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Aggregation theorems for allocation problems. (English) Zbl 0538.05006

When \(n\) individuals assign values to m decision-variables such that:
(i) All variables are numerical ones,
(ii) each individual assigns m values corresponding to a standard form (sequence) of these variables,
(iii) all values must be non-negative and sum to some fixed positive \(\sigma\), then such a problem is generally called “allocation problem”.
In practice individuals will differ in the values that they assign to the decision-variables, and so one will be faced with the problem of aggregating their individual assignments in order to produce consensual values of these variables. In the case when \(m\ge 3\), a method of aggregation assigns consensual values to decision variables in such a way that (i) the value assigned to the \(j\)-th variable is \(\Phi_j(z_ j)\in [0,\sigma]\) where \(z_j\) is the sequence values assigned to the \(j\)-th variable, and (ii) the consensual value is zero if all individuals assign that variable the value zero if and only if the method is based on weighted arithmetic averaging, with weights invariant over the \(m\) variables.
In this paper extensions to these results are given, allowing individual and consensual values to be restricted to some subinterval of \([0,\sigma]\).

MSC:

05A99 Enumerative combinatorics
68R99 Discrete mathematics in relation to computer science
90C99 Mathematical programming
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References:

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