Multi-attribute utility¶

A utility representation assigning values to outcomes described by several attributes while encoding tradeoffs, interactions and uncertainty preferences.

Version: v1 · 2026-09-08 · History
Domain-specific #: 5685
Origin domain: decision analysis
Subdomain: decision analysis
Aliases: MAUT

Core Idea¶

Additive, multiplicative and multilinear forms require distinct preferential-independence assumptions, and utility under uncertainty differs from a deterministic value score.^[1] Single-attribute utilities scale each consequence dimension, elicited tradeoff weights and interaction terms combine them and expected utility ranks uncertain alternatives under the declared model. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

The load-bearing residual is not the broad topic of decision analysis. It is the domain-specific identity fixed by the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit. That residual remains recognizable when examples, notation, scale, or implementation change, but it disappears if the carrier is mistyped, the condition that the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test. This gives the entry an operational identity rather than merely a historical label.

A useful analysis keeps three layers separate. The constitutive layer says what must be true: the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit. The evidential layer asks what observation or proof warrants the claim: type the carrier, state every parameter and convention in the definition, test that the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases. The use layer asks what reasoning becomes available once the identity is established: recognizing and comparing instances of Multi-attribute utility, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions. Conflating the layers is the most common source of scope inflation.

Structural Signature¶

Carrier: the typed decision analysis carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets
Inputs or antecedent state: the exact decision analysis carrier, defining parameters and conventions, boundary conditions, source evidence, comparison cases, and any measurement or proof assumptions needed to evaluate Multi-attribute utility
Constitutive operation: Single-attribute utilities scale each consequence dimension, elicited tradeoff weights and interaction terms combine them and expected utility ranks uncertain alternatives under the declared model.
Invariant: the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit
Recognition test: type the carrier, state every parameter and convention in the definition, test that the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases
Output or consequence: recognizing and comparing instances of Multi-attribute utility, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions
Failure boundary: the carrier is mistyped, the condition that the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test

What It Is Not¶

It is not the whole field of decision analysis. The field contains many questions and methods that do not instantiate Multi-attribute utility.
It is not its most familiar example. A canonical instance directly demonstrates that the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit. exhibits the structure, but the example is evidence for the abstraction rather than its definition.
It is not the neighboring catalog concept Multi-criteria decision analysis. MCDA is the broader family of multi-criterion methods; multi-attribute utility imposes a utility-theoretic preference representation and, under uncertainty, expected-utility structure.
It is not a claim that every boundary case has one uncontested classification. a generalized or degenerate case may change existence, uniqueness, measurement, or naming conventions, so the exact definition of Multi-attribute utility must control the decision
It is not an unrestricted metaphor for any process that seems similar. Outside decision analysis, the vocabulary and validity conditions do not transfer literally.

Scope of Application¶

Multi-attribute utility belongs to decision analysis and is useful where the analyst can specify the typed decision analysis carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit. The scope is broad within that domain but bounded by the need for the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.^[n1]

Definition and recognition. Determine whether a proposed instance satisfies the constitutive conditions rather than merely sharing terminology.
Construction or evolution. Track how the exact decision analysis carrier, defining parameters and conventions, boundary conditions, source evidence, comparison cases, and any measurement or proof assumptions needed to evaluate Multi-attribute utility are converted, constrained, or organized by Single-attribute utilities scale each consequence dimension, elicited tradeoff weights and interaction terms combine them and expected utility ranks uncertain alternatives under the declared model..
Comparison. Compare instances using carrier, parameters, convention, domain, scale, boundary conditions, evidence, exact versus approximate form, and limiting behavior, without treating convenience measures as the definition.
Boundary analysis. Diagnose cases where a generalized or degenerate case may change existence, uniqueness, measurement, or naming conventions, so the exact definition of Multi-attribute utility must control the decision and state which convention or theorem controls the decision.
Downstream reasoning. Use the established identity to support recognizing and comparing instances of Multi-attribute utility, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions while preserving the assumptions under which the inference is valid.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Multi-attribute utility can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated. The disciplined statement is: given the exact decision analysis carrier, defining parameters and conventions, boundary conditions, source evidence, comparison cases, and any measurement or proof assumptions needed to evaluate Multi-attribute utility, the structure counts as Multi-attribute utility exactly when the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit.

This format also separates identity from measurement. Empirical, computational, or documentary proxies support recognition only under declared validity and uncertainty assumptions; formal cases require proof rather than measurement. Measurements can be noisy, implementations can approximate, and proofs can use equivalent characterizations; none of those facts licenses changing the object being measured. When reports disagree, first check scope and convention, then data or proof, and only then interpret the disagreement as substantive.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Multi-attribute utility. Multi-attribute utility compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

The compression has a price. A single label can hide canonical, generalized, restricted, approximate, computational, empirical, and historically variant formulations of Multi-attribute utility. Good use therefore carries a small declaration of assumptions alongside the name. The abstraction manages complexity when it reduces the state space of the question while keeping the failure boundary visible; it mismanages complexity when the label substitutes for that boundary analysis.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed decision analysis carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit independently of one notation or implementation. This step prevents the canonical example from becoming the definition.
Derive consequences. From the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit, infer recognizing and comparing instances of Multi-attribute utility, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions. Record each assumption used so that a later change of setting does not silently preserve an invalid conclusion.
Test adversarial cases. Examine a generalized or degenerate case may change existence, uniqueness, measurement, or naming conventions, so the exact definition of Multi-attribute utility must control the decision and an object that resembles Multi-attribute utility in purpose or vocabulary but does not satisfy its invariant is outside the class. A robust identity explains why the first is convention-sensitive and why the second is outside the class.
Compare and refine. Use carrier, parameters, convention, domain, scale, boundary conditions, evidence, exact versus approximate form, and limiting behavior to compare legitimate instances, and refine the model when discrepancies reflect hidden variation rather than failure of the abstraction itself.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of decision analysis because they reuse the typed decision analysis carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, Single-attribute utilities scale each consequence dimension, elicited tradeoff weights and interaction terms combine them and expected utility ranks uncertain alternatives under the declared model., and type the carrier, state every parameter and convention in the definition, test that the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases. A theorem, diagnostic, or modeling warning can travel when those roles remain literal. For example, the distinction between constitutive identity and a convenient observable transfers from A canonical instance directly demonstrates that the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit. to An applied instance preserves the invariant under changed notation, scale, dataset, jurisdiction, or implementation..^[n2]

Transfer outside the home domain is weaker. The skeletal pattern—type the carrier, apply the defining mechanism of Multi-attribute utility, preserve its invariant, and derive only consequences licensed by the stated boundary—may suggest an analogy, but the domain-specific mechanisms, admissible evidence, and consequences do not come along automatically. The safe transfer procedure maps each role explicitly, checks the invariant again, and refuses the name when only a superficial resemblance remains.

Examples¶

Canonical¶

A canonical instance directly demonstrates that the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit. The example exposes the carrier and directly tests that the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit; changing incidental notation preserves the identity, while removing that condition destroys it. This example is canonical because every role can be inspected: the carrier is the typed decision analysis carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets; the operative rule is Single-attribute utilities scale each consequence dimension, elicited tradeoff weights and interaction terms combine them and expected utility ranks uncertain alternatives under the declared model.; the invariant is the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit; and the result supports recognizing and comparing instances of Multi-attribute utility, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions.^[1] Changing incidental notation or scale leaves the structure intact, while removing the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit destroys the classification.

Mapped back: the typed decision analysis carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets → Single-attribute utilities scale each consequence dimension, elicited tradeoff weights and interaction terms combine them and expected utility ranks uncertain alternatives under the declared model. → the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit → recognizing and comparing instances of Multi-attribute utility, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions

Applied / In Practice¶

An applied instance preserves the invariant under changed notation, scale, dataset, jurisdiction, or implementation. The applied case qualifies only because the same invariant and boundary test remain literal under changed parameters or implementation. The applied case is not licensed merely by vocabulary. It qualifies because the same recognition test—type the carrier, state every parameter and convention in the definition, test that the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases—can be run and because the same failure boundary—the carrier is mistyped, the condition that the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test—remains meaningful.^[n1] The case also shows why practical outputs should report assumptions, resolution, and uncertainty instead of a naked label.

Mapped back: declared instance → recognition test → boundary check → qualified use

Structural Tensions¶

T1: Axiomatic identity vs. operational recognition. The defining conditions may be exact while empirical or computational recognition is approximate. Neither pole can be removed without changing the analytical task. Diagnostic: Can the reviewer state both the exact condition and the evidence used to infer it?
T2: Local roles vs. global consequence. The mechanism is enacted through local relations, but the abstraction is usually valued for a global classification or prediction. Neither pole can be removed without changing the analytical task. Diagnostic: Does the claimed global result actually follow from the declared local conditions?
T3: Ideal form vs. finite representation. Theory states a clean invariant while data structures, measurements, or proofs expose only finite representations. Neither pole can be removed without changing the analytical task. Diagnostic: Would increasing resolution converge toward the same classification?
T4: Canonical convention vs. legitimate variants. A standard formulation supports communication, while variants may preserve the same core under changed assumptions. Neither pole can be removed without changing the analytical task. Diagnostic: Which role is invariant across variants, and which convention-specific conclusion changes?
T5: Compression vs. hidden assumptions. The name compresses a complex argument but can conceal prerequisites. Neither pole can be removed without changing the analytical task. Diagnostic: Can each downstream inference be traced to an explicit assumption?
T6: Autonomous residual vs. reduction to catalog neighbors. The candidate uses broader structures but adds an identity-bearing residual. Neither pole can be removed without changing the analytical task. Diagnostic: After subtracting the proposed parent and named neighbors, does the constitutive residual still support independent diagnostics?

Structural–Framed Character¶

The entry is structurally mixed but domain-framed. Its portable skeleton is type the carrier, apply the defining mechanism of Multi-attribute utility, preserve its invariant, and derive only consequences licensed by the stated boundary. Its identity-bearing terms—Multi-attribute utility, carrier, parameter, invariant, boundary, evidence, model, transformation, and application—derive their meaning from decision analysis and cannot be replaced by generic systems language without losing the tests that distinguish valid from invalid instances.

This mixed character explains why the abstraction is reusable inside the domain yet does not meet the Prime bar. The structure organizes reasoning, but its claims still depend on domain-specific objects, evidence, and intervention semantics.

Structural Core vs. Domain Accent¶

The structural core consists of a carrier, Single-attribute utilities scale each consequence dimension, elicited tradeoff weights and interaction terms combine them and expected utility ranks uncertain alternatives under the declared model., a recognition invariant, and a consequence. That skeleton may resemble patterns elsewhere, especially type the carrier, apply the defining mechanism of Multi-attribute utility, preserve its invariant, and derive only consequences licensed by the stated boundary. The domain accent is not decorative: Multi-attribute utility, carrier, parameter, invariant, boundary, evidence, model, transformation, and application determine what counts as an admissible carrier, a valid transition, and successful evidence.

The abstraction therefore remains domain-specific. A cross-domain reuse that preserves only words such as 'balance,' 'cut,' 'sequence,' 'loss,' or 'simulation' is metaphor. Literal transfer requires the original role structure and diagnostics, which in this case remain anchored in decision analysis.

The proposed strict upward parent is prime:expected_utility. prime:expected_utility is the nearest broader Prime while the source-domain carrier and invariant supply the autonomous residual. This is a proposal-only workspace relationship: the accepted Prime supplies a genuinely instantiated structural prerequisite or superclass, while Multi-attribute utility adds domain-specific constraints.

The entry does not collapse into that parent because the domain-specific identity fixed by the decision-maker and alternatives, attributes and feasible outcome space, preference relation, certainty or uncertainty, single-attribute utility scales, weights and interaction form, independence assumptions, normalization, elicitation evidence and sensitivity analysis are explicit It also declines a nearby thematic catalog node: the neighbor does not literally subsume the constitutive identity of Multi-attribute utility. This explicit assert-and-decline pattern keeps the proposed DAG narrow and prevents a merely thematic edge.

The prospective workspace queue contains one strict upward edge to prime:expected_utility. No live DAG mutation is authorized.

Relationships to Other Abstractions¶

Current abstraction Multi-attribute utility Domain-specific

Parents (1) — more general patterns this builds on

Multi-attribute utility is a kind of Expected Utility Prime

The proposed strict upward parent is prime:expected_utility.

Hierarchy paths (4) — routes to 3 parentless roots

Multi-attribute utility → Expected Utility → Expected Value → Aggregation → Micro Macro Linkage

Show alternative paths (3)

Neighborhood in Abstraction Space¶

Multi-attribute utility sits in a crowded region of the domain-specific corpus (34^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Welfare, Production & Economic Choice (45 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

Multi-criteria decision analysis. MCDA is the broader family of multi-criterion methods; multi-attribute utility imposes a utility-theoretic preference representation and, under uncertainty, expected-utility structure.
One canonical example. An instance demonstrates the structure but does not define the whole abstraction.
Measurement or implementation of Multi-attribute utility. A proxy or realization is evidence for the abstraction, not the abstraction itself.
Generalized Multi-attribute utility. An extension qualifies only when its changed axioms and retained invariant are stated.

Notes¶

[n1] Such functions are the focus of the current article. The goal is to calculate a utility function u(x_1,...,x_n) which represents the person's preferences on lotteries of bundles. I.e, lottery A is preferred over lottery B if and only if the expectation of the function u is higher under A than under B: : E_A[u(x_1,...,x_n)] > E_B[u(x_1,...,x_n)] == Assessing a multi-attribute cardinal utility function == If the number of possible bundles is finite, u can be constructed directly as explained by von Neumann and Morgenstern (VNM): order the bundles from least preferred to most preferred, assign utility 0 to the former and utility 1 to the latter, and assign to each bundle in between a utility equal to the probability of an equivalent lottery. If the number of bundles is infinite, one option is to start by ignoring the randomness, and assess an ordinal utility function v(x_1,...,x_n) which represents the person's utility on sure bundles. I.e, a bundle x is preferred over a bundle y if and only if the function v is higher for x than for y: : v(x_1,...,x_n) > v(y_1,...,y_n) This function, in effect, converts the multi-attribute problem to a single-attribute problem: the attribute is v . Then, VNM can be used to construct the function u . Note that u must be a positive monotone transformation of v. This means that there is a monotonically increasing function r: \mathbb{R}\to \mathbb{R} , such that: : u(x_1,...,x_n) = r(v(x_1,...,x_n)) The problem with this approach is that it is not easy to assess the function r. When assessing a single-attribute cardinal utility function using VNM, we ask questions such as: "What probability to win $2 is equivalent to $1?". So to assess the function r, we have to ask a question such as: "What probability to win 2 units of value is equivalent to 1 value?". The latter question is much harder to answer than the former, since it involves "value", which is an abstract quantity. A possible solution is to calculate n one-dimensional cardinal utility functions - one for each attribute. For example, suppose there are two attributes: apples ( x_1 ) and bananas ( x_2 ), both range between 0 and 99. Using VNM, we can calculate the following 1-dimensional utility functions: * u(x_1,0) - a cardinal utility on apples when there are no bananas (the southern boundary of the domain); * u(99,x_2) - a cardinal utility on bananas when apples are at their maximum (the eastern boundary of the domain). Using linear transformations, scale the functions such that they have the same value on (99,0). Then, for every bundle (x_1',x_2') , find an equivalent bundle (a bundle with the same v) which is either of the form (x_1,0) or of the form (99,x_2) , and set its utility to the same number. Often, certain independence properties between attributes can be used to make the construction of a utility function easier. Some such independence properties are described below. == Additive independence == The strongest independence property is called additive independence. Two attributes, 1 and 2, are called additive independent, if the preference between two lotteries (defined as joint probability distributions on the two attributes) depends only on their marginal probability distributions (the marginal PD on attribute 1 and the marginal PD on attribute 2). This means, for example, that the following two lotteries are equivalent: * L : An equal-chance lottery between (x_1,x_2) and (y_1,y_2) ; * M : An equal-chance lottery between (x_1,y_2) and (y_1,x_2) . In both these lotteries, the marginal PD on attribute 1 is 50% for x_1 and 50% for y_1 . Similarly, the marginal PD on attribute 2 is 50% for x_2 and 50% for y_2 . Hence, if an agent has additive-independent utilities, he must be indifferent between these two lotteries. A fundamental result in utility theory is that, two attributes are additive-independent, if and only if their two-attribute utility function is additive and has the form: :::: u(x_1,x_2)=u_1(x_1) + u_2(x_2) PROOF: \longrightarrow If the attributes are additive-independent, then the lotteries L and M , defined above, are equivalent. This means that their expected utility is the same, i.e.: E_L[u]=E_M[u] . Multiplying by 2 gives: : u(x_1,x_2)+u(y_1,y_2)=u(x_1,y_2)+u(y_1,x_2) This is true for any selection of the x_i and y_i . Assume now that y_1 and y_2 are fixed. Arbitrarily set u(y_1,y_2)=0 . Write: u_1(x_1) = u(x_1,y_2) and u_2(x_2) = u(y_1,x_2) . The above equation becomes: : u(x_1,x_2) = u_1(x_1)+u_2(x_2) \longleftarrow If the function u is additive, then by the rules of expectation, for every lottery L : : E_L[u(x_1,x_2)] = E_L[u_1(x_1)] + E_L[u_2(x_2)] This expression depends only on the marginal probability distributions of L on the two attributes. This result generalizes to any number of attributes: if preferences over lotteries on attributes 1,...,n depend only on their marginal probability distributions, then the n-attribute utility function is additive: :::: u(x_1,\dots,x_n) = \sum_{i=1}^n{k_i u_i(x_i)} where u and the u_i are normalized to the range [0,1] , and the k_i are normalization constants. Much of the work in additive utility theory has been done by Peter C. Fishburn. == Utility independence == A slightly weaker independence property is utility independence. Attribute 1 is utility-independent of attribute 2, if the conditional preferences on lotteries on attribute 1 given a constant value of attribute 2, do not depend on that constant value. This means, for example, that the preference between a lottery and a lottery is the same, regardless of the value of x_2 . Note that utility independence (in contrast to additive independence) is not symmetric: it is possible that attribute 1 is utility-independent of attribute 2 and not vice versa. If attribute 1 is utility-independent of attribute 2, then the utility function for every value of attribute 2 is a linear transformation of the utility function for every other value of attribute 2. Hence it can be written as: : u(x_1,x_2)=c_1(x_2)+c_2(x_2)\cdot u(x_1,x_2^0) when x_2^0 is a constant value for attribute 2. Similarly, If attribute 2 is utility-independent of attribute 1: : u(x_1,x_2)=d_1(x_1)+d_2(x_1)\cdot u(x_1^0,x_2) If the attributes are mutually utility independent, then the utility function u has the following multi-linear form: : u(x_1,x_2)=u_1(x_1)+u_2(x_2)+k\cdot u_1(x_1)\cdot u_2(x_2) Where k is a constant which can be positive, negative or 0. * When k=0 , the function u is additive and the attributes are additive-independent. * When k\neq 0 , the utility function is multiplicative, since it can be written as: : [k u(x_1,x_2)+1]=[k u_1(x_1)+1] \cdot [k u_2(x_2)+1] :where each term is a linear transformation k\cdot+1 of a utility function. These results can be generalized to any number of attributes. Given attributes 1,...,n, if any subset of the attributes is utility-independent of its complement, then the n-attribute utility function is multi-linear and has one of the following forms: * 'Additive', or - * 'Multiplicative': :::: 1 + k u(x_1,\dots,x_n) = \prod_{i=1}^n{1+k k_i u_i(x_i)} where: * The u and the u_i are normalized to the range [0,1] ; * The k_i are constants in [0,1] ; * k is a constant which is either in (-1,0) or in (0,\infty) (note that the limit when k\to 0 is the additive form). == Comparison of independence concepts == It is useful to compare three different concepts related to independence of attributes: Additive-independence (AI), Utility-independence (UI) and Preferential-independence (PI). AI and UI both concern preferences on lotteries and are explained above. PI concerns preferences on sure outcomes and is explained in the article on ordinal utility. Their implication order is as follows: ::: AI ⇒ UI ⇒ PI AI is a symmetric relation (if attribute 1 is AI of attribute 2 then attribute 2 is AI of attribute 1), while UI and PI are not. AI implies mutual UI. The opposite is, in general, not true; it is true only if k=0 in the multi-linear formula for UI attributes. But if, in addition to mutual UI, there exist x_1,x_2,y_1,y_2 for which the two lotteries L and M , defined above, are equivalent - then k must be 0, and this means that the preference relation must be AI. UI implies PI. The opposite is, in general, not true. But if: * there are at least 3 essential attributes, and: * all pairs of attributes {1,i} are PI of their complement, and: * attribute 1 is UI of its complement, then all attributes are mutually UI. Moreover, in that case there is a simple relation between the cardinal utility function u representing the preferences on lotteries, and the ordinal utility function v representing the preferences on sure bundles. The function u must have one of the following forms: This idea is attributed to Richard F. Meyer and John W. Pratt. ↩a ↩b

[n2] Ralph L. Keeney and Howard Raiffa, Decisions with Multiple Objectives, Cambridge University Press. ↩

References¶

[1] Ralph L Keeney, Howard Raiffa, 'Decisions with Multiple Objectives', 1993. registry ↩a ↩b