bernoulli published a treatise in which he posits that rational behavior can be described as maximizing the expectation of a function
u, which in particular need not be monetary-valued, thus accounting for risk aversion. This is the
expected utility hypothesis. As stated, the hypothesis may appear to be a bold claim. The aim of the
expected utility theorem is to provide "modest conditions" (i.e. axioms) describing when the expected utility hypothesis holds, which can be evaluated directly and intuitively: "The axioms should not be too numerous, their system is to be as simple and transparent as possible, and each axiom should have an immediate intuitive meaning by which its appropriateness may be judged directly. In a situation like ours this last requirement is particularly vital, in spite of its vagueness: we want to make an intuitive concept amenable to mathematical treatment and to see as clearly as possible what hypotheses this requires." –
VNM 1953 § 3.5.2, p. 25 As such, claims that the expected utility hypothesis does not characterize rationality must reject one of the VNM axioms. A variety of
generalized expected utility theories have arisen, most of which drop or relax the independence axiom.
Implications for ethics and moral philosophy Because the theorem assumes nothing about the nature of the possible outcomes of the gambles, they could be morally significant events, for instance involving the life, death, sickness, or health of others. By constructing/defining a suitable utility function, a von Neumann–Morgenstern rational agent is capable of taking such outcomes into account, sacrificing personal wealth or well-being for the benefit of the collective. In other words, both what is naturally perceived as "personal gain", and what is naturally perceived as "altruism", can be included in the VNM-utility function of a VNM-rational individual. Therefore, the full range of
agent-focused to agent-neutral behaviors is possible with various VNM-utility functions. If the utility of N is the same as that of pM, a von Neumann–Morgenstern rational agent must be indifferent between 1N and pM+(1-p)0. An agent-focused von Neumann–Morgenstern rational agent therefore cannot favor more equal, or "fair", distributions of utility between its own possible future selves.
Distinctness from other notions of utility Some
utilitarian moral theories are concerned with quantities called the "total utility" and "average utility" of collectives, and characterize morality in terms of favoring the utility or happiness of others with disregard for one's own. These notions can be related to, but are distinct from, VNM-utility: • 1) VNM-utility is a
decision utility: it is that according to which one decides, and thus by definition cannot be something which one disregards. • 2) VNM-utility is not canonically additive across multiple individuals (see Limitations), so "total VNM-utility" and "average VNM-utility" are not immediately meaningful (some sort of normalization assumption is required). The term
E-utility for "experience utility" has been coined to refer to the types of "hedonistic" utility like that of
Bentham's
greatest happiness principle. Since morality affects decisions, a VNM-rational agent's morals will affect the definition of its own utility function (see above). Thus, the morality of a VNM-rational agent can be characterized by
correlation of the agent's VNM-utility with the VNM-utility, E-utility, or "happiness" of others, among other means, but not by
disregard for the agent's own VNM-utility, a contradiction in terms. == Limitations ==