Concepts inTight information-theoretic lower bounds for welfare maximization in combinatorial auctions
Combinatorial auction
A combinatorial auction is a type of smart market in which participants can place bids on combinations of discrete items, or ¿packages,¿ rather than just individual items or continuous quantities. Simple combinatorial auctions have been used for many years in estate auctions, where a common procedure is to accept bids for packages of items.
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Information theory
Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and communicating data.
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Upper and lower bounds
In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set (P, ¿) is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is less than or equal to every element of S. A set with an upper bound is said to be bounded from above by that bound, a set with a lower bound is said to be bounded from below by that bound.
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Superadditivity
In mathematics, a sequence { an }, n ¿ 1, is called superadditive if it satisfies the inequality for all m and n. The major reason for the use of superadditive sequences is the following lemma due to Michael Fekete. Lemma: (Fekete) For every superadditive sequence { an }, n ¿ 1, the limit lim an/n exists and equal to sup an/n. (The limit may be positive infinity, for instance, for the sequence an = log n!. ) Similarly, a function f(x) is superadditive if for all x and y in the domain of f.
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Subadditivity
In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive functions in various areas of mathematics, particularly norms and square roots. Additive functions are special cases of subadditive functions.
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Codomain
In mathematics, the codomain or target set of a function is the set Y into which all of the output of the function is constrained to fall. It is the set Y in the notation f: X ¿ Y. The codomain is also sometimes referred to as the range but that term is ambiguous as it may also refer to the image. The codomain is part of the modern definition of a function f as a triple (X, Y, F), with F a subset of the Cartesian product X × Y.
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Submodular set function
In mathematics, submodular functions are set functions which usually appear in approximation algorithms, functions modeling user preferences in game theory. These functions have a natural diminishing returns property which makes them suitable for many applications.
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E (mathematical constant)
is the unique value of a, such that the derivative of f(x) = a at the point x = 0 is equal to 1. The blue curve illustrates this case, e. For comparison, functions 2 (dotted curve) and 4 (dashed curve) are shown; they are not tangent to the line of slope 1 and y-intercept 1 (red). ]] The number e is an important mathematical constant, approximately equal to 2.71828, that is the base of the natural logarithm.
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