Empirical methods Prescriptive and policy In decision theory, the von Neumann–Morgenstern (VNM) utility theorem demonstrates that rational choice under uncertainty involves making decisions that take the form of maximizing the expected value of some cardinal utility function.
In 1947, John von Neumann and Oskar Morgenstern proved that any individual whose preferences satisfied four axioms has a utility function, where such an individual's preferences can be represented on an interval scale and the individual will always prefer actions that maximize expected utility.
[1] That is, they proved that an agent is (VNM-)rational if and only if there exists a real-valued function u defined by possible outcomes such that every preference of the agent is characterized by maximizing the expected value of u, which can then be defined as the agent's VNM-utility (it is unique up to affine transformations i.e. adding a constant and multiplying by a positive scalar).
No claim is made that the agent has a "conscious desire" to maximize u, only that u exists.
[2] In the theorem, an individual agent is faced with options called lotteries.
Continuity assumes that there is a "tipping point" between being better than and worse than a given middle option: where the notation on the left side refers to a situation in which L is received with probability p and N is received with probability (1–p).
Instead of continuity, an alternative axiom can be assumed that does not involve a precise equality, called the Archimedean property.
[3] It says that any separation in preference can be maintained under a sufficiently small deviation in probabilities: Only one of (3) or (3′) need to be assumed, and the other will be implied by the theorem.
For any VNM-rational agent (i.e. satisfying axioms 1–4), there exists a function u which assigns to each outcome A a real number u(A) such that for any two lotteries, where E(u(L)), or more briefly Eu(L) is given by As such, u can be uniquely determined (up to adding a constant and multiplying by a positive scalar) by preferences between simple lotteries, meaning those of the form pA + (1 − p)B having only two outcomes.
Conversely, any agent acting to maximize the expectation of a function u will obey axioms 1–4.
Here we outline the construction process for the case in which the number of sure outcomes is finite.
Hence, by the Completeness and Transitivity axioms, it is possible to order the outcomes from worst to best: We assume that at least one of the inequalities is strict (otherwise the utility function is trivial—a constant).
We use these two extreme outcomes—the worst and the best—as the scaling unit of our utility function, and define: For every probability
is the expectation of u: To see why this utility function makes sense, consider a lottery
Hence: Von Neumann and Morgenstern anticipated surprise at the strength of their conclusion.
But according to them, the reason their utility function works is that it is constructed precisely to fill the role of something whose expectation is maximized: "Many economists will feel that we are assuming far too much ... Have we not shown too much?
... As far as we can see, our postulates [are] plausible ... We have practically defined numerical utility as being that thing for which the calculus of mathematical expectations is legitimate."
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 However, if the person is VNM-rational, such facts are automatically accounted for in their utility function u.
In 1738, Daniel Bernoulli published a treatise[7] 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.
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.
A variety of generalized expected utility theories have arisen, most of which drop or relax the independence axiom.
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.
A von Neumann–Morgenstern rational agent is capable of acting with great concern for such events, sacrificing much personal wealth or well-being, and all of these actions will factor into the construction/definition of the agent's VNM-utility function.
Therefore, the full range of agent-focused to agent-neutral behaviors are possible with various VNM-utility functions[clarification needed].
An agent-focused von Neumann–Morgenstern rational agent therefore cannot favor more equal, or "fair", distributions of utility between its own possible future selves.
These notions can be related to, but are distinct from, VNM-utility: The term E-utility for "experience utility" has been coined[2] to refer to the types of "hedonistic" utility like that of Bentham's greatest happiness principle.
Von Neumann and Morgenstern recognized this limitation: "...concepts like a specific utility of gambling cannot be formulated free of contradiction on this level.
[1]Since for any two VNM-agents X and Y, their VNM-utility functions uX and uY are only determined up to additive constants and multiplicative positive scalars, the theorem does not provide any canonical way to compare the two.
The expected utility hypothesis has been shown to have imperfect predictive accuracy in a set of lab based empirical experiments, such as the Allais paradox.