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In 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 they are 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; 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. 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 (experienced utilities), notions of utility intended to measure happiness such as that of Bentham's Greatest Happiness Principle. Set-up In the theorem, an individual agent is faced with options called lotteries. Given some mutually exclusive outcomes, a lottery is a scenario where each outcome will happen with a given probability, all probabilities summing to one. For example, for two outcomes A and B, L = 0.25 A + 0.75 B {\displaystyle L=0.25A+0.75B} denotes a scenario where P(A) = 25% is the probability of A occurring and P(B) = 75% (and exactly one of them will occur). More generally, for a lottery with many possible outcomes Ai, we write: L = ∑ p i A i , {\displaystyle L=\sum p_{i}A_{i},} with the sum of the p i {\displaystyle p_{i}} s equalling 1. The outcomes in a lottery can themselves be lotteries between other outcomes, and the expanded expression is considered an equivalent lottery: 0.5(0.5A + 0.5B) + 0.5C = 0.25A + 0.25B + 0.50C. If lottery M is preferred over lottery L, we write M ≻ L {\displaystyle M\succ L} , or equivalently, L ≺ M {\displaystyle L\prec M} . If the agent is indifferent between L and M, we write the indifference relation L ∼ M . {\displaystyle L\sim M.} If M is either preferred over or viewed with indifference relative to L, we write L ⪯ M . {\displaystyle L\preceq M.} The axioms The four axioms of VNM-rationality are then completeness, transitivity, continuity, and independence. Completeness assumes that an individual has well defined preferences: Axiom 1 (Completeness) For any lotteries L,M, at least one of the following holds: L ⪰ M {\displaystyle \,L\succeq M} , M ⪰ L {\displaystyle \,M\succeq L} (the individual must express some preference or indifference). Note that this implies reflexivity. Transitivity assumes that preferences are consistent across any three options: Axiom 2 (Transitivity) If L ⪰ M {\displaystyle \,L\succeq M} and M ⪰ N {\displaystyle \,M\succeq N} , then L ⪰ N {\displaystyle \,L\succeq N} . Continuity assumes that there is a "tipping point" between being better than and worse than a given middle option: Axiom 3 (Continuity): If L ⪯ M ⪯ N {\displaystyle \,L\preceq M\preceq N} , then there exists a probability p ∈ [ 0 , 1 ] {\displaystyle \,p\in [0,1]} such that p L + ( 1 − p ) N ∼ M {\displaystyle \,pL+(1-p)N\,\sim \,M} 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. It says that any separation in preference can be maintained under a sufficiently small deviation in probabilities: Axiom 3′ (Archimedean property): If L ≺ M ≺ N {\displaystyle \,L\prec M\prec N} , then there exists a probability ε ∈ ( 0 , 1 ) {\displaystyle \,\varepsilon \in (0,1)} such that ( 1 − ε ) L + ε N ≺ M ≺ ε L + ( 1 − ε ) N . {\displaystyle \,(1-\varepsilon )L+\varepsilon N\,\prec \,M\,\prec \,\varepsilon L+(1-\varepsilon )N.} Only one of (3) or (3′) need to be assumed, and the other will be implied by the theorem. Independence of irrelevant alternatives assumes that a preference holds independently of the possibility of another outcome: Axiom 4 (Independence): For any N {\displaystyle \,N} and p ∈ ( 0 , 1 ] {\displaystyle \,p\in (0,1]} , L ⪯ .... Discover the John Von Neumann Oskar Morgenstern popular books. Find the top 100 most popular John Von Neumann Oskar Morgenstern books.