Business intelligence (BI) consists of strategies, methodologies, and technologies used by enterprises for data analysis and management of business information to inform business strategies and business operations. Common functions of BI technologies include reporting, online analytical processing, analytics, dashboard development, data mining, process mining, complex event processing, business performance management, benchmarking, text mining, predictive analytics, and prescriptive analytics. BI tools can handle large amounts of structured and sometimes unstructured data to help organizations identify, develop, and otherwise create new strategic business opportunities. They aim to allow for the easy interpretation of these big data. Identifying new opportunities and implementing an effective strategy based on insights is assumed to potentially provide businesses with a competitive market advantage and long-term stability, and help them take strategic decisions. Business intelligence can be used by enterprises to support a wide range of business decisions ranging from operational to strategic. Basic operating decisions include product positioning or pricing. Strategic business decisions involve priorities, goals, and directions at the broadest level. In all cases, business intelligence is considered most effective when it combines data from the market in which a company operates (external data) with data from internal company sources, such as financial and operational information. When integrated, external and internal data provide a comprehensive view that creates ‘intelligence’ not possible from any single data source alone. Among their many uses, business intelligence tools empower organizations to gain insight into new markets, to assess demand and suitability of products and services for different market segments, and to gauge the impact of marketing efforts. BI applications use data gathered from a data warehouse (DW) or from a data mart, and the concepts of BI and DW combine as "BI/DW" or as "BIDW". A data warehouse contains a copy of analytical data that facilitates decision support. == History == The earliest known use of the term business intelligence is in Richard Millar Devens' Cyclopædia of Commercial and Business Anecdotes (1865). Devens used the term to describe how the banker Sir Henry Furnese gained profit by receiving and acting upon information about his environment, prior to his competitors: Throughout Holland, Flanders, France, and Germany, he maintained a complete and perfect train of business intelligence. The news of the many battles fought was thus received first by him, and the fall of Namur added to his profits, owing to his early receipt of the news. The ability to collect and react accordingly based on the information retrieved, Devens says, is central to business intelligence. When Hans Peter Luhn, a researcher at IBM, used the term business intelligence in an article published in 1958, he employed the Webster's Dictionary definition of intelligence: "the ability to apprehend the interrelationships of presented facts in such a way as to guide action towards a desired goal." In 1989, Howard Dresner (later a Gartner analyst) proposed business intelligence as an umbrella term to describe "concepts and methods to improve business decision making by using fact-based support systems." It was not until the late 1990s that this usage was widespread. == Definition == According to Solomon Negash and Paul Gray, business intelligence (BI) can be defined as systems that combine: Data gathering Data storage Knowledge management with analysis to evaluate complex corporate and competitive information for presentation to planners and decision makers, with the objective of improving the timeliness and the quality of the input to the decision process." According to Forrester Research, business intelligence is "a set of methodologies, processes, architectures, and technologies that transform raw data into meaningful and useful information used to enable more effective strategic, tactical, and operational insights and decision-making." Under this definition, business intelligence encompasses information management (data integration, data quality, data warehousing, master-data management, text- and content-analytics, et al.). Therefore, Forrester refers to data preparation and data usage as two separate but closely linked segments of the business-intelligence architectural stack. Some elements of business intelligence are: Multidimensional aggregation and allocation Denormalization, tagging, and standardization Realtime reporting with analytical alert A method of interfacing with unstructured data sources Group consolidation, budgeting, and rolling forecasts Statistical inference and probabilistic simulation Key performance indicators optimization Version control and process management Open item management Forrester distinguishes this from the business-intelligence market, which is "just the top layers of the BI architectural stack, such as reporting, analytics, and dashboards." === Compared with competitive intelligence === Though the term business intelligence is sometimes a synonym for competitive intelligence (because they both support decision making), BI uses technologies, processes, and applications to analyze mostly internal, structured data and business processes while competitive intelligence gathers, analyzes, and disseminates information with a topical focus on company competitors. If understood broadly, competitive intelligence can be considered as a subset of business intelligence. === Compared with business analytics === Business intelligence and business analytics are sometimes used interchangeably, but there are alternate definitions. Thomas Davenport, professor of information technology and management at Babson College argues that business intelligence should be divided into querying, reporting, Online analytical processing (OLAP), an "alerts" tool, and business analytics. In this definition, business analytics is the subset of BI focusing on statistics, prediction, and optimization, rather than the reporting functionality. == Unstructured data == Business operations can generate a very large amount of data in the form of emails, memos, notes from call centers, news, user groups, chats, reports, web pages, presentations, image files, video files, and marketing material. According to Merrill Lynch, more than 85% of all business information exists in these forms; a company might only use such a document a single time. Because of the way it is produced and stored, this information is either unstructured or semi-structured. The management of semi-structured data is an unsolved problem in the information technology industry. According to projections from Gartner (2003), white-collar workers spend 30–40% of their time searching, finding, and assessing unstructured data. BI uses both structured and unstructured data. The former is easy to search, and the latter contains a large quantity of the information needed for analysis and decision-making. Because of the difficulty of properly searching, finding, and assessing unstructured or semi-structured data, organizations may not draw upon these vast reservoirs of information, which could influence a particular decision, task, or project. This can ultimately lead to poorly informed decision-making. Therefore, when designing a business intelligence/DW solution, the specific problems associated with semi-structured and unstructured data must be accommodated, as well as those associated with structured data. === Limitations of semi-structured and unstructured data === There are several challenges to developing BI with semi-structured data. According to Inmon & Nesavich, some of those are: Physically accessing unstructured textual data – unstructured data is stored in a huge variety of formats. Terminology – Among researchers and analysts, there is a need to develop standardized terminology. Volume of data – As stated earlier, up to 85% of all data exists as semi-structured data. Couple that with the need for word-to-word and semantic analysis. Searchability of unstructured textual data – A simple search on some data, e.g. apple, results in links where there is a reference to that precise search term. (Inmon & Nesavich, 2008) gives an example: "a search is made on the term felony. In a simple search, the term felony is used, and everywhere there is a reference to felony, a hit to an unstructured document is made. But a simple search is crude. It does not find references to crime, arson, murder, embezzlement, vehicular homicide, and such, even though these crimes are types of felonies". === Metadata === To solve problems with searchability and assessment of data, it is necessary to know something about the content. This can be done by adding context through the use of metadata. Many systems already capture some metadata (e.g. filename, author, size, etc.), but more usef
Dating app
An online dating application, commonly known as a dating app, is an online dating service presented through a mobile phone application. These apps often take advantage of a smartphone's GPS location capabilities, always on-hand presence, and access to mobile wallets. These apps aim to speed up the online dating process of sifting through potential dating partners, chatting, flirting, and potentially meeting or becoming romantically involved. Online dating apps are now mainstream in the United States. As of 2017, online dating (which included both apps and other online dating services) was the principal method by which new couples in the U.S. met. The percentage of couples meeting online is predicted to increase to 70% by 2040. == Origins == The first computerized dating service was launched in 1964, the St. James Computer Dating Service, which became known as Com-Pat. The first U.S. dating service that used computerized match making was Operation Match. It required men and women to complete a questionnaire and was launched in 1965. Operation Match inspired the methodology of Dateline, which became popular in the 1970s and 1980s. Match.com was launched in 1995 and turned computerized match making into a profitable business. Grindr targeted gay and bisexual men at launch. Tinder, launched in 2012, led to a growth of online dating applications by both new providers and existing online dating services that expanded into the mobile app market. == Usage by demographic group == Online dating applications typically target a younger demographic group, though some apps, like Senior Match and Silver Singles are geared toward the 50 and up demographic. In 2016, almost 50% of people knew of someone who use the services or had met their loved one through the service. After the iPhone launch in 2007, online dating data has mushroomed as application usage increased. In 2005, only 10% of 18-24 year olds reported to have used online dating services; this number quickly grew to over 27%, making this target demographic the largest number of users for most applications. When Pew Research Center conducted a study in 2016, they found that 59% of U.S. adults agreed that online dating is a good way to meet people compared to 44% in 2005. This explosion in usage can be explained by the increased use of smartphones. By the end of 2022, it is expected there will be 413 million active users of online dating services worldwide. A 2023 Pew Research Center survey of 6,034 American adults found that 30% had ever used an online dating site or app, including 53% of those aged 18 to 29, 37% of those aged 30 to 49, and 17% of those aged 50 and over. Lesbian, gay and bisexual respondents reported using dating apps at nearly twice the rate of straight respondents (51% versus 28%), and 36% of divorced, separated or widowed adults had used one, compared with 16% of married adults. The increased use of smartphones by those 65 and older has also driven that population to the use dating apps. The Pew Research Center found that usage increase by 8 points since last surveyed in 2012. A study in 2021 found that more than one-third of seniors have dated in the past 5 years, and roughly one-third of those dating seniors have turned to dating apps. During the COVID-19 pandemic, Morning Consult found that more Americans were using online dating apps than ever before. In one survey in April 2020, the company discovered that 53% of U.S. adults who use online dating apps have been using them more during the pandemic. As of February 2021, that share increased to 71 percent. Research using Hofstede's cultural dimensions theory has indicated that norms about online dating applications tend to differ across cultures. A study published in the Journal of Creative Communications looked into the relationships between dating-app advertisements from over 51 countries and the cultural dimensions of these countries. The results revealed that dating-app advertisements appealed to multiple cultural needs, including the needs for relationships, friendship, entertainment, sex, status, design and identity. The use of these appeals was found to be 'congruent with ... the individualism/collectivism and uncertainty avoidance cultural dimensions.' == Popular applications == Following Tinder's success, other companies released dating applications. Raya was released in 2015 as a membership-based dating app, allowing entrance only through referrals, which was marketed as a dating app for celebrities. In early 2026, Hily surpassed Bumble to become the third most-used dating application in the United States and the fifth highest-grossing overall, driven largely by growing popularity among Generation Z users, while remaining behind Tinder and Hinge, both owned by Match Group. A number of dating apps have been created targeting adherents of particular religions seeking partners of the same religion, such as Muzmatch for Muslims, Christian Mingle, SALT, and Christian Connection for Christians, and JSwipe and JDate for Jews. === VR Dating === VR Dating is an application of Social VR where people can exist, collaborate, and perform various activities together. Virtual reality apps use virtual and augmented realities to make the dating experience more lifelike and more effective, as well as allow people to expand what is already possible in the world of online dating. There are several online platforms of VR Dating. The VR dating app Nevermet is the VR equivalent of Tinder, where people can search and find on dates. However, instead of actual real-life pictures, users will update pictures of virtual selves and will be interacting with avatars rather than real faces. Flirtual is a self-contained social VR app that serves to match users who then decide where and how to meet in VR. Flirtual hosts speed dating and social events in VR. == Effects on dating == The usage of online dating applications can have both advantages and disadvantages: === Advantages === Many of the applications provide personality tests for matching or use algorithms to match users. These factors enhance the possibility of users getting matched with a compatible candidate. Users are in control; they are provided with many options so there are enough matches that fit their particular type. Users can simply choose to not match the candidates that they know they are not interested in. Narrowing down options is easy. Once users think they are interested, they are able to chat and get to know the potential candidate. This form of communication can reduce the time, cost, and uncertainty often associated with traditional dating methods. Online dating offers convenience; people want dating to work around their schedules. Online dating can also increase self-confidence; even if users get rejected, they know there are hundreds of other candidates that will want to match with them so they can simply move on to the next option. In fact, 60% of U.S. adults agree that online dating is a good way to meet people and 66% say they have gone on a real date with someone they met through an application. Today, 5% of married Americans or Americans in serious relationships said they met their significant other online. The 39% of online dating users (representing 12% of all U.S. adults) say they have been in a committed relationship or married someone they met on a dating site or app. ==== Rejection sensitive individuals ==== Individuals high in rejection sensitivity are more likely to use online dating applications. As they tend to expect, perceive and overreact to rejection, rejection sensitive individuals struggle with traditional dating. Online dating applications allow for them to better reveal their true selves, potentially increasing their dating success. Online dating applications also obscure rejection cues, alleviating the rejection-related distress experienced by rejection sensitive individuals. ==== Anxiously attached individuals ==== Individuals high in attachment anxiety are also more likely to use online dating applications. While they harbour negative self-views, anxiously attached individuals are also more eager to enter and commit to relationships. Online dating applications allow for them to present "an authentic yet ideal version of themselves", mitigating their tendencies to view themselves as undesirable. This increases their romantic confidence, and potentially alleviates their anxiety when searching for a romantic partner. === Disadvantages === Sometimes having too many options can be overwhelming. With so many options available, users can get lost in their choices and end up spending too much time looking for the "perfect" candidate instead of using that time to start a real relationship. In addition, the algorithms and matching systems put in place may not always be as accurate as users think. There is no perfect system that can match two people's personalities perfectly every time. Communication online also lacks the physical chemistry aspec
Agent mining
Agent mining is a research field that combines two areas of computer science: multiagent systems and data mining. It explores how intelligent computer agents can work together to discover, analyze, and learn from large amounts of data more effectively than traditional methods. == Historical context == The interaction and the integration between multiagent systems and data mining have a long history. The very early work on agent mining focused on agent-based knowledge discovery, agent-based distributed data mining, and agent-based distributed machine learning, and using data mining to enhance agent intelligence. The International Workshop on Agents and Data Mining Interaction has been held for more than 10 times, co-located with the International Conference on Autonomous Agents and Multi-Agent Systems. Several proceedings are available from Springer Lecture Notes in Computer Science.
It's the Most Terrible Time of the Year
It's the Most Terrible Time of the Year is an AI-generated television commercial created for McDonald's Netherlands by TBWA\Neboko and The Sweetshop. It was released on 6 December 2025 before being pulled four days later due to negative reception over its use of generative artificial intelligence and its cynical, negative depiction of the holiday season. == Plot == On a bleak, snowy day, various people in the city experience different kinds of mishaps during the Christmas season. Among other incidents, families struggle with their huge loads of presents; Santa Claus gets stuck in traffic; a Christmas tree "redecorates" a man's home, sending him through the window; another family puts up with annoying relatives and a burnt Christmas dinner. Because of all this chaos, a man decides to find refuge in a McDonald's outlet. A Christmas choir finishes singing the jingle "It's the Most Terrible Time of the Year" with the call to action to "hide out in McDonald's till January's here". == Campaign == It's the Most Terrible Time of the Year is a 45-second television commercial made by Dutch agency TBWA\Neboko with involvement of United States-based film production studio The Sweetshop. The advertisement was produced heavily with generative artificial intelligence (AI) following the trend set by other brands such as Coca-Cola and Toys "R" Us. McDonald's Netherlands, the client, released a statement that the commercial was meant to depict "the stressful moments during the holidays in the Netherlands". The commercial also used Andy Williams's "It's the Most Wonderful Time of the Year" with lyrics changed to fit with the concept of the advertisement. According to The Sweetshop, the production of the advertisement took "seven weeks". It also added that much effort was put into the commercial compared to the traditional process. Ten people of its in-house AI engine The Gardening Club worked on the project. Los Angeles-based directors Mark Potoka and Matt Spicer were initially credited to be involved in the film but they resigned due to being sidelined from the production process. == Reception == The advertisement was released on McDonald's Netherlands' YouTube channel on 6 December 2025. It had a negative reception over the use of generative AI and the "cynical" concept of the work's story. The video was made private on 9 December 2025. The Sweetshop stated that the production of the advertisement took human effort. McDonald's Netherlands, while stating the original intent of the commercial, released a statement after its pullout that, for many of its customers, the holiday season is the "most wonderful time of the year".
On a Red Station, Drifting
On a Red Station, Drifting is a 2012 science fiction novella by Aliette de Bodard. Set in her Xuya Universe, it focuses on two women aboard a space station with a failing artificial intelligence. It received critical acclaim, becoming a finalist for the 2012 Nebula Award for Best Novella, the 2013 Hugo Award for Best Novella, and the 2013 Locus Award for Best Novella. == Plot == Lê Thi Linh is a magistrate of the Dai Viet Empire who is forced to flee her planet after criticizing the Emperor’s wartime policies. At the same time, rebel groups seize control of her planet and kill most of her subordinates. Linh seeks refuge with her distant relatives on Prosper Station. Prosper is controlled by an artificial intelligence called the Honoured Ancestress. Lê Thi Quyen, Linh’s cousin by marriage, manages the day-to-day operations of Prosper while her husband is away at war. Quyen and Linh immediately fall into conflict. Quyen’s brother-in-law Huu Hieu sells his mem-implants, which are copies of their ancestors’ consciousnesses. Meanwhile, the Honoured Ancestress experiences increasingly severe technical problems. Hieu and Linh become close. Hieu plans use the money from the sale of the implants to leave Prosper and marry his lover on a different station. Linh is upset knowing that she will never be able to leave. A visiting cousin, Lady Oahn, provides schematics for the repair of the Honoured Ancestress. In an effort to hurt Quyen, Linh writes an unflattering poem at a banquet honoring Oanh. In doing so, she reveals that Hieu is trying to leave Prosper. Hieu attempts suicide out of shame, but Linh rescues him. Quyen is able to repair the Honoured Ancestress, restoring her functionality at the expense of erasing many of her memories. The Emperor’s Embroidered Guard arrives at Prosper Station in search of Linh. Linh finds the missing mem-implants and returns them to Quyen. Quyen and Linh briefly reconcile before Linh is arrested and removed from Prosper Station. == Major themes == A review in Kirkus wrote that the novel's "familiar setting" was a "departure point" for the novel to explore its themes. The novel explores family ties; almost everyone on Prosper Station is related in some fashion. Additionally, the use of ancestors' mem-implants further explores the concept of family ties, with some descendants being considered more "worthy" than others due to their higher number of implants. The novel also explores questions of worth, as those who fail at ability tests are often forced to become the "lesser partners" in marriages and are discriminated against due to their perceived lack of achievement. The author notes that it is interesting that gender plays no role in the question of worth, and that the majority of the men in the story are actually the "lesser partner" in their marriage. == Style == The novel is divided into three sections. Liz Bourke wrote that each section builds thematically "towards an emotional crescendo". == Reception == Writing for Locus, Liz Bourke praised the novel's exploration of interpersonal conflict between Linh and Quyen, writing that "essentially subverts the popularly-understood derogatory overtones of 'domestic conflict'". Bourke also praised the story's tension, calling it "so well-strung the prose practically vibrates under its influence". A review for Kirkus stated that the novel is a "beautifully realized story and the characters, plot, theme and writing are expertly crafted." === Awards ===
AI-assisted software development
AI-assisted software development is the use of artificial intelligence (AI) to augment software development. It uses large language models (LLMs), AI agents and other AI technologies to assist software developers. It helps in a range of tasks of the software development life cycle, from code generation to debugging, editing, testing, UI design, understanding the code, and documentation. Agentic coding denotes the use of AI agents for software development. == Technologies == === Source code generation === Large language models trained or fine-tuned on source-code corpora can generate source code from natural-language descriptions, comments, or docstrings. Research on code-generation systems often evaluates generated programs by functional correctness, such as whether the output passes automated test cases, rather than by syntax alone. Such tools can be features or extensions of integrated development environments (IDEs). === Intelligent code completion === AI agents using pre-trained and fine-tuned LLMs can predict and suggest code completions based on context. According to Husein, Aburajouh & Catal in a 2025 literature review in Computer Standards & Interfaces, "LLMs significantly enhance code completion performance across several programming languages and contexts, and their capability to predict relevant code snippets based on context and partial input boosts developer productivity substantially." === Testing, debugging, code review and analysis === AI is used to automatically generate test cases, identify potential bugs and security vulnerabilities, and suggest fixes. AI can also be used to perform static code analysis and suggest potential performance improvements. == Limitations == Both ownership of and responsibility for AI-generated code is disputed. According to a report from the German Federal Office for Information Security, the use of AI coding assistants without careful oversight from experienced developers can introduce both minor and major security vulnerabilities, and any potential gain in productivity should be weighed against the cost of additional quality control and security measures. According to Deloitte, outputs from AI-assisted software development must be validated through a combination of automated testing, static analysis tools and human review, creating a governance layer to improve quality and accountability. == Vibe coding ==
Fuzzy measure theory
In mathematics, fuzzy measure theory considers generalized measures in which the additive property is replaced by the weaker property of monotonicity. The central concept of fuzzy measure theory is the fuzzy measure (also capacity, see ), which was introduced by Choquet in 1953 and independently defined by Sugeno in 1974 in the context of fuzzy integrals. There exists a number of different classes of fuzzy measures including plausibility/belief measures, possibility/necessity measures, and probability measures, which are a subset of classical measures. == Definitions == Let X {\displaystyle \mathbf {X} } be a universe of discourse, C {\displaystyle {\mathcal {C}}} be a class of subsets of X {\displaystyle \mathbf {X} } , and E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} . A function g : C → R {\displaystyle g:{\mathcal {C}}\to \mathbb {R} } where ∅ ∈ C ⇒ g ( ∅ ) = 0 {\displaystyle \emptyset \in {\mathcal {C}}\Rightarrow g(\emptyset )=0} E ⊆ F ⇒ g ( E ) ≤ g ( F ) {\displaystyle E\subseteq F\Rightarrow g(E)\leq g(F)} is called a fuzzy measure. A fuzzy measure is called normalized or regular if g ( X ) = 1 {\displaystyle g(\mathbf {X} )=1} . == Properties of fuzzy measures == A fuzzy measure is: additive if for any E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} such that E ∩ F = ∅ {\displaystyle E\cap F=\emptyset } , we have g ( E ∪ F ) = g ( E ) + g ( F ) . {\displaystyle g(E\cup F)=g(E)+g(F).} ; supermodular if for any E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} , we have g ( E ∪ F ) + g ( E ∩ F ) ≥ g ( E ) + g ( F ) {\displaystyle g(E\cup F)+g(E\cap F)\geq g(E)+g(F)} ; submodular if for any E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} , we have g ( E ∪ F ) + g ( E ∩ F ) ≤ g ( E ) + g ( F ) {\displaystyle g(E\cup F)+g(E\cap F)\leq g(E)+g(F)} ; superadditive if for any E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} such that E ∩ F = ∅ {\displaystyle E\cap F=\emptyset } , we have g ( E ∪ F ) ≥ g ( E ) + g ( F ) {\displaystyle g(E\cup F)\geq g(E)+g(F)} ; subadditive if for any E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} such that E ∩ F = ∅ {\displaystyle E\cap F=\emptyset } , we have g ( E ∪ F ) ≤ g ( E ) + g ( F ) {\displaystyle g(E\cup F)\leq g(E)+g(F)} ; symmetric if for any E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} , we have | E | = | F | {\displaystyle |E|=|F|} implies g ( E ) = g ( F ) {\displaystyle g(E)=g(F)} ; Boolean if for any E ∈ C {\displaystyle E\in {\mathcal {C}}} , we have g ( E ) = 0 {\displaystyle g(E)=0} or g ( E ) = 1 {\displaystyle g(E)=1} . Understanding the properties of fuzzy measures is useful in application. When a fuzzy measure is used to define a function such as the Sugeno integral or Choquet integral, these properties will be crucial in understanding the function's behavior. For instance, the Choquet integral with respect to an additive fuzzy measure reduces to the Lebesgue integral. In discrete cases, a symmetric fuzzy measure will result in the ordered weighted averaging (OWA) operator. Submodular fuzzy measures result in convex functions, while supermodular fuzzy measures result in concave functions when used to define a Choquet integral. == Möbius representation == Let g be a fuzzy measure. The Möbius representation of g is given by the set function M, where for every E , F ⊆ X {\displaystyle E,F\subseteq X} , M ( E ) = ∑ F ⊆ E ( − 1 ) | E ∖ F | g ( F ) . {\displaystyle M(E)=\sum _{F\subseteq E}(-1)^{|E\backslash F|}g(F).} The equivalent axioms in Möbius representation are: M ( ∅ ) = 0 {\displaystyle M(\emptyset )=0} . ∑ F ⊆ E | i ∈ F M ( F ) ≥ 0 {\displaystyle \sum _{F\subseteq E|i\in F}M(F)\geq 0} , for all E ⊆ X {\displaystyle E\subseteq \mathbf {X} } and all i ∈ E {\displaystyle i\in E} A fuzzy measure in Möbius representation M is called normalized if ∑ E ⊆ X M ( E ) = 1. {\displaystyle \sum _{E\subseteq \mathbf {X} }M(E)=1.} Möbius representation can be used to give an indication of which subsets of X interact with one another. For instance, an additive fuzzy measure has Möbius values all equal to zero except for singletons. The fuzzy measure g in standard representation can be recovered from the Möbius form using the Zeta transform: g ( E ) = ∑ F ⊆ E M ( F ) , ∀ E ⊆ X . {\displaystyle g(E)=\sum _{F\subseteq E}M(F),\forall E\subseteq \mathbf {X} .} == Simplification assumptions for fuzzy measures == Fuzzy measures are defined on a semiring of sets or monotone class, which may be as granular as the power set of X, and even in discrete cases the number of variables can be as large as 2|X|. For this reason, in the context of multi-criteria decision analysis and other disciplines, simplification assumptions on the fuzzy measure have been introduced so that it is less computationally expensive to determine and use. For instance, when it is assumed the fuzzy measure is additive, it will hold that g ( E ) = ∑ i ∈ E g ( { i } ) {\displaystyle g(E)=\sum _{i\in E}g(\{i\})} and the values of the fuzzy measure can be evaluated from the values on X. Similarly, a symmetric fuzzy measure is defined uniquely by |X| values. Two important fuzzy measures that can be used are the Sugeno- or λ {\displaystyle \lambda } -fuzzy measure and k-additive measures, introduced by Sugeno and Grabisch respectively. === Sugeno λ-measure === The Sugeno λ {\displaystyle \lambda } -measure is a special case of fuzzy measures defined iteratively. It has the following definition: ==== Definition ==== Let X = { x 1 , … , x n } {\displaystyle \mathbf {X} =\left\lbrace x_{1},\dots ,x_{n}\right\rbrace } be a finite set and let λ ∈ ( − 1 , + ∞ ) {\displaystyle \lambda \in (-1,+\infty )} . A Sugeno λ {\displaystyle \lambda } -measure is a function g : 2 X → [ 0 , 1 ] {\displaystyle g:2^{X}\to [0,1]} such that g ( X ) = 1 {\displaystyle g(X)=1} . if A , B ⊆ X {\displaystyle A,B\subseteq \mathbf {X} } (alternatively A , B ∈ 2 X {\displaystyle A,B\in 2^{\mathbf {X} }} ) with A ∩ B = ∅ {\displaystyle A\cap B=\emptyset } then g ( A ∪ B ) = g ( A ) + g ( B ) + λ g ( A ) g ( B ) {\displaystyle g(A\cup B)=g(A)+g(B)+\lambda g(A)g(B)} . As a convention, the value of g at a singleton set { x i } {\displaystyle \left\lbrace x_{i}\right\rbrace } is called a density and is denoted by g i = g ( { x i } ) {\displaystyle g_{i}=g(\left\lbrace x_{i}\right\rbrace )} . In addition, we have that λ {\displaystyle \lambda } satisfies the property λ + 1 = ∏ i = 1 n ( 1 + λ g i ) {\displaystyle \lambda +1=\prod _{i=1}^{n}(1+\lambda g_{i})} . Tahani and Keller as well as Wang and Klir have shown that once the densities are known, it is possible to use the previous polynomial to obtain the values of λ {\displaystyle \lambda } uniquely. === k-additive fuzzy measure === The k-additive fuzzy measure limits the interaction between the subsets E ⊆ X {\displaystyle E\subseteq X} to size | E | = k {\displaystyle |E|=k} . This drastically reduces the number of variables needed to define the fuzzy measure, and as k can be anything from 1 (in which case the fuzzy measure is additive) to X, it allows for a compromise between modelling ability and simplicity. ==== Definition ==== A discrete fuzzy measure g on a set X is called k-additive ( 1 ≤ k ≤ | X | {\displaystyle 1\leq k\leq |\mathbf {X} |} ) if its Möbius representation verifies M ( E ) = 0 {\displaystyle M(E)=0} , whenever | E | > k {\displaystyle |E|>k} for any E ⊆ X {\displaystyle E\subseteq \mathbf {X} } , and there exists a subset F with k elements such that M ( F ) ≠ 0 {\displaystyle M(F)\neq 0} . == Shapley and interaction indices == In game theory, the Shapley value or Shapley index is used to indicate the weight of a game. Shapley values can be calculated for fuzzy measures in order to give some indication of the importance of each singleton. In the case of additive fuzzy measures, the Shapley value will be the same as each singleton. For a given fuzzy measure g, and | X | = n {\displaystyle |\mathbf {X} |=n} , the Shapley index for every i , … , n ∈ X {\displaystyle i,\dots ,n\in X} is: ϕ ( i ) = ∑ E ⊆ X ∖ { i } ( n − | E | − 1 ) ! | E | ! n ! [ g ( E ∪ { i } ) − g ( E ) ] . {\displaystyle \phi (i)=\sum _{E\subseteq \mathbf {X} \backslash \{i\}}{\frac {(n-|E|-1)!|E|!}{n!}}[g(E\cup \{i\})-g(E)].} The Shapley value is the vector ϕ ( g ) = ( ψ ( 1 ) , … , ψ ( n ) ) . {\displaystyle \mathbf {\phi } (g)=(\psi (1),\dots ,\psi (n)).}