Risk, Value, and Utility—DRAFT
Introduction
In ordinary language, risk, value, and utility name three related but different ideas. The OED includes these definitions:
risk (noun) (Exposure to) the possibility of loss, injury, or other adverse or unwelcome circumstance; a chance or situation involving such a possibility.
value (noun) Worth or quality as measured by a standard of equivalence.
utility (noun) The intrinsic property of anything that leads an individual to choose it rather than something else
Risk points toward the possibility of loss, injury, or some other adverse outcome. Value points toward worth, assessed against a standard of equivalence. Utility points toward desirability to a chooser. The three notions overlap in practice, especially when consequences are measured in money, but they need not coincide. A position may have high value and still carry substantial risk; a position may have utility to one decision-maker and little to another; and a position may have low risk simply because its possible outcomes are compressed, even if its value is unremarkable.
Finance and insurance bring these ideas onto a common monetary scale. Once outcomes are expressed in units of cash, value becomes the amount one is willing to pay or receive to accept or remove a risk. That common unit does not erase the conceptual difference, but it does make a close mathematical relation possible. Value is naturally ordered in the direction “more is better,” while risk is naturally ordered in the direction “more is worse.” If both are measured in the same units, the passage from one to the other is therefore expected to involve a reversal of sign.
That sign reversal is not merely a formal trick. It reflects two complementary ways of reading the same monetary position. From one side, we ask: what consideration must be added to make assuming this position acceptable? From the other, we ask: how much certain cash can be extracted from this position while leaving behind something acceptable? The first question is phrased in the language of risk; the second in the language of value. In a monetary setting the two questions are dual to one another, and much of the theory of monetary risk measures rests on that duality.
Utility enters from a different direction. Utility does not begin by asking for a cash equivalent standard of acceptability. It begins with preference: faced with uncertain prospects, which does a decision-maker choose? A utility function is then a numerical realization of that preference ordering. Utility therefore belongs first to the language of choice, not to the language of capital requirement or market worth. In special settings the three notions can be linked, and sometimes partially identified, but conceptually they start from different questions: risk asks about adverse possibility, value asks about monetary worth, and utility asks about preference.
Risk Measures and Valuation Functions
In the theory of monetary risk measures, risk and value are usually the same nonlinear functional viewed from opposite sides. A monetary risk measure \(\rho\) assigns to a position \(X\) the amount of certain cash that must be added to make \(X\) acceptable. If \(\rho(X)\) is large, the position is bad in the relevant sense: it requires a large capital injection. The negative quantity \(-\rho(X)\) therefore behaves like a value. A position with larger \(-\rho(X)\) is better, because less capital is needed to support it, or equivalently more certain cash can be extracted from it.
The algebra makes the point immediately. A monetary risk measure satisfies cash invariance, \[ \rho(X+m)=\rho(X)-m, \] for every constant \(m \in \mathbb{R}\). If we define \[ V(X)=-\rho(X), \] then \[ V(X+m)=V(X)+m. \] Thus \(V\) treats certain cash exactly as a valuation rule should: adding \(m\) units of cash increases value by \(m\). Monotonicity of \(\rho\) likewise turns into monotonicity of \(V\): if \(X \le Y\) pointwise, then \(\rho(X) \ge \rho(Y)\) and hence \(V(X) \le V(Y)\). In that precise sense, a monetary risk measure is a value functional written with the sign reversed.
The acceptance-set formulation says the same thing geometrically. Given an acceptance set \(A\), one defines \[ \rho(X)=\inf \{ m \in \mathbb{R} : X+m \in A \}. \] The number \(\rho(X)\) is the least cash amount that must be injected to move \(X\) into the acceptable region. Turning the sign around, \(-\rho(X)\) is the largest cash amount that can be withdrawn from \(X\) while leaving behind an acceptable position. One can therefore read the same object in two equivalent ways: as required capital, or as extractable value.
In the finance setting of Föllmer and Schied (2016), that dual reading is often implicit throughout. The functional \(\rho\) is introduced as a capital requirement or reserve, which makes the risk interpretation natural. But exactly the same functional, with the sign changed, can be read as a conservative valuation rule or bid price. In a complete market such a value would typically be linear, coming from expectation under a pricing measure. In an incomplete market, or under model uncertainty, one is led instead to nonlinear valuations, and monetary risk measures provide one important class of such functionals.
Convex duality reinforces the same interpretation. A convex monetary risk measure often has a representation of the form \[ \rho(X)=\sup_{Q \in \mathcal{Q}} \left( E_Q[-X] - \alpha(Q) \right). \] Negating gives \[ -\rho(X)=\inf_{Q \in \mathcal{Q}} \left( E_Q[X] + \alpha(Q) \right). \] In the first form we see a worst-case capital requirement; in the second we see a conservative value obtained by taking the least favorable adjusted expectation. The mathematics is unchanged, but the economic reading shifts from liability to asset.
That equivalence is helpful because utility theory begins from the value side, while monetary risk theory begins from the capital side. Utility asks how desirable a position is. Risk-measure theory asks how much capital must be added to support it. Under cash invariance, the two viewpoints are separated mainly by sign. The deeper differences appear elsewhere, especially in the treatment of mixing, diversification, and law-based evaluation. Those differences motivate the comparison developed in the next sections.
Utility and Dual Utility
Utility theory starts from preferences over uncertain consequences. Let \(C\) denote the consequence space; for present purposes one may take \(C=\mathbb{R}\), interpreted as monetary outcomes. A lottery is a probability distribution on \(C\). If \(C\) is finite, a lottery is simply a list of consequences and associated probabilities. More generally, a lottery is a probability measure \(\mu\) on \(C\). One may realize such a lottery by a random variable \(X:\Omega\to C\) on some probability space, but in the utility-theoretic formulation the law is the primitive object and the underlying sample space is secondary.
A preference relation on lotteries is a binary relation \(\succcurlyeq\) indicating weak preference. We write \(\mu \succcurlyeq \nu\) when lottery \(\mu\) is judged at least as desirable as lottery \(\nu\). The usual axioms are completeness and transitivity, often with suitable continuity and independence assumptions. Completeness means that any two lotteries can be compared; transitivity means that the comparisons are consistent. Continuity rules out preference jumps, and the independence axiom controls how preference behaves under randomization with a common third lottery.
A utility representation assigns a real number to each consequence, via a function \(u:C\to\mathbb{R}\), and then evaluates a lottery \(\mu\) by expected utility \[ U(\mu)=\int_C u(x)\,\mu(dx). \] The representation theorem says, roughly, that if the preference relation satisfies the von Neumann–Morgenstern axioms, then there exists such a function \(u\) with \[ \mu \succcurlyeq \nu \quad \Longleftrightarrow \quad U(\mu)\ge U(\nu). \] The utility function is unique up to a positive affine transformation. In that sense, utility is not primitive; what is primitive is the preference ordering, and utility is a numerical realization of it.
The key axiom for present purposes is independence. Given lotteries \(\mu,\nu,\eta\) and \(\lambda\in(0,1)\), the mixture \[ \lambda \mu \oplus (1-\lambda)\eta \] means: with probability \(\lambda\) play lottery \(\mu\), and with probability \(1-\lambda\) play lottery \(\eta\). The independence axiom says that if \(\mu \succcurlyeq \nu\), then \[ \lambda \mu \oplus (1-\lambda)\eta \succcurlyeq \lambda \nu \oplus (1-\lambda)\eta. \] Thus expected utility is linear with respect to mixture of lotteries, that is, with respect to convex combination of probability laws on the consequence space. (Given a representation, independence mirrors the fact that integrals are linear in the measure.)
At that point a distinction appears that is easy to miss because the same convex coefficients occur elsewhere. In monetary risk theory, one usually starts not with lotteries as laws, but with positions \(X,Y\) defined on a common probability space. Then the combination \[ \lambda X + (1-\lambda)Y \] means a statewise portfolio combination: hold proportion \(\lambda\) of position \(X\) and proportion \(1-\lambda\) of position \(Y\). That operation is not the same as the lottery mixture above, which is why we use the \(\oplus\) notation. A lottery mixture randomizes between whole positions; a portfolio combination holds fractions of both positions in every state. Expected utility is built around the first operation. Convex risk measures are built around the second.
Dual utility occupies a middle ground. Like expected utility, it works at the level of laws rather than statewise combinations. But unlike expected utility, it does not treat mixture of lotteries linearly. Instead, it evaluates the law nonlinearly, typically by distorting probabilities or, equivalently in many formulations, by integrating quantiles against a non-uniform weight. In that sense, dual utility keeps the law-based viewpoint of expected utility while abandoning the independence axiom that forces linearity in probabilities.
A standard example is rank-dependent or spectral evaluation. Let \(X\) be a payoff with distribution function \(F_X\) and quantile function \(q_X\). A dual-utility or spectral functional takes a form such as \[ U_g(X)=\int_0^1 q_X(p)\,dg(p), \] or an equivalent distortion form written in terms of \(F_X\). The function \(g\) distorts probability rank, so bad or good parts of the distribution receive more or less weight than under ordinary expectation. When \(g(p)=p\), one recovers linear expectation. For a general distortion, one obtains a law-invariant but nonlinear evaluation of the lottery.
The conceptual split is therefore as follows. Expected utility is linear in mixture of lotteries. Monetary risk measures are typically convex under statewise combination of positions. Dual utility, and likewise spectral risk measures, evaluate laws directly but do so nonlinearly, usually through distorted probabilities or weighted quantiles. The same symbols \(\lambda\) and \(1-\lambda\) appear in all three settings, but they act on different objects and encode different economic ideas. That distinction drives much of the difference among utility, value, and risk.
A Two-State Toy Example
Let the world have two states, \(H\) and \(T\), each with probability \(1/2\). Define two payoffs
| state | \(X\) | \(Y\) |
|---|---|---|
| \(H\) | 10 | 0 |
| \(T\) | 0 | 10 |
So \(X\) pays on heads, \(Y\) pays on tails. Now form two different “half and half” constructions.
External Lottery Mixture
Flip an independent coin first. If it lands A, you get the whole bet \(X\). If it lands B, you get the whole bet \(Y\). Call that object \[ M = \tfrac12 X \oplus \tfrac12 Y. \] Its final payoff is 10 half the time and 0 half the time. In distribution, \[ M \sim \begin{cases} 10 & \text{with prob. } 1/2, \\ 0 & \text{with prob. } 1/2. \end{cases} \]
State-By-State Combination
Instead, hold half of each bet: \[ C = \tfrac12 X + \tfrac12 Y. \]
Then
| state | \(C\) |
|---|---|
| \(H\) | 5 |
| \(T\) | 5 |
So \(C\) is the sure payoff 5.
Same coefficients, completely different result: the external lottery mixture gives a 0/10 gamble; the statewise combination gives sure 5. That distinction is the whole story in miniature.
Utility Evaluation
Take a simple concave utility, say \(u(x)=\sqrt{x}\). Then, under the external lottery mixture \[ EU(M)=\tfrac12 u(10)+\tfrac12 u(0)=\tfrac12\sqrt{10}\approx 1.581. \] Under state-by-state combination \[ EU(C)=u(5)=\sqrt{5}\approx 2.236. \] So utility prefers the statewise combination to the external lottery mixture.
But notice what utility theory treats as primitive: the lottery mixture operation \(\oplus\). The von Neumann–Morgenstern independence axiom speaks about that operation, not about the statewise combination \(X+Y\).
Spectral-Risk Evaluation
Now rewrite the same example as losses instead of payoffs. Let
| state | \(L_1\) | \(L_2\) |
|---|---|---|
| \(H\) | 10 | 0 |
| \(T\) | 0 | 10 |
Again form
- external lottery mixture: choose \(L_1\) or \(L_2\) with probability \(1/2\);
- statewise sharing: \(\tfrac12 L_1+\tfrac12 L_2\).
Take the spectral risk measure \(\mathrm{TVaR}_{0.5}\). The mixed loss is 10 with probability \(1/2\) and 0 with probability \(1/2\). Its worst 50% is all at loss 10, so \[ \mathrm{TVaR}_{0.5}(M)=10. \] Under statewise sharing the combined loss is sure 5, so \[ \mathrm{TVaR}_{0.5}(C)=5. \] Again: the external lottery mixture does not reduce the risk; statewise sharing does reduce the risk. That contrast is exactly what convexity of a monetary risk measure is about.
The Independence Axiom
In expected utility, if \(X \succcurlyeq Y\), then for any third lottery \(Z\) and any \(\lambda\in(0,1)\), \[ \lambda X \oplus (1-\lambda) Z \succcurlyeq \lambda Y \oplus (1-\lambda) Z. \] That is: linearity with respect to lottery mixing.
A spectral risk measure does not satisfy an analogous linearity in external lottery mixtures. It is nonlinear in the distribution. Instead, a convex monetary risk measure satisfies \[ \rho\bigl(\lambda X + (1-\lambda)Y\bigr) \le \lambda \rho(X)+(1-\lambda)\rho(Y), \] which concerns statewise portfolio combination.
So the two theories put convex structure on different operations.
Appendix: Terminology
The language of utility theory, risk theory, and probability overlaps heavily, but the underlying objects are not always the same. The same symbol may denote a consequence, a lottery, or a random variable representing that lottery, and many texts pass among those viewpoints without comment. For probabilists that can feel slippery. The present appendix fixes the terminology used in this note.
Consequence and State
A consequence is an outcome in the payoff space. In the simplest monetary setting the consequence space is \(C=\mathbb{R}\), and a consequence is just a number \(x \in \mathbb{R}\). It records what is finally received or paid. A state, by contrast, is an element \(\omega\) of an underlying sample space \(\Omega\). States describe which contingency occurs; consequences describe what payoff results.
A random payoff is therefore a function \[ X:\Omega \to C. \] The state \(\omega\) goes in, and the consequence \(X(\omega)\) comes out. The probability measure \(P\) lives on \(\Omega\), not on \(C\). Once \(X\) is given, it induces a probability law on the consequence space.
| object | meaning |
|---|---|
| \(\omega \in \Omega\) | state of the world |
| \(x \in C\) | consequence or payoff |
| \(X:\Omega\to C\) | random payoff or position |
| \(P\) on \(\Omega\) | probability on states |
Lottery and Random Variable
A lottery is a probability distribution on the consequence space \(C\). If \(C=\mathbb{R}\), a lottery is a probability measure on \(\mathbb{R}\). In a finite example, a lottery can be written as a list of consequences with associated probabilities, such as \[ L=\begin{cases} 10 & \text{with prob. } 1/2,\\ 0 & \text{with prob. } 1/2. \end{cases} \]
A random variable \(X:\Omega\to C\) is one concrete realization of such a lottery. Many different random variables, possibly on different probability spaces, can have the same distribution and hence represent the same lottery. Thus, strictly speaking, a lottery is a law, whereas a random variable is a map from states to consequences.
In much of expected-utility theory, only the distribution matters, and authors routinely identify a lottery with any random variable that has that law. That identification is usually harmless there, but it becomes important to distinguish the two once we start adding or coupling positions on a common space.
| term | precise meaning |
|---|---|
| lottery | probability law on \(C\) |
| random variable / position | measurable map \(X:\Omega\to C\) |
| same lottery | same distribution on \(C\) |
| same random variable | same statewise map, up to the intended notion of equality |
The Law of \(X\)
Given a random variable \(X:\Omega\to C\), its law is the induced probability measure on \(C\), \[ P_X = P \circ X^{-1}. \] For a measurable set \(A \subseteq C\), \[ P_X(A)=P\{\omega \in \Omega : X(\omega)\in A\}. \] Thus the law records how likely each set of consequences is, forgetting the underlying states that produced them.
For example, let \(\Omega=\{H,T\}\) with \(P(H)=P(T)=1/2\), and define \[ X(H)=10, \qquad X(T)=0. \] Then the law of \(X\) is \[ P_X = \tfrac12 \delta_{10} + \tfrac12 \delta_0. \] That law is the corresponding lottery. Another random variable \(Y\) may have the same law and hence represent the same lottery, while differing from \(X\) as a function on the state space.
Mixture of Laws and Statewise Combination
The most important terminological distinction in the main text concerns two different operations that are often described informally using the same words, such as “take half of each.”
First, one can mix lotteries. If \(\mu\) and \(\nu\) are probability laws on \(C\), then \[ \lambda \mu \oplus (1-\lambda)\nu \] is the lottery that plays \(\mu\) with probability \(\lambda\) and \(\nu\) with probability \(1-\lambda\). This is a convex combination of probability measures. It is the operation that appears in the independence axiom of expected utility.
Second, one can combine positions state by state. If \(X\) and \(Y\) are random variables on the same probability space, then \[ \lambda X + (1-\lambda)Y \] is the random payoff obtained by holding proportions \(\lambda\) and \(1-\lambda\) of the two positions in every state. This is a pointwise linear combination of functions on \(\Omega\). It is the operation that appears in the convexity axiom for monetary risk measures.
These are not the same operation. A mixture of laws randomizes between whole positions. A statewise combination holds fractions of both positions in every state. The distinction is invisible if one looks only at a single marginal distribution, but it becomes decisive as soon as one studies diversification, dependence, or convexity.
A tiny example makes the point. Let \(\Omega=\{H,T\}\) with equal probabilities, and define \[ X(H)=10,\quad X(T)=0,\qquad Y(H)=0,\quad Y(T)=10. \] Then \(X\) and \(Y\) have the same law, namely the lottery paying 10 with probability \(1/2\) and 0 with probability \(1/2\). As lotteries they are identical. But as random variables on a common state space they are different, since they pay in opposite states.
The statewise combination is \[ \tfrac12 X + \tfrac12 Y = 5 \] in every state, so it is the sure consequence 5. By contrast, the mixture of laws \[ \tfrac12 P_X + \tfrac12 P_Y \] is still the lottery paying 10 with probability \(1/2\) and 0 with probability \(1/2\), since \(P_X=P_Y\). Thus the same coefficients \(1/2\) and \(1/2\) produce two different objects: - a sure payoff under statewise combination; - a two-point lottery under mixture of laws.
That small example is the main conceptual warning. In expected utility, convex coefficients act on lotteries, that is, on laws. In monetary risk theory, convex coefficients usually act on positions, that is, on random variables on a common space. In dual utility and spectral theory, one again works primarily at the law level, but the evaluation of the law is nonlinear rather than linear in probabilities.
| operation | input | output | interpretation |
|---|---|---|---|
| \(\lambda \mu \oplus (1-\lambda)\nu\) | two lotteries (laws) | a lottery | randomize between whole lotteries |
| \(\lambda X + (1-\lambda)Y\) | two positions on same space | a position | hold fractions of both in every state |
Keeping those distinctions in view removes much of the apparent jargon. The theories are not disagreeing about elementary probability; they are organizing different primitives for different purposes.
Risk Measures and Valuation FUnctions
In the theory of monetary risk measures, risk and value are usually the same nonlinear functional viewed from opposite sides. A monetary risk measure \(\rho\) assigns to a position \(X\) the amount of certain cash that must be added to make \(X\) acceptable. If \(\rho(X)\) is large, the position is bad in the relevant sense: it requires a large capital injection. The negative quantity \(-\rho(X)\) therefore behaves like a value. A position with larger \(-\rho(X)\) is better, because less capital is needed to support it, or equivalently more certain cash can be extracted from it.
The algebra makes the point immediately. A monetary risk measure satisfies cash invariance, \[ \rho(X+m)=\rho(X)-m, \] for every constant \(m \in \mathbb{R}\). If we define \[ V(X)=-\rho(X), \] then \[ V(X+m)=V(X)+m. \] Thus \(V\) treats certain cash exactly as a valuation rule should: adding \(m\) units of cash increases value by \(m\). Monotonicity of \(\rho\) likewise turns into monotonicity of \(V\): if \(X \le Y\) pointwise, then \(\rho(X) \ge \rho(Y)\) and hence \(V(X) \le V(Y)\). In that precise sense, a monetary risk measure is a value functional written with the sign reversed.
The acceptance-set formulation says the same thing geometrically. Given an acceptance set \(A\), one defines \[ \rho(X)=\inf \{ m \in \mathbb{R} : X+m \in A \}. \] The number \(\rho(X)\) is the least cash amount that must be injected to move \(X\) into the acceptable region. Turning the sign around, \(-\rho(X)\) is the largest cash amount that can be withdrawn from \(X\) while leaving behind an acceptable position. One can therefore read the same object in two equivalent ways: as required capital, or as extractable value.
In the finance setting of Föllmer and Schied (2016), that dual reading is often implicit throughout. The functional \(\rho\) is introduced as a capital requirement or reserve, which makes the risk interpretation natural. But exactly the same functional, with the sign changed, can be read as a conservative valuation rule or bid price. In a complete market such a value would typically be linear, coming from expectation under a pricing measure. In an incomplete market, or under model uncertainty, one is led instead to nonlinear valuations, and monetary risk measures provide one important class of such functionals.
Convex duality reinforces the same interpretation. A convex monetary risk measure often has a representation of the form \[ \rho(X)=\sup_{Q \in \mathcal{Q}} \left( E_Q[-X] - \alpha(Q) \right). \] Negating gives \[ -\rho(X)=\inf_{Q \in \mathcal{Q}} \left( E_Q[X] + \alpha(Q) \right). \] In the first form we see a worst-case capital requirement; in the second we see a conservative value obtained by taking the least favorable adjusted expectation. The mathematics is unchanged, but the economic reading shifts from liability to asset.
That equivalence is helpful because utility theory begins from the value side, while monetary risk theory begins from the capital side. Utility asks how desirable a position is. Risk-measure theory asks how much capital must be added to support it. Under cash invariance, the two viewpoints are separated mainly by sign. The deeper differences appear elsewhere, especially in the treatment of mixing, diversification, and law-based evaluation. Those differences motivate the comparison developed in the next sections.
Utility and Dual Utility
Utility theory starts from preferences over uncertain consequences. Let \(C\) denote the consequence space; for present purposes one may take \(C=\mathbb{R}\), interpreted as monetary outcomes. A lottery is a probability distribution on \(C\). If \(C\) is finite, a lottery is simply a list of consequences and associated probabilities. More generally, a lottery is a probability measure \(\mu\) on \(C\). One may realize such a lottery by a random variable \(X:\Omega\to C\) on some probability space, but in the utility-theoretic formulation the law is the primitive object and the underlying sample space is secondary.
A preference relation on lotteries is a binary relation \(\succcurlyeq\) indicating weak preference. We write \(\mu \succcurlyeq \nu\) when lottery \(\mu\) is judged at least as desirable as lottery \(\nu\). The usual axioms are completeness and transitivity, often with suitable continuity and independence assumptions. Completeness means that any two lotteries can be compared; transitivity means that the comparisons are consistent. Continuity rules out preference jumps, and the independence axiom controls how preference behaves under randomization with a common third lottery.
A utility representation assigns a real number to each consequence, via a function \(u:C\to\mathbb{R}\), and then evaluates a lottery \(\mu\) by expected utility \[ U(\mu)=\int_C u(x)\,\mu(dx). \] The representation theorem says, roughly, that if the preference relation satisfies the von Neumann–Morgenstern axioms, then there exists such a function \(u\) with \[ \mu \succcurlyeq \nu \quad \Longleftrightarrow \quad U(\mu)\ge U(\nu). \] The utility function is unique up to a positive affine transformation. In that sense, utility is not primitive; what is primitive is the preference ordering, and utility is a numerical realization of it.
The key axiom for present purposes is independence. Given lotteries \(\mu,\nu,\eta\) and \(\lambda\in(0,1)\), the mixture \[ \lambda \mu \oplus (1-\lambda)\eta \] means: with probability \(\lambda\) play lottery \(\mu\), and with probability \(1-\lambda\) play lottery \(\eta\). The independence axiom says that if \(\mu \succcurlyeq \nu\), then \[ \lambda \mu \oplus (1-\lambda)\eta \succcurlyeq \lambda \nu \oplus (1-\lambda)\eta. \] Thus expected utility is linear with respect to mixture of lotteries, that is, with respect to convex combination of probability laws on the consequence space. (Given a representation, independence mirrors the fact that integrals are linear in the measure.)
At that point a distinction appears that is easy to miss because the same convex coefficients occur elsewhere. In monetary risk theory, one usually starts not with lotteries as laws, but with positions \(X,Y\) defined on a common probability space. Then the combination \[ \lambda X + (1-\lambda)Y \] means a statewise portfolio combination: hold proportion \(\lambda\) of position \(X\) and proportion \(1-\lambda\) of position \(Y\). That operation is not the same as the lottery mixture above, which is why we use the \(\oplus\) notation. A lottery mixture randomizes between whole positions; a portfolio combination holds fractions of both positions in every state. Expected utility is built around the first operation. Convex risk measures are built around the second.
Dual utility occupies a middle ground. Like expected utility, it works at the level of laws rather than statewise combinations. But unlike expected utility, it does not treat mixture of lotteries linearly. Instead, it evaluates the law nonlinearly, typically by distorting probabilities or, equivalently in many formulations, by integrating quantiles against a non-uniform weight. In that sense, dual utility keeps the law-based viewpoint of expected utility while abandoning the independence axiom that forces linearity in probabilities.
A standard example is rank-dependent or spectral evaluation. Let \(X\) be a payoff with distribution function \(F_X\) and quantile function \(q_X\). A dual-utility or spectral functional takes a form such as \[ U_g(X)=\int_0^1 q_X(p)\,dg(p), \] or an equivalent distortion form written in terms of \(F_X\). The function \(g\) distorts probability rank, so bad or good parts of the distribution receive more or less weight than under ordinary expectation. When \(g(p)=p\), one recovers linear expectation. For a general distortion, one obtains a law-invariant but nonlinear evaluation of the lottery.
The conceptual split is therefore as follows. Expected utility is linear in mixture of lotteries. Monetary risk measures are typically convex under statewise combination of positions. Dual utility, and likewise spectral risk measures, evaluate laws directly but do so nonlinearly, usually through distorted probabilities or weighted quantiles. The same symbols \(\lambda\) and \(1-\lambda\) appear in all three settings, but they act on different objects and encode different economic ideas. That distinction drives much of the difference among utility, value, and risk.
A Two-State Toy Example
Let the world have two states, \(H\) and \(T\), each with probability \(1/2\). Define two payoffs
| state | \(X\) | \(Y\) |
|---|---|---|
| \(H\) | 10 | 0 |
| \(T\) | 0 | 10 |
So \(X\) pays on heads, \(Y\) pays on tails. Now form two different “half and half” constructions.
External Lottery Mixture
Flip an independent coin first. If it lands A, you get the whole bet \(X\). If it lands B, you get the whole bet \(Y\). Call that object \[ M = \tfrac12 X \oplus \tfrac12 Y. \] Its final payoff is 10 half the time and 0 half the time. In distribution, \[ M \sim \begin{cases} 10 & \text{with prob. } 1/2, \\ 0 & \text{with prob. } 1/2. \end{cases} \]
State-By-State Combination
Instead, hold half of each bet: \[ C = \tfrac12 X + \tfrac12 Y. \]
Then
| state | \(C\) |
|---|---|
| \(H\) | 5 |
| \(T\) | 5 |
So \(C\) is the sure payoff 5.
Same coefficients, completely different result: the external lottery mixture gives a 0/10 gamble; the statewise combination gives sure 5. That distinction is the whole story in miniature.
Utility Evaluation
Take a simple concave utility, say \(u(x)=\sqrt{x}\). Then, under the external lottery mixture \[ EU(M)=\tfrac12 u(10)+\tfrac12 u(0)=\tfrac12\sqrt{10}\approx 1.581. \] Under state-by-state combination \[ EU(C)=u(5)=\sqrt{5}\approx 2.236. \] So utility prefers the statewise combination to the external lottery mixture.
But notice what utility theory treats as primitive: the lottery mixture operation \(\oplus\). The von Neumann–Morgenstern independence axiom speaks about that operation, not about the statewise combination \(X+Y\).
Spectral-Risk Evaluation
Now rewrite the same example as losses instead of payoffs. Let
| state | \(L_1\) | \(L_2\) |
|---|---|---|
| \(H\) | 10 | 0 |
| \(T\) | 0 | 10 |
Again form
- external lottery mixture: choose \(L_1\) or \(L_2\) with probability \(1/2\);
- statewise sharing: \(\tfrac12 L_1+\tfrac12 L_2\).
Take the spectral risk measure \(\mathrm{TVaR}_{0.5}\). The mixed loss is 10 with probability \(1/2\) and 0 with probability \(1/2\). Its worst 50% is all at loss 10, so \[ \mathrm{TVaR}_{0.5}(M)=10. \] Under statewise sharing the combined loss is sure 5, so \[ \mathrm{TVaR}_{0.5}(C)=5. \] Again: the external lottery mixture does not reduce the risk; statewise sharing does reduce the risk. That contrast is exactly what convexity of a monetary risk measure is about.
The Independence Axiom
In expected utility, if \(X \succcurlyeq Y\), then for any third lottery \(Z\) and any \(\lambda\in(0,1)\), \[ \lambda X \oplus (1-\lambda) Z \succcurlyeq \lambda Y \oplus (1-\lambda) Z. \] That is: linearity with respect to lottery mixing.
A spectral risk measure does not satisfy an analogous linearity in external lottery mixtures. It is nonlinear in the distribution. Instead, a convex monetary risk measure satisfies \[ \rho\bigl(\lambda X + (1-\lambda)Y\bigr) \le \lambda \rho(X)+(1-\lambda)\rho(Y), \] which concerns statewise portfolio combination.
So the two theories put convex structure on different operations.
Appendix: Terminology
The language of utility theory, risk theory, and probability overlaps heavily, but the underlying objects are not always the same. The same symbol may denote a consequence, a lottery, or a random variable representing that lottery, and many texts pass among those viewpoints without comment. For probabilists that can feel slippery. The present appendix fixes the terminology used in this note.
Consequence and State
A consequence is an outcome in the payoff space. In the simplest monetary setting the consequence space is \(C=\mathbb{R}\), and a consequence is just a number \(x \in \mathbb{R}\). It records what is finally received or paid. A state, by contrast, is an element \(\omega\) of an underlying sample space \(\Omega\). States describe which contingency occurs; consequences describe what payoff results.
A random payoff is therefore a function \[ X:\Omega \to C. \] The state \(\omega\) goes in, and the consequence \(X(\omega)\) comes out. The probability measure \(P\) lives on \(\Omega\), not on \(C\). Once \(X\) is given, it induces a probability law on the consequence space.
| object | meaning |
|---|---|
| \(\omega \in \Omega\) | state of the world |
| \(x \in C\) | consequence or payoff |
| \(X:\Omega\to C\) | random payoff or position |
| \(P\) on \(\Omega\) | probability on states |
Lottery and Random Variable
A lottery is a probability distribution on the consequence space \(C\). If \(C=\mathbb{R}\), a lottery is a probability measure on \(\mathbb{R}\). In a finite example, a lottery can be written as a list of consequences with associated probabilities, such as \[ L=\begin{cases} 10 & \text{with prob. } 1/2,\\ 0 & \text{with prob. } 1/2. \end{cases} \]
A random variable \(X:\Omega\to C\) is one concrete realization of such a lottery. Many different random variables, possibly on different probability spaces, can have the same distribution and hence represent the same lottery. Thus, strictly speaking, a lottery is a law, whereas a random variable is a map from states to consequences.
In much of expected-utility theory, only the distribution matters, and authors routinely identify a lottery with any random variable that has that law. That identification is usually harmless there, but it becomes important to distinguish the two once we start adding or coupling positions on a common space.
| term | precise meaning |
|---|---|
| lottery | probability law on \(C\) |
| random variable / position | measurable map \(X:\Omega\to C\) |
| same lottery | same distribution on \(C\) |
| same random variable | same statewise map, up to the intended notion of equality |
The Law of \(X\)
Given a random variable \(X:\Omega\to C\), its law is the induced probability measure on \(C\), \[ P_X = P \circ X^{-1}. \] For a measurable set \(A \subseteq C\), \[ P_X(A)=P\{\omega \in \Omega : X(\omega)\in A\}. \] Thus the law records how likely each set of consequences is, forgetting the underlying states that produced them.
For example, let \(\Omega=\{H,T\}\) with \(P(H)=P(T)=1/2\), and define \[ X(H)=10, \qquad X(T)=0. \] Then the law of \(X\) is \[ P_X = \tfrac12 \delta_{10} + \tfrac12 \delta_0. \] That law is the corresponding lottery. Another random variable \(Y\) may have the same law and hence represent the same lottery, while differing from \(X\) as a function on the state space.
Mixture of Laws and Statewise Combination
The most important terminological distinction in the main text concerns two different operations that are often described informally using the same words, such as “take half of each.”
First, one can mix lotteries. If \(\mu\) and \(\nu\) are probability laws on \(C\), then \[ \lambda \mu \oplus (1-\lambda)\nu \] is the lottery that plays \(\mu\) with probability \(\lambda\) and \(\nu\) with probability \(1-\lambda\). This is a convex combination of probability measures. It is the operation that appears in the independence axiom of expected utility.
Second, one can combine positions state by state. If \(X\) and \(Y\) are random variables on the same probability space, then \[ \lambda X + (1-\lambda)Y \] is the random payoff obtained by holding proportions \(\lambda\) and \(1-\lambda\) of the two positions in every state. This is a pointwise linear combination of functions on \(\Omega\). It is the operation that appears in the convexity axiom for monetary risk measures.
These are not the same operation. A mixture of laws randomizes between whole positions. A statewise combination holds fractions of both positions in every state. The distinction is invisible if one looks only at a single marginal distribution, but it becomes decisive as soon as one studies diversification, dependence, or convexity.
A tiny example makes the point. Let \(\Omega=\{H,T\}\) with equal probabilities, and define \[ X(H)=10,\quad X(T)=0,\qquad Y(H)=0,\quad Y(T)=10. \] Then \(X\) and \(Y\) have the same law, namely the lottery paying 10 with probability \(1/2\) and 0 with probability \(1/2\). As lotteries they are identical. But as random variables on a common state space they are different, since they pay in opposite states.
The statewise combination is \[ \tfrac12 X + \tfrac12 Y = 5 \] in every state, so it is the sure consequence 5. By contrast, the mixture of laws \[ \tfrac12 P_X + \tfrac12 P_Y \] is still the lottery paying 10 with probability \(1/2\) and 0 with probability \(1/2\), since \(P_X=P_Y\). Thus the same coefficients \(1/2\) and \(1/2\) produce two different objects: - a sure payoff under statewise combination; - a two-point lottery under mixture of laws.
That small example is the main conceptual warning. In expected utility, convex coefficients act on lotteries, that is, on laws. In monetary risk theory, convex coefficients usually act on positions, that is, on random variables on a common space. In dual utility and spectral theory, one again works primarily at the law level, but the evaluation of the law is nonlinear rather than linear in probabilities.
| operation | input | output | interpretation |
|---|---|---|---|
| \(\lambda \mu \oplus (1-\lambda)\nu\) | two lotteries (laws) | a lottery | randomize between whole lotteries |
| \(\lambda X + (1-\lambda)Y\) | two positions on same space | a position | hold fractions of both in every state |
Keeping those distinctions in view removes much of the apparent jargon. The theories are not disagreeing about elementary probability; they are organizing different primitives for different purposes.