Given the von Neumann-Morgenstern utility function of an individual, U (W) = where W stands for amount of money and ln is the natural logarithm. Comment upon attitude towards risk of such an individual with the help of a diagram.
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n decision theory, the von Neumann–Morgenstern (VNM) utility theorem shows that, under certain axioms of rational behavior, a decision-maker faced with risky (probabilistic) outcomes of different choices will behave as if he or she is maximizing the expected value of some function defined over the potential outcomes at some specified point in the future. This function is known as the von Neumann–Morgenstern utility function. The theorem is the basis for expected utility theory.
In 1947, John von Neumann and Oskar Morgenstern proved that any individual whose preferences satisfied four axioms has a utility function;[1] such an individual's preferences can be represented on an interval scale and the individual will always prefer actions that maximize expected utility. 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 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.
The expected utility hypothesis is that rationality can be modeled as maximizing an expected value, which given the theorem, can be summarized as "rationality is VNM-rationality". However, the axioms themselves have been critiqued on various grounds, resulting in the axioms being given further justification.[2]
VNM-utility is a decision utility in that it is used to describe decision preferences. It is related but not equivalent to so-called E-utilities[3] (experience utilities), notions of utility intended to measure happiness such as that of Bentham's Greatest Happiness Principle.
Given the von Neumann-Morgenstern utility function of an individual,
The von Neumann-Morgenstern (VNM) utility theorem in decision theory demonstrates that, under certain axioms of rational behaviour, an individual faced with risky (probabilistic) outcomes of different choices will act as though he or she is maximising the expected value of some function defined over the potential outcomes at some specified point in the future. The von Neumann-Morgenstern utility function is the name of this function. Expected utility theory is founded on the theorem.
John von Neumann and Oskar Morgenstern demonstrated in 1947 that any person whose preferences met four axioms had a utility function; this person would always favour behaviours that maximise predicted utility. Such a person's preferences may be expressed on an interval scale. In other words, they demonstrated that an agent is (VNM-)rational if and only if a real-valued function u defined by potential outcomes exists, such that each of the agent's preferences is described by maximising the anticipated value of u, which can then be defined as the agent's VNM-utility (it is unique up to adding a constant and multiplying by a positive scalar). It is merely said that u exists, not that the actor "consciously desires" to maximise it.
Given the theorem, it can be said that "rationality is VNM-rationality," which is the anticipated utility hypothesis that rationality may be characterised as maximising an expected value. However, the axioms have come under fire for several reasons, leading to additional explanations.
VNM-utility is a decision utility in that it is used to describe decision preferences. It is related but not equivalent to so-called E-utilities[3] (experience utilities), notions of utility intended to measure happiness such as that of Bentham's Greatest Happiness Principle.
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