Rationing with baselines: the composition extension operator

Abstract

We introduce a new operator for general rationing problems in which, besides conflicting claims, individual baselines play an important role in the rationing process. The operator builds onto ideas of composition, which are not only frequent in rationing, but also in related problems such as bargaining, choice, and queuing. We characterize the operator and show how it preserves some standard axioms in the literature on rationing. We also relate it to recent contributions in such literature.

Appendix: Proof of the results

Proof of Theorem 1

It is straightforward to see that, for any (standard) rule R, R ^c satisfies the four axioms. Thus, we focus on the converse implication. Let S be a rule satisfying the four axioms and let (N,b,c,E) be a problem with baselines. If E≥T(b,c), then, by baseline invariance, and baseline truncation, S(N,b,c,E)=S(N,0,c−t(b,c),E−T(b,c)). If E≤T(b,c), by baseline truncation and truncation of excessive claims, S(N,b,c,E)=t(b,c)+S(N,t(b,c),t(b,c),E). Moreover, by polar baseline equivalence, S(N,t(b,c),t(b,c),E)=S(N,0,t(b,c),E).

Let $R:\mathcal{D}\to\bigcup_{N\in\mathcal{N}}\mathbb{R}^{n}$ be such that, for any $(N,c,E )\in\mathcal{D}$,

$$ R (N,c,E ) =S(N,0,c,E). $$

In other words, R assigns to each problem the allocation that S yields for the corresponding problem with baselines in which baselines are null. Then,

$$ S (N,b,c,E ) = \left \{ \begin{array}{l@{\quad}l} R(N,t(b,c), E) & \mbox{if} \ E\le T(b,c), \\ t(b,c)+R(N, c-t(b,c), E - T(b,c)) & \mbox{if} \ E>T(b,c),\end{array} \right . $$

which implies that S≡R ^c. □

Proof of Theorem 2

Note first that, by Theorem 1 in Hougaard et al. (2013), the three properties are preserved by the baselines-first extension operator. As this operator coincides with the composition extension operator for the case in which truncated baselines are collectively feasible, we only need to focus on the opposite case to prove the statements of the theorem.

Let R be a rule satisfying resource monotonicity. Let $(N,c,E )\in\mathcal{D}$ and each E′>E, with E′≤∑c _i . Let $b\in\mathbb{R}^{n}$ be a baselines profile and let i∈N be a given agent. If E<E′≤T(b,c), then $R^{c}_{i}(N,b,c,E)=R_{i}(N,t(b,c),E)$ and $R^{c}_{i}(N,b,c,E^{\prime })=R_{i}(N,t(b,c),E^{\prime })$. Now, as R satisfies resource monotonicity, the desired inequality follows.

Let R be a rule satisfying consistency. Let $(N,c,E)\in\mathcal{D}$ and $b\in\mathbb{R}^{n}_{+}$. Let x=R ^c(N,b,c,E). The aim is to show that, for any M⊂N,

$$ R^{c}\biggl(M,b_{M},c_{M},\sum _{i\in M}x_{i}\biggr)=x_{M}. $$

Fix M⊂N and let E′=∑ _j∈M x _j and T′(b,c)=∑ _j∈M t _j (b,c). Then, it is straightforward to show that E≤T(b,c) if and only if E′≤T′(b,c). If E≤T(b,c), then x _i =R _i (N,t(b,c),E) for each i∈N, and thus E′=∑ _i∈M R _i (N,t(b,c),E). Therefore, $R^{c}_{i}(M,b_{M},c_{M},E^{\prime })=R_{i}(M,t_{M}(b,c),E^{\prime})$ for each i∈M. Now, as R is consistent, it follows that R _i (N,t(b,c),E)=R _i (M,t _M (b,c),E′), for each i∈M, as desired.

To conclude, the statement on resource-population uniformity follows from the fact that such axiom is equivalent to the combination of resource monotonicity and consistency (e.g., Hougaard et al. 2013). □

Proof of Theorem 3

Again, by Theorem 3 in Hougaard et al. (2013), we only need to focus on the case in which truncated baselines are collectively unfeasible to prove the statements of the theorem.

Let R be a rule satisfying claims monotonicity and linked claims-resource monotonicity. Our aim is to show that R ^c satisfies the general versions of the two properties.

Claims monotonicity Let $(N,c,E )\in\mathcal{D}$ and i∈N, such that $c_{i}\leq c_{i}^{\prime }$. Let $b\in\mathbb{R}^{n}_{+}$ be a baselines profile, T(b,c)=∑ _j∈N t _j (b,c), and T′(b,c)=∑ _j∈N t _j (b,c′).¹²

If E≤T(b,c), then, $R^{c}_{i}(N,b,c,E)=R_{i}(N,t(b,c),E)$ and

$$ R^{c}_i\bigl(N,b,c^{\prime },E\bigr)=R_i \bigl(N,t\bigl(b,c^{\prime }\bigr),E\bigr)=R_i\bigl(N, \bigl(t_{N\setminus\{i\}}(b,c),t_{i}\bigl(b,c^{\prime }\bigr)\bigr),E \bigr). $$

As R satisfies claims monotonicity, the desired inequality follows.

If T(b,c)<E<T′(b,c), then, $R^{c}_{i}(N,b,c,E)= t_{i} (b,c)+ R_{i}(N,c -t(b,c), E- T(b,c))$, and $R^{c}_{i}(N,b,c^{\prime },E)=R_{i}(N,t(b,c^{\prime }),E)$. Now, this case implies that t _i (b,c)=c _i (otherwise, t _i (b,c)=b _i and hence T(b,c)=T′(b,c)). Thus, by boundedness, $R^{c}_{i}(N,b,c,E)= c_{i}$. Now, by resource monotonicity and claims monotonicity of R, R _i (N,t(b,c′),E)≥R _i (N,t(b,c),T(b,c))=t _i (b,c)=c _i , from where the desired inequality follows.

Linked claims-resource monotonicity Let $(N,c,E )\in \mathcal{D}$ and i∈N. Let $b\in\mathbb{R}^{n}_{+}$ be a baselines profile, ε>0 and c′=(c _i +ε,c _N∖{i}). Let T′(b,c)=T(b,c)+t _i (b,c′)−t _i (b,c). Then, t _i (b,c′)≤t _i (b,c)+ε and T(b,c)≤T′(b,c)≤T(b,c)+ε.

If E≤T′(b,c)−ε, then $R^{c}_{i}(N,b,c,E)=R_{i}(N,t(b,c),E)$ and $R^{c}_{i}(N,b,c^{\prime },E+\varepsilon)=R_{i}(N,t(b,c^{\prime }),E+\varepsilon)=R_{i}(N,(t_{i}(b,c^{\prime }),t_{-i}(b,c)),E+\varepsilon)$. By claims monotonicity of R, R _i (N,t(b,c′),E+ε)≤R _i (N,(t _i (b,c)+ε,t _−i (b,c)),E+ε). By linked claims-resource monotonicity of R, R _i (N,(t _i (b,c)+ε,t _−i (b,c)),E+ε)≤R _i (N,t(b,c),E)+ε, from where the desired inequality follows.

If T′(b,c)−ε<E<T(b,c), then $R^{c}_{i}(N,b,(c_{i}+\varepsilon ,c_{N\setminus\{i\}}),E+\varepsilon)=t_{i}(b,c^{\prime })+R_{i}(N,(c_{i}+\varepsilon-t_{i}(b,c^{\prime }) ,(c-t(b,c))_{-i}),E+\varepsilon-T^{\prime }(b,c))$, and $R^{c}_{i}(N,b,c,E)=t_{i}(b,c^{\prime })-R^{d}_{i}(N,t(b,c),T(b,c) -E)$. Thus, the desired inequality becomes

$$\begin{aligned} \varepsilon-t_{i}\bigl(b,c^{\prime } \bigr)+t_{i}(b,c) \ge& R^{d}_{i} \bigl(N,t(b,c),T(b,c) -E\bigr) \\ &{}+R_{i}\bigl(N,\bigl(c_{i}+ \varepsilon-t_{i}\bigl(b,c^{\prime }\bigr) ,\bigl(c-t(b,c) \bigr)_{-i}\bigr),E+\varepsilon-T^{\prime }(b,c)\bigr) \end{aligned}$$

(3)

Now, by balance and boundedness, the right hand side of (3) is bounded above by T(b,c)−E+E+ε−T′(b,c), which is precisely the left hand side of (3).

As for the second statement of the theorem, let R be a rule satisfying resource monotonicity, population monotonicity and linked resource-population monotonicity. By Theorem 2, R ^c satisfies the general property of resource monotonicity. Our aim is to show that R ^c also satisfies the general properties of population monotonicity and linked resource-population monotonicity.

Population monotonicity Let $(N,c,E )\in\mathcal{D}$ and $(N^{\prime},c^{\prime},E^{\prime} )\in\mathcal{D}$ be such that N⊆N′, $c_{N}^{\prime}=c$ and E=E′. Let $b\in \mathbb{R}^{n}_{+}$ and $b^{\prime}\in\mathbb{R}^{n^{\prime}}_{+}$ be two baselines profiles such that $b_{N}^{\prime}=b$. Note that t _j (b′,c′)=t _j (b,c) for each j∈N. Finally, let T(b,c)=∑ _j∈N t _j (b,c) and T′(b,c)=∑ _j∈N′ t _j (b′,c′). If E≤T(b,c), $R^{c}_{i}(N^{\prime},b^{\prime},c^{\prime},E)=R_{i}(N^{\prime},t(b^{\prime},c^{\prime}),E)$ and $R^{c}_{i}(N,b,c,E)=R_{i}(N,t(b,c),E)$. As R satisfies population monotonicity, the desired inequality follows.

Linked resource-population monotonicity Let $(N,c,E )\in \mathcal{D}$ and $(N^{\prime},c^{\prime},E )\in\mathcal{D}$ be such that N⊆N′ and $c_{N}^{\prime}=c$. Let $b\in\mathbb{R}^{n}_{+}$ and $b^{\prime}\in\mathbb{R}^{n^{\prime}}_{+}$ be two baselines profiles such that $b_{N}^{\prime}=b$. Note that t _j (b′,c′)=t _j (b,c) for each j∈N. Finally, let T(b,c)=∑ _j∈N t _j (b,c), and T′(b,c)=∑ _j∈N′ t _j (b′,c′). If $E\le T(b,c)-\sum_{N^{\prime}\setminus N} (c^{\prime}_{j}-t_{j}(b^{\prime},c^{\prime}) )$, then E≤T(b,c) and E′≤T′(b,c) and, therefore, $R^{c}_{i}(N^{\prime},b^{\prime},c^{\prime},E^{\prime})=R_{i}(N^{\prime},t(b^{\prime},c^{\prime}),E^{\prime})$ and $R^{c}_{i}(N,b,c,E)=R_{i}(N,t(b,c),E)$. By resource monotonicity and population monotonicity of R, the desired inequality follows. □

Proof of Theorem 4

In order to prove the first statement, let R be a rule satisfying order preservation and let (N,b,c,E) be an extended problem for which baselines are ordered like claims. Let i,j∈N be such that c _i ≤c _j . As b _i ≤b _j it follows that t _i (b,c)≤t _j (b,c). Now, if E≤T(b,c), $R^{c}_{i}(N,b,c,E)=R_{i}(N, t(b,c),E)$ and $R^{c}_{j}(N,b,c,E)=R_{j}(N,t(b,c),E)$. As R is order preserving, and t _i (b,c)≤t _j (b,c), it follows that $R^{c}_{i}(N,b,c,E)\le R^{c}_{j}(N,b,c,E)$, as desired. If, on the other hand, E≥T(b,c), the result follows from the proof in Hougaard et al. (2013).

As for the second statement, let R be a rule satisfying order preservation and let (N,b,c,E) be an extended problem for which baselines, and claim-baseline differences, are ordered like claims. Let i,j∈N be such that c _i ≤c _j . It then follows that t _i (b,c)≤t _j (b,c) and that c _i −t _i (b,c)≤c _j −t _j (b,c).

Now, if E≤T(b,c), $R^{c}_{i}(N,b,c,E)=R_{i}(N, t(b,c),E)$ and $R^{c}_{j}(N,b,c,E)= R_{j}(N, t(b,c),E)$.

As R is order preserving, and t _i (b,c)≤t _j (b,c), it follows that $R^{c}_{i}(N,b,c,E)\le R^{c}_{j}(N,b,c,E)$.

As R ^d is order preserving, and c _i −t _i (b,c)≤c _j −t _j (b,c), it follows that

$$c_{i}-t_{i}(b,c)+R^{d}_i\bigl(N, t(b,c),T(b,c)-E\bigr)\le c_{j}-t_{j}(b,c)+R^{d}_j\bigl(N, t(b,c),T(b,c)-E\bigr), $$

i.e., $c_{i}-R^{c}_{i}(N,b,c,E)\le c_{j}-R^{c}_{j}(N,b,c,E)$, as desired.

If, on the other hand, E≥T(b,c), $R^{c}_{i}(N,b,c,E)=t_{i}(b,c)+R_{i}(N, c-t(b,c),E-T(b,c))$ and $R^{c}_{j}(N,b,c,E)=t_{j}(b,c)+R_{j}(N,c-t(b,c),E-T(b,c))$. As t _i (b,c)≤t _j (b,c), c _i −t _i (b,c)≤c _j −t _j (b,c), and R is order preserving, it follows that $R^{c}_{i}(N,b, c,E)\le R^{c}_{j}(N,b,c,E)$ and $c_{i}-R^{c}_{i}(N,b,c,E)\le c_{j}-R^{c}_{j}(N,b,c,E)$, as desired. □