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...f cardinal utility functions. An analogous property of [[ordinal utility]] functions is [[weakly additive]]. ...submodular]] and [[Supermodular_function#Supermodular_functions_of_subsets|supermodular]]. ...

...s from a set <math>A</math> is written <math>u(A)</math>. Cardinal utility functions are often normalized such that <math>u(\emptyset)=0</math>, where <math>\em ...> and <math>u(A)\geq u(B)</math> implies <math>A \succeq B</math>. Utility functions can have several properties.<ref name=gs99>{{Cite journal | doi = 10.1006/j ...

...are tight; this follows from properties of upper-bounded supermodular set functions.<ref name="lp" />{{rp|Lem.1.3.3}} In a non-bipartite graph, the deficiency function is, in general, not supermodular. ...

...h> and let <math>f(\cdot; s):X\rightarrow \mathbb{R}</math> be a family of functions parameterized by <math>s\in S</math>, where <math>(S, \geq_S)</math> is a [ ...rossing Property,” ''Econometrica'', 80(5), 2333–2348.</ref> The family of functions <math>\{f(\cdot;s)\}_{s\in S}</math>, <math>f:X \times S\to \mathbb{R}</mat ...

...deling user preferences) and [[electrical network]]s. Recently, submodular functions have also found utility in several real world problems in [[machine learnin ...here <math>2^\Omega</math> denotes the [[Power set#Representing subsets as functions|power set]] of <math>\Omega</math>, which satisfies one of the following eq ...

...r generalization is the [[Ahlswede–Daykin inequality|Ahlswede–Daykin "four functions" theorem (1978)]]. Furthermore, it has the same conclusion as the [[Griffit ...dition''' (sometimes a function satisfying this condition is called '''log supermodular''') i.e., ...

...ong agents with different [[Utility functions on indivisible goods|utility functions]], such that the welfare – defined as the sum of the agents' utilities – is ...to every possible subset of items. It is usually assumed that the utility functions are [[monotone set function]]s, that is, <math>Z_1\supseteq Z_2</math> impl ...