for all , , where denotes the componentwise maximum and the componentwise minimum of and .
If −f is supermodular then f is called submodular, and if the inequality is changed to an equality the function is modular.
If f is twice continuously differentiable, then supermodularity is equivalent to the condition[1]
^The equivalence between the definition of supermodularity and its calculus formulation is sometimes called Topkis' characterization theorem. See Milgrom, Paul; Roberts, John (1990). "Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities". Econometrica. 58 (6): 1255–1277 [p. 1261]. doi:10.2307/2938316. JSTOR 2938316.
and 18 Related for: Supermodular function information
In mathematics, a function f : R k → R {\displaystyle f\colon \mathbb {R} ^{k}\to \mathbb {R} } is supermodular if f ( x ↑ y ) + f ( x ↓ y ) ≥ f ( x )...
coverage. Supermodularfunction Matroid, Polymatroid Utility functions on indivisible goods H. Lin and J. Bilmes, A Class of Submodular Functions for Document...
possible utility function for this case is given at the right. A utility function is additive if and only if it is both submodular and supermodular. Subadditivity...
nonnegative function on it, that is assumed to satisfy the (FKG) lattice condition (sometimes a function satisfying this condition is called log supermodular) i...
Indeed, they show that their concept of quasi-supermodularity (a generalization of supermodularfunction) along with the single-crossing property, is necessary...
{\displaystyle i\neq j} . Equivalently, this means that the function Π {\displaystyle \,\Pi } is supermodular. On the other hand, the decisions are strategic substitutes...
f(x)-f(x\wedge x').} Every supermodularfunction is quasisupermodular. As in the case of single crossing differences, and unlike supermodularity, quasisupermodularity...
Specifically, a game is convex if its characteristic function v {\displaystyle v} is supermodular: v ( S ∪ T ) + v ( S ∩ T ) ≥ v ( S ) + v ( T ) , ∀ ...
utility functions is weakly additive. A utility function is additive if and only if it is both submodular and supermodular. Utility functions on indivisible...
Submodular fuzzy measures result in convex functions, while supermodular fuzzy measures result in concave functions when used to define a Choquet integral...
The method uses lattice theory and introduces the notions of quasi-supermodularity and the single-crossing condition. The wide application of monotone...
submodular function f {\displaystyle f} such that f ( ∅ ) = 0 {\displaystyle f(\emptyset )=0} and E P f = E P {\displaystyle EP_{f}=EP} . For a supermodular f...
valuations. When agents' utilities are superadditive set functions (more general than supermodular), a ( log m ) 1 + ϵ m {\displaystyle {\frac {(\log m)^{1+\epsilon...
complementarity is due to Masahiko Aoki and relies on the theory of supermodular games developed by Paul Milgrom and John Roberts. The basic structure...
{\displaystyle X} is a deterministic function of Y {\displaystyle Y} and Y {\displaystyle Y} is a deterministic function of X {\displaystyle X} then all information...
lattice-theoretic methods to analyze games of strategic complementarities (or supermodular games), and in general complementarities, in economics. His contribution...
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