The reverse TAL-family of rules for bankruptcy problems

Abstract

This paper analyzes a family of rules for bankruptcy problems that generalizes the so-called reverse Talmud rule and encompasses both the constrained equal-awards rule and the constrained equal-losses rule. The family, introduced by van den Brink et al. (Eur J Oper Res 228:413–417, 2013), is a counterpart to the so-called TAL-family of rules, introduced and studied by Moreno-Ternero and Villar (Soc Choice Welf 27:231–249, 2006a), and it is included within the so-called CIC-family of rules introduced by Thomson (Soc Choice Welf 31:667–692, 2008). We provide a systematic study of the structural properties of the rules within the family, as well as its connections with the existing related literature.

Appendix: Proofs of the results

In what follows, we omit all straightforward steps in the proofs.

Proof of Proposition 2

As all rules within the reverse TAL-family are parametric, we obtain, following Young (1987), that all rules within the reverse TAL-family satisfy equal treatment of equals, continuity, and consistency.¹⁸ By Lemma 3 in Chambers and Thomson (2002) they also satisfy anonymity.

As for order preservation, let $\theta \in [ 0,1]$ and $( N,c,E) \in {\mathcal {D}}$ be given. We distinguish two cases.

Case 1 $E\le \theta C$.

In this case, $\textit{RT}_{i}^{\theta }( N,c,E) =\max \{ \theta c_{i}-\lambda ,0 \}$, for each $i\in N$. Let $i,j\in N$ be such that $c_{i}\ge c_{j}$. Then, $\textit{RT}_{i}^{\theta }( N,c,E) \ge \textit{RT}_{j}^{\theta }( N,c,E)$. Now, suppose that $\lambda <\theta c_{j}\le \theta c_{i}$. Then, $c_{i}-\textit{RT}_{i}^{\theta }( N,c,E) =c_{i}-(\theta c_{i}-\lambda ) \ge c_{j}-(\theta c_{j}-\lambda ) =c_{j}-\textit{RT}_{j}^{\theta }( N,c,E)$. On the other hand, if $\theta c_{i}\ge \lambda \ge \theta c_{j}$, then $c_{i}-\textit{RT}_{i}^{\theta }( N,c,E) =(1-c_{i})+\lambda \ge c_{j}=c_{j}-\textit{RT}_{j}^{\theta }( N,c,E)$.¹⁹ Finally, if $\lambda >\theta c_{i}$ then $c_{i}-\textit{RT}_{i}^{\theta }( N,c,E) =c_{i}\ge c_{j}=c_{j}-\textit{RT}_{j}^{\theta }( N,c,E)$.

Case 2 $E\ge \theta C$.

In this case, $\textit{RT}_{i}^{\theta }( N,c,E) =\theta c_{i}+\min \{ (1-\theta ) c_{i}, \mu \}$, for each $i\in N$. Let $i,j\in N$ be such that $c_{i}\ge c_{j}$. Then, $\textit{RT}_{i}^{\theta }( N,c,E) \ge \textit{RT}_{j}^{\theta }( N,c,E)$. Now, suppose that $\mu <(1-\theta )c_{j} \le (1-\theta )c_i$. In this case, $c_{i}-\textit{RT}_{i}^{\theta }( N,c,E) =(1-\theta )c_{i}-\mu \ge (1-\theta )c_{j}-\mu =c_{j}-\textit{RT}_{j}^{\theta }( N,c,E)$. On the other hand, if $(1-\theta )c_{i}\ge \mu \ge (1-\theta )c_{j}$ then $c_{i}-\textit{RT}_{i}^{\theta }( N,c,E) =(1-\theta )c_{i}-\mu \ge 0=c_{j}-\textit{RT}_{j}^{\theta }( N,c,E)$. Finally, if $\mu >(1-\theta )c_{i}$, then $c_{i}-\textit{RT}_{i}^{\theta }( N,c,E) =0=c_{j}-\textit{RT}_{j}^{\theta }( N,c,E)$.

Finally, we turn to scale invariance. Let $( N,c,E) \in {\mathcal {D}}$ be given. We distinguish three cases:

Case 1 $E<\theta C$.

As $\alpha E<\alpha \theta C$ for each $\alpha >0,$ it follows that $\textit{RT}_{i}^{\theta }( N,c,E) =\max \{ \theta c_{i}-\lambda ,0 \}$, and $\textit{RT}_{i}^{\theta }( N,\alpha c,\alpha E) =\max \{ \theta \alpha c_{i}-\lambda ^{\prime },0 \}$, for each $i\in N$, where $\lambda$ and $\lambda ^{\prime }$ are such that $\sum _{i\in N}{} \textit{RT}_{i}^{\theta }( N,c,E) =E$ and $\sum _{i\in N}{} \textit{RT}_{i}^{\theta }( N,\alpha c,\alpha E) =\alpha E$, respectively. Now, there exists $\overline{\lambda }>0$, such that $\lambda ^{\prime }=\alpha \overline{\lambda }$. Thus, $\max \{ \theta \alpha c_{i}-\lambda ^{\prime },0\} =\alpha \max \{ \theta c_{i}-\overline{\lambda }, 0 \}$. Assume, without loss of generality, that $c_{n}=\max _{i\in N}\{c_{i}\}.$ Then, the function $H^{\theta }:[ 0,\theta c_{n}] \rightarrow {\mathbb {R}}_{+}$ such that $H^{\theta }(\lambda )=\sum _{i\in N}\max \{ \theta c_{i}-\lambda ,0 \}$ is piecewise linear and strictly increasing. Moreover, its image corresponds to $[0,\theta C]$. Thus, for each $E\in [0,\theta C)$ there exists a unique $\lambda _{0}$ such that $H^{\theta }(\lambda _{0})=E$ . This implies $\lambda =\overline{\lambda }=\lambda _{0}$, and therefore, $\textit{RT}_{i}^{\theta }( N,\alpha c,\alpha E) =\alpha \textit{RT}_{i}^{\theta }( N,c,E)$, for each $i\in N$.

Case 2 $E>\theta C$.

As $\alpha E>\alpha \theta C$ for each $\alpha >0$, we have $\textit{RT}_{i}^{\theta }( N,c,E) =\theta c_{i}+\min \{ (1-\theta ) c_{i}, \mu \}$, and $\textit{RT}_{i}^{\theta }( N,\alpha c,\alpha E) =\theta \alpha c_{i}+\min \{ (1-\theta ) \alpha c_{i}, \mu ^{\prime } \}$, for each $i\in N$, where $\mu$ and $\mu ^{\prime }$ are such that $\sum _{i\in N}{} \textit{RT}_{i}^{\theta }( N,c,E) =E$ and $\sum _{i\in N}{} \textit{RT}_{i}^{\theta }( N,\alpha c,\alpha E) =\alpha E$, respectively. Now, there exists $\overline{\mu }>0$, such that $\mu ^{\prime }=\alpha \overline{\mu }$. Thus, $\alpha \theta c_{i}+\min \{ (1-\theta ) \alpha c_{i}, \mu ^{\prime } \} =\alpha (\theta c_{i}+\min \{ (1-\theta ) c_{i}, \overline{\mu } \})$. Take again $c_{n}=\max _{i\in N}\{c_{i}\}.$ Then, the function $H^{\theta }:[ 0,( 1-\theta ) c_{n}] \rightarrow {\mathbb {R}}_{+}$ such that $H^{\theta }(\mu )=\sum _{i\in N}\min \{ (1-\theta ) c_{i},\mu \}$ is piecewise linear and strictly increasing. Moreover, its image corresponds to $[\theta C,C]$. Thus, for each $E\in (\theta C,C]$ there exists a unique $\mu _{0}$ such that $H^{\theta }(\mu _{0})=E$. This implies $\mu =\overline{\mu }=\mu _{0}$, and, therefore, $\textit{RT}_{i}^{\theta }( N,\alpha c,\alpha E) =\alpha \textit{RT}_{i}^{\theta }( N,c,E)$, for each $i\in N$.

Case 3 $E=\theta C$.

Then $\textit{RT}_{i}^{\theta }(N,c,E) =\theta c_{i}$ for each $i\in N$ and therefore, the property trivially holds. $\square$

Proof of Proposition 3

All rules within the reverse TAL-family satisfy resource monotonicity, claims monotonicity, and population monotonicity. Thus, by Proposition 1, they all satisfy linked claim-resource monotonicity (the dual property of claims monotonicity) and linked resource-population monotonicity (the dual of population monotonicity). Finally, Chun (1999) shows that resource monotonicity and consistency together imply resource-and-population uniformity. As, by Proposition 2, all rules within the reverse TAL-family are consistent, it follows that they all satisfy resource-and-population uniformity. $\square$

Proof of Proposition 4

By duality, it suffices to show one of the statements. It is well known that $\textit{RT}^{0}=A$ satisfies claims truncation invariance whereas $\textit{RT}^{1}=L$ does not. Let us see that no intermediate rule within the reverse TAL-family satisfies it either. Let $\theta \in (0,1)$ and consider the two-claimant problem $B=(\{1,2\},( E,\frac{E}{\theta }), E)$. Then, $\textit{RT}^{\theta }( B) =(\frac{\theta E}{2},E-\frac{\theta E}{2})$. The associated problem in which claims are truncated is $(\{1,2\},( E,E) ,E)$, whose solution is $(\frac{E}{2 },\frac{E}{2})$. Therefore, $\textit{RT}^{\theta }( \{1,2\},c,E) \ne \textit{RT}^{\theta }( \{1,2\},t(N,E,c),E).$ $\square$

Proof of Proposition 5

By duality, it suffices to show one of the statements. It is well known that $\textit{RT}^{0}=A$ satisfies average truncated lower bound on awards whereas $\textit{RT}^{1}=L$ does not (e.g., Moreno-Ternero and Villar 2004). Let us see that no intermediate rule within the reverse TAL-family satisfies it either. Let $\theta \in (0,1)$ and consider the two-claimant problem $B=(\{1,2\},( E,\frac{E}{\theta }),E )$. Then, $\textit{RT}^{\theta }( B) =(\frac{\theta E}{2},E-\frac{\theta E}{2})\ne (\frac{E}{2},\frac{E}{2})$. On the other hand, assume, by contradiction, that $\textit{RT}^{\theta }$ satisfies average truncated lower bound on awards, for some $\theta \in (0,1)$. Then, $\textit{RT}^{\theta }( B) =(\frac{E}{2},\frac{E}{2})$, which represents a contradiction. $\square$

Proof of Proposition 6

• (i) The rules $\textit{RT}^{1}=L$ and $\textit{RT}^{0}=A$ satisfy composition up (e.g., Moulin 2000). Let us see that there is no other rule within the reverse TAL-family for which this happens. Let $\theta \in (0,1)$ be given. Consider the two-claimant problem

  $$\begin{aligned} (N,c,E)=\left( \{1,2\},\left( \frac{1}{3\theta },\frac{1}{\theta }\right) ,1 \right) , \end{aligned}$$

  and let $E_{1}=\frac{1}{2}=E_{2}$. Then, $E_{1}<E=1<\frac{1}{3}+1=\theta \cdot ( c_{1}+c_{2})$. Thus, $\textit{RT}^{\theta }(N,c,E)=L(N,\theta c,E)=( \frac{1}{6},\frac{5}{6})$ and $\textit{RT}^{\theta }( N,c,E_{1}) =L(N,\theta c, E_{1})=( 0,\frac{1}{2})$.

  Let $c^{\prime }=c-\textit{RT}^{\theta }( N,c,E_{1}) =( \frac{1}{ 3\theta },\frac{1}{\theta }-\frac{1}{2})$. Then, $E_{2}-\theta \cdot ( c_{1}^{\prime }+c_{2}^{\prime }) =\frac{1}{2}- ( \frac{8-3\theta }{6})<0$, so that

  $$\begin{aligned} \textit{RT}^{\theta }\left( N,c^{\prime },E_{2}\right) =L\left( N,\theta c^{\prime },E_{2}\right) =\left\{ \begin{array}{ll} \left( 0,\frac{1}{2}\right) &{}\quad \hbox {if}\, \theta \le \frac{1}{3} \\ \left( \frac{1}{3}-\frac{5-3\theta }{12},\frac{2-\theta }{2}-\frac{5-3\theta }{12}\right) &{}\quad \hbox {if}\, \theta >\frac{1}{3} \end{array} \right. . \end{aligned}$$

  Thus, $\textit{RT}^{\theta }( N,c,E) \ne \textit{RT}^{\theta }( N,c,E_{1}) +\textit{RT}^{\theta }( N,c^{\prime },E_{2})$ for each $\theta \in (0,1)$.

• (ii) Both A and L satisfy composition down (e.g., Moulin 2000). Now, suppose that there exists some $\theta \in (0,1)$ such that $\textit{RT}^{{}\theta }$ satisfies composition down. Then $\textit{RT}^{1-\theta }$, the dual rule of $\textit{RT}^{\theta }$, should satisfy composition up, which contradicts part (i) of this theorem. $\square$

Proof of Proposition 7

Yeh (2006) shows that the constrained equal-awards rule is the only rule that satisfies sustainability and claims monotonicity. By duality, the constrained equal losses rule is the only rule that satisfies independence of residual claims and linked claim-resource monotonicity. Yeh (2006) also shows that the constrained equal-awards rule is the only rule satisfying exemption, order preservation, and consistency. By duality, the constrained equal-losses rule is the only rule satisfying exclusion, order preservation and consistency.²⁰ As all rules within the reverse TAL-family are consistent and order preserving (Proposition 2), and satisfy claims monotonicity and linked claim-resource monotonicity (Proposition 3) the proofs of all the statements follow from Yeh’s results and duality. $\square$