The Choquet Integral is a mathematical aggregation method that can account for synergy and redundancy among evaluation criteria, rather than considering only the importance of individual criteria. It uses a non-additive measure to represent interactions among criteria that are difficult to capture through conventional weighted averages.
The mathematical foundations of the Choquet Integral date back to the work of French mathematician Gustave Choquet, who introduced the theory of capacities in 1954. The approach was subsequently developed through research on fuzzy measures and fuzzy integrals and has found applications in Multi-Criteria Decision Analysis (MCDA), pattern recognition, information fusion, reliability assessment, and business evaluation.
In a conventional Decision Matrix, each evaluation criterion is generally assigned a weight, and the overall score is calculated by multiplying each criterion’s score by its weight and summing the results. However, real-world evaluation criteria are not necessarily independent.
Consider evaluating a new product according to technological capability and manufacturing capability. Even if a company possesses advanced technology, commercialization may be difficult without the ability to manufacture the product reliably. Conversely, combining advanced technology with strong manufacturing capabilities may create a competitive advantage greater than the value of either capability considered separately.
Another problem arises when evaluation criteria overlap. For example, market growth and profitability may partially reflect the same underlying information. Adding their weighted scores independently may effectively count the same advantage more than once.
The Choquet Integral addresses these problems by assigning importance not only to individual criteria but also to combinations of criteria.
The defining characteristic of the Choquet Integral is its ability to evaluate not only how well individual criteria are satisfied, but also the additional value—or redundancy—that arises when those criteria are considered together.
However, the Choquet Integral cannot represent every possible nonlinear relationship. Understanding its mathematical structure is important for recognizing both its capabilities and its limitations.
Relationships That Weighted Averages Cannot Adequately Represent
A conventional weighted Decision Matrix calculates the overall evaluation of an alternative using the following formula:
V(a) = Σ wᵢ × vᵢ(a)
Here, wᵢ is the weight of criterion i, and vᵢ(a) is the evaluation score of alternative a under that criterion. The weights are typically normalized so that their sum equals 1.
Suppose technological capability and manufacturing capability are considered equally important, with each assigned a weight of 0.5.
Consider the following two alternatives.
| Alternative | Technological Capability | Manufacturing Capability |
|---|---|---|
| A | 1.00 | 0.20 |
| B | 0.55 | 0.55 |
The evaluation scores range from 0 to 1 and are assumed to represent comparable value scales.
Using a weighted average, the overall evaluations are:
A = 0.5 × 1.00 + 0.5 × 0.20 = 0.60
B = 0.5 × 0.55 + 0.5 × 0.55 = 0.55
Alternative A receives the higher score.
Its exceptionally high technological capability compensates for its weak manufacturing capability.
However, suppose successful commercialization requires both capabilities to reach a reasonable level. Under such conditions, is Alternative A necessarily preferable to Alternative B?
A company with outstanding technology but insufficient manufacturing capabilities may struggle to bring its product to market. Another company with moderately strong capabilities in both areas may be better positioned to commercialize its technology.
In a conventional weighted average, strong performance under one criterion can compensate for weak performance under another. This is known as compensatory evaluation.
The Choquet Integral makes it possible to represent preferences that place additional value on having multiple complementary capabilities simultaneously.
Non-Additive Measures and Capacities
The central concept underlying the Choquet Integral is the capacity.
In MCDA, capacities are also commonly referred to as non-additive measures or fuzzy measures.
A conventional weighted average assigns a weight to each individual criterion. In contrast, the Choquet Integral uses a set function that assigns values to combinations of criteria.
For example, let T represent technological capability and M represent manufacturing capability.
Suppose we define the following capacity values.
| Set of Criteria | Capacity μ |
|---|---|
| Empty set ∅ | 0.00 |
| Technological capability {T} | 0.30 |
| Manufacturing capability {M} | 0.30 |
| Both capabilities {T, M} | 1.00 |
In this example, the capacity assigned to technological capability alone is 0.30, and the capacity assigned to manufacturing capability alone is also 0.30.
However, the capacity assigned to their combination is 1.00.
The sum of their individual capacities is only 0.60. The additional contribution reflects a preference for the combination of technological and manufacturing capabilities.
For a conventional additive measure, the following relationship holds for disjoint sets A and B:
μ(A ∪ B) = μ(A) + μ(B)
A capacity used in the Choquet Integral does not need to satisfy this additivity requirement.
Instead, a standard normalized capacity satisfies the following conditions:
μ(∅) = 0
μ(N) = 1
μ(A) ≤ μ(B) whenever A ⊆ B
Here, N represents the set of all evaluation criteria.
The final condition is called monotonicity. It requires that the capacity assigned to a set of criteria cannot decrease when additional criteria are included.
A capacity represents the relative importance or contribution associated with a set of criteria within the evaluation model. It should not automatically be interpreted as a probability of an event.
The term fuzzy measure can also be misleading. The essential mathematical feature is not simply the numerical representation of vague judgments, but the use of a monotone set function that does not require additivity.
A Numerical Example of the Choquet Integral
Let us calculate the Choquet Integral for the technological and manufacturing capabilities example.
We use the following capacity:
μ({T}) = 0.30
μ({M}) = 0.30
μ({T, M}) = 1.00
For two nonnegative evaluation scores x and y, the Choquet Integral can be calculated by applying the capacity of the entire set to the lower score and the capacity of the higher-scoring criterion to the difference between the scores.
If x ≥ y, where x represents technological capability and y represents manufacturing capability, the formula becomes:
Cμ(x, y) = y × μ({T, M}) + (x − y) × μ({T})
Alternative A
Alternative A has a technological capability score of 1.00 and a manufacturing capability score of 0.20.
Therefore:
Cμ(A) = 0.20 × 1.00 + (1.00 − 0.20) × 0.30
= 0.20 + 0.24
= 0.44
Alternative B
Alternative B has a technological capability score of 0.55 and a manufacturing capability score of 0.55.
Because both scores are equal:
Cμ(B) = 0.55 × 1.00
= 0.55
The results can now be compared.
| Alternative | Weighted Average | Choquet Integral |
|---|---|---|
| A | 0.60 | 0.44 |
| B | 0.55 | 0.55 |
| Preferred Alternative | A | B |
The weighted average favors Alternative A, while the Choquet Integral favors Alternative B.
This difference arises because the Choquet model places additional value on possessing both technological and manufacturing capabilities.
It is particularly interesting that Alternative B’s score has not increased. Instead, the score of Alternative A has decreased because its outstanding technological capability is not matched by sufficient manufacturing capability.
In a model that values complementarity, achieving a high score under only one criterion provides less benefit than it would under an additive model.
Thus, even when individual evaluation criteria are treated symmetrically, their combined importance can produce different rankings from those obtained through a weighted average.
Of course, this calculation does not prove that Alternative B is objectively superior. The conclusion depends on whether the specified capacity appropriately represents the decision-maker’s preferences.
Synergy and Redundancy
The Choquet Integral can represent both positive interactions and overlapping contributions among criteria.
Consider two criteria, A and B.
Synergy or Complementarity
A positive interaction is represented when:
μ({A, B}) > μ({A}) + μ({B})
In this case, the contribution assigned to the combination exceeds the sum of the individual contributions.
The technological and manufacturing capabilities example illustrates this relationship.
Redundancy or Substitutivity
A negative interaction is represented when:
μ({A, B}) < μ({A}) + μ({B})
In this case, the criteria have overlapping or substitutable contributions.
For example, an R&D project may be evaluated using both market growth potential and market attractiveness. These criteria may partially reflect the same information.
Treating them as independent components of a weighted sum may overstate the importance of market-related advantages.
A non-additive capacity can represent this redundancy.
Additive Contributions
When:
μ({A, B}) = μ({A}) + μ({B})
the two criteria have no interaction in this particular additive sense.
However, when three or more criteria are involved, evaluating the interaction between two criteria may require considering their relationships with other criteria.
A simple comparison of two individual capacities is not always sufficient to characterize interactions throughout a more complex evaluation model.
The General Formula of the Choquet Integral
When there are n evaluation criteria, the discrete Choquet Integral can be calculated by arranging their scores in ascending order.
Let:
x(1) ≤ x(2) ≤ … ≤ x(n)
represent the sorted nonnegative evaluation scores.
Let A(i) denote the set of criteria whose scores are at least x(i), with x(0) = 0.
The discrete Choquet Integral is then:
Cμ(x) = Σ [x(i) − x(i−1)] × μ(A(i))
where the sum is taken from i = 1 to n.
The calculation evaluates successive increases in the ordered scores, weighting each increase according to the capacity assigned to the remaining set of criteria.
When the capacity is additive and the capacities of individual criteria correspond to conventional weights, the Choquet Integral reduces to the ordinary weighted sum.
In this sense, the Choquet Integral can be understood as a generalization of weighted averaging.
However, it is not a universal method for representing arbitrary nonlinear functions.
For a fixed capacity, the discrete Choquet Integral is linear within regions where the ordering of evaluation scores remains unchanged. Across the entire evaluation space, it is a piecewise-linear function whose expression changes according to the relative ordering of the scores.
This property is important when evaluating the expressive power of the Choquet Integral.
Representing Interactions with the Möbius Transform
A capacity can also be represented using the Möbius Transform.
The Möbius Transform decomposes the capacity assigned to sets of criteria into contributions associated with individual criteria and their combinations.
Let μ represent the capacity and m its Möbius representation.
Their relationship is:
μ(A) = Σ m(B), for all B ⊆ A
Conversely, the Möbius coefficients are calculated as:
m(A) = Σ (−1)^(|A|−|B|) μ(B), for all B ⊆ A
Consider the earlier two-criterion example.
μ({T}) = 0.30
μ({M}) = 0.30
μ({T, M}) = 1.00
The corresponding Möbius coefficients are:
m({T}) = 0.30
m({M}) = 0.30
m({T, M}) = 1.00 − 0.30 − 0.30 = 0.40
The final coefficient, 0.40, represents a positive interaction associated with the combination of technological and manufacturing capabilities.
For this two-criterion model, the Choquet Integral can be written as:
Cμ(xT, xM) = 0.30xT + 0.30xM + 0.40min(xT, xM)
Here, min(xT, xM) represents the lower of the two evaluation scores.
This expression makes the interaction particularly clear.
The additional contribution depends on the extent to which both capabilities are simultaneously present. If either capability is weak, the interaction term is limited.
However, Möbius coefficients are not identical to ordinary criterion weights.
They may be negative, and assigning coefficients independently can produce capacities that violate monotonicity.
Normalization and monotonicity constraints must therefore be checked when constructing a model using Möbius coefficients.
Interpreting Criterion Importance with Shapley Values
In a Choquet Integral model, examining the capacity assigned to an individual criterion does not necessarily reveal its overall importance.
Consider the earlier example, where technological capability and manufacturing capability each have a capacity of 0.30, but their combination has a capacity of 1.00.
How should the additional contribution of 0.40 be attributed to the two criteria?
One approach is to use the Shapley Value.
Originally developed in cooperative game theory, the Shapley Value calculates the average marginal contribution of an element across the different combinations in which it can participate.
In Choquet Integral models, Shapley Values can be used as indicators of the overall importance of individual criteria, taking their interactions with other criteria into account.
In the technological and manufacturing capabilities example, the two criteria are symmetric.
Their Shapley Values are therefore both 0.50.
| Criterion | Individual Capacity | Shapley Value |
|---|---|---|
| Technological capability | 0.30 | 0.50 |
| Manufacturing capability | 0.30 | 0.50 |
Although each criterion has an individual capacity of 0.30, its overall importance becomes 0.50 after accounting for its contribution to the combination.
Shapley Values can make it easier to interpret Choquet models and compare them with conventional weighting approaches.
An additional tool is the Interaction Index, which helps characterize synergy and redundancy among criteria.
These indices provide useful descriptions of the preference structure encoded in a capacity.
However, they should not be interpreted as proof of causal relationships among the corresponding real-world factors.
The Problem of Parameter Growth
One of the greatest practical challenges in using the Choquet Integral is the number of capacity values that may need to be specified.
For n evaluation criteria, the number of subsets is:
2ⁿ
In a normalized capacity, the empty set has capacity 0 and the full set has capacity 1.
Therefore, the number of capacity values that are not fixed by these two conditions is:
2ⁿ − 2
For example, five evaluation criteria require 30 such capacity values, while ten criteria require 1,022.
| Number of Criteria | Weights in a Weighted Sum | Non-Fixed Capacity Values |
|---|---|---|
| 2 | 2 | 2 |
| 3 | 3 | 6 |
| 5 | 5 | 30 |
| 10 | 10 | 1,022 |
Note: The counts assume that the capacities of the empty set and the full set are fixed. Conventional weights are also normalized to sum to 1, leaving one fewer independent degree of freedom than the number of criteria. Capacity values are additionally subject to monotonicity constraints.
As the number of criteria increases, manually specifying all the capacity values quickly becomes impractical.
To address this problem, researchers have developed methods for simplifying capacity models and estimating capacities from preference information.
The 2-Additive Choquet Integral
A particularly important practical approach is the 2-additive capacity model.
A general Choquet Integral can represent higher-order interactions involving three or more evaluation criteria.
However, in many practical decision problems, examining pairwise synergy and redundancy may already provide substantial improvements over a conventional weighted sum.
In a 2-additive model, Möbius coefficients associated with sets containing three or more criteria are constrained to zero.
The model therefore represents contributions from individual criteria and interactions between pairs of criteria.
For nonnegative evaluation scores, the resulting Choquet Integral can be expressed as:
Cμ(x) = Σ m({i})xᵢ + Σ m({i,j})min(xᵢ,xⱼ)
The first term represents contributions from individual criteria.
The second term represents pairwise interactions among criteria.
The number of coefficients becomes:
n + n(n−1)/2
For ten criteria, this consists of ten individual coefficients and 45 pairwise interaction coefficients, giving 55 coefficients in total.
Because normalization imposes one equality constraint, the model has 54 independent degrees of freedom before considering the additional monotonicity constraints.
This is substantially smaller than the 1,022 non-fixed capacity values required by a fully general normalized model with ten criteria.
However, even a 2-additive model cannot be specified by assigning arbitrary values to all coefficients.
The resulting capacity must satisfy monotonicity, and particular care is needed when positive and negative interaction coefficients coexist.
A 2-additive model also excludes interactions that are specific to combinations of three or more criteria.
Nevertheless, focusing initially on pairwise interactions can be a reasonable compromise between expressive power and practical usability.
How Are Capacities Determined?
The mathematical calculation of the Choquet Integral is relatively straightforward once a capacity has been specified.
The more difficult problem is determining the capacity itself.
One approach is to ask decision-makers to express the importance of individual criteria and the interactions among them.
For example, they might indicate that technological and manufacturing capabilities are strongly complementary, while market potential and expected profitability contain overlapping information.
However, statements such as “strong synergy” do not directly determine numerical capacity values.
One solution is to present decision-makers with hypothetical alternatives and ask them to express preferences.
For example, a decision-maker might compare an alternative with very high technological capability but weak manufacturing capability against another with moderate performance in both areas.
By collecting multiple preference comparisons, an analytical system can estimate capacities consistent with the observed judgments.
Historical decision data may also be useful.
If previous project evaluations and the decisions made from them are available, the model may be fitted to reproduce aspects of those preferences.
Because the discrete Choquet Integral is linear in the capacity parameters when evaluation scores are fixed, capacity estimation can, under appropriate formulations, be performed using methods such as linear programming.
However, reproducing historical decisions does not necessarily mean that the resulting model is suitable for future decisions.
Past choices may have involved biases, incomplete information, or strategic objectives that no longer apply.
Capacity estimation must therefore consider interpretability, monotonicity, model complexity, generalization, and the risk of overfitting.
Differences from AHP, ANP, and Decision Matrices
The Choquet Integral belongs to the broader field of MCDA, but it addresses a different modeling problem from AHP and ANP.
A Decision Matrix organizes alternatives and evaluation criteria into a table and typically combines scores through weighted aggregation.
The Analytic Hierarchy Process (AHP) derives relative priorities through pairwise comparisons.
The Analytic Network Process (ANP) extends AHP by representing dependencies and feedback among decision elements and calculating priorities using a supermatrix.
The Choquet Integral, by contrast, represents how the combined value of evaluation criteria may differ from the sum of their individual contributions.
| Method | Central Question | Main Mathematical Mechanism |
|---|---|---|
| Decision Matrix | How do alternatives compare across criteria? | Scores and weights |
| AHP | How important are the criteria relative to one another? | Pairwise comparisons and eigenvectors |
| ANP | How do decision elements influence one another? | Networks and supermatrices |
| Choquet Integral | How does value change when criteria are combined? | Non-additive capacities and integration |
| MAUT | How are preferences over multiple attributes and uncertain outcomes represented? | Utility functions and expected utility |
A particularly important distinction concerns interactions in ANP and the Choquet Integral.
ANP can represent the relative influence of one decision element on another.
The Choquet Integral represents whether satisfying several evaluation criteria together creates more or less value than would be expected from their separate contributions.
For example, the statement that technical feasibility influences development costs describes a relationship between decision factors.
The statement that technological and manufacturing capabilities create additional value when both are present describes an interaction in the evaluation of alternatives.
These are different modeling problems.
The Choquet Integral is particularly suited to the second.
Limitations of the Choquet Integral
Although the Choquet Integral offers greater flexibility than a weighted average, several limitations must be considered.
First, the evaluation scores must be expressed on appropriate, comparable value scales.
Combining quantities measured in unrelated units without suitable transformations is not meaningful. Even carefully estimated capacities cannot compensate for poorly constructed evaluation scales.
Second, specifying or estimating capacities can be difficult.
The number of capacity values increases rapidly with the number of criteria, making additional preference information or training data necessary.
Third, the expressive power of the standard Choquet Integral is limited.
For a fixed capacity, it is a piecewise-linear aggregation model. It cannot directly represent every possible functional dependency, nonlinear threshold, or temporal relationship.
Fourth, interactions in a Choquet model should not be confused with causal relationships.
Assigning a positive interaction between two criteria does not demonstrate that the corresponding real-world factors interact physically or causally.
The capacity represents a structure of preferences within the evaluation model.
Fifth, certain requirements are better treated as mandatory constraints rather than ordinary evaluation criteria.
For example, compliance with safety regulations or legal requirements may be a condition for accepting an alternative, regardless of its performance under other criteria.
The purpose of using the Choquet Integral is not to increase mathematical complexity for its own sake.
It is to represent important interactions among criteria when those interactions cannot be adequately captured by a simpler additive model.
Applications to R&D and New Business Evaluation
In research and development and new business planning, value often depends on combinations of capabilities and conditions.
For example, the evaluation of a new business opportunity may involve technological expertise, manufacturing capabilities, sales channels, brand strength, and customer relationships.
Evaluating these factors independently may fail to capture the conditions required for commercialization.
Advanced technology combined with manufacturing capability, competitive products combined with effective distribution channels, and customer needs combined with technological capabilities may create forms of value that depend on multiple factors being present simultaneously.
The Choquet Integral provides a way to incorporate such complementary relationships into an evaluation model.
It can also address redundancy.
Criteria such as market growth and expected profitability, or market share and competitive strength, may partially reflect overlapping information.
If they are treated as entirely independent criteria, a weighted sum may overestimate the contribution of certain advantages.
However, identifying possible synergy or redundancy is only the first step.
The strength of the interaction must also be established using reasonable assumptions, preference information, or empirical evidence.
In R&D evaluation, where uncertainty is often substantial, it is particularly important to conduct sensitivity analysis using alternative capacity settings.
Such analysis can reveal whether the preferred project changes when interactions are assumed to be stronger or weaker.
Another distinction is necessary when considering R&D portfolios.
Interactions among the criteria used to evaluate an individual project are not the same as interactions among separate R&D projects.
The former can be modeled through a conventional multi-criteria Choquet Integral.
The latter concerns portfolio selection and may require a different model to represent synergies, dependencies, or shared resources among projects.
The Choquet Integral in the Age of AI
Large Language Models (LLMs) create new possibilities for addressing some of the practical difficulties associated with the Choquet Integral.
Traditionally, analysts had to identify synergy and redundancy among evaluation criteria and translate those relationships into numerical capacities.
For many decision-makers, both tasks are difficult.
LLMs can help identify potential interactions from natural-language descriptions of decision problems.
For example, when evaluating new product development projects, an LLM might examine questions such as:
- Do technological and manufacturing capabilities create complementary value?
- Are market potential and expected profitability partially redundant?
- Does the value of an advanced technology depend on the availability of suitable production facilities?
- Are important interactions among evaluation criteria missing from the model?
LLMs could also retrieve supporting information from historical product development cases, technical literature, market research, and corporate documents.
However, an interaction suggested by an LLM is not necessarily an established fact.
General knowledge may suggest that two capabilities are complementary, but it does not automatically establish the strength of that relationship for a specific company or project.
A suitable division of responsibilities is therefore to use LLMs to suggest potential interactions and retrieve relevant evidence, let users confirm the assumptions and preferences, and use a mathematical engine for capacity estimation and Choquet Integral calculations.
Sensitivity analysis can then be used to examine how the results depend on the assumed interactions.
For example, an R&D project may receive the highest ranking when the synergy between technological and manufacturing capabilities is strong, but rank lower when that synergy is assumed to be weak.
Identifying such relationships helps reveal which judgments have the greatest influence on the final decision.
Another promising direction is estimating capacities from historical decisions or pairwise preference comparisons.
This connects the Choquet Integral with research on Preference Learning, interactive decision support, and incremental preference elicitation.
In such systems, the decision model could be progressively refined as users provide additional judgments.
The Choquet Integral in ThinkNavi
ThinkNavi regards the Choquet Integral as an advanced decision-making method for examining the value created by combinations of evaluation criteria.
The Decision Matrix planned for ThinkNavi Free provides a basic framework for assigning scores and weights to criteria and calculating overall evaluations.
Nonlinear value functions and sensitivity analysis are also planned to extend this framework.
However, nonlinear value functions for individual criteria do not, by themselves, represent interactions among criteria when their results are combined additively.
The Choquet Integral offers a way to address this limitation.
For a practical implementation, a 2-additive Choquet Integral would be a reasonable starting point.
Rather than requiring users to specify a large number of capacity values directly, the interface could allow them to define the importance of individual criteria and identify important pairwise interactions.
For example, users could indicate whether pairs of criteria are complementary, redundant, or approximately independent in their contributions to value.
A mathematical engine would translate the supplied preference information into a valid capacity model, checking normalization and monotonicity constraints.
The system could then compare the results of a conventional weighted sum and the Choquet Integral side by side.
This would help users understand which interactions are responsible for changes in the rankings of alternatives.
Shapley Values could be used to display the overall importance of individual criteria, while interaction indices could be presented through a matrix or network visualization.
Such visualizations would make the interpretation of a mathematically sophisticated model more accessible.
In an LLM-assisted version, AI could help identify potential interactions, retrieve supporting information, and guide users through the process of specifying or refining capacities.
The Choquet Integral could also be integrated into ConceptMiner R&D for evaluating research themes and hypotheses generated through cross-domain concept exploration.
For example, a project evaluation could account for combinations of technical feasibility, market potential, compatibility with existing technologies, and manufacturing capability.
However, the Choquet Integral need not become the standard aggregation method for every decision problem.
When interactions among criteria are negligible or do not materially affect the decision, a conventional weighted sum may be easier to understand and entirely sufficient.
The significance of the Choquet Integral does not lie in using a more sophisticated formula to calculate overall scores. It lies in making explicit the synergy and redundancy among evaluation criteria that conventional weighted sums often leave unexamined.
This allows decision-makers to consider not only which criteria are important, but also which combinations of criteria create value.
As a mathematical approach to representing human preferences, the Choquet Integral offers valuable possibilities for decision-support systems in the age of AI.
Related Thinking Methods:
- Multi-Criteria Decision Analysis (MCDA) | Mathematical Approaches to Integrating Multiple Criteria and Value Judgments
- Decision Matrix | Comparing Alternatives Across Multiple Criteria and Evaluating Decision Validity
- Analytic Hierarchy Process (AHP) | Quantifying Evaluation Criteria Through Pairwise Comparisons
- Analytic Network Process (ANP) | Decision-Making with Interdependencies and Feedback
- Multi-Attribute Value Theory (MAVT),
- Multi-Attribute Utility Theory (MAUT),
- Shapley Values, Sensitivity Analysis,
- Preference Learning,
- R&D Portfolio Analysis
References and Further Reading
- Gustave Choquet, “Theory of Capacities,” Annales de l’Institut Fourier, Vol. 5, pp. 131–295, 1954. Original Paper
- Michel Grabisch, “The Application of Fuzzy Integrals in Multicriteria Decision Making,” European Journal of Operational Research, Vol. 89, No. 3, pp. 445–456, 1996. Paper
- Michel Grabisch, “k-Order Additive Discrete Fuzzy Measures and Their Representation,” Fuzzy Sets and Systems, Vol. 92, No. 2, pp. 167–189, 1997. Paper
- Michel Grabisch and Christophe Labreuche, “A Decade of Application of the Choquet and Sugeno Integrals in Multi-Criteria Decision Aid,” Annals of Operations Research, Vol. 175, pp. 247–286, 2010. Paper
- M. Bottero, V. Ferretti, J. R. Figueira, S. Greco, and B. Roy, “On the Choquet Multiple Criteria Preference Aggregation Model: Theoretical and Practical Insights from a Real-World Application,” European Journal of Operational Research, Vol. 271, No. 1, pp. 120–140, 2018. Paper
- Nawal Benabbou, Patrice Perny, and Paolo Viappiani, “Incremental Elicitation of Choquet Capacities for Multicriteria Choice, Ranking and Sorting Problems,” Artificial Intelligence, Vol. 246, pp. 152–180, 2017. Paper