Abstract
This chapter introduces and discusses the basic notions of decision-making in a multicriteria environment, namely, the set of feasible alternatives, the vector criterion and the preference relation of a decision-maker. Here we formulate the multicriteria choice problem. In addition, Chap. 1 defines a pair of fundamentally important notions, the set of nondominated alternatives and the Pareto set, which are vital for the statement and rigorous substantiation of the Edgeworth-Pareto principle.
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Notes
- 1.
Some different alternatives may have same vector values. And so, it seems correct to replace “for \(x^{{\prime }} ,\;x^{{\prime \prime }} \in X\)” with “for all \(x_{1} \in \tilde{x}_{1} ,\;x_{2} \in \tilde{x}_{2} ,\;\tilde{x}_{1} ,\tilde{x}_{2} \in \tilde{X}\), where \(\tilde{X}\) represents the aggregate of the equivalence classes induced by the preference relation \(x_{1} \sim x_{2}\) \(\Leftrightarrow\) \(f(x_{1} ) = f(x_{2} )\) on the set X.” Here \(\tilde{x}_{i}\) denotes the equivalence class induced by the element \(x_{i}\), \(i = 1,\;2\).
- 2.
As easily verified, the inverse Condorcet condition [1] implies Axiom 1, but not vice versa.
- 3.
Note, that for the Edgeworth-Pareto principle to be true (see Theorem 1.1), in this axiom we may consider the extension \(\succ\) not on the whole space \(R^{m}\) but merely on the Cartesian product \(Y_{1} \times Y_{2} \times \ldots \times Y_{m}\), where \(Y_{i}\) is the smallest interval including \(f_{i} (x),\;i = 1,\;2, \ldots ,\;m\).
- 4.
We remark that further exposition of this chapter can be generalized to the case of “individual” preference relations.
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Noghin, V.D. (2018). Edgeworth-Pareto Principle. In: Reduction of the Pareto Set. Studies in Systems, Decision and Control, vol 126. Springer, Cham. https://doi.org/10.1007/978-3-319-67873-3_1
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DOI: https://doi.org/10.1007/978-3-319-67873-3_1
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