Nonlinear Utility Theory refers to mathematical approaches that represent the idea that changes in the quantity, performance, or characteristics of something do not necessarily produce proportional changes in the value or satisfaction people derive from it. These approaches are relevant to decision theory, economics, Multi-Criteria Decision Analysis (MCDA), investment evaluation, product development, business strategy, and many other fields.
For example, a person may perceive a substantial difference in convenience between apartments located five and ten minutes from a railway station, while attaching relatively little importance to the difference between 25 and 30 minutes. Similarly, improving a product’s performance may create considerable value until a certain level is reached, after which further improvements provide diminishing benefits.
These examples illustrate that changes in measurable characteristics do not necessarily correspond to proportional changes in perceived value.
Conventional Decision Matrices often assign numerical scores, such as values from 0 to 10, to different evaluation criteria and calculate overall scores using weighted averages. However, if the relationship between a characteristic and its perceived value is nonlinear, a simple proportional scoring scale may not adequately represent the decision-maker’s preferences.
Nonlinear value and utility functions provide mathematical tools for addressing this problem.
Strictly speaking, Nonlinear Utility Theory is not a single, independently established theoretical framework. Rather, it encompasses several related concepts, including nonlinear value functions for certain outcomes, utility functions for decisions under uncertainty, and multi-attribute utility models that represent interactions among evaluation criteria.
The essential idea is to distinguish measurable characteristics from the value people assign to them, and to represent that relationship through explicit mathematical models.
The Difference Between Value Functions and Utility Functions
An important starting point is the distinction between a Value Function and a Utility Function.
Although these terms are sometimes used interchangeably in practice, they have different roles in formal decision theory.
A value function generally represents preferences over outcomes under certainty. For example, it may express how desirable different commuting times, product performance levels, or development periods are to a decision-maker.
A utility function, in the context of expected utility theory, represents preferences over choices involving uncertain outcomes. For example, it can be used to compare an investment offering a guaranteed return with another investment that offers a potentially larger return but also involves a risk of loss.
| Aspect | Value Function | Utility Function |
|---|---|---|
| Main focus | Evaluation of outcomes under certainty | Preferences involving uncertain outcomes |
| Related theories | Multi-Attribute Value Theory (MAVT) | Expected Utility Theory, Multi-Attribute Utility Theory (MAUT) |
| Main purpose | Convert characteristics into values | Represent preferences over outcomes and risks |
| Example | Relationship between product performance and satisfaction | Relationship between financial outcomes and risk aversion |
For ranking certain outcomes, a value function can generally be transformed through a strictly increasing function without changing the ordering of those outcomes.
In expected utility theory, however, a utility function representing preferences that satisfy the relevant axioms is unique only up to a positive affine transformation.
This difference matters because expected utility depends on the mathematical structure of the utility scale, not merely on the ranking of outcomes.
Furthermore, when several value functions are combined in a multi-criteria model, their scales must be consistent with the aggregation procedure and its weights. Individual value functions cannot be transformed arbitrarily without potentially changing the overall evaluation.
Understanding these distinctions is essential for constructing meaningful nonlinear evaluation models.
Why Value Is Not Proportional to Performance
Doubling the performance of a product does not necessarily double its value to the user.
Consider computer processing speed.
When a computer is too slow to perform essential tasks, improving its performance may create substantial value. However, once processing speed is sufficient for the intended tasks, further improvements may provide relatively little additional benefit.
Similar relationships can arise with living space, transportation convenience, product durability, production capacity, and numerous other characteristics.
Nonlinear value functions provide mathematical representations of these relationships.
A simple family of functions is:
v(x) = x^α
Here, x is a characteristic normalized to the interval from 0 to 1, and α determines the shape of the curve.
When α = 1, the relationship is linear:
v(x) = x
When 0 < α < 1, the function is concave. Improvements at lower levels produce relatively large increases in value, while additional improvements at higher levels produce diminishing returns.
For example:
v(x) = √x
When α > 1, the function is convex. Improvements at lower levels have relatively little effect, while improvements at higher levels produce increasingly large gains in value.
For example:
v(x) = x²
The differences can be illustrated numerically.
| Characteristic x | Linear: x | Concave: √x | Convex: x² |
|---|---|---|---|
| 0.00 | 0.000 | 0.000 | 0.000 |
| 0.25 | 0.250 | 0.500 | 0.063 |
| 0.50 | 0.500 | 0.707 | 0.250 |
| 0.75 | 0.750 | 0.866 | 0.563 |
| 1.00 | 1.000 | 1.000 | 1.000 |
Suppose a decision-maker believes that reaching 25% of the maximum performance level already provides approximately half of the total possible value. A square-root function might be a reasonable candidate.
Conversely, if substantial value emerges only when performance approaches its highest levels, a convex function such as a quadratic curve may be more appropriate.
These are illustrative mathematical forms rather than universal descriptions of human preferences.
The appropriate function depends on the decision context and the preferences being represented.
Common Types of Nonlinear Value Functions
Several characteristic forms are useful for representing nonlinear value relationships.
Diminishing Returns
A diminishing-returns function represents a situation in which additional improvements become progressively less valuable.
For example, increasing the floor area of an apartment from 20 to 40 square meters may provide substantial additional value, while increasing it from 100 to 120 square meters may be considerably less important to the same person.
Concave functions, including logarithmic and square-root functions, can represent such relationships.
Increasing Returns
An increasing-returns function represents a situation in which improvements become more valuable as performance approaches higher levels.
For example, increasing the heat resistance of a material may provide limited value until its performance approaches the requirements of a high-temperature manufacturing process.
Convex functions can approximate such relationships.
However, if the material becomes usable only after satisfying a specific temperature requirement, a threshold model or mandatory constraint may be more appropriate than a continuously increasing value function.
S-Shaped Value Functions
An S-shaped function represents a relationship in which value increases slowly at first, rises rapidly over an intermediate range, and eventually approaches saturation.
For example, improvements in measurement accuracy may have limited practical value until a minimum level of precision is approached. Once the system becomes sufficiently accurate for a new application, its value may increase rapidly.
Further improvements beyond an already adequate precision level may provide diminishing additional benefits.
Logistic functions and other sigmoid models can represent such relationships.
Threshold Functions
A threshold function represents a significant change in value when a characteristic crosses a particular level.
For example, a product that does not satisfy a mandatory safety standard may be legally prohibited from entering the market.
In such circumstances, the value of compliance differs fundamentally from ordinary incremental improvements in performance.
However, mandatory legal or safety requirements are often better represented as hard constraints than as steep value functions.
When selecting a nonlinear function, the important question is not merely which curve looks appropriate, but what kind of judgment that curve is intended to represent.
Nonlinear Utility and Risk Preferences
Nonlinear utility functions are particularly important when decisions involve uncertainty.
Consider two alternatives.
Alternative A: Receive ¥500,000 with certainty.
Alternative B: Receive ¥1,000,000 with a probability of 50%, and receive nothing with a probability of 50%.
Both alternatives have the same expected monetary value:
Expected Value = ¥500,000
However, some people may prefer the guaranteed payment, while others may be willing to accept the uncertain outcome.
Expected Utility Theory can represent such differences through the shape of a utility function.
Suppose the decision-maker has the following utility function:
u(x) = √(x / 100)
Here, x represents the payment in units of ¥10,000, ranging from 0 to 100. Thus, x = 100 corresponds to ¥1,000,000.
The utility of ¥1,000,000 is 1, and the utility of receiving nothing is 0.
For Alternative A:
u(50) = √0.5 ≈ 0.707
For Alternative B, expected utility is:
EU(B) = 0.5 × u(100) + 0.5 × u(0)
= 0.5 × 1 + 0.5 × 0
= 0.5
Under this utility function, Alternative A is preferred because its utility exceeds the expected utility of Alternative B.
A concave utility function generally represents risk aversion over the relevant range of monetary outcomes.
Conversely, a convex utility function can represent risk-seeking preferences, while a linear utility function represents risk neutrality with respect to monetary outcomes.
It is important to recognize that expected utility itself is calculated as a probability-weighted sum:
EU = Σ pᵢu(xᵢ)
The nonlinearity lies in the relationship between the monetary outcome x and its utility u(x).
Maximizing expected monetary value is not necessarily the same as maximizing expected utility.
The square-root function in this example is an illustrative assumption. It does not imply that human preferences generally follow this particular mathematical form.
Certainty Equivalents and Risk Premiums
In utility theory, the Certainty Equivalent (CE) is the amount of a guaranteed outcome that provides the same utility as an uncertain alternative.
In the previous example, the expected utility of Alternative B is 0.5.
The utility function is:
u(x) = √(x / 100)
To find the certainty equivalent, we solve:
u(CE) = 0.5
The result is:
CE = ¥250,000
Thus, under the assumed utility function, the decision-maker evaluates the uncertain opportunity to receive ¥1,000,000 with a 50% probability as equivalent in utility to receiving ¥250,000 with certainty.
The expected monetary value of the uncertain alternative is ¥500,000.
The difference between the expected monetary value and the certainty equivalent is called the Risk Premium.
Risk Premium = Expected Monetary Value − Certainty Equivalent
In this example:
Risk Premium = ¥500,000 − ¥250,000 = ¥250,000
The risk premium indicates how much expected monetary value the decision-maker would be willing to sacrifice to eliminate the uncertainty associated with the outcome.
However, risk preferences depend on factors such as wealth, financial circumstances, tolerance for losses, and the purpose of the decision.
A utility function estimated for one range of outcomes or decision context should not automatically be assumed to apply universally.
Expected Utility Theory and Prospect Theory
Expected Utility Theory provides a fundamental framework for analyzing rational choices involving uncertain outcomes.
John von Neumann and Oskar Morgenstern established an axiomatic foundation showing that preferences satisfying certain conditions can be represented through expected utility.
However, observed human decisions do not always conform to the predictions of classical Expected Utility Theory.
In response to such findings, Daniel Kahneman and Amos Tversky introduced Prospect Theory in 1979.
Prospect Theory emphasizes that people often evaluate outcomes as gains or losses relative to a reference point, rather than solely in terms of their final wealth.
Its characteristic value function is generally concave in the domain of gains and convex in the domain of losses.
It also incorporates Loss Aversion, the tendency for losses to have a greater psychological impact than gains of the same magnitude.
Another important feature is that decision weights do not necessarily correspond directly to objective probabilities.
For example, people may overweight small probabilities or respond differently to mathematically equivalent choices depending on how they are presented.
However, Prospect Theory is not simply Expected Utility Theory with a different nonlinear function.
It introduces additional concepts, including reference dependence, loss aversion, and nonlinear probability weighting.
Prospect Theory is primarily a descriptive framework intended to explain patterns in actual human decision-making.
It should not automatically be treated as a normative model that decision-makers ought to follow in every situation.
Multi-Attribute Value Theory and Multi-Attribute Utility Theory
Real-world decisions usually involve more than one evaluation criterion.
For example, selecting an R&D project may require consideration of technical feasibility, market potential, development costs, and strategic importance.
Multi-Attribute Value Theory (MAVT) and Multi-Attribute Utility Theory (MAUT) provide mathematical frameworks for representing preferences involving multiple attributes.
In an additive multi-attribute value model, each criterion is associated with a value function, and the resulting values are combined using weights.
V(a) = Σ wᵢvᵢ(xᵢ(a))
Each individual value function vᵢ may be nonlinear.
For example, development costs could be represented through a function in which value decreases increasingly rapidly beyond a certain cost level.
Market potential might be represented through a function in which value rises substantially after the market reaches a commercially attractive scale.
Such functions can represent the characteristics of individual evaluation criteria more appropriately than simple proportional scoring.
However, nonlinear value functions for individual criteria do not automatically represent interactions among those criteria.
When the individual values are combined additively, the overall model retains an additive structure.
Certain preferential independence conditions are required to justify this form of aggregation.
For example, if the value of proximity to a railway station depends strongly on commuting frequency, a simple additive model may fail to represent the decision-maker’s preferences adequately.
In MAUT, which deals with uncertain outcomes, preferential independence and utility independence must also be distinguished.
The mathematical conditions supporting an additive utility model under uncertainty are not identical to those used for an additive value model under certainty.
Nonlinear Interactions Among Multiple Criteria
When the value of an alternative depends on combinations of criteria, nonlinear functions for individual criteria may not be sufficient.
Consider evaluating an R&D project according to technological capability and manufacturing capability.
A project may possess advanced technology but lack the manufacturing capability required for commercialization.
Conversely, when both capabilities are present, their combination may create substantial additional value.
A simple mathematical representation of such an interaction is:
V(x₁, x₂) = 0.25x₁ + 0.25x₂ + 0.50x₁x₂
Here, x₁ represents technological capability, x₂ represents manufacturing capability, and both range from 0 to 1.
The final term,
0.50x₁x₂
represents an additional contribution associated with the combination of the two criteria.
Consider two alternatives.
| Alternative | Technological Capability | Manufacturing Capability |
|---|---|---|
| A | 1.00 | 0.20 |
| B | 0.55 | 0.55 |
Under a simple weighted average with equal weights, Alternative A scores 0.60 and Alternative B scores 0.55.
Alternative A ranks first.
Under the interaction model:
V(A) = 0.25 × 1.00 + 0.25 × 0.20 + 0.50 × 1.00 × 0.20
= 0.400
For Alternative B:
V(B) = 0.25 × 0.55 + 0.25 × 0.55 + 0.50 × 0.55 × 0.55
= 0.42625
Alternative B now ranks first.
This occurs because the model assigns additional value to alternatives that possess both capabilities.
The formula is an illustrative multi-attribute value function containing an interaction term. It does not imply that every multi-attribute utility model can be represented in this form.
The important distinction is between nonlinearity within individual evaluation criteria and interactions among different criteria.
These are separate modeling problems and may require different mathematical approaches.
Relationship with the Choquet Integral
The Choquet Integral is another method for representing interactions among evaluation criteria.
It uses a non-additive measure, called a capacity, to assign importance not only to individual criteria but also to combinations of criteria.
For example, a Choquet model can represent synergy between technological and manufacturing capabilities or redundancy between market growth and expected profitability.
In the previous example, the interaction was represented by the product:
x₁x₂
In a 2-additive Choquet Integral model, pairwise interactions can instead be represented using terms involving:
min(x₁, x₂)
Both approaches can represent relationships that cannot be captured by a simple weighted sum, but they have different mathematical properties.
A discrete Choquet Integral with a fixed capacity is a piecewise-linear aggregation function. It does not provide an unrestricted representation of arbitrary smooth nonlinear relationships.
Multi-attribute value and utility models may use multiplicative forms, interaction terms, or other functional structures under appropriate assumptions.
However, the more flexible the model becomes, the more difficult it may be to determine its parameters and interpret the resulting evaluations.
Neither approach is universally superior.
The appropriate method depends on the preference relationships that need to be represented.
How Are Nonlinear Value Functions Determined?
An important practical challenge is deciding which functional form and parameters should be used.
For example, suppose shorter development time is considered preferable.
Even if everyone agrees that a six-month development period is better than a twelve-month period, it may be difficult to determine how much more valuable the shorter period is.
One approach is to ask decision-makers to evaluate selected performance levels.
For example, suppose a development period of six months is assigned a value of 1, while a period of 24 months is assigned a value of 0.
The decision-maker can then be asked to evaluate a twelve-month development period relative to those reference points.
Further judgments at nine, fifteen, or eighteen months can help determine the shape of the value function.
However, simply assigning arbitrary numerical ratings is not always sufficient.
A meaningful value function should reflect judgments about differences in value.
For example, the decision-maker might compare the loss in value associated with extending development from six to twelve months with the loss associated with extending it from twelve to eighteen months.
Such comparisons can provide information about the curvature of the value function.
For utility functions involving uncertainty, preferences can instead be elicited by comparing certain outcomes with lotteries.
For example, a decision-maker might compare receiving ¥500,000 with certainty against a 50% chance of receiving ¥1,000,000 and a 50% chance of receiving nothing.
By repeating such comparisons with different amounts and probabilities, it becomes possible to estimate aspects of the decision-maker’s risk preferences.
However, people do not necessarily provide perfectly consistent responses to complex comparisons.
Judgments may also be influenced by framing, presentation order, or the context in which questions are asked.
Therefore, practical utility elicitation should emphasize interpretable functions, consistency checks, and sensitivity analysis rather than unnecessary mathematical complexity.
Learning Nonlinear Utility from Data
When historical decisions or preference judgments are available, value and utility functions may also be estimated from data.
For example, an organization may have records of previous equipment selections, including prices, technical specifications, maintenance costs, and the alternatives ultimately chosen.
Analyzing such decisions may reveal patterns in how the organization evaluates competing characteristics.
This approach is related to Preference Learning and statistical choice modeling.
Methods such as Generalized Additive Models (GAMs), splines, and monotonic regression can also be useful for estimating nonlinear relationships between characteristics and evaluation outcomes.
However, observing that an alternative was selected does not prove that it was the most desirable alternative.
Past decisions may have been affected by budget restrictions, incomplete information, organizational constraints, or established practices.
Furthermore, a model that accurately predicts historical choices is not necessarily a model that appropriately represents the decision-maker’s underlying values.
For practical decision support, learned preference models should be interpretable and open to review and revision by their users.
Bayesian Networks and Influence Diagrams
Nonlinear utility functions are also relevant to decisions involving uncertain technological or market outcomes.
For example, evaluating an R&D project may require estimating the probability of technical success, the probability of market growth, and the investment required for commercialization.
A Bayesian Network represents probabilistic dependencies among events and can calculate conditional probabilities based on available evidence.
However, the R&D project with the highest probability of success is not necessarily the most desirable project.
A project with a lower probability of success may offer much greater strategic or financial value if successful.
This creates a need to combine probabilistic models with value or utility models.
One established framework is the Influence Diagram.
Influence Diagrams represent chance variables, decisions, and utilities within a unified graphical model.
They can be used to compare available actions according to expected utility under uncertainty.
By incorporating nonlinear utility functions, the model can represent not only the magnitude of possible outcomes but also the decision-maker’s preferences concerning risk.
However, probability and utility are fundamentally different concepts.
The probability that an event will occur does not indicate how desirable that event is.
Similarly, specifying a nonlinear utility function does not establish that the underlying probabilistic model is correct.
Both the probability model and the utility model require appropriate assumptions and supporting evidence.
Applications to R&D and New Business Development
Nonlinear value and utility models can be particularly important in R&D and new business evaluation.
For example, the value generated by improving a product’s performance depends on its intended application.
A product may be unable to enter a particular market until its performance satisfies a minimum requirement.
Once that requirement is met, new applications may become possible and the product’s value may increase rapidly.
Further performance improvements beyond a commercially sufficient level may then provide diminishing benefits.
A value function with thresholds or saturation may represent this situation more appropriately than a simple linear score.
Development time provides another example.
If being first to market is critical, a relatively small delay beyond a particular date may sharply reduce the expected value of a project.
In contrast, for long-term fundamental research, a difference of several months may have relatively little strategic significance.
These situations involve more than different criterion weights.
They may require fundamentally different value functions for the same evaluation criterion.
In new business development, value may also depend on combinations of technological capabilities, manufacturing resources, sales channels, and customer relationships.
Such interactions may require multi-attribute models with explicit interaction terms or non-additive aggregation.
Risk preferences may also matter.
A new business opportunity may offer substantial returns if successful but involve major losses if it fails.
A nonlinear utility function can help represent the decision-maker’s attitude toward these uncertain outcomes.
However, evaluating innovative R&D projects is not always simply a matter of maximizing expected utility based on known probabilities.
For genuinely novel research, the probability distributions of important outcomes may themselves be poorly understood.
In such circumstances, identifying critical uncertainties and designing experiments to resolve them may be more valuable than attempting to construct an artificially precise expected utility calculation.
Validating Utility Models Through Sensitivity Analysis
After a nonlinear utility or value function has been specified, it is important to examine how strongly the results depend on the function’s shape.
For example, using a square-root value function to evaluate product performance may produce different rankings from using a quadratic function.
Similarly, in an S-shaped function, changing the point at which value begins to increase rapidly can significantly affect the resulting evaluations.
Sensitivity Analysis can be used to investigate these relationships.
Parameters such as curvature, thresholds, criterion weights, and interaction coefficients can be modified to examine their influence on overall scores and rankings.
For decisions involving uncertainty, the degree of risk aversion can also be varied to determine whether the preferred alternative changes.
If a decision depends strongly on a particular functional assumption, that assumption deserves careful examination.
Conversely, if the same alternative remains preferable across a wide range of plausible functional forms and parameter values, the decision may be considered relatively robust.
However, numerical stability does not prove that the underlying model is appropriate.
Sensitivity Analysis should therefore be combined with examination of criterion definitions, the interpretation of value functions, and the decision-maker’s actual preferences.
Nonlinear Utility Theory in the Age of AI
Large Language Models (LLMs) create new possibilities for making nonlinear value and utility models more accessible.
Traditionally, users needed to choose mathematical functions and adjust numerical parameters directly.
For many decision-makers, specifying a utility function through mathematical formulas is difficult.
LLMs could instead help translate natural-language preferences into candidate mathematical models.
For example, a user might explain:
“Meeting the minimum performance requirement is critical, but improvements beyond that level provide relatively little additional value.”
An LLM could suggest threshold-based or saturating value functions as possible representations.
Similarly, a statement such as:
“If development takes longer than twelve months, the commercial value decreases sharply.”
could lead to a function that changes rapidly around the specified threshold.
However, identifying a plausible functional form is not the same as determining its numerical parameters accurately.
The parameters should be refined through comparisons, reference points, or other preference judgments supplied by the user.
A practical interactive process could involve an LLM proposing questions, the user expressing preferences, and a mathematical engine estimating the corresponding function.
Historical decisions and evaluation records could provide additional information for progressively refining the preference model.
However, learned preferences may not perfectly represent the user’s current objectives or values.
Users should therefore be able to inspect, modify, and reject proposed functions.
Nonlinear Utility Theory in ThinkNavi
ThinkNavi regards nonlinear value and utility models as important extensions of conventional Decision Matrices and Multi-Criteria Decision Analysis.
ThinkNavi Free is planned to provide tools that allow users to enter evaluation criteria, alternatives, scores, and weights manually and calculate overall evaluations.
Introducing nonlinear value functions would allow users to represent the relationship between each criterion’s measured characteristics and its perceived value, rather than relying solely on proportional scoring.
For example, users could select linear, diminishing-returns, increasing-returns, S-shaped, or threshold functions for individual evaluation criteria.
An interactive graph could allow them to adjust parameters and immediately examine how the value function changes.
Alternatively, users could assign values to several reference points, and the system could construct a monotonic curve that reflects those judgments.
This would allow users to work with nonlinear evaluation models without necessarily understanding the underlying mathematical formulas.
The system could also compare the results of linear and nonlinear evaluation side by side, showing which value functions are responsible for changes in the rankings of alternatives.
When interactions among criteria are important, the evaluation model could be extended using the Choquet Integral or other multi-attribute value functions.
When outcomes are uncertain, additional functions could support risk preferences, expected utility, certainty equivalents, and risk premiums.
These capabilities could be implemented as shared mathematical components used by Decision Matrices, AHP, MCDA, Sensitivity Analysis, and other ThinkNavi tools.
In an LLM-assisted version, AI could suggest value functions based on natural-language descriptions of user preferences, formulate questions to elicit additional judgments, and help users examine the meaning and validity of their models.
Nonlinear evaluation models could also be integrated into ConceptMiner R&D.
For example, R&D themes and business hypotheses generated through concept exploration could be evaluated using nonlinear relationships involving performance, development time, market size, technical feasibility, and other criteria.
However, introducing a nonlinear utility model does not automatically make a decision more rational.
The essential requirement is that users understand the meaning of the functions being applied and can revise them when necessary.
The significance of Nonlinear Utility Theory does not lie in calculating evaluations through more complicated functions. It lies in recognizing that physical performance and economic quantities do not necessarily correspond proportionally to human value—and making those relationships explicit and open to examination.
Computers and AI can make such mathematical representations easier to construct and use, supporting more transparent, systematic, and critically examined decision-making.
Related Thinking Methods:
- Multi-Criteria Decision Analysis (MCDA), Multi-Criteria Decision Analysis (MCDA) | Mathematical Approaches to Integrating Multiple Criteria and Value Judgments
- Decision Matrix, Decision Matrix | Comparing Alternatives Across Multiple Criteria and Evaluating Decision Validity
- Analytic Hierarchy Process (AHP), Analytic Hierarchy Process (AHP) | Quantifying Evaluation Criteria Through Pairwise Comparisons
- Analytic Network Process (ANP),
- Choquet Integral,
- Multi-Attribute Value Theory (MAVT),
- Multi-Attribute Utility Theory (MAUT),
- Expected Utility Theory,
- Prospect Theory,
- Bayesian Networks,
- Influence Diagrams,
- Sensitivity Analysis,
- Preference Learning
References and Further Reading
- John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, 1944.
- Ralph L. Keeney and Howard Raiffa, Decisions with Multiple Objectives: Preferences and Value Tradeoffs, Wiley, 1976. Reprinted by Cambridge University Press, 1993.
- Ralph L. Keeney, “Utility Independence and Preferences for Multiattributed Consequences,” Operations Research, 19(4), pp. 875–893, 1971. Paper
- Ralph L. Keeney, “Multiplicative Utility Functions,” Operations Research, 22(1), pp. 22–34, 1974. Paper
- Daniel Kahneman and Amos Tversky, “Prospect Theory: An Analysis of Decision under Risk,” Econometrica, 47(2), pp. 263–291, 1979. Paper
- Ali E. Abbas, “Constructing Multiattribute Utility Functions for Decision Analysis,” INFORMS Tutorials in Operations Research, pp. 62–98, 2010. Paper
- Stanford Encyclopedia of Philosophy — Normative Theories of Rational Choice: Expected Utility
- Fifty Years of Multiple Criteria Decision Analysis: From Classical Methods to Robust Ordinal Regression