1 Introduction

A bankruptcy problem is a fundamental allocation problem in which multiple claimants have individual claims on an estate. The question arises which of the possible estate allocations should be selected. For this, bankruptcy theory studies appropriate bankruptcy rules that assign to each bankruptcy problem a feasible allocation, i.e.  an estate allocation such that the individual payoffs are bounded by the corresponding claims. Many axiomatic studies are devoted to bankruptcy problems with transferable utility, or TU-bankruptcy problems (cf. O’Neill, 1982), where the estate and the claims are of a monetary nature. An extensive survey is provided by Thomson (2019).

An elementary bankruptcy rule for bankruptcy problems with transferable utility is the constrained equal awards rule, which divides the estate as equally as possible in such a way that the claimants are not allocated more than their claims. This rule originates from old Jewish legal literature and forms the basis of the Talmud rule introduced by Aumann and Maschler (1985). Several axiomatic characterizations of the constrained equal awards rule in the transferable utility context are provided by Dagan (1996), Herrero and Villar (2001, 2002), Yeh (2004), and Yeh (2006).

Bankruptcy problems with nontransferable utility, or NTU-bankruptcy problems (cf. Orshan et al., 2003), arise when the claimants have utility functions over their monetary payoffs. Since each monetary allocation corresponds to a utility allocation, the attainable monetary allocations of the estate are expressed in a set of induced utility allocations which are assumed to be normalized in such a way that zero money corresponds to zero utility. The claim vector represents the individual utility claims on the estate. NTU-bankruptcy problems form a natural generalization of TU-bankruptcy problems where the underlying utility functions are not assumed to be linear, and consequently, the estate does not necessarily correspond to a simplex in the payoff space.

The constrained equal awards rule for bankruptcy problems with nontransferable utility allocates utility payoffs as equally as possible in such a way that the claimants are not allocated more than their utility claims. We study generalizations of axiomatic characterizations of the constrained equal awards rule for bankruptcy problems with transferable utility. In general, we show that many of the axiomatic results from the transferable utility context carry over to the nontransferable utility context by adequately reformulating the underlying properties on the extended domain. Interestingly, we also reveal new structures in particular cases.

We focus on axiomatic characterizations which are based on the properties symmetry, sustainability, and consistency. In a bankruptcy problem with transferable utility, claimants only differ in their claims. In a bankruptcy problem with nontransferable utility, claimants not only differ in their claims, but also in their utility measures. On the one hand, axioms become stronger when they are imposed on a larger domain. On the other hand, uniqueness can be harder to achieve on a larger domain. The strong symmetry property which requires that claimants with equal claims get equal payoffs is occasionally necessary to generalize axiomatic characterizations. The sustainability property, which requires that claimants with small enough claims are fully reimbursed, plays a central role in several other results. An elevator lemma (cf. Thomson, 2011) is exploited to extend several axiomatic characterizations for bankruptcy problems with two claimants to any number of claimants using consistency.

Alternatively, bankruptcy problems with nontransferable utility can be interpreted as bargaining problems with claims (cf. Chun & Thomson, 1992). Within this context, the constrained equal awards rule can be considered as a constrained version of the Kalai (1977) solution for bargaining problems without claims.

This paper is organized in the following way. Section 2 formally introduces bankruptcy problems with nontransferable utility and the constrained equal awards rule. In Sects. 3, 4, and 5, we derive several axiomatic characterizations based on the properties symmetry, sustainability, and consistency, respectively. Section 6 concludes.

2 Preliminaries

Let \({\mathcal {N}}\) with \(|{\mathcal {N}}|\ge 3\) be a finite set. The collection of all subsets of \({\mathcal {N}}\) is denoted by \(2^{\mathcal {N}}=\{N\mid N\subseteq {\mathcal {N}}\}\). Let \(N\in 2^{\mathcal {N}}\setminus \{\emptyset \}\). For all \(x\in {\mathbb {R}}_+^N\) and all \(S\in 2^N\setminus \{\emptyset \}\), \(x_S\in {\mathbb {R}}_+^S\) denotes \(x_S=(x_i)_{i\in S}\). For all \(x,y\in {\mathbb {R}}_+^N\), \(x\le y\) denotes \(x_i\le y_i\) for all \(i\in N\), and \(x<y\) denotes \(x_i<y_i\) for all \(i\in N\). For each \(E\subseteq {\mathbb {R}}_+^N\),

  • the comprehensive hull is given by \(\mathrm{comp}(E)=\{x\in {\mathbb {R}}_+^N\mid \exists _{y\in E}:y\ge x\}\);

  • the strong Pareto set is given by \(\mathrm{SP}(E)=\{x\in E\mid \lnot \exists _{y\in E,y\ne x}:y\ge x\}\);

  • the weak Pareto set is given by \(\mathrm{WP}(E)=\{x\in E\mid \lnot \exists _{y\in E}:y>x\}\);

  • the weak upper contour set is given by \(\mathrm{WUC}(E)=\{x\in {\mathbb {R}}_+^N\mid \lnot \exists _{y\in E}:y>x\}\).

Note that \(\mathrm{SP}(E)\subseteq \mathrm{WP}(E)\subseteq \mathrm{WUC}(E)\). A set \(E\subseteq {\mathbb {R}}_+^N\) is comprehensive if \(E=\mathrm{comp}(E)\), and is strictly comprehensive if \(E=\mathrm{comp}(E)\) and \(\mathrm{SP}(E)=\mathrm{WP}(E)\).

A bankruptcy problem with nontransferable utility is a triple (N, E, c), where \(N\in 2^{\mathcal {N}}\setminus \{\emptyset \}\) is a set of claimants, \(E\subseteq {\mathbb {R}}_+^N\) with \(E\cap {\mathbb {R}}_{++}^N\ne \emptyset\) is a compact and strictly comprehensive estate,Footnote 1 and \(c\in \mathrm{WUC}(E)\cap {\mathbb {R}}_{++}^N\) is a vector of claims of N on E. Let \(\mathrm{BR}^{\mathcal {N}}\) denote the class of all bankruptcy problems with nontransferable utility. Note that each TU-bankruptcy problem gives rise to an NTU-bankruptcy problem \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) with \(E=\{x\in {\mathbb {R}}_+^N\mid \sum _{i\in N}x_i\le e\}\) for some \(e\in {\mathbb {R}}_{++}\).

A bankruptcy rule f assigns to each bankruptcy problem \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) a unique payoff allocation \(f(N,E,c)\in \mathrm{WP}(E)\) such that \(f(N,E,c)\le c\). The constrained equal awards rule, \(\mathrm{CEA}\), is the bankruptcy rule that assigns to each \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) the unique payoff allocation

$$\begin{aligned}\mathrm{CEA}(N,E,c)=\left( \min \{c_i,a^{E,c}\}\right) _{i\in N},\end{aligned}$$

where \(a^{E,c}\in {\mathbb {R}}_{++}\) is such that \(\mathrm{CEA}(N,E,c)\in \mathrm{WP}(E)\). In other words, the constrained equal awards rule for bankruptcy problems with nontransferable utility allocates utility payoffs as equally as possible under the restriction that the claimants are not allocated more than their claims.

Example 1

Let \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) be such that \(N=\{1,2\}\), \(E=\{x\in {\mathbb {R}}_+^N\mid 2x_1^2+9x_2\le 72\}\), and \(c=(2,9)\). Then, \(a^{E,c}=7\frac{1}{9}\) and \(\mathrm{CEA}(N,E,c)=(2,7\frac{1}{9})\). This is illustrated as follows.

figure a

3 Symmetry

This section studies axiomatic characterizations of the constrained equal awards rule based on symmetry. In a bankruptcy problem with nontransferable utility, claimants not only differ in their claims, but also differ in their utility measures. The symmetry property states that claimants with identical characteristics should get equal payoffs. The constrained equal awards rule even satisfies the stronger symmetry property which requires that claimants with equal claims get equal payoffs. This property is called claim symmetry.

Symmetry \(f_i(N,E,c)=f_j(N,E,c)\) for all \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) and all \(i,j\in N\) with \(E=\{x\in {\mathbb {R}}_+^N\mid \exists _{y\in E}:y_i=x_j,y_j=x_i,y_{N\setminus \{i,j\}}=x_{N\setminus \{i,j\}}\}\) and \(c_i=c_j\).

Claim Symmetry \(f_i(N,E,c)=f_j(N,E,c)\) for all \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) and all \(i,j\in N\) with \(c_i=c_j\).

Note that both symmetry and claim symmetry generalize the symmetry property for TU-bankruptcy rules. The derivations of properties of the constrained equal awards rule are presented in Appendix.

Dagan (1996) characterized the constrained equal awards rule for TU-bankruptcy problems by symmetry, truncation invariance, and composition up. The truncation invariance property states that it is not relevant to claim more than your utopia value, i.e.  your maximal individual payoff within the estate. This is supported by the fact that you cannot get more than your utopia value in each estate allocation.

Let \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\). The vector of utopia values \(u^E\in {\mathbb {R}}_{++}^N\) is defined by

$$\begin{aligned}u^E=\left( \max \{x_i\mid x\in E\}\right) _{i\in N}.\end{aligned}$$

The vector of truncated claims \({\hat{c}}^E\in {\mathbb {R}}_{++}^N\) is defined by

$$\begin{aligned}{\hat{c}}^E=\left( \min \{c_i,u_i^E\}\right) _{i\in N}.\end{aligned}$$

Truncation Invariance \(f(N,E,c)=f(N,E,{\hat{c}}^E)\) for all \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\).

Example 2

Let \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) be such that \(N=\{1,2\}\), \(E=\{x\in {\mathbb {R}}_+^N\mid 2x_1^2+9x_2\le 72\}\), and \(c=(2,9)\) as in Example 1. Then, \(u^E=(6,8)\) and \({\hat{c}}^E=(2,8)\). This is illustrated as follows.

figure b

We formulate an axiomatic characterization of the constrained equal awards rule for NTU-bankruptcy problems in terms of symmetry, truncation invariance, and estate monotonicity. Estate monotonicity imposes that no claimant is worse off when the estate expands. Such a monotonicity property was also explored by Kalai (1977) in the context of bargaining problems.

Estate Monotonicity \(f(N,E,c)\le f(N,E',c)\) for all \((N,E,c),(N,E',c)\in \mathrm{BR}^{\mathcal {N}}\) with \(E\subseteq E'\).

Note that we do not need a generalization of the stronger composition up property used by Dagan (1996) in the form of step-by-step negotiations formulated by Kalai (1977). The property composition up would be based on the following idea. Suppose that the estate turns out to be larger than initially thought. Then, there are two ways of proceeding. Either one cancels the initial allocation and solves the new problem, or one interprets the initial allocation as origin for the new problem. If a bankruptcy rule satisfies composition up, then both ways of proceeding lead to the same payoff allocation. Note that composition up would imply estate monotonicity.

Theorem 3.1

The constrained equal awards rule is the unique bankruptcy rule satisfying symmetry, truncation invariance, and estate monotonicity.

Proof

By Lemma A. 1, Lemma A. 2, and Lemma A. 3, the constrained equal awards rule satisfies symmetry, truncation invariance, and estate monotonicity. Let f be a bankruptcy rule satisfying symmetry, truncation invariance, and estate monotonicity. Let \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\). Denote \(N=\{1,\ldots ,|N|\}\) such that \(c_1\le \ldots \le c_{|N|}\). Let \(k\in N\) be such that \(c_i\le a^{E,c}\) for all \(i\in N\) with \(i\le k\) and \(c_i>a^{E,c}\) for all \(i\in N\) with \(i>k\). For all \(i\in N\) with \(i\le k\) and all small \(\varepsilon >0\), define \(E_i^\varepsilon \subseteq {\mathbb {R}}_+^N\) by \(E_i^\varepsilon =\mathrm{comp}(\mathrm{conv}(A_i^\varepsilon ))\), where

$$A_{i}^{\varepsilon } = \left\{ {\left( {\left( {c_{j} + \frac{{\left| N \right| - \left| S \right|}}{{\left| N \right| - 1}}\varepsilon } \right)_{{j \in S:c_{j} < c_{i} }} ,\left( {c_{i} - \frac{{\left| S \right| - 1}}{{\left| N \right| - 1}}\varepsilon } \right)_{{j \in S:c_{j} \ge c_{i} }} ,\left( 0 \right)_{{j \in N\backslash S}} } \right)\left| S \right. \in 2^{N} \backslash \left\{ \phi \right\}} \right\}$$

For all small \(\varepsilon >0\), \(E_1^\varepsilon \subseteq \ldots \subseteq E_k^\varepsilon\), and for all \(i\in N\) with \(i\le k\),

$$\begin{aligned} u_j^{E_i^\varepsilon }= {\left\{ \begin{array}{ll} c_j+\varepsilon &{} \text {if } c_j<c_i; \\ c_i &{} \text {if } c_j\ge c_i, \end{array}\right. } \quad \text {and}\quad {\hat{c}}_j^{E_i^\varepsilon }= {\left\{ \begin{array}{ll} c_j &{} \text {if } c_j<c_i; \\ c_i &{} \text {if } c_j\ge c_i. \end{array}\right. } \end{aligned}$$
figure c

By symmetry, \(f(N,E_1^\varepsilon ,{\hat{c}}^{E_1^\varepsilon })=(c_1-\varepsilon )_{j\in N}\) for all small \(\varepsilon >0\). By truncation invariance, \(f(N,E_1^\varepsilon ,c)=f(N,E_1^\varepsilon ,{\hat{c}}^{E_1^\varepsilon })=(c_1-\varepsilon )_{j\in N}\) for all small \(\varepsilon >0\). By estate monotonicity, \(f_1(N,E_i^\varepsilon ,c)=c_1\) for all \(i\in N\) with \(i\le k\) and \(c_i>c_1\) and all small \(\varepsilon >0\). By sequentially applying symmetry, truncation invariance, and estate monotonicity, for all small \(\varepsilon >0\), \(f(N,E_k^\varepsilon ,c)=((c_i)_{i\in N:c_i<c_k},(c_k-\varepsilon )_{i\in N:c_i\ge c_k})\).

Let \(\varepsilon >0\) be small and define \(E'\subseteq {\mathbb {R}}_+^N\) by

$$E^{\prime} = {\text{comp}}\left( {{\text{conv}}\left( {\left\{ {\left( {\left( {{\text{CEA}}_{i} (N,E,c) + \tfrac{{|N| - |S|}}{{|N| - 1}}\varepsilon } \right)_{{i \in S}} (0)_{{i \in N\backslash S}} } \right)\left| {S \in 2^{N} \backslash \left\{ \phi \right\}} \right.} \right\}} \right)} \right).$$

Then, \(E_k^\varepsilon \subseteq E'\subseteq E\) and for all \(i\in N\), \(u_i^{E'}=\mathrm{CEA}_i(N,E,c)+\varepsilon\) and

$$\begin{aligned} {\hat{c}}_i^{E'}= {\left\{ \begin{array}{ll} c_i &{} \text {if } i\le k; \\ a^{E,c}+\varepsilon &{} \text {if } i>k. \end{array}\right. } \end{aligned}$$

By estate monotonicity, \(f_i(N,E',c)=c_i\) for all \(i\in N\) with \(i\le k\). By symmetry and truncation invariance, \(f_i(N,E',c)=a^{E,c}\) for all \(i\in N\) with \(i>k\). Hence, by estate monotonicity, \(f(N,E,c)=\mathrm{CEA}(N,E,c)\). \(\square\)

It follows immediately that the constrained equal awards rule is the unique bankruptcy rule satisfying claim symmetry, truncation invariance, and estate monotonicity. The estate monotonicity property which concerns changes in the estate can be replaced by the weak claims linearity property which concerns changes in the claims. This property requires that all linear combinations of the claim vector and its outcome lead to the same estate allocation.

Weak Claims Linearity \(f(N,E,c)=f(N,E,\theta c+(1-\theta )f(N,E,c))\) for all \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) and all \(\theta \in {\mathbb {R}}_+\).

A result of Dietzenbacher and Peters (2020) implies that the constrained equal awards rule for TU-bankruptcy problems can be characterized by symmetry, truncation invariance, and weak claims linearity. For NTU-bankruptcy problems, the constrained equal awards rule and the constrained relative equal awards rule (cf. Dietzenbacher et al., 2021), \(\mathrm{CREA}\), the bankruptcy rule that assigns to each \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) the unique payoff allocation

$$\begin{aligned}\mathrm{CREA}(N,E,c)=\left( \min \{c_i,\alpha ^{E,c}u_i^E\}\right) _{i\in N},\end{aligned}$$

where \(\alpha ^{E,c}\in {\mathbb {R}}_{++}\) is such that \(\mathrm{CREA}(N,E,c)\in \mathrm{WP}(E)\), both satisfy symmetry, truncation invariance, and weak claims linearity. However, the constrained equal awards rule is the unique rule satisfying claim symmetry, truncation invariance, and weak claims linearity.

Theorem 3.2

The constrained equal awards rule is the unique bankruptcy rule satisfying claim symmetry, truncation invariance, and weak claims linearity.

Proof

By Lemma A. 1, Lemma A. 2, and Lemma A. 4, the constrained equal awards rule satisfies claim symmetry, truncation invariance, and weak claims linearity. Let f be a bankruptcy rule satisfying claim symmetry, truncation invariance, and weak claims linearity. Let \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\). If \(c\in E\), then \(f(N,E,c)=c=\mathrm{CEA}(N,E,c)\). Suppose that \(c\notin E\). Let \(S=\{i\in N\mid f_i(N,E,c)<c_i\}\). Then, \(S\ne \emptyset\). Let \(x\in {\mathbb {R}}_+^N\) with \(x=\theta c+(1-\theta )f(N,E,c)\) for some \(\theta \in {\mathbb {R}}_+\) be such that \(x_i\ge u_i^E\) for all \(i\in S\), and define \(y\in {\mathbb {R}}_+^N\) by \(y_i=\max _{j\in S}\{u_j^E\}\) for all \(i\in S\) and \(y_i=c_i\) for all \(i\in N\setminus S\). Then, for all \(i\in S\),

$$\begin{aligned}{\hat{x}}_i^E=\min \{x_i,u_i^E\}=u_i^E=\min \{y_i,u_i^E\}={\hat{y}}_i^E,\end{aligned}$$

and \(x_i=c_i=y_i\) for all \(i\in N\setminus S\), so \({\hat{x}}^E={\hat{y}}^E\). By claim symmetry, there exists \(z\in {\mathbb {R}}_+\) such that \(f_i(N,E,y)=z\) for all \(i\in S\). By truncation invariance, for all \(i\in S\),

$$\begin{aligned}f_i(N,E,x)=f_i(N,E,{\hat{x}}^E)=f_i(N,E,{\hat{y}}^E)=f_i(N,E,y)=z.\end{aligned}$$

By weak claims linearity, \(f_i(N,E,c)=f_i(N,E,x)=z\) for all \(i\in S\). Then, \(z\le a^{E,c}\), since otherwise \(f(N,E,c)\ge \mathrm{CEA}(N,E,c)\) and \(f(N,E,c)\ne \mathrm{CEA}(N,E,c)\), which contradicts that E is strictly comprehensive.

Suppose that there exists \(i\in N\setminus S\) such that \(f_i(N,E,c)>a^{E,c}\). Then, \(f_j(N,E,c)\le a^{E,c}<f_i(N,E,c)\) for all \(j\in S\). Let \(x\in {\mathbb {R}}_+^N\) with \(x=\theta c+(1-\theta )f(N,E,c)\) for some \(\theta \in {\mathbb {R}}_+\) be such that \(x_j=f_i(N,E,c)=x_i\) for some \(j\in S\). By claim symmetry, \(f_i(N,E,x)=f_j(N,E,x)\). By weak claims linearity, \(f_i(N,E,c)=f_j(N,E,c)\). This is a contradiction, so \(f_i(N,E,c)\le \min \{c_i,a^{E,c}\}=\mathrm{CEA}_i(N,E,c)\) for all \(i\in N\). Hence, since E is strictly comprehensive, \(f(N,E,c)=\mathrm{CEA}(N,E,c)\). \(\square\)

Albizuri et al. (2020) characterized the constrained equal awards rule for TU-bankruptcy problems by symmetry and independence of higher claims, which requires that the payoff to a claimant does not change when the higher claim of another claimant increases (see also Moulin, 2004).

Independence of Higher Claims \(f_i(N,E,c)=f_i(N,E,(c_j',c_{N\setminus \{j\}}))\) for all \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) and all \(i,j\in N\) with \(i\ne j\) and \(c_i\le c_j\le c_j'\).

For NTU-bankruptcy problems, the constrained equal awards rule and the bankruptcy rule f that assigns to each \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) the payoff allocation

$$\begin{aligned} f(N,E,c)= {\left\{ \begin{array}{ll} (6,0) &{} \text {if } N=\{1,2\}, E=\{x\in {\mathbb {R}}_+^N\mid 2x_1^2+9x_2\le 72\}, \\ &{} \text {and } \min \{c_1,c_2\}\ge 6; \\ \mathrm{CEA}(N,E,c) &{} \text {otherwise,} \end{array}\right. } \end{aligned}$$

both satisfy symmetry and independence of higher claims. However, the constrained equal awards rule is the unique rule satisfying claim symmetry and independence of higher claims.

Theorem 3.3

The constrained equal awards rule is the unique bankruptcy rule satisfying claim symmetry and independence of higher claims.

Proof

By Lemma A. 1 and Lemma A. 5, the constrained equal awards rule satisfies claim symmetry and independence of higher claims. Let f be a bankruptcy rule satisfying claim symmetry and independence of higher claims. Let \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\). Denote \(N=\{1,\ldots ,|N|\}\) such that \(c_1\le \ldots \le c_{|N|}\). Let \(k\in N\) be such that \(c_i\le a^{E,c}\) for all \(i\in N\) with \(i<k\) and \(c_i>a^{E,c}\) for all \(i\in N\) with \(i\ge k\). By independence of higher claims, for all \(i\in N\) with \(i<k\),

$$\begin{aligned}f_i(N,E,c)=f_i(N,E,\mathrm{CEA}(N,E,c))=\mathrm{CEA}_i(N,E,c)=c_i.\end{aligned}$$

For all \(i\in N\) with \(i\ge k\), define \(x^i\in {\mathbb {R}}_{++}^N\) by \(x^i=(\min \{c_i,c_j\})_{j\in N}\). For all \(i\in N\) with \(i\ge k\), \(\mathrm{CEA}(N,E,c)\le x^i\le c\) and by claim symmetry and independence of higher claims,

$$\begin{aligned}f_k(N,E,c)=f_k(N,E,x^k)=a^{E,c}=\mathrm{CEA}_k(N,E,x^k)=\mathrm{CEA}_k(N,E,c).\end{aligned}$$

By sequentially applying this argument, \(f_i(N,E,c)=\mathrm{CEA}_i(N,E,c)\) for all \(i\in N\) with \(i\ge k\). Hence, \(f(N,E,c)=\mathrm{CEA}(N,E,c)\). \(\square\)

4 Sustainability

This section studies axiomatic characterizations of the constrained equal awards rule based on sustainability. Herrero and Villar (2002) introduced this property to describe TU-bankruptcy rules which fully reimburse claimants with small enough claims. In this way, sustainability protects claimants with small enough claims. A claim is considered small enough if all claims could be covered by the estate when the claimants would not claim more than this specific claim. In particular, this means that claimants who are fully reimbursed by the constrained equal awards rule have small enough claims.

Sustainability \(f_i(N,E,c)=c_i\) for all \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) and all \(i\in N\) with

$$\begin{aligned}\left( \min \{c_i,c_j\}\right) _{j\in N}\in E.\end{aligned}$$

Lemma 4.1

Let f be a bankruptcy rule satisfying sustainability. Then, \(f_i(N,E,c)=c_i\) for all \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) and all \(i\in N\) with \(\mathrm{CEA}_i(N,E,c)=c_i\).

Proof

Let \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\), and let \(i\in N\) be such that \(\mathrm{CEA}_i(N,E,c)=c_i\). For all \(j\in N\), \(\min \{c_i,c_j\}\le \min \{a^{E,c},c_j\}=\mathrm{CEA}_j(N,E,c)\). Since E is comprehensive, \((\min \{c_i,c_j\})_{j\in N}\in E\). By sustainability, \(f_i(N,E,c)=c_i\). \(\square\)

Example 3

Let \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) be such that \(N=\{1,2\}\), \(E=\{x\in {\mathbb {R}}_+^N\mid 2x_1^2+9x_2\le 72\}\), and \(c=(2,9)\) as in Example 1 and Example 2. Then, \((2,2)\in E\) and \((2,9)\notin E\). This means that \(f(N,E,c)=(2,7\frac{1}{9})=\mathrm{CEA}(N,E,c)\) for each bankruptcy rule f satisfying sustainability. This is illustrated as follows.

figure d

Herrero and Villar (2002) and Yeh (2004) characterized the constrained equal awards rule for TU-bankruptcy problems by sustainability and composition down. The property composition down is based on the following idea. Suppose that the estate turns out to be smaller than initially thought. Then, there are two ways of proceeding. Either one cancels the initial allocation and solves the new problem, or one interprets the initial allocation as claims for the new problem. If a bankruptcy rule satisfies composition down, then both ways of proceeding lead to the same payoff allocation.

Composition Down \(f(N,E,c)=f(N,E,f(N,E',c))\) for all \((N,E,c),(N,E',c)\in \mathrm{BR}^{\mathcal {N}}\) with \(E\subseteq E'\).

Note that composition down implies estate monotonicity. The constrained equal awards rule and the bankruptcy rule f that assigns to each \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) the payoff allocation

$$\begin{aligned} f(N,E,c)= {\left\{ \begin{array}{ll} ({\hat{c}}_1^E,\max \{x\mid ({\hat{c}}_1^E,x)\in E\}) &{} \text {if } N=\{1,2\} \text {and } c_1<c_2; \\ \mathrm{CEA}(N,E,c) &{} \text {otherwise,} \end{array}\right. } \end{aligned}$$

both satisfy sustainability and estate monotonicity. However, the constrained equal awards rule is the unique bankruptcy rule satisfying sustainability and composition down.

Theorem 4.2

The constrained equal awards rule is the unique bankruptcy rule satisfying sustainability and composition down.

Proof

By Lemma A. 6 and Lemma A. 7, the constrained equal awards rule satisfies sustainability and composition down. Let f be a bankruptcy rule satisfying sustainability and composition down. Let \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\). Suppose that \(f(N,E,c)\ne \mathrm{CEA}(N,E,c)\). Since E is strictly comprehensive, there exists \(i\in N\) such that \(f_i(N,E,c)<\mathrm{CEA}_i(N,E,c)\). By Lemma 4.1,

$$\begin{aligned}f_i(N,E,c)<\mathrm{CEA}_i(N,E,c)=a^{E,c}<c_i.\end{aligned}$$

Since composition down implies estate monotonicity, there exists \((N,E',c)\in \mathrm{BR}^{\mathcal {N}}\) with \(E\subseteq E'\) such that \(f_i(N,E',c)=a^{E,c}\). For all \(j\in N\),

$$\begin{aligned}\min \{f_i(N,E',c),f_j(N,E',c)\}\le \min \{a^{E,c},c_j\}=\mathrm{CEA}_j(N,E,c).\end{aligned}$$

Since E is comprehensive,

$$\begin{aligned}\left( \min \{f_i(N,E',c),f_j(N,E',c)\}\right) _{j\in N}\in E.\end{aligned}$$

By sustainability,

$$\begin{aligned}f_i(N,E,f(N,E',c))=f_i(N,E',c)=a^{E,c}>f_i(N,E,c).\end{aligned}$$

This contradicts composition down. Hence, \(f(N,E,c)=\mathrm{CEA}(N,E,c)\). \(\square\)

The composition down property which concerns a strong form of estate monotonicity can be replaced by claim monotonicity. This property imposes that a claimant cannot be worse off when its claim increases, ceteris paribus.

Claim Monotonicity \(f_i(N,E,c)\le f_i(N,E,(c_i',c_{N\setminus \{i\}}))\) for all \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) and all \(i\in N\) with \(c_i\le c_i'\).

Inspired by Yeh (2006), we show that the constrained equal awards rule is axiomatically characterized by sustainability and claim monotonicity.

Theorem 4.3

The constrained equal awards rule is the unique bankruptcy rule satisfying sustainability and claim monotonicity.

Proof

By Lemma A. 6 and Lemma A. 8, the constrained equal awards rule satisfies sustainability and claim monotonicity. Let f be a bankruptcy rule satisfying sustainability and claim monotonicity. Let \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\). Suppose that \(f(N,E,c)\ne \mathrm{CEA}(N,E,c)\). Since E is strictly comprehensive, there exists \(i\in N\) such that \(f_i(N,E,c)<\mathrm{CEA}_i(N,E,c)\). By Lemma 4.1,

$$\begin{aligned}f_i(N,E,c)<\mathrm{CEA}_i(N,E,c)=a^{E,c}<c_i.\end{aligned}$$

Define \(c_i'=a^{E,c}\). For all \(j\in N\),

$$\begin{aligned}\min \{c_i',c_j\}=\min \{a^{E,c},c_j\}=\mathrm{CEA}_j(N,E,c).\end{aligned}$$

This means that

$$\begin{aligned}\left( \min \{c_i',c_j\}\right) _{j\in N}\in E.\end{aligned}$$

By sustainability,

$$\begin{aligned}f_i(N,E,(c_i',c_{N\setminus \{i\}}))=c_i'=a^{E,c}>f_i(N,E,c).\end{aligned}$$

This contradicts claim monotonicity. Hence, \(f(N,E,c)=\mathrm{CEA}(N,E,c)\). \(\square\)

Yeh (2006) also showed that the constrained equal awards rule is the only TU-bankruptcy rule satisfying sustainability and supermodularity. Supermodularity states that claimants with higher claims should benefit more when the estate expands. This is stronger than the order preservation property which states that claimants with higher claims should get higher payoffs.

Supermodularity \(f_i(N,E',c)-f_i(N,E,c)\le f_j(N,E',c)-f_j(N,E,c)\) for all \((N,E,c),(N,E',c)\in \mathrm{BR}^{\mathcal {N}}\) with \(E\subseteq E'\) and all \(i,j\in N\) with \(c_i\le c_j\).

Order Preservation \(f_i(N,E,c)\le f_j(N,E,c)\) for all \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) and all \(i,j\in N\) with \(c_i\le c_j\).

Note that order preservation implies claim symmetry. The constrained equal awards rule and the bankruptcy rule f that assigns to each \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) the payoff allocation

$$f(N,E,c) = \left\{ {\begin{array}{*{20}l} {(\max \{ x {\mid }(\hat{c}_{2}^{E} ,x) \in E\} ,\hat{c}_{2}^{E} )} \hfill & {{\text{if }}N = \{1,2\}, c_{1}<c_{2},{\mkern 1mu} \,{\text{and}}\,(c_{1} ,c_{1} ) \notin E;} \hfill \\ {{\text{CEA}}(N,E,c)} \hfill & {{\text{otherwise,}}} \hfill & {} \hfill \\ \end{array} } \right.$$

both satisfy sustainability and order preservation. However, the constrained equal awards rule is the unique bankruptcy rule satisfying sustainability and supermodularity.

Theorem 4.4

The constrained equal awards rule is the unique bankruptcy rule satisfying sustainability and supermodularity.

Proof

By Lemma A. 6 and Lemma A. 9, the constrained equal awards rule satisfies sustainability and supermodularity. Let f be a bankruptcy rule satisfying sustainability and supermodularity. Then, f satisfies order preservation. Let \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\). Denote \(N=\{1,\ldots ,|N|\}\) such that \(c_1\le \ldots \le c_{|N|}\). Let \(k\in N\) be such that \(c_i\le a^{E,c}\) for all \(i\in N\) with \(i<k\) and \(c_i>a^{E,c}\) for all \(i\in N\) with \(i\ge k\). By sustainability, Lemma 4.1 implies that \(f_i(N,E,c)=c_i=\mathrm{CEA}_i(N,E,c)\) for all \(i\in N\) with \(i<k\). If \(f_k(N,E,c)\ge a^{E,c}\), then order preservation implies that \(f_i(N,E,c)\ge a^{E,c}=\mathrm{CEA}_i(N,E,c)\) for all \(i\in N\) with \(i\ge k\), which means that \(f(N,E,c)=\mathrm{CEA}(N,E,c)\) since E is strictly comprehensive. Suppose that \(f_k(N,E,c)<a^{E,c}\). Since E is strictly comprehensive, there exists \(j\in N\) with \(j>k\) such that

$$\begin{aligned}f_k(N,E,c)<a^{E,c}<f_j(N,E,c).\end{aligned}$$

Let \((N,E',c)\in \mathrm{BR}^{\mathcal {N}}\) be such that \(E\subseteq E'\) and \(a^{E',c}=c_k\). By sustainability, Lemma 4.1 implies that \(f_i(N,E',c)=c_i=\mathrm{CEA}_i(N,E',c)\) for all \(i\in N\) with \(i\le k\). By order preservation, \(f_i(N,E',c)\ge c_k=a^{E',c}=\mathrm{CEA}_i(N,E',c)\) for all \(i\in N\) with \(i>k\). Since \(E'\) is strictly comprehensive, this means that \(f_i(N,E',c)=\mathrm{CEA}_i(N,E',c)\) for all \(i\in N\). In particular,

$$\begin{aligned} f_k(N,E',c)-f_k(N,E,c)&= c_k-f_k(N,E,c) \\&= a^{E',c}-f_k(N,E,c) \\&= f_j(N,E',c)-f_k(N,E,c) \\&> f_j(N,E',c)-f_j(N,E,c). \end{aligned}$$

This contradicts supermodularity. Hence, \(f(N,E,c)=\mathrm{CEA}(N,E,c)\). \(\square\)

5 Consistency

This section studies axiomatic characterizations of the constrained equal awards rule based on consistency. In contrast to all properties considered so far, consistency concerns changes in population. As surveyed by Thomson (2011), the consistency notion is studied for several types of allocation problems. For our purposes, consider a bankruptcy problem and the corresponding payoff allocation prescribed by a particular bankruptcy rule. Suppose that some claimants leave with their allocated payoffs and the remaining claimants reevaluate their payoffs in the reduced bankruptcy problem. A bankruptcy rule satisfies consistency if it assigns to the remaining claimants the same payoffs in the reduced problem as within the original problem.

The estate of such a reduced bankruptcy problem can be modeled as the projection of the part of the original estate where all leaving claimants are allocated their corresponding payoffs. Formally, for each bankruptcy problem \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\), each set of remaining claimants \(S\in 2^N\setminus \{\emptyset \}\), and each payoff allocation \(x\in {\mathbb {R}}_+^N\), the reduced estate \(E_S^x\subseteq {\mathbb {R}}_+^S\) is defined by

$$E_{S}^{x} = \left\{ {y \in \mathbb{R}_{ + }^{S} \left| {\left( {y,x_{{N\backslash S}} } \right) \in E} \right.} \right\}.$$

Bilateral consistency describes the consistency property where only reduced bankruptcy problems with two claimants are considered.

Bilateral Consistency \(f_S(N,E,c)=f(S,E_S^{f(N,E,c)},c_S)\) for all \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) and all \(S\in 2^N\) with \(|S|=2\).

The bilateral consistency principle can also be conversed. Consider a bankruptcy problem and a feasible payoff allocation. Suppose that a certain bankruptcy rule prescribes for each reduced bankruptcy problem with two claimants the corresponding payoffs within this allocation. If the bankruptcy rule satisfies converse consistency, then it prescribes this payoff allocation for the original bankruptcy problem.

Converse Consistency \(f(N,E,c)=x\) for all \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) and all \(x\in \mathrm{WP}(E)\) with \(x\le c\) and \(x_S=f(S,E_S^x,c_S)\) for all \(S\in 2^N\) with \(|S|=2\).

If a bilateral consistent bankruptcy rule coincides with a conversely consistent bankruptcy rule for bankruptcy problems with two claimants, then the two rules coincide for each bankruptcy problem. This type of result is known as an elevator lemma. Thomson (2011) formulated the elevator lemma for a rich class of allocation problems. We exploit the elevator lemma as formulated by Dietzenbacher et al. (2020) on the specific domain of bankruptcy problems with nontransferable utility.

Lemma 5.1

(Elevator Lemma) Let f and g be two bankruptcy rules. If f satisfies bilateral consistency, g satisfies converse consistency, and \(f(N,E,c)=g(N,E,c)\) for all \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) with \(|N|=2\), then \(f=g\).

If a bankruptcy rule satisfies both bilateral consistency and converse consistency, then each axiomatic characterization for bankruptcy problems with two claimants can be lifted to bankruptcy problems with any number of claimants by additionally requiring bilateral consistency or converse consistency. This means that we can focus on axiomatic characterizations of the constrained equal awards rule for bankruptcy problems with two claimants. For this class, the sustainability property coincides with the weaker exemption property. Exemption also states that claimants with small enough claims should be fully compensated, but now a claim is considered to be small enough if it is smaller than the maximal payoff in a feasible equal awards allocation. We show that exemption coincides with sustainability for bankruptcy problems with two claimants and apply the Elevator Lemma to derive new axiomatic characterizations of the constrained equal awards rule using bilateral consistency and converse consistency.

Exemption \(f_i(N,E,c)=c_i\) for all \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) and all \(i\in N\) with \(c_i\le \rho ^E\), where \(\rho ^E=\max \{x\in {\mathbb {R}}_+\mid (x)_{j\in N}\in E\}\).

Note that sustainability implies exemption.

Example 4

Let \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) be such that \(N=\{1,2\}\), \(E=\{x\in {\mathbb {R}}_+^N\mid 2x_1^2+9x_2\le 72\}\), and \(c=(2,9)\) as in Example 1, Example 2, and Example 3. Then, \(\rho ^E=\frac{3}{4}\sqrt{73}-2\frac{1}{4}\). This is illustrated as follows.

figure e

Theorem 5.2

  1. (i)

    The constrained equal awards rule is the unique bankruptcy rule satisfying bilateral consistency, exemption, and composition down.

  2. (ii)

    The constrained equal awards rule is the unique bankruptcy rule satisfying bilateral consistency, exemption, and claim monotonicity.

  3. (iii)

    The constrained equal awards rule is the unique bankruptcy rule satisfying bilateral consistency, exemption, and supermodularity.

  4. (iv)

    The constrained equal awards rule is the unique bankruptcy rule satisfying converse consistency, exemption, and composition down.

  5. (v)

    The constrained equal awards rule is the unique bankruptcy rule satisfying converse consistency, exemption, and claim monotonicity.

  6. (vi)

    The constrained equal awards rule is the unique bankruptcy rule satisfying converse consistency, exemption, and supermodularity.

Proof

By Lemma A. 6, Lemma A. 7, Lemma A. 8, Lemma A. 9, Lemma A. 10, and Lemma A. 11, the constrained equal awards rule satisfies bilateral consistency, converse consistency, exemption, composition down, claim monotonicity, and supermodularity. Let f be a bankruptcy rule satisfying bilateral consistency or converse consistency, exemption, and composition down or claim monotonicity or supermodularity.

Let \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) with \(|N|=2\). Let \(i\in N\) be such that \((\min \{c_i,c_j\})_{j\in N}\in E\). If \(c_i\ge c_{N\setminus \{i\}}\), then \(c\in E\), so \(f_i(N,E,c)=c_i\) since E is strictly comprehensive. Suppose that \(c_i\le c_{N\setminus \{i\}}\). Then, \(c_i\le \rho ^E\). By exemption, \(f_i(N,E,c)=c_i\). Hence, f satisfies sustainability on the domain of bankruptcy problems with two claimants.

By composition down, claim monotonicity, or supermodularity, reformulating Theorem 4.2, Theorem 4.3, or Theorem 4.4 on the domain of bankruptcy problems with two claimants implies that \(f(N,E,c)=\mathrm{CEA}(N,E,c)\) for all \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) with \(|N|=2\). By bilateral consistency or converse consistency, Lemma 5.1 implies that \(f=\mathrm{CEA}\) since \(\mathrm{CEA}\) satisfies both bilateral consistency and converse consistency. \(\square\)

The results of Theorem 5.2\((i)-(iv)\) were also derived by Herrero and Villar (2001), Yeh (2004), and Yeh (2006) for TU-bankruptcy problems. Surprisingly, in the particular characterization based on bilateral consistency, exemption, and supermodularity, supermodularity can be weakened to order preservation. Yeh (2006) discovered a similar relation for TU-bankruptcy problems.

Theorem 5.3

The constrained equal awards rule is the unique bankruptcy rule satisfying bilateral consistency, exemption, and order preservation.

Proof

By Lemma A. 6, Lemma A. 9, and Lemma A. 10, the constrained equal awards rule satisfies bilateral consistency, exemption, and order preservation. Let f be a bankruptcy rule satisfying bilateral consistency, exemption, and order preservation. Let \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) with \(|N|=2\). Denote \(N=\{1,2\}\) such that \(c_1\ge c_2\). If \(c_2\le \rho ^E\), then exemption implies that \(f_2(N,E,c)=c_2=\mathrm{CEA}_2(N,E,c)\), and \(f_1(N,E,c)=\mathrm{CEA}_1(N,E,c)\) since E is strictly comprehensive. Suppose that \(c_2>\rho ^E\). Define \((N',E',c')\in \mathrm{BR}^{\mathcal {N}}\) such that \(N'=\{1,2,3\}\), \(E=\{x\in {\mathbb {R}}_+^N\mid (x,\rho ^E)\in E'\}\), and \(c'=(c_1,c_2,\rho ^E)\). Then, \(c_1'\ge c_2'>c_3'=\rho ^E=\rho ^{E'}\).

By exemption and order preservation,

$$\begin{aligned}f_1(N',E',c')\ge f_2(N',E',c')\ge f_3(N',E',c')=c_3'=\rho ^{E'}.\end{aligned}$$

Since \(E'\) is strictly comprehensive, \(f_i(N',E',c')=\rho ^{E'}=\rho ^E\) for all \(i\in N'\). By bilateral consistency,

$$\begin{aligned}f(N,E,c)=f(N,E_N'^{f(N',E',c')},c_N')=f_N(N',E',c')=\left( \rho ^E\right) _{i\in N}=\mathrm{CEA}(N,E,c).\end{aligned}$$

Hence, \(f(N,E,c)=\mathrm{CEA}(N,E,c)\) for all \((N,E,c)\in \mathrm{BR}^{\mathcal {N}}\) with \(|N|=2\). By bilateral consistency, Lemma 5.1 implies that \(f=\mathrm{CEA}\) since \(\mathrm{CEA}\) satisfies converse consistency. \(\square\)

6 Concluding remarks

This paper showed that many axiomatic characterizations of the constrained equal awards rule apply in a much more general context than for which they are originally formulated. In particular, we generalized several axiomatic characterizations based on symmetry, sustainability, and consistency to the domain of bankruptcy problems with nontransferable utility. An overview of the results and corresponding properties is provided in the following table.

 

T3.1

T3.2

T3.3

T4.2

T4.3

T4.4

T5.2

T5.3

symmetry

\(\star\)

       

claim symmetry

 

\(\star\)

\(\star\)

     

truncation invariance

\(\star\)

\(\star\)

      

estate monotonicity

\(\star\)

       

weak claims linearity

 

\(\star\)

      

independence of higher claims

  

\(\star\)

     

sustainability

   

\(\star\)

\(\star\)

\(\star\)

  

composition down

   

\(\star\)

  

\(\star ^a\)

 

claim monotonicity

    

\(\star\)

 

\(\star ^a\)

 

supermodularity

     

\(\star\)

\(\star ^a\)

 

bilateral consistency

      

\(\star ^b\)

\(\star\)

converse consistency

      

\(\star ^b\)

 

exemption

      

\(\star\)

\(\star\)

order preservation

       

\(\star\)

  1. aComposition down, claim monotonicity, or supermodularity
  2. bBilateral consistency or converse consistency

Recall that supermodularity implies order preservation, order preservation implies claim symmetry, and claim symmetry implies symmetry. Moreover, composition down implies estate monotonicity, and sustainability implies exemption.