Piece-wise division of objects
A piecewise-constant valuation is a kind of a function that represents the utility of an agent over a continuous resource, such as land. It occurs when the resource can be partitioned into a finite number of regions, and in each region, the value-density of the agent is constant. A piecewise-uniform valuation is a piecewise-constant valuation in which the constant is the same in all regions.
Piecewise-constant and piecewise-uniform valuations are particularly useful in algorithms for fair cake-cutting.
Formal definition.
There is a "resource" represented by a set "C." There is a "valuation" over the resource, defined as a continuous measure formula_1. The measure "V" can be represented by a "value-density function" formula_2. The value-density function assigns, to each point of the resource, a real value. The measure "V" of each subset "X" of "C" is the integral of "v" over "X".
A valuation "V" is called piecewise-constant, if the corresponding value-density function "v" is a piecewise-constant function. In other words: there is a partition of the resource "C" into finitely many regions, "C"1...,"Ck", such that for each "j" in 1...,"k", the function "v" inside "Cj" equals some constant "Uj".
A valuation "V" is called piecewise-uniform if the constant is the same for all regions, that is, for each "j" in 1...,"k", the function "v" inside "Cj" equals some constant "U".
Generalization.
A piecewise-linear valuation is a generalization of piecewise-constant valuation in which the value-density in each region "j" is a linear function, "ajx"+"bj" (piecewise-constant corresponds to the special case in which "aj"=0 for all "j").
References.
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