Benutzer:JonskiC/Von Neuman-Morgenstern Erwartungsnutzen-Hypothese

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Von-Neumann-Morgenstern-Erwartungsnutzenfunktion

Entscheidungen unter Unsicherheit werden mikroökonomisch oft als Lotterie modelliert. Der Nutzen der Wahl einer Alternative ist hier nicht unmittelbar bekannt. Statt einer Nutzenfunktion wird daher eine Erwartungsnutzenfunktion (auch VNM-Nutzenfunktion) für die Modellierung der Präferenzen eines Akteurs eingesetzt.

Dabei wird der Erwartungswert über eine (typischerweise eindimensionale) Nutzenfunktion für die einzelnen Alternativen als Nutzenwert definiert. Die Nutzenfunktion der jeweiligen Alternativen und deren Wahrscheinlichkeitsverteilung bestimmen daher den Nutzen einer Lotterie: Erwartungsnutzen ist einfach der Erwartungswert des Nutzens der Alternativen. Eine solche Nutzenfunktion wird auch als Von-Neumann-Morgenstern-(Erwartungs)-Nutzenfunktion bezeichnet.

$V(z)=\operatorname {E} [u(z)]=\sum _{i=1}^{N}p_{i}\cdot u(z_{i})$

$V$ bezeichnet die Erwartungsnutzenfunktion über die Zufallsvariable $z$ ($i$ Zustände, die mit unterschiedlichen Wahrscheinlichkeiten $p$ eintreten) und $u$ ist die sogenannte Bernoulli-Nutzenfunktion in Abhängigkeit von $z$. Die Von-Neumann-Morgenstern-Nutzenfunktion ist somit nichts anderes als der mit den Wahrscheinlichkeiten gewichtete Nutzen aus den verschiedenen Zuständen, die aus der Lotterie resultieren können.

Die Existenz einer Erwartungsnutzenfunktion setzt jedoch stärkere Annahmen voraus, insbesondere das umstrittene Unabhängigkeitsaxiom, gemäß dem irrelevante Alternativen keinen Einfluss auf das Ergebnis haben dürfen. Unabhängig von der Zulässigkeit einer Erwartungsnutzenformulierung können ökonomisch Handelnde als risikofreudig, risikoneutral oder risikoscheu eingestuft werden.

n decision theory, the von Neumann-Morgenstern 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 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; 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.

Any individual whose preferences violate von Neumann and Morgenstern's axioms would agree to a Dutch book, which is a set of bets that necessarily leads to a loss. Therefore, it is arguable that any individual who violates the axioms is irrational. 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".

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 (experience 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,

$L=0.25A+0.75B\,$

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 A_i, we write

$L=\sum p_{i}A_{i},\,$

with the sum of the $p_{i}$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 $L\prec M.$ If M is either preferred over or viewed with indifference relative to L, we write $L\preceq M.$ If the agent is indifferent between L and M, we have the indifference relation $L\sim M.$

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, exactly one of the following holds:

$\,L\prec M\,$, $\,M\prec L\,$, or Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle \, L \sim M }

(either M is preferred, L is preferred, or the individual is indifferent).

Transitivity assumes that preference is consistent across any three options:

Axiom 2 (Transitivity) If $\,L\preceq M\,$ and $\,M\preceq N\,$, then $\,L\preceq N\,$.

Continuity assumes that there is a "tipping point" between being better than and worse than a given middle option:

Axiom 3 (Continuity): If $\,L\preceq M\preceq N\,$, then there exists a probability $\,p\in [0,1]\,$ such that

$\,pL+(1-p)N\,\sim \,M\,$

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 $\,L\prec M\prec N\,$, then there exists a probability $\,\varepsilon \in [0,1]$ such that

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Only one of (3) and (3′) need 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): If Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle \,L\prec M\,} , then for any Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle \,N\,} and Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle \,p\in[0,1]\,} ,

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The independence axiom implies the axiom on reduction of compound lotteries:

Axiom 4′ (Reduction of compound lotteries): For any Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle Z, \,W, } any Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle p, q, r \in [0,1]} such that Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle rq=p,} and any lottery Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle X=qZ+(1-q)W,}

Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle pZ+(1-p)W \sim rX +(1-r)W.}

The theorem

For any VNM-rational agent (i.e. satisfying 1–4), there exists a function u assigning to each outcome A a real number u(A) such that for any two lotteries,

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where E(u(L)) denotes the expected value of u in L (we'll abbreviate E(u(L)) to Eu(L)):

Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle Eu(p_1A_1 + \ldots + p_nA_n) = p_1u(A_1) + \cdots + p_nu(A_n). \, }

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. Such a function is called the agent's von Neumann–Morgenstern (VNM) utility.

Proof sketch

The proof is constructive: it shows how the desired function Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle u} can be built. Here we outline the construction process for the case in which the number of sure outcomes is finite.

Suppose there are n sure outcomes, Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle A_1\dots A_n} . Note that every sure outcome can be seen as a lottery: it is a degenerate lottery in which the outcome is selected with probability 1. Hence, by the Completeness axiom, it is possible to order the outcomes from worst to best:

Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle A_1\preceq A_2\preceq \cdots \preceq A_n}

We assume that at least one of the inequalities is strict (otherwise the utility function is trivial - a constant). So Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle A_1\prec A_n} . We use these two extreme outcomes - the worst and the best - as the scaling unit of our utility function, and define:

Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle u(A_1)=0} and Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle u(A_n)=1}

For every probability Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle p\in[0,1]} , define a lottery that selects the best outcome with probability Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle p} and the worst outcome otherwise:

Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle L(p) = p\cdot A_n + (1-p)\cdot A_1}

Note that Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle L(0)\sim A_1} and Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle L(1)\sim A_n} .

By the Continuity axiom, for every sure outcome Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle A_i} , there is a probability Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle q_i} such that:

Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle L(q_i) \sim A_i}

And:

Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle 0 = q_1\leq q_2\leq \cdots \leq q_n = 1}

For every Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle i} , the utility function for outcome Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle A_i} is defined as:

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so the utility of every lottery Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle M=\sum_i{p_i A_i}} is the expectation of u:

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Why does this utility function make sense?

Consider a lottery Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle M = \sum_i{p_i A_i}} , which selects outcome Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle A_i} with probability Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle p_i} . But, by our assumption, the decision maker is indifferent between the sure outcome Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle A_i} and the lottery Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle q_i\cdot A_n + (1-q_i)\cdot A_1} . So, by the Reduction axiom, he is indifferent between the lottery Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle M} and the following lottery:

Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle M' = \sum_i{p_i [q_i\cdot A_n + (1-q_i)\cdot A_1]}}

Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle M' = (\sum_i{p_i q_i})\cdot A_n + (\sum_i{p_i(1-q_i)})\cdot A_1}

Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle M' = u(M)\cdot A_n + (1-u(M))\cdot A_1}

The lottery Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle M'} is, in effect, a lottery in which the best outcome is won with probability Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle u(M)} , and the worst outcome otherwise.

Hence, if Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle u(M)>u(L)} , a rational decision maker would prefer the lottery Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle M} over the lottery Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle L} , because it gives him a larger chance to win the best outcome.

Hence:

Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle L\prec M \; \mathrm{iff} \; E(u(L)) < E(u(M))}

Reaction

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." – VNM 1953, § 3.1.1 p.16 and § 3.7.1 p. 28

Thus, the content of the theorem is that the construction of u is possible, and they claim little about its nature.

Consequences

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

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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

Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle 20\%u(\$10,000)+80\%u(\$0) < u(\$1000)}

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. 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. In other words, both what is naturally perceived as "personal gain", and what is naturally perceived as "altruism", are implicitly balanced in the VNM-utility function of a VNM-rational individual. Therefore, the full range of agent-focussed to agent-neutral behaviors are Vorlage:Clarify.

If the utility of Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle N} is Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle pM} , a von Neumann–Morgenstern rational agent must be indifferent between Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle 1N} and Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle 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

Nested gambling

Since if L and M are lotteries, then pL + (1 − p)M is simply "expanded out" and considered a lottery itself, the VNM formalism ignores what may be experienced as "nested gambling". This is related to the Ellsberg problem where people choose to avoid the perception of risks about risks. Von Neumann and Morgenstern recognized this limitation:

"...concepts like a specific utility of gambling cannot be formulated free of contradiction on this level. This may seem to be a paradoxical assertion. But anybody who has seriously tried to axiomatize that elusive concept, will probably concur with it." – VNM 1953 § 3.7.1, p. 28.

Incomparability between agents

Since for any two VNM-agents X and Y, their VNM-utility functions u_X and u_Y are only determined up to additive constants and multiplicative positive scalars, the theorem does not provide any canonical way to compare the two. Hence expressions like u_X(L) + u_Y(L) and u_X(L) − u_Y(L) are not canonically defined, nor are comparisons like u_X(L) < u_Y(L) canonically true or false. In particular, the aforementioned "total VNM-utility" and "average VNM-utility" of a population are not canonically meaningful without normalization assumptions.

Applicability to economics

The expected utility hypothesis, as applied to economics, has limited predictive accuracy, simply because in practice, humans do not always behave VNM-rationally. This is manifested in several experimental outcomes such as the Allais paradox. This can be interpreted as evidence that

• humans are not always rational, or
• VNM-rationality is not an appropriate characterization of rationality, or
• some combination of both, or
• humans do behave VNM-rationally but the objective evaluation of u or the construction of u are intractable problems.

Austrian economics criticism

Vorlage:Importance section Austrian economists criticize the vNM theory on several grounds:

• It leans on an assumption that utilities are constant over time, so they can be revealed by action over time.
• It relies on the concept of indifference of utilities, which is invalid.
• It rests on the fallacious application of a theory of numerical probability to an area where it cannot apply. Numerical probability can be assigned only to situations where there is a class of entities, such that nothing is known about the members except they are members of this class, and where successive trials reveal an asymptotic tendency toward a stable proportion, or frequency of occurrence, of a certain event in that class Vorlage:Disputed inline. Yet, in human action, there are no classes of homogeneous members. Each event is a unique event and is different from other unique events.
• The neo-cardinalists Vorlage:Definition needed admit that their theory is not even applicable to gambling if the individual has either a like or a dislike for gambling itself. Since the fact that a man gambles demonstrates that he likes to gamble, it is clear that the Neumann-Morgenstern utility doctrine fails even in this tailor-made case Vorlage:Disputed inline.

References and further reading

• Nash Jr., John F. The Bargaining Problem. Econometrica 18:155 1950
• Anand, Paul. Foundations of Rational Choice Under Risk Oxford, Oxford University Press. 1993 reprinted 1995, 2002
• Fishburn, Peter C. Utility Theory for Decision Making. Huntington, NY. Robert E. Krieger Publishing Co. 1970. ISBN 978-0-471-26060-8
• Sixto Rios (1998) Some problems and developments in decision science, Revista Matematica Complutense 11(1):113–41.

n economics, game theory, and decision theory the expected utility hypothesis, concerning people's preferences with regard to choices that have uncertain outcomes (gambles), states that if specific axioms are satisfied, the subjective value associated with an individual's gamble is the statistical expectation of that individual's valuations of the outcomes of that gamble. Initiated by Daniel Bernoulli in 1738, this hypothesis has proved useful to explain some popular choices that seem to contradict the expected value criterion (which takes into account only the sizes of the payouts and the probabilities of occurrence), such as occur in the contexts of gambling and insurance. Until the mid-twentieth century, the standard term for the expected utility was the moral expectation, contrasted with "mathematical expectation" for the expected value.

The von Neumann–Morgenstern utility theorem provides necessary and sufficient conditions under which the expected utility hypothesis holds. From relatively early on, it was accepted that some of these conditions would be violated by real decision-makers in practice but that the conditions could be interpreted nonetheless as 'axioms' of rational choice.

Expected value and choice under risk

In the presence of risky outcomes, a human decision maker does not always choose the option with higher expected value investments. For example, suppose there is a choice between a guaranteed payment of $1, and a gamble in which the probability of getting a $100 payment is 1 in 80 and the alternative, far more likely outcome is getting nothing (the expected value is then $1.25); according to expected value theory people should choose the $100-or-nothing gamble, but as stressed by expected utility theory, some people are risk averse enough to prefer the sure thing, even though it has a lower expected value, while other less risk averse people would still choose the riskier, higher-mean gamble.

Bernoulli's formulation

Nicolas Bernoulli described the St. Petersburg paradox (involving infinite expected values) in 1713, prompting two Swiss mathematicians to develop expected utility theory as a solution. The theory can also more accurately describe more realistic scenarios (where expected values are finite) than expected value alone. In 1728, Gabriel Cramer, in a letter to Nicolas Bernoulli, wrote, "the mathematicians estimate money in proportion to its quantity, and men of good sense in proportion to the usage that they may make of it."

In 1738, Nicolas' cousin Daniel Bernoulli, published the canonical 18th Century description of this solution in Specimen theoriae novae de mensura sortis or Exposition of a New Theory on the Measurement of Risk. Daniel Bernoulli proposed that a mathematical function of probability should be used to correct the expected value, accounting for risk aversion, where the risk premium is higher for low-probability events than the difference between the payout level of a particular outcome and its expected value.

Bernoulli's paper was the first formalization of marginal utility, which has broad application in economics in addition to expected utility theory. He used this concept to formalize the idea that the same amount of additional money was less useful to an already-wealthy person than it would be to a poor person.

Infinite expected value — St. Petersburg paradox

The St. Petersburg paradox (named after the journal in which Bernoulli's paper was published) arises when there is no upper bound on the potential rewards from very low probability events. Because some probability distribution functions have an infinite expected value, an expected-wealth maximizing person would pay an infinite amount to take this gamble. In real life, people do not do this.

Bernoulli proposed a solution to this paradox in his paper: the utility function used in real life means that the expected utility of the gamble is finite, even if its expected value is infinite. (Thus he hypothesized diminishing marginal utility of increasingly larger amounts of money.) It has also been resolved differently by other economists by proposing that very low probability events are neglected, by taking into account the finite resources of the participants, or by noting that one simply cannot buy that which is not sold (and that sellers would not produce a lottery whose expected loss to them were unacceptable).

Von Neumann–Morgenstern formulation

The von Neumann–Morgenstern axioms

There are four axioms of the expected utility theory that define a rational decision maker. They are completeness, transitivity, independence and continuity.

Completeness assumes that an individual has well defined preferences and can always decide between any two alternatives.

• Axiom (Completeness): For every A and B either Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle A \succeq B} or Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle A \preceq B} .

This means that the individual either prefers A to B, or is indifferent between A and B, or prefers B to A.

Transitivity assumes that, as an individual decides according to the completeness axiom, the individual also decides consistently.

• Axiom (Transitivity): For every A, B and C with Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle A \succeq B} and Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle B \succeq C} we must have Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle A \succeq C} .

Independence of irrelevant alternatives pertains to well-defined preferences as well. It assumes that two gambles mixed with an irrelevant third one will maintain the same order of preference as when the two are presented independently of the third one. The independence axiom is the most controversial axiom.Vorlage:Citation needed.

• Axiom (Independence of irrelevant alternatives):

Let A, B, and C be three lotteries with Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle A \succeq B} , and let Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle t} be the probability that a third choice is present: Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle t \in [0, 1]} ;
if
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then the third choice, C, is irrelevant, and the order of preference for A before B holds, independently of the presence of C.

Continuity assumes that when there are three lotteries (A, B and C) and the individual prefers A to B and B to C, then there should be a possible combination of A and C in which the individual is then indifferent between this mix and the lottery B.

• Axiom (Continuity): Let A, B and C be lotteries with Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle A \succeq B \succeq C} ; then there exists a probability p such that B is equally good as Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle pA+(1-p)C} .

If all these axioms are satisfied, then the individual is said to be rational and the preferences can be represented by a utility function, i.e. one can assign numbers (utilities) to each outcome of the lottery such that choosing the best lottery according to the preference Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle \succeq} amounts to choosing the lottery with the highest expected utility. This result is called the von Neumann–Morgenstern utility representation theorem.

In other words: if an individual's behavior always satisfies the above axioms, then there is a utility function such that the individual will choose one gamble over another if and only if the expected utility of one exceeds that of the other. The expected utility of any gamble may be expressed as a linear combination of the utilities of the outcomes, with the weights being the respective probabilities. Utility functions are also normally continuous functions. Such utility functions are also referred to as von Neumann–Morgenstern (vNM) utility functions. This is a central theme of the expected utility hypothesis in which an individual chooses not the highest expected value, but rather the highest expected utility. The expected utility maximizing individual makes decisions rationally based on the axioms of the theory.

The von Neumann–Morgenstern formulation is important in the application of set theory to economics because it was developed shortly after the Hicks–Allen "ordinal revolution" of the 1930s, and it revived the idea of cardinal utility in economic theory.Vorlage:Citation needed Note, however, that while in this context the utility function is cardinal, in that implied behavior would be altered by a non-linear monotonic transformation of utility, the expected utility function is ordinal because any monotonic increasing transformation of it gives the same behavior.

Risk aversion

Vorlage:Further information The expected utility theory takes into account that individuals may be risk-averse, meaning that the individual would refuse a fair gamble (a fair gamble has an expected value of zero). Risk aversion implies that their utility functions are concave and show diminishing marginal wealth utility. The risk attitude is directly related to the curvature of the utility function: risk neutral individuals have linear utility functions, while risk seeking individuals have convex utility functions and risk averse individuals have concave utility functions. The degree of risk aversion can be measured by the curvature of the utility function.

Since the risk attitudes are unchanged under affine transformations of u, the first derivative u' is not an adequate measure of the risk aversion of a utility function. Instead, it needs to be normalized. This leads to the definition of the Arrow–Pratt measure of absolute risk aversion:

Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle \mathit{ARA}(w) =-\frac{u''(w)}{u'(w)}}

The Arrow–Pratt measure of relative risk aversion is:

Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle \mathit{RRA}(w) =-\frac{wu''(w)}{u'(w)}}

Special classes of utility functions are the CRRA (constant relative risk aversion) functions, where RRA(w) is constant, and the CARA (constant absolute risk aversion) functions, where ARA(w) is constant. They are often used in economics for simplification.

A decision that maximizes expected utility also maximizes the probability of the decision's consequences being preferable to some uncertain threshold (Castagnoli and LiCalzi,1996; Bordley and LiCalzi,2000;Bordley and Kirkwood, ). In the absence of uncertainty about the threshold, expected utility maximization simplifies to maximizing the probability of achieving some fixed target. If the uncertainty is uniformly distributed, then expected utility maximization becomes expected value maximization. Intermediate cases lead to increasing risk-aversion above some fixed threshold and increasing risk-seeking below a fixed threshold.

Examples of von Neumann-Morgenstern utility functions

The utility function Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle u(w)=\log(w)} was originally suggested by Bernoulli (see above). It has relative risk aversion constant and equal to one, and is still sometimes assumed in economic analyses. The utility function Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle u(w)= -e^{-aw}} exhibits constant absolute risk aversion, and for this reason is often avoided, although it has the advantage of offering substantial mathematical tractability when asset returns are normally distributed. Note that, as per the affine transformation property alluded to above, the utility function Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle K-e^{-aw}} gives exactly the same preferences orderings as does Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle -e^{-aw}} ; thus it is irrelevant that the values of Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle -e^{-aw}} and its expected value are always negative: what matters for preference ordering is which of two gambles gives the higher expected utility, not the numerical values of those expected utilities.

The class of constant relative risk aversion utility functions contains three categories. Bernoulli's utility function

Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle u(w) = \log(w)}

has relative risk aversion equal to unity. The functions

Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle u(w) = w^{\alpha}}

for Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle \alpha \in (0,1)} have relative risk aversion equal to Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle 1-\alpha} . And the functions

Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle u(w) = -w^{\alpha}}

for Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle \alpha < 0} also have relative risk aversion equal to Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle 1-\alpha} .

See also the discussion of utility functions having hyperbolic absolute risk aversion (HARA).

Measuring risk in the expected utility context

Often people refer to "risk" in the sense of a potentially quantifiable entity. In the context of mean-variance analysis, variance is used as a risk measure for portfolio return; however, this is only valid if returns are normally distributed or otherwise jointly elliptically distributed, or in the unlikely case in which the utility function has a quadratic form. However, David E. Bell proposed a measure of risk which follows naturally from a certain class of von Neumann-Morgenstern utility functions. Let utility of wealth be given by Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle u(w)= w-be^{-aw}} for individual-specific positive parameters a and b. Then expected utility is given by

Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle \begin{align} \operatorname{E}[u(w)]&=\operatorname{E}[w]-b\operatorname{E}[e^{-aw}]\\ &=\operatorname{E}[w]-b\operatorname{E}[e^{-a\operatorname{E}[w]-a(w-\operatorname{E}[w])}]\\ &=\operatorname{E}[w]-be^{-a\operatorname{E}[w]}\operatorname{E}[e^{-a(w-\operatorname{E}[w])}]\\ &= \text{Expected wealth} - b \cdot e^{-a\cdot \text{Expected wealth}}\cdot \text{Risk}. \end{align} }

Thus the risk measure is Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle \operatorname{E}(e^{-a(w-\operatorname{E}w)})} , which differs between two individuals if they have different values of the parameter Fehler beim Parsen (MathML mit SVG- oder PNG-Rückgriff (empfohlen für moderne Browser und Barrierefreiheitswerkzeuge): Ungültige Antwort („Math extension cannot connect to Restbase.“) von Server „https://wikimedia.org/api/rest_v1/“:): {\displaystyle a} , allowing different people to disagree about the degree of risk associated with any given portfolio. See also Entropic risk measure.

For general utility functions, however, expected utility analysis does not permit the expression of preferences to be separated into two parameters with one representing the expected value of the variable in question and the other representing its risk.

Criticism

Expected utility theory is a theory about how to make optimal decisions under risk. It has a normative interpretation which economists particularly used to think applies in all situations to rational agents but now tend to regard as a useful and insightful first order approximation. In empirical applications, a number of violations have been shown to be systematic and these falsifications have deepened understanding of how people actually decide. For example, in 2000 behavioral economist Matthew Rabin argued that mathematically the utility of wealth cannot explain loss aversion and attempts to so use it will fail. Bernoulli's theory on the utility of wealth assumed that if two people have the same wealth all other things being equal the people should be equally happy. However, where two people have US$1m but one has just prior to that had US$2m but lost US$1m whereas the other had US$500k and had just gained US$500k they will not be equally happy. Bernoulli's theory thus lacked a reference point. Nevertheless, it remained a dominant theory for over 250 years. Daniel Kahneman and Amos Tversky in 1979 presented their prospect theory which showed empirically, among other things, how preferences of individuals are inconsistent among same choices, depending on how those choices are presented.

Like any mathematical model, expected utility theory is an abstraction and simplification of reality. The mathematical correctness of expected utility theory and the salience of its primitive concepts do not guarantee that expected utility theory is a reliable guide to human behavior or optimal practice.

The mathematical clarity of expected utility theory has helped scientists design experiments to test its adequacy, and to distinguish systematic departures from its predictions. This has led to the field of behavioral finance, which has produced deviations from expected utility theory to account for the empirical facts.

Conservatism in updating beliefs

It is well established that humans find logic hard, mathematics harder, and probability even more challengingVorlage:Citation needed. Psychologists have discovered systematic violations of probability calculations and behavior by humans.Vorlage:Citation needed Consider, for example, the Monty Hall problem.

In updating probability distributions using evidence, a standard method uses conditional probability, namely the rule of Bayes. An experiment on belief revision has suggested that humans change their beliefs faster when using Bayesian methods than when using informal judgment.

Irrational deviations

Behavioral finance has produced several generalized expected utility theories to account for instances where people's choices deviate from those predicted by expected utility theory. These deviations are described as "irrational" because they can depend on the way the problem is presented, not on the actual costs, rewards, or probabilities involved.

Particular theories include prospect theory, rank-dependent expected utility and cumulative prospect theory and SP/A theory.

Preference reversals over uncertain outcomes

Starting with studies such as Lichtenstein & Slovic (1971), it was discovered that subjects sometimes exhibit signs of preference reversals with regard to their certainty equivalents of different lotteries. Specifically, when eliciting certainty equivalents, subjects tend to value "p bets" (lotteries with a high chance of winning a low prize) lower than "$ bets" (lotteries with a small chance of winning a large prize). When subjects are asked which lotteries they prefer in direct comparison, however, they frequently prefer the "p bets" over "$ bets." Many studies have examined this "preference reversal," from both an experimental (e.g., Plott & Grether, 1979) and theoretical (e.g., Holt, 1986) standpoint, indicating that this behavior can be brought into accordance with neoclassical economic theory under specific assumptions.

Uncertain probabilities

If one is using the frequentist notion of probability, where probabilities are considered to be facts, then applying expected value and expected utility to decision-making requires knowing the probability of various outcomes. However, in practice there will be many situations where the probabilities are unknown, one is operating under uncertainty. In economics, one talks of Knightian uncertainty or Ambiguity. Thus one must make assumptions about the probabilities, but then the expected value of various decisions can be very sensitive to the assumptions. This is particularly a problem when the expectation is dominated by rare extreme events, as in a long-tailed distribution.

Alternative decision techniques are robust to uncertainty of probability of outcomes, either not depending on probabilities of outcomes and only requiring scenario analysis (as in minimax or minimax regret), or being less sensitive to assumptions.

Bayesian approaches to probability treat it as a degree of belief and thus they do not draw a distinction between risk and a wider concept of uncertainty: they deny the existence of Knightian uncertainty. They would model uncertain probabilities with hierarchical models, i.e. where the uncertain probabilities are modelled as distributions whose parameters are themselves drawn from a higher-level distribution (hyperpriors).

See also

• Allais paradox
• Ambiguity aversion
• Bayesian probability
• Behavioral economics
• Decision theory
• Generalized expected utility
• Indifference price
• Loss function
• Lottery (probability)
• Marginal utility
• Prospect theory
• Rank-dependent expected utility
• Risk aversion
• Risk in psychology
• Subjective expected utility
• Two-moment decision models

Further reading

• Anand P.: Foundations of Rational Choice Under Risk. Oxford University Press, Oxford 1993, ISBN 0-19-823303-5.
• Arrow K.J.: Uncertainty and the Welfare Economics of Medical Care. In: American Economic Review. 53, 1963, S. 941–73.
• de Finetti, Bruno. "Probabilism: A Critical Essay on the Theory of Probability and on the Value of Science," (translation of 1931 article) in Erkenntnis, volume 31, September 1989.
• de Finetti, Bruno. 1937, “La Prévision: ses lois logiques, ses sources subjectives,” Annales de l'Institut Henri Poincaré,

de Finetti, Bruno. "Foresight: its Logical Laws, Its Subjective Sources," (translation of the 1937 article in French) in H. E. Kyburg and H. E. Smokler (eds), Studies in Subjective Probability, New York: Wiley, 1964.

• de Finetti, Bruno. Theory of Probability, (translation by AFM Smith of 1970 book) 2 volumes, New York: Wiley, 1974-5.
• Morgenstern, Oskar: Some Reflections on Utility. In: Andrew Schotter (Hrsg.): Selected Economic Writings of Oskar Morgenstern. New York University Press, 1976, ISBN 0-8147-7771-6, S. 65–70.
• Charles Sanders Peirce and Joseph Jastrow: On Small Differences in Sensation. In: Memoirs of the National Academy of Sciences. 3, 1885, S. 73–83. http://psychclassics.yorku.ca/Peirce/small-diffs.htm
• Pfanzagl, J: Subjective Probability Derived from the Morgenstern-von Neumann Utility Theory. In: Martin Shubik (Hrsg.): Essays in Mathematical Economics In Honor of Oskar Morgenstern. Princeton University Press, 1967, S. 237–251.
• Pfanzagl, J. in cooperation with V. Baumann and H. Huber: Events, Utility and Subjective Probability. In: Theory of Measurement. Wiley, 1968, S. 195–220.
• Scott Plous (1993) "The psychology of judgment and decision making", Chapter 7 (specifically) and 8,9,10, (to show paradoxes to the theory).
• Ramsey, Frank Plumpton; “Truth and Probability” (PDF), Chapter VII in The Foundations of Mathematics and other Logical Essays (1931).
• Schoemaker PJH: The Expected Utility Model: Its Variants, Purposes, Evidence and Limitations. In: Journal of Economic Literature. 20, 1982, S. 529–563.
• Donald Davidson, Patrick Suppes and Sidney Siegel: Decision-Making: An Experimental Approach. Stanford University Press, 1957.