Extended proportionality in division problems with multiple references

Abstract

We consider an extension of the classic problem of division with claims to situations in which the characteristics of the agents are multi-dimensional. The proportional rule is the most prominent in the one-dimensional case, however there is no obvious way to define proportionality for the extended model. In this paper we introduce a property which generalizes proportionality, identify the family of rules satisfying it, and characterize a particular rule within this family on the basis of a property of symmetry.

Appendix

Proof Theorem 3.2

Without loss of generality, assume that N={1,…,n} and M={1,…,m}. Let R be a ratio efficient rule on $\mathcal{C}^{M}_{N}$. Hence, given $(C,E)\in\mathcal{C}^{M}_{N}$, x=R(C,E) is a Pareto-optimal solution⁷ to the vector optimization problem:

$$ \everymath{\displaystyle }\begin{array} {l@{\ }l} \operatorname{Max} & \biggl \{ \min_{i\in N} \biggl\{ \frac{x_i}{c_i^1}\biggr\},\ldots, \min _{i\in N} \biggl\{ \frac{x_i}{c_i^m}\biggr\} \biggr\} \\[5pt] \mathrm{s}.\mathrm{t}.& x\in X(E), \\ \end{array} $$

(6.1)

This problem is equivalent to the vector linear problem:

$$ \everymath{\displaystyle }\begin{array} {l@{\ }l} \operatorname{Max} & \bigl \{t^1, \ldots, t^m\bigr\} \\[3pt] \mathrm{s}.\mathrm{t}.& t^j \leq \frac{x_i}{c_i^j},\quad i\in N,\ j\in M \\[5pt] & x\in X(E). \end{array} $$

(6.2)

in the sense that if x is Pareto-optimal to (6.1), there exists $t\in \mathbb{R}^{M}$, such that (x,t) is Pareto-optimal to (6.2). Conversely, if (x,t) is Pareto-optimal to (6.2), then x is Pareto-optimal to (6.1).

We now rely on the parametric characterization of solutions to linear vector optimization problems as the solutions of weighted maximin problems (see Yano and Sakawa 1989) to state that (x,t) is Pareto-optimal to (6.2) if and only if there exists a vector of weights $\gamma(C,E)\in\Delta^{M}_{++}$, such that (x,t) is an optimal solution to

$$\everymath{\displaystyle }\begin{array} {l@{\ }l} \operatorname{Max} & \operatorname{Min}_{j\in M} \bigl\{\gamma^j(C,E)t^j\bigr\} \\[3pt] \mbox{s.t}.& t^j \leq \frac{x_i}{c_i^j},\quad i\in N,\ j\in M \\[8pt] & x\in X(E). \end{array} $$

Now, by denoting z ^j=γ(C,E)^j t ^j for j∈M, this last problem can be written as:

$$ \everymath{\displaystyle }\begin{array} {l@{\ }l} \operatorname{Max} & \operatorname{Min}_{j\in M}\bigl\{z^j\bigr\} \\[3pt] \mbox{s.t.}& z^j \leq \frac{x_i}{(c_i^j / \gamma^j(C,E))},\quad i\in N,\ j\in M \\[8pt] & x\in X(E). \end{array} $$

(6.3)

It is easy to see that if (x,z) is an optimal solution to (6.3), then $(x, \tilde{z})$, where for all j∈M, $\tilde{z}^{j}=\min_{k\in M} \{z^{k}\}$ is also optimal to (6.3). It follows that the set of optimal allocations for (6.3) coincide with the set of optimal allocations to the following linear problem.

$$\everymath{\displaystyle }\begin{array}{l@{\ }l} \operatorname{Max} & s\\[3pt] \mbox{s.t.}& s \leq \frac{x_i}{(c_i^j / \gamma^j(C,E))},\quad i\in N,\ j\in M\\[8pt] & x\in X(E). \end{array} $$

Let $K(C, E)= \frac{1}{\sum_{j}\frac{1}{\gamma^{j}(C,E)}}$. For j∈M, denote $\alpha^{j}(C,E)= \frac{k(C,E)}{\gamma^{j}(C,E)}$, and note that for i∈N, the set of constraints $s \leq \frac{x_{i}}{(c_{i}^{j} / \gamma^{j}(C,E))}=\frac{x_{i}}{\alpha^{j}(C,E)c_{i}^{j}}, j\in M$ is equivalent to $\frac{s}{K} \leq \frac{x_{i}}{\overline{c^{\alpha}}_{i}}$, where $\overline{c^{\alpha}}_{i}= \max_{j\in M}\{\alpha^{j}(C,E)c^{j}_{i}\}$ for all i∈N, and the problem can be written as:

$$ \everymath{\displaystyle }\begin{array}{l@{\ }l} \operatorname{Max} & s\\[3pt] \mbox{s.t.}& \frac{s}{K} \leq \frac{x_i}{\overline{c^\alpha}_i},\quad i\in N,\\[8pt] & x\in X(E). \end{array} $$

(6.4)

Since for each $(C,E)\in\mathcal{C}_{N}^{M}$, K is a constant, the optimal allocations to this problem coincide with the optimal allocations to problem

$$ \everymath{\displaystyle }\begin{array}{l@{\ }l} \operatorname{Max} & t\\[3pt] \mbox{s.t.}& t \leq \frac{x_i}{\overline{c^\alpha}_i},\quad i\in N,\\[8pt] & x\in X(E). \end{array} $$

(6.5)

Hence, if R is ratio-efficient, given x ^∗=R(C,E), there exists t ^∗ such that (x ^∗,t ^∗) is the optimal solution to (6.5), and therefore,⁸ $x^{*}=p(\overline{c^{\alpha}}, E)$. As defined in Sect. 2, $p(\overline{c^{\alpha}}, E)=\overline{P} (C^{\alpha}, E)=\overline{P}^{\alpha}(C, E)$.

The converse also follows from the characterization of Pareto-optimal solutions to linear vector optimization problems as the solutions to weighted maximin problems. □

Proof Proposition 4.2

Let $(C,E)\in\mathcal {C}^{M}$, and $x=\overline{P}^{\omega}(C,E)=\overline{P}(C^{\omega},E)=p(\overline {c^{\omega}},E)$, where $\overline{c^{\omega}}=(\max_{j\in M}\{{c^{\omega}}^{j}_{i}\})_{i\in N}$. Consider N′⊂N. From the consistency of the classic proportional rule, it follows that $x_{N'}=p(\overline{c^{\omega}}_{N'}, x(N'))$. On the other hand, $\overline{P}^{\omega}(C_{N'}, x(N'))=\overline{P}(C_{N'}^{\omega},E)= p(\overline{c_{N'}^{\omega}},E)=p(\overline{c^{\omega}}_{N'}, x(N'))=x_{N'}$. □

Proof Theorem 4.3

It is straightforward to prove that the max-proportional rule satisfies neutrality.

We will prove that if the regular weighted max-proportional rule $\overline{P}^{\omega}$ satisfies neutrality, then ω is such that ω ^j=ω ^k, for all j,k∈M. It is easy to see that in this case the rule coincides with the max-proportional rule as defined in Sect. 2.

Consider $(C, E) \in\mathcal{C}^{M}_{N}$, $C\in \mathbb{R}^{N\times M}$, such that $c_{i}^{j}=1$ for i=j, $c_{i}^{j}=\varepsilon$ if i≠j, with $0<\varepsilon< \frac{\omega^{j}}{\omega^{k}}$, for all j,k∈M, and let m and n be |M| and |N| respectively.

Case 1: m≤n. In this case,

$$C=\left (\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 1 & \varepsilon& \cdots& \varepsilon\\ \varepsilon& 1 & \cdots& \varepsilon\\ \vdots & \vdots & \ddots& \vdots\\ \varepsilon& \varepsilon&\cdots & 1\\ \varepsilon& \varepsilon&\cdots & \varepsilon\\ \vdots & \vdots & \ddots& \vdots\\ \varepsilon& \varepsilon&\cdots& \varepsilon\\ \end{array} \right ). $$

For each i∈N, $\omega^{j} c_{i}^{j}= \omega^{j}$ if j=i, and $\omega^{j} c_{i}^{j}= \omega^{j} \varepsilon$ if j≠i. Therefore, for i∈N, such that i≤m, $\max_{j\in M}\{\omega^{j} c_{i}^{j}\}= \omega^{i}$. If, on the other hand, i>m, then $\max_{j\in M}\{\omega^{j} c_{i}^{j}\}=\overline{\omega}\varepsilon$, where $\overline{\omega}= \max_{j} \{\omega^{j}\}$. It follows that

$$\overline{P}^{\omega}(C, E)= p\bigl(\bigl(\omega^1, \omega^2,\ldots,\omega^m, \overline{\omega}\varepsilon, \ldots, \overline{\omega}\varepsilon\bigr), E\bigr). $$

Consider σ ^{1,2}∈Π ^M, such that σ ^{1,2}(1)=2, σ ^{1,2}(2)=1, σ ^{1,2}(j)=j, if j≠1,2, and denote by D the permuted matrix:

$$D=C^{\sigma^{\{1,2\}}}=\left (\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \varepsilon& 1 & \cdots& \varepsilon\\ 1 & \varepsilon& \cdots& \varepsilon\\ \vdots & \vdots & \ddots& \vdots\\ \varepsilon& \varepsilon&\cdots & 1\\ \varepsilon& \varepsilon&\cdots & \varepsilon\\ \vdots & \vdots & \ddots& \vdots\\ \varepsilon& \varepsilon&\cdots& \varepsilon\\ \end{array} \right ). $$

If i≠1,2, then it is easy to see that $\max_{j\in M}\{\omega^{j} d_{i}^{j}\}=\max_{j\in M}\{\omega^{j} c_{i}^{j}\}$. For i=1, $\max_{j\in M}\{\omega^{j} d_{1}^{j}\}=\omega^{2}$ and for i=2, $\max_{j\in M}\{\omega^{j} d_{2}^{j}\}=\omega^{1}$. Therefore:

$$\overline{P}^{\omega}(D, E)= p\bigl(\bigl(\omega^2, \omega^1,\ldots,\omega^m, \overline{\omega}\varepsilon, \ldots, \overline{\omega}\varepsilon\bigr), E\bigr). $$

Now, by applying neutrality, $\overline{P}^{\omega}(D, E)=\overline{P}^{\omega}(C,E)$, and it follows that ω ₁=ω ₂. Similarly, by considering permutations σ ^{1,3},…,σ ^{1,m}∈Π ^M, the equality of all weights is proved.

Case 2: m>n. Suppose m≥3, since otherwise there is at most one agent and there is not a real division problem. In this case matrix C is:

$$C=\left (\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 1 & \varepsilon& \cdots& \varepsilon& \varepsilon& \cdots& \varepsilon\\\varepsilon& 1 & \cdots& \varepsilon& \varepsilon& \cdots& \varepsilon\\ \vdots & \vdots & \ddots& \vdots & \vdots & \ddots& \vdots\\ \varepsilon& \varepsilon&\cdots & 1 & \varepsilon& \cdots& \varepsilon\\ \end{array} \right ). $$

By considering $D=C^{\sigma^{\{1,2\}}}$, a similar reasoning applies to obtain

$$\overline{P}^{\omega}(C, E)= p\bigl(\bigl(\omega^1, \omega^2,\ldots,\omega^m\bigr), E\bigr) = \overline{P}^{\omega}(D, E)= p\bigl(\bigl(\omega^1,\omega^2, \ldots,\omega^m,\bigr), E\bigr) $$

which implies ω ₁=ω ₂.

Again, by considering permutations σ ^{1,3},…,σ ^{1,m}∈Π ^M, the equality of all weights is proved. □