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Convex cones, assessment functions, balanced attributes

Ignacy Kaliszewski

Abstract

We investigate a class of polyhedral convex cones, with $R^k_+$ (the nonegative orthant in $\mathbb{R}^k$) as a special case. We start with the observation that for convex cones contained in $\mathbb{R}^k$, the respective cone efficiency is inconsistent with the Pareto efficiency, the latter being deeply rooted in economics, the decision theory, and the multiobjective optimization theory. Despite that, we argue that convex cones contained in $\mathbb{R}^k$ and the respective cone efficiency are also relevant to these domains. To demonstrate this, we interpret polyhedral convex cones of the investigated class in terms of assessment functions, i.e., functions that aggregate multiple numerical attributes into single numbers. Further, we observe that all assessment functions in the current use share the same limitation; that is, they do not take explicitly into account attribute proportionality. In consequence, the issue of {\em attribute balance} (meaning {\it the balance of attribute values}) escapes them. In contrast, assessment functions defined by polyhedral convex cones of the investigated class, contained in $\mathbb{R}^k$, enforce the attribute balance. However, enforcing the attribute balance is, in general, inconsistent with the well-established paradigm of Pareto efficiency. We give a practical example where such inconsistency is meaningful.

Convex cones, assessment functions, balanced attributes

Abstract

We investigate a class of polyhedral convex cones, with (the nonegative orthant in ) as a special case. We start with the observation that for convex cones contained in , the respective cone efficiency is inconsistent with the Pareto efficiency, the latter being deeply rooted in economics, the decision theory, and the multiobjective optimization theory. Despite that, we argue that convex cones contained in and the respective cone efficiency are also relevant to these domains. To demonstrate this, we interpret polyhedral convex cones of the investigated class in terms of assessment functions, i.e., functions that aggregate multiple numerical attributes into single numbers. Further, we observe that all assessment functions in the current use share the same limitation; that is, they do not take explicitly into account attribute proportionality. In consequence, the issue of {\em attribute balance} (meaning {\it the balance of attribute values}) escapes them. In contrast, assessment functions defined by polyhedral convex cones of the investigated class, contained in , enforce the attribute balance. However, enforcing the attribute balance is, in general, inconsistent with the well-established paradigm of Pareto efficiency. We give a practical example where such inconsistency is meaningful.
Paper Structure (10 sections, 4 theorems, 56 equations, 13 figures, 1 table)

This paper contains 10 sections, 4 theorems, 56 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Basic definitions and notation
  3. Properly $R^k_+$-efficient elements
  4. Properly $K^{Poly}_\rho$-efficient elements
  5. Assessment functions
  6. Assessment functions in the production theory and the utility theory
  7. Pareto dominance-consistent assessment functions
  8. Pareto dominance-inconsistent assessment functions
  9. Rankings with Pareto inconsistent assessment functions
  10. Discussion and concluding remarks

Key Result

Theorem 3.1

An element $\bar{y}\in Z$ is properly $R^{k}_+$-efficient if and only if there exists $\lambda \in R^k_>$ and $\rho>0 \, ,$ such that $\bar{y}$ solves For each properly efficient element $y\in Z$ there exists $\lambda \in R^k_>$ such that $y$ solves $P^{\infty}_{R^{k}_+}$ uniquely for every $\rho>0$ satisfying $M\leq((k-1)\rho)^{-1} \, .$ For each element $y \in Z$ which solves $P^{\infty}_{R^{k}

Figures (13)

  • Figure 1: The thick curve indicates elements that can be $K$-efficient under an appropriate selection of a convex cone $K$. In particular, all of them are $K^{Poly}_{-\frac{1}{k}}$-efficient. From those elements, all elements that belong to the dotted square are $R^k_+$-efficient, whereas the remaining ones are not $R^k_+$-efficient (are not Pareto efficient). The latter are $K^{Poly}_{\rho}$-efficient for some $-\frac{1}{k} \leq \rho < 0$.
  • Figure 2: The thick curve indicates elements that are $K^{Poly}_{\rho}$-efficient for some $\rho > 0$.
  • Figure 3: Elements: improperly $R^k_+$-efficient (circles) and improperly $K^{Poly}_{\rho}$-efficient (discs) for some $\rho < 0$.
  • Figure 4: An illustration to Example \ref{['example']}: element $\bullet$ is improperly $K^{Poly}_{\rho}$-efficient.
  • Figure 5: Graph and contours of the Leontief function (\ref{['Leontief_function']}) in $\mathbb{R}^2$; a PDCA function.
  • ...and 8 more figures

Theorems & Definitions (10)

  • Definition 1.1
  • Definition 2.1
  • Definition 3.1
  • Theorem 3.1
  • Definition 4.1
  • Example 4.1
  • Theorem 4.1
  • Lemma 4.1
  • Example 5.1
  • Lemma A.3.1