2.1. Broomean problems and their allocations

In this section, we present the core elements of our theory in a formal framework.

A Broomean problem is a structure

${\cal B} = \left( {E,N,a,s} \right),$

where the estate $E \gt 0$ specifies the amount of the good-to-be-divided amongst the individuals in $N = \left\{ {1, \ldots, n} \right\}$ . An individual $i \in N$ has a claim $\left( {{a_i},{s_i}} \right)$ with amount ${a_i} \ge 0$ and strength ${s_i} \gt 0$ , as specified by amounts-vector $a$ and strengths-vector $s$ , the amounts-vector being such that $\mathop \sum \nolimits_{i \in N} \,{a_i} \ge E$ : the sum of claims (weakly) exceeds the estate. ^{Footnote 5} As claim-strengths are strictly comparative, they are only determined up to an arbitrary positive multiplicative constant $\rho $ : if $s = \rho \cdot s{\rm{'}}$ then vectors $s$ and $s{\rm{'}}$ determine the same claim-strengths. However, for sake of definiteness we will typically ^{Footnote 6} normalize claim strengths and assume that $\mathop \sum \nolimits_{i \in N} {s_i} = 1$ . As an example of a Broomean problem, consider the representation of ${\cal O}$ wing Money:

$${\cal O} = \left( {40,\left\{ {A,B} \right\},\left( {20,{\rm{\;}}60} \right),\left( {{1 \over 2},{1 \over 2}} \right)} \right)$$

Indeed, the claims of $A$ (bram) and $B$ (envolio) in Owing Money have different amounts ( $20$ and $60$ ) but they are equally strong.

An allocation $x$ for ${\cal B}$ allots an amount ${x_i} \ge 0$ to each individual $i \in N$ and respects the estate: $\mathop \sum \nolimits_{i \in N} \,{x_i} \le E$ . With slight abuse of notation, we will write $x \in {\cal B}$ to indicate that $x$ is an allocation for ${\cal B}$ . We say that:

Efficiency: $x \in {\cal B}$ is efficient when $\mathop \sum \nolimits_{i \in N} \,{x_i} = E$ .

Claims-respecting: $x \in {\cal B}$ is claims-respecting when ${x_i} \le {a_i}$ for each $i \in N$ .

When an agent has a claim with an amount of ${a_i}$ and receives ${x_i}$ , the satisfaction of that claim may be expressed as:

$${\rm{Sat}}\left( {{x_i},{a_i}} \right) = {\rm{min}}\left\{ {{{{x_i}} \over {{a_i}}},{\rm{\;}}1} \right\} \cdot 100{\rm{\% }}$$

That is, claim satisfaction is a constrained (by the amount of the claim) and linear function of receipt. ^{Footnote 7} An allocation $x$ for a Broomean problem is said to satisfy claims in proportion to their strength just in case, for any two individuals $i$ and $j$ : if $i$ ’s claims is $\rho $ times as strong as $j$ ’s claim, then $i$ ’s claim receives $\rho $ times as much satisfaction. That is:

Satisfies claims in proportion to their strength: $x \in {\cal B}$ satisfies claims in proportion to their strength when ${\rm{Sat}}\left( {{x_i},{a_i}} \right) = {{{s_i}} \over {{s_j}}} \cdot {\rm{Sat}}\left( {{x_j},{a_j}} \right){\rm{\;for\;all\;}}i,{\rm{\;}}j \in N$ .

2.2. The Fairness formula for Broomean problems

We study the implications of applying the Fairness formula (FF) to Broomean problems. ^{Footnote 8} For Broomean problems, as we demonstrate in the Appendix, the requirements of the FF afford the following concise and simple presentation:

Fairness formula for ${\cal B}$ roomean problems: FF ${\cal B}$ . For a Broomean problem ${\cal B} = \left( {E,N,a,s} \right)$ , fairness requires one to realize an allocation $x$ which is:

1. (i)

   1. (a) Efficient, i.e $x$ allocates the entire estate.

   2. (b) Claims-respecting, i.e. $x$ does not award anyone more than their claim.

2. (ii) Satisfies claims in proportion to their strength.

3. (iii) Whenever (i) and (ii) conflict: $x$ should respect (i) and satisfy claims in proportion to their strength to as large a degree as possible.

Whereas FF ${\cal B}$ (i) and FF ${\cal B}$ (ii) unambiguously define properties for allocations, the meaning of FF ${\cal B}$ (iii) is underspecified. We will elaborate on and make precise the content of FF ${\cal B}$ (iii) in section 3, where we define the absolute priority rule.

For ${\cal O}$ wing Money we do not need to rely on FF ${\cal B}$ (iii): allocation $\left( {10,{\rm{\;}}30} \right)$ satisfies FF ${\cal B}$ (i) and FF ${\cal B}$ (ii) so that there is no conflict between absolute and comparative fairness for ${\cal O}$ wing Money. However, not all problems are like that. To see that it may be impossible to simultaneously respect FF ${\cal B}$ (i) and FF ${\cal B}$ (ii), consider the following problem.

Needing Owed Money. Romeo owes $20$ to Abram and $60$ to Benvolio and has $80$ left. Abram needs his money twice as strongly as Benvolio. Romeo is bound to care for the needs of Abram and Benvolio, such that Romeo’s reason for reimbursing Abram is twice as strong as his reason for repaying Benvolio. How, in order to be fair, should Romeo divide the $80$ ?

${\cal{N}}\!\:$ eeding Owed Money is represented as follows:

$${\cal N} = \left( {80,\left\{ {A,B} \right\},\left( {20,{\rm{\;}}60} \right),\left( {{2 \over 3},{1 \over 3}} \right)} \right)$$

It is readily verified that $\left( {20,{\rm{\;}}60} \right)$ is the only allocation for ${\cal N}\!$ which respects FF ${\cal B}$ (i). As claims are not satisfied in proportion to their strength in $\left( {20,{\rm{\;}}60} \right)$ , ${\cal N}\!$ illustrates that FF ${\cal B}$ (i) and FF ${\cal B}$ (ii) may conflict. But also, as $\left( {20,{\rm{\;}}60} \right)$ is the only allocation for ${\cal N}\!\!$ which respects FF ${\cal B}$ (i), any theory which seeks to resolve this conflict by prioritizing absolute fairness must recommend $\left( {20,{\rm{\;}}60} \right)$ for ${\cal N}\!$ .

However, when there is a conflict between absolute and comparative fairness, there are typically many allocations which satisfy FF ${\cal B}$ (i). For instance, consider the following Broomean problem ${\cal M}$ , to which we will refer later on as the ${\cal M}$ ore money problem:

$${\cal M} = \left( {80,\left\{ {A,B,C} \right\},\left( {20,{\rm{\;}}60,{\rm{\;}}40} \right),\left( {{1 \over 2},{1 \over 4},{1 \over 4}} \right)} \right)$$

Indeed, as the reader may care to verify, and as a direct consequence of Proposition 2 below, there is no allocation for ${\cal M}$ which respects both FF ${\cal B}$ (i) and FF ${\cal B}$ (ii). Now there are many allocations for ${\cal M}$ which respect FF ${\cal B}$ (i), such as $\left( {0,{\rm{\;}}60,{\rm{\;}}20} \right)$ , $\left( {20,{\rm{\;}}20,{\rm{\;}}40} \right)$ or $\left( {20,{\rm{\;}}36,{\rm{\;}}24} \right)$ . Owing to its underspecified meaning, it is not clear which of these allocations is recommended by FF ${\cal B}$ (iii). In the next section we will introduce the absolute priority rule ${P^\unicode{x2020} }$ , a division rule for Broomean problems whose properties makes precise the content of FF ${\cal B}$ (iii). For ${\cal M}$ , ${P^\unicode{x2020} }$ recommends $\left( {20,{\rm{\;}}36,{\rm{\;}}24} \right)$ .

3.1. Division rules and their properties

A division rule $f$ maps any Broomean problem ${\cal B}$ to an allocation $f\left( {\cal B} \right)$ . In section 2.1, we defined what it means for an allocation $x$ to be efficient, claims-respecting and to satisfy claims in proportion to their strength. Each of these three properties gives rise to a corresponding property of a division rule: a division rule $f$ is efficient/claims-respecting/satisfies claims in proportion to their strength when, for any Broomean problem ${\cal B}$ , $f\left( {\cal B} \right)$ is efficient/claims-respecting/satisfies claims in proportion to their strength.

Our discussion of allocations for ${\cal N}\!$ eeding Owed Money readily translates into the following impossibility result for division rules.

It will be useful to define the weighted proportional rule $P$ , which divides the estate proportional to the strength-weighted amounts, ${s_i}{a_i}$ , of the individuals.

(1) $$P{({\cal B})_i} = {{{s_i}{a_i}} \over {\mathop \sum \nolimits_{j \in N} \,{s_j}{a_j}}}E$$

Alternatively, we can define $P$ as follows:

(2) $$P{({\cal B})_i} = \lambda {s_i}{a_i}{\rm{\;with\;}}\lambda \gt {0\ \rm s.t.} \mathop \sum \limits_{i \in N} P{({\cal B})_i} = E$$

To see that (1) and (2) are, indeed, equivalent, observe that the value of $\lambda $ which solves (2), call it ${\lambda ^P}$ , is equal to ${E \over {\mathop \sum \nolimits_{j \in N} \,{s_j}{a_j}}}$ so that each individual $i$ is awarded ${\lambda ^P}{s_i}{a_i}$ according to both definitions of the weighted proportional rule.

Although $P$ is efficient, it is neither claims-respecting nor does it satisfy claims in proportion to their strengths, as its recommendation $P\left( {\cal N} \right) = \left( {32,{\rm{\;}}48} \right)$ for ${\cal N}\!$ eeding Owed Money testifies. Hence, an allocation that is recommended by $P$ may violate both FF ${\cal B}$ (i) and FF ${\cal B}$ (ii). Although we do not want to recommend $P$ as a rule of fair division, $P$ conveniently allows us to characterize the conditions for which there are allocations which simultaneously satisfy FF ${\cal B}$ (i) and FF ${\cal B}$ (ii):

To see that (a) implies (b) note that as $x$ is claims-respecting it follows that $Sat\left( {{x_i},{a_i}} \right) = {{{x_i}} \over {{a_i}}}$ for all $i \in N$ . Thus when $x$ satisfies claims in proportion to their strength it follows that ${{{x_i}} \over {{a_i}}} = {{{s_i}} \over {{s_j}}}{{{x_j}} \over {{a_j}}}$ so that ${x_j} = {{{s_j}{a_j}} \over {{s_i}{a_i}}}{x_i}$ . So then, as $x$ is efficient, we have that $\mathop \sum \nolimits_{j \in N} \,{{{s_j}{a_j}} \over {{s_i} \cdot {a_i}}}{x_i} = E$ , so that for all $i \in N$ :

$${x_i} = {\lambda ^P}{s_i}{a_i},{\rm{\;with}}\;{\lambda ^P} = {E \over {\mathop \sum \nolimits_{j \in N} \,{s_j}{a_j}}} \gt 0$$

Thus, (a) implies (b).

Now let $x$ be an efficient claims-respecting allocation for which (b) holds. It then follows that $Sat\left( {{x_i},{a_i}} \right) = {{{x_i}} \over {{a_i}}} = \lambda \cdot {s_i}$ for all $i \in N$ . From $Sat\left( {{x_i},{a_i}} \right) = \lambda \cdot {s_i}$ and $Sat\left( {{x_j},{a_j}} \right) = \lambda \cdot {s_j}$ it follows that $Sat\left( {{x_i},{a_i}} \right) = {{{s_i}} \over {{s_j}}} \cdot Sat\left( {{x_j},{a_j}} \right)$ so that $x$ satisfies claims in proportion to their strength. Thus, (b) implies (a).

The left-to-right direction of (i) now follows from the proof that (a) implies (b): an allocation with ${\lambda ^P}{s_i}{a_i}$ for all $i \in N$ is equal to $P\left( {\cal B} \right)$ . The right-to-left direction of (i) is immediate: if $P{({\cal B})_i} \le {a_i}$ for all $i \in N$ then $P\left( {\cal B} \right)$ is efficient, claims-respecting and satisfies claims in proportion to their strength. Claim (ii) follows from (i).

So, Proposition 2 tells us that it is perfectly fine to use $P$ whenever $P{({\cal B})_i} \le {a_i}$ for all individuals $i$ . The question, which we will answer in section 3.2, thus becomes how $P$ should be extended to Broomean problems with individuals for which $P{({\cal B})_i} \gt {a_i}$ .

Now, although $P$ does not satisfy claims in proportion to their strength, it does satisfy all claims that are partially reimbursed in proportion to their strength. We say that an allocation $x \in {\cal B}$ satisfies partially reimbursed claims in proportion to their strength when:

$${\rm{Sat}}\left( {{x_i},{a_i}} \right) = {{{s_i}} \over {{s_j}}} \cdot {\rm{Sat}}\left( {{x_j},{a_j}} \right){\rm{\;for\;all\;}}i,j{\rm{\;such\;that\;}}{x_i} \lt {a_i}{\rm{\;and\;}}{x_j} \lt {a_j}$$

The corresponding property for division rules then reads as follows.

Satisfies partially reimbursed claims in proportion to their strength. A division rule $f$ satisfies partially reimbursed claims in proportion to their strength when, for any Broomean problem ${\cal B}$ , $f\left( {\cal B} \right)$ satisfies partially reimbursed claims in proportion to their strength.

Indeed, as an immediate consequence of its definition, $P$ satisfies partially reimbursed claims in proportion to their strength. Another division rule which does so is the absolute priority rule ${P^\unicode{x2020} }$ , to which we will turn next.

3.2 The absolute priority rule ${P^\unicode{x2020} }$

The absolute priority rule ${P^\unicode{x2020} }$ is defined as follows ^{Footnote 9} :

(3) $${P^\unicode{x2020} }{({\cal B})_i} = {\rm{min}}\left\{ {\lambda {s_i}{a_i},{a_i}} \right\}{\rm{\;with\;}}\lambda \gt 0\ {\rm s.t.}\,\mathop \sum \limits_{i \in N} {P^\unicode{x2020} }{({\cal B})_i} = E$$

It is an immediate consequence of definition (3) that ${P^\unicode{x2020} }$ is efficient and claims-respecting. Moreover, it readily follows that ${P^\unicode{x2020} }$ recommends an efficient and claims-respecting allocation which satisfies claims in proportion to their strength whenever such an allocation exists:

Thus, the upshot of proposition 3 is that ${P^\unicode{x2020} }$ respects both FF ${\cal B}$ (i) and FF ${\cal B}$ (ii) whenever doing so is possible. Let us now turn to the sense in which ${P^\unicode{x2020} }$ gives substance to FF ${\cal B}$ (iii). In order to do so, let us define, for any Broomean problem ${\cal B} = \left( {E,N,a,s} \right)$ and subset $J \subseteq N$ of individuals, the remainder problem ${{\cal B}^J}$ as follows:

$${{\cal B}^J} = \left( {E - \mathop \sum \limits_{j \in J} {a_j},{\rm{\;}}N\backslash J,{\rm{\;}}{a_{N\backslash J}},{\rm{\;}}{s_{N\backslash J}}} \right)$$

Thus ${{\cal B}^J}$ is the problem that results from ${\cal B}$ when the individuals in $J$ leave with their claim amounts so that, per definition, ${{\cal B}^\emptyset } = {\cal B}$ . In remainder problem ${{\cal B}^J}$ , the remaining estate $E - \mathop \sum \nolimits_{j \in J} \,{a_j}$ has to be divided amongst the remaining individuals in $N\backslash J$ whose claim amounts and strengths are specified by the restrictions of, respectively, $a$ and $s$ to $N\backslash J$ . As an example, with respect to ${\cal M}$ ore money as defined in section 2.2, we have:

$${{\cal M}^{\left\{ A \right\}}} = \left( {60,{\rm{\;}}\left\{ {B,C} \right\},\left( {60,40} \right),\left( {{1 \over 4},{1 \over 4}} \right)} \right)$$

We now follow Casas-Méndez et al. (Reference Casas-Méndez, Fragnelli and García-Jurado2011) and define the set ${R_{\cal B}}$ of fully proportionally reimbursable individuals in a Broomean problem ${\cal B}$ .

Fully proportionally reimbursable individuals. Let ${\cal B}$ be a Broomean problem. The set ${R_{\cal B}}$ of fully proportionally reimbursable individuals in ${\cal B}$ is defined recursively, by repeated applications of $P$ to ${\cal B}$ and its remainder problems ${{\cal B}^{{J_k}}}$ , where:

$${J_0} = \emptyset, {\rm{\;\;}}{J_{k + 1}} = {J_n} \cup \{ i \in N\backslash {J_n}|P{({{\cal B}^{{J_k}}})_i} \ge {a_i}\} $$

$${R_{\cal B}} = {J_{{k^ \star }}},{\rm{where}}\;{k^ \star } \ge 0{\rm{\;is\;such\;that}}\;{J_{{k^ \star }}} = {J_{{k^ \star } + 1}}$$

To illustrate the definition of ${R_{\cal B}}$ we consider ${\cal M}$ ore money. Applying $P$ to ${{\cal M}^{{J_0}}} = {\cal M}$ yields $\left( {22{6 \over 7},34{2 \over 7},22{6 \over 7}} \right)$ in which only $A$ is allotted more than his claim amount so that ${J_1} = {J_0} \cup \left\{ A \right\} = \left\{ A \right\}$ . Applying $P$ to ${{\cal M}^{{J_1}}} = {{\cal M}^{\left\{ A \right\}}}$ yields $\left( {36,24} \right)$ in which no individual gets more than their claim amount. Hence ${J_2} = \left\{ A \right\} \cup \emptyset = \left\{ A \right\}$ as well, so that ${R_{\cal M}} = \left\{ A \right\}$ .

The definition of ${R_{\cal B}}$ allows for the following alternative definition of ${P^\unicode{x2020} }$ which, as we prove below, is equivalent to definition (3).

(4) $${P^\unicode{x2020} }{({\cal B})_i} = \left\{ {\matrix{ {{a_i}} \hfill & {{\rm{if\;}}i \in {R_{\cal B}}} \hfill \cr {P{{({{\cal B}^{{R_{\cal B}}}})}_i}} \hfill & {{\rm{if\;}}i \in N\backslash {R_{\cal B}}} \hfill \cr } } \right.$$

As ${R_{\cal B}}$ is obtained by repeated applications of $P$ to ${\cal B}$ and its remainder problems ${{\cal B}^J}$ , an inspection of definition (4) reveals that ${P^\unicode{x2020} }\left( {\cal B} \right)$ can also be obtained as such. Indeed, definition (4) give rise to the following algorithm for computing the ${P^\unicode{x2020} }$ allocation:

Absolute Priority Rule Algorithm. Award each individual their $P$ roportional division of the estate—unless at least one individual’s $P$ roportional division is larger than their claim, in which case award such individuals their entire claim, remove them from the set of individuals under consideration and their claim from the estate, and repeat.

Let us illustrate this algorithm, i.e. definition (4), by applying it to the following problem:

$${\cal S} = \left( {50,\left\{ {A,B,C} \right\},\left( {20,{\rm{\;}}15,{\rm{\;}}30} \right),\left( {{1 \over 2},{7 \over {20}},{3 \over {20}}} \right)} \right)$$

First we apply $P$ to ${\cal S}$ , which yields $\left( {25.32,{\rm{\;}}13.29,{\rm{\;}}11.39} \right)$ . As $P$ allots more to $A$ than his claim amount, $A$ is fully reimbursed, i.e. allotted $20$ and removed, resulting in the following remainder problem:

$${{\cal S}^{\left\{ A \right\}}} = \left( {30,\left\{ {B,C} \right\},\left( {15,{\rm{\;}}30} \right),\left( {{7 \over {20}},{3 \over {20}}} \right)} \right)$$

Next, apply $P$ to ${{\cal S}^{\left\{ A \right\}}}$ , which yields $\left( {16.15,{\rm{\;}}13.85} \right)$ . Now $P$ allots more to $B$ than his claim amount of $15$ so that $B$ is fully reimbursed and removed, resulting in remainder problem:

$${{\cal S}^{\left\{ {A,B} \right\}}} = \left( {5,\left\{ C \right\},30,{3 \over {20}}} \right)$$

Finally, applying $P$ to ${{\cal S}^{\left\{ {A,B} \right\}}}$ yields $5$ for $C$ , which does not exceed his claim amount. Hence, ${P^\unicode{x2020} }\left( {\cal S} \right) = \left( {20,{\rm{\;}}15,5} \right)$ and ${R_{\cal S}} = \left\{ {A,B} \right\}$ .

Let us now prove, as promised, that definition (3) and (4) are equivalent.

Proposition 4 Definitions (3) and (4) of ${P^\unicode{x2020} }$ are equivalent.

Proof: Let ${\cal B} = \left( {E,N,a,s} \right)$ be a Broomean problem. For each $k \ge 0$ , let ${J_k} \subseteq N$ be defined as in the definition of the set of fully proportionally reimbursable individuals ${R_{\cal B}}$ . For each ${J_k}$ , we define ${\lambda _k}$ as the value of $\lambda $ for which:

$$\mathop \sum \limits_{j \in {J_k}} {a_j} + \mathop \sum \limits_{j \in N\backslash {J_k}} \lambda {s_j}{a_j} = E$$

It readily follows from the definition of ${J_k}$ and ${\lambda _k}$ that:

1. (i) For all $k \ge 0$ , for all $j \in N\backslash {J_k}:{\lambda _k}{s_j}{a_j} = P{({{\cal B}^{{J_k}}})_j}$

2. (ii) For all $k \ge 1$ : ${J_k} = \{ j \in N|{\lambda _{k - 1}}{s_j}{a_j} \ge {a_j}\} $

Let ${k^ \star }$ the (smallest) value of $k$ for which ${J_{{k^ \star }}} = {J_{{k^ \star } + 1}}$ and remember that ${J_{{k^ \star }}} = {R_{\cal B}}$ , the set of fully proportionally reimbursable individuals in ${\cal B}$ . We claim that:

$$\mathop \sum \limits_{i \in N} {\rm{min}}\left\{ {{\lambda _{{k^ \star }}}{s_i}{a_i},{a_i}} \right\} = \mathop \sum \limits_{i \in {J_{{k^ \star }}}} {a_i} + \mathop \sum \limits_{i \in N\backslash {J_{{k^ \star }}}} {\lambda _{{k^ \star }}}{s_i}{a_i} = E$$

The first equality of the claim follows from the fact that $J_k^ \star = \{ j \in N|{\lambda _{{k^ \star }}}{s_j}{a_j} \ge {a_j}\} $ , which directly follows from (ii) and the definition of ${\lambda _{{k^ \star }}}$ . The second equality follows from the definition of ${\lambda _{{k^ \star }}}$ . Hence ${\lambda _{{k^ \star }}}$ is the value of $\lambda $ for which $\mathop \sum \nolimits_{i \in N} \,\,{\rm{min}}\left\{ {\lambda {s_i}{a_i},{a_i}} \right\} = E$ so that, according to definition (3), each individual $i$ receives ${\rm{min}}\left\{ {{\lambda _{{k^ \star }}}{s_i}{a_i},{a_i}} \right\}$ . As ${J_{{k^ \star }}} = {R_{\cal B}}$ , it follows from (i), (ii) and the definition of ${\lambda ^ \star }$ , that:

$${\rm{min}}\left\{ {{\lambda _{{k^ \star }}}{s_i}{a_i},{a_i}} \right\} = \left\{ {\matrix{ {{a_i}} \hfill & {{\rm{if\;}}i \in {R_{\cal B}}} \hfill \cr {P{{({{\cal B}^{{R_{\cal B}}}})}_i}} \hfill & {{\rm{if\;}}i \in N\backslash {R_{\cal B}}} \hfill \cr } } \right.$$

Thus, definitions (3) and (4) of ${P^\unicode{x2020} }$ are equivalent.□

It is immediately clear from definition (4) that ${P^\unicode{x2020} }$ is fully proportionally reimbursing, a property which Casas-Méndez et al. (Reference Casas-Méndez, Fragnelli and García-Jurado2011) define as follows.

Fully proportionally reimbursing. A division rule $f$ is fully proportionally reimbursing when, for each Broomean problem ${\cal B}$ , $f\left( {\cal B} \right)$ is fully proportionally reimbursing: $f{({\cal B})_i} = {a_i}$ for each $i \in {R_{\cal B}}$ .

Conjointly with efficiency and satisfying partially reimbursed claims in proportion to their strength, the fully proportionally reimbursing property can be used to characterize ${P^\unicode{x2020} }$ , as attested by the following proposition.

The properties of proposition 5 are logically independent, where a set ${\rm{\Gamma }}$ of properties for division rules is said to be logically independent if for each property $\gamma $ in ${\rm{\Gamma }}$ , there is a division rule which violates $\gamma $ but which satisfies all other properties in ${\rm{\Gamma }}$ . To see that efficiency, fully proportionally reimbursing and satisfying partially reimbursed claims in proportion to their strength are logically independent, observe that:

• Allocation $\left( {20,{\rm{\;}}24,{\rm{\;}}16} \right)$ for ${\cal M} = \left( {80,\left\{ {A,B,C} \right\},\left( {20,{\rm{\;}}60,{\rm{\;}}40} \right),\left( {{1 \over 2},{1 \over 4},{1 \over 4}} \right)} \right)$ is fully proportionally reimbursing, satisfies partially reimbursed claims in proportion to their strength, but is not efficient. Hence, as ${P^\unicode{x2020} }$ is efficient, claims-respecting and satisfies partially reimbursed claims in proportion to their strength, division rule $g$ is claims-respecting and satisfies partially reimbursed claims in proportion to their strength but is not efficient:

$$g\left( {\cal B} \right) = \left\{ {\matrix{ {\left( {20,24,16} \right)} \hfill & {{\rm{if\;}}{\cal B} = {\cal M}} \hfill \cr {{P^\unicode{x2020} }\left( {\cal B} \right)} \hfill & {{\rm{otherwise}}.} \hfill \cr } } \right.$$

Thus, $g$ establishes that efficiency is logically independent of being fully proportionally reimbursing and satisfying partially reimbursed claims in proportion to their strength.

• Similarly, to see that being fully proportionally reimbursing is independent of efficiency and satisfying partially reimbursed claims in proportion to their strength, observe that $\left( {10,{\rm{\;}}60,{\rm{\;}}10} \right)$ for ${\cal M}$ is efficient, satisfies partially reimbursed claims in proportion to their strength, but is not fully proportionally reimbursing (as ${R_{\cal M}} = \left\{ A \right\}$ and as $A$ does not get fully reimbursed).

• Similarly, to see that satisfying partially reimbursed claims in proportion to their strength is independent of being fully proportionally reimbursing and efficiency, observe that $\left( {20,{\rm{\;}}30,{\rm{\;}}30} \right)$ for ${\cal M}$ is fully proportionally reimbursing and efficient but does not satisfy partially reimbursed claims in proportion to their strength.

The characterization of ${P^\unicode{x2020} }$ that is provided by proposition 5 is normatively appealing: it is in virtue of proposition 5 and proposition 3 that we claim that ${P^\unicode{x2020} }$ captures and makes precise the content of FF ${\cal B}$ . Proposition 3 tells us that ${P^\unicode{x2020} }$ selects an allocation which respects both FF ${\cal B}$ (i) and FF ${\cal B}$ (ii) whenever doing so is possible. Whenever such is not possible, FF ${\cal B}$ (iii) says that fairness requires to select an allocation which respects the requirements of absolute fairness, i.e. FF ${\cal B}$ (i), and which ‘satisfies claims in proportion to their strength to as large a degree as possible’. By selecting an efficient allocation which fully reimburses all fully proportionally reimbursable individuals and in which the claims of all other individuals are satisfied in proportion to their strength, the absolute priority ${P^\unicode{x2020} }$ does exactly that: the properties of ${P^\unicode{x2020} }$ make precise the content of the Fairness formula for ${\cal B}$ roomean problems.

3.3. The comparative priority rule ${P^\unicode{x2021} }$

In this section, we will define the comparative priority rule ${P^\unicode{x2021} }$ . Although we do not think that ${P^\unicode{x2021} }$ is normatively appealing, it is interesting to study ${P^\unicode{x2021} }$ as it is the counterpart of the absolute priority rule ${P^\unicode{x2020} }$ : in case of a conflict between comparative and absolute fairness, the comparative priority rule ${P^\unicode{x2021} }$ prioritizes the former over the latter while it does ‘as much as possible to respect the requirements of absolute fairness’.

According to the comparative priority rule ${P^\unicode{x2021} }$ an individual $i$ receives:

(5) $${P^\unicode{x2021} }{({\cal B})_i} = {\lambda ^\unicode{x2021} }{s_i}{a_i},{\rm{with}}\;{{\rm{\lambda }}^\unicode{x2021} }{\rm{the\;solution\;to\;the\;following\;problem}}:$$

${\rm{\;max\;\;\;\;}}\lambda $

$${\rm{subject\;to}}:{\rm{\;\;\;\;}}\lambda {s_i}{a_i} \le {a_i}{\rm{\;\;\;\;\;\;for\;all\;}}i \in N,{\rm{\;\;\;\;}}\left( {amount{\rm{\;}}constraint} \right)$$

$$\mathop \sum \limits_{i \in N} \,\lambda {s_i}{a_i} \le E{\rm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}}\left ( {estate{\rm{\;}}constraint} \right)$$

It readily follows from its definition that ${P^\unicode{x2021} }$ is maximally claims-respecting proportional where:

Maximally claims-respecting proportional. A division rule $f$ is maximally claims-respecting proportional when, for any Broomean problem ${\cal B}$ , $f\left( {\cal B} \right)$ is maximally claims-respecting proportional, i.e.:

1. (i) $f\left( {\cal B} \right)$ satisfies claims in proportion to their strength, and

2. (ii) there is no $y \in {\cal B}$ which is claims-respecting, which satisfies claims in proportion to their strength and which is such that $\mathop \sum \nolimits_{i \in N} {y_i} \gt \mathop \sum \nolimits_{i \in N} f{({\cal B})_i}$ .

In fact, the maximally claims-respecting proportional property by itself characterizes ${P^\unicode{x2021} }$ , as recorded by the following proposition.

Although ${P^\unicode{x2021} }$ always recommends maximally claims-respecting proportional allocations, ${P^\unicode{x2021} }$ does not always recommend maximally proportional allocations, where an allocation $x \in {\cal B}$ is maximally proportional just in case:

1. (i) $x$ satisfies claims in proportion to their strength, and

2. (ii) there is no $y \in {\cal B}$ which satisfies claim in proportion to their strength and which is such that $\mathop \sum \nolimits_{i \in N} {y_i} \gt \mathop \sum \nolimits_{i \in N} \,{x_i}$ .

To see that ${P^\unicode{x2021} }$ need not yield maximally proportional allocations, consider Broomean problem ${\cal T}$ :

$${\cal T} = \left( {14,\left\{ {A,B} \right\},\left( {10,{\rm{\;}}6} \right),\left( {{2 \over 3},{1 \over 3}} \right)} \right)$$

Note that allocation $\left( {11,{\rm{\;}}3} \right)$ for ${\cal T}$ is maximally proportional: the claims of $A$ and $B$ receive $100{\rm{\% }}$ and $50{\rm{\% }}$ satisfaction respectively so that $\left( {11,{\rm{\;}}3} \right)$ satisfies claims in proportion to their strength and, as $\left( {11,{\rm{\;}}3} \right)$ is efficient, is maximally proportional as well. But $\left( {11,{\rm{\;}}3} \right)$ is not a claims-respecting allocation and thus not recommended by ${P^\unicode{x2021} }$ . Indeed, $\left( {10,{\rm{\;}}3} \right)$ is maximally claims-respecting proportional (but not maximally proportional) and thus recommended by ${P^\unicode{x2021} }$ .

Our characterization of ${P^\unicode{x2021} }$ in terms of a single property is basically a restatement of ${P^\unicode{x2021} }$ ’s definition as given by (5). This definition requires, in order to determine ${P^\unicode{x2021} }\left( {\cal B} \right)$ , to solve a constrained optimization problem. However, ${P^\unicode{x2021} }$ also allows for an alternative, computationally much simpler definition that we will present next. In order to do so remember that ${\lambda ^P}$ is the value of $\lambda $ which solves equation (4) so that, per definition, $P{({\cal B})_i} = {\lambda ^P}{s_i}{a_i}$ . Also, let ${s^{max}} = {\rm{max}}\left\{ {{s_1}, \ldots, {s_n}} \right\}$ be the maximal claim-strength in ${\cal B}$ . Then, an alternative definition of ${P^\unicode{x2021} }$ is as follows:

(6) $${P^\unicode{x2021} }{({\cal B})_i} = {\lambda ^\unicode{x2021} }{s_i}{a_i},{\rm{with}}\;{\lambda ^\unicode{x2021} } = {\rm{min}}\left\{ {{1 \over {{s^{max}}}},{\lambda ^P}} \right\}$$

Clearly then, when ${1 \over {{s^{max}}}} \le {\lambda ^P}$ we are in case (i) and when ${\lambda ^P} \le {1 \over {{s^{max}}}}$ we are in case (ii), so that ${\lambda ^\unicode{x2021} }$ is the minimum of ${1 \over {{s^{max}}}}$ and ${\lambda ^P}$ , which is what definition (6) says.□

As we remarked above, we do not think that ${P^\unicode{x2021} }$ is normatively appealing. However, in the philosophical debate on absolute and comparative fairness, Hooker (Reference Hooker2005: 341) seems to suggest otherwise when he writes that ‘fairness requires the greatest possible proportionate satisfaction of claims’. Now to say that an allocation should realize ‘the greatest possible proportionate satisfaction of claims’ seems to be tantamount to saying that an allocation should be maximally claims-respecting proportional. Hence, ${P^\unicode{x2021} }$ arguably makes precise Hooker’s intuitive sketch of an account of absolute and comparative fairness. Or, if not, ${P^\unicode{x2021} }$ forces Hooker to re-articulate his account in different terms.

3.4. Three proportionality rules

As the reader may have observed already, the three ‘proportionality rules’ $P$ , ${P^\unicode{x2020} }$ and ${P^\unicode{x2021} }$ allow for a uniform presentation. Indeed, with ${\lambda ^P}$ , ${\lambda ^\unicode{x2020} }$ and ${\lambda ^\unicode{x2021} }$ the values of $\lambda $ which solve, respectively, (2), (3) and (5), we have:

$$P{({\cal B})_i} = \lambda _i^P{s_i}{a_i},{\rm{\;\;\;\;}}{P^\unicode{x2020} }{({\cal B})_i} = min\left\{ {{\lambda ^\unicode{x2020} }{s_i}{a_i},{a_i}} \right\},{\rm{\;\;\;\;}}{P^\unicode{x2021} }{({\cal B})_i} = \lambda _i^\unicode{x2021} {s_i}{a_i}$$

These ‘ $\lambda $ -definitions’ of the proportionality rules may be helpful for understanding the relations between $P$ , ${P^\unicode{x2020} }$ and ${P^\unicode{x2021} }$ . In particular, one readily verifies that:

• For any Broomean problem ${\cal B}$ : ${\lambda ^\unicode{x2021} } \le {\lambda ^P} \le {\lambda ^\unicode{x2020} }$ .

• If there is no $x \in {\cal B}$ which is efficient, claims-respecting and which satisfies claims in proportion to their strength, then ${\lambda ^\unicode{x2021} } \lt {\lambda ^P} \lt {\lambda ^\unicode{x2020} }$ .

The second claim is illustrated by Table 1, which records the recommendations of the three proportionality rules for ${\cal N}\!$ eeding Owed Money:

$${\cal N} = \left( {80,\left\{ {A,B} \right\},\left( {20,{\rm{\;}}60} \right),\left( {{2 \over 3},{1 \over 3}} \right)} \right)$$

Table 1. Allocations for ${\cal N}\!$ eeding Owed Money as a function of $\lambda $

4.1. Bankruptcy problems

Fair division problems such as Owing Money are paradigmatic for the mathematical and economic literature (Thomson Reference Thomson2019) on so-called bankruptcy problems. A bankruptcy problem is a triple

$${\mathfrak{B}} = \left( {E,N,a} \right)$$

where $E \gt 0$ , $N = \left\{ {1, \ldots, n} \right\}$ , ${a_i} \ge 0$ and $\mathop \sum \nolimits_{i \in N} \,\,{a_i} \ge E$ . Indeed, there are important parallels between these fairness structures: a bankruptcy problem is, formally, a Broomean problem without claim strengths. ^{Footnote 10}

As for its interpretation, the individuals in a bankruptcy problem are sometimes referred to as ‘claimants’ and the entries in $a$ as representing their ‘claims’. Indeed, it is not uncommon in the literature to refer to bankruptcy problems as ‘claims problems’. However, the notion of a ‘claim’ is basically a primitive term here, with a much broader meaning than it has in the philosophical literature and often times the $a$ vector is interpreted as specifying demands or wants.

Nevertheless, fair division problems which involve (Broomean) claims that are all of equal strength, such as Owing Money, are conveniently represented as bankruptcy problems. Indeed, we can either represent Owing Money as Broomean problem ${\cal O}$ , as we did before, or as a bankruptcy problem ${\mathfrak{O}}$ .

$${\cal O} = \left( {40,\left\{ {A,B} \right\},\left( {20,{\rm{\;}}60} \right),\left( {{1 \over 2},{1 \over 2}} \right)} \right),\;\;\; {{\mathfrak{O}}} = \left( {40,\left\{ {A,B} \right\},\left( {20,{\rm{\;}}60} \right)} \right)$$

The proportional rule $\mathbb{P}$ divides the estate in any bankruptcy problem proportional to claims:

$${\mathbb{P}}{({\mathfrak B})_i} = {{{a_i}} \over {\mathop \sum \nolimits_{j \in N} \,{a_j}}} \cdot E$$

For Owing Money, we have that ${\mathbb{P}}\left( {\mathfrak{O}} \right) = {P^\unicode{x2020} }\left( {\cal O} \right)$ : the recommendation of $\mathbb{P}$ coincides with that of the absolute priority rule ${P^\unicode{x2020} }$ which is induced by the FF. It readily follows, as recorded by the following proposition, that this result holds for any problem in which all claims are equally strong.

Now, the proportional rule $\mathbb{P}$ is but one of the many bankruptcy rules that are studied in the economic literature (Herrero and Villar Reference Herrero and Villar2001). Consider Table 2, which displays allocations for ${\mathfrak{O}}$ wing Money for three different allocation rules.

Table 2. Allocations for $\mathfrak{O}$ wing Money, in different bankruptcy rules

Whereas $\mathbb{P}$ awards shares proportional to claims, $CEA$ equalizes awards as much as possible, without giving any agent more than their claim. $CEL$ first calculates the difference between the sum of all claims ( $80$ ) and the estate ( $40$ ) to determine the joint loss $L$ ( $40$ ), which is then equally shared between all individuals without awarding any individual a negative amount. More generally, $CEA$ , $CEL$ and $\mathbb{P}$ can be defined as follows:

$$CEA{(\mathfrak {B})_i} = {\rm{min}}\left\{ {{a_i},\lambda } \right\}\ {\rm{with\;}}\lambda \gt 0\ {\rm s.t.}\mathop \sum \limits_{i \in N} CEA{({\mathfrak{B}})_i} = E$$

$$CEL{({\mathfrak{B}})_i} = {\rm{max}}\left\{ {{a_i} - \lambda, 0} \right\}\ {\rm{with\;}}\lambda \gt 0\ {\rm s.t.}\mathop \sum \limits_{i \in N} CEL{({\mathfrak{B}})_i} = E$$

$${\mathbb{P}}{({\mathfrak{B}})_i} = \lambda {a_i}{\rm{\;with\;}}\lambda {\gt} {0\, \rm s. t.}\mathop \sum \limits_{i \in N} \mathbb{P}{({\mathfrak{B}})_i} = {\it E}$$

Apart from $\mathbb{P}$ , $CEA$ and $CEL$ , a multitude of other bankruptcy rules have been proposed in the literature. Indeed, proportionality is, in contrast to the philosophical literature, not afforded a special normative status:

Indeed, in the literature on bankruptcy problems, alternative rules are studied and compared on the basis of their elementary properties or axioms. An important aspect of the study of bankruptcy problems characterize bankruptcy rules, i.e. to single out a bankruptcy rule as the only one satisfying a set set of axioms. Preferably, the axioms occurring in a characterization are logically independent from one another and plausible in the sense that they embody clear ethical or operational criteria. Indeed, we have adopted this approach in section 3.2, where we characterized the absolute priority rule ${P^\unicode{x2020} }$ in terms of three logically independent properties, which are plausible in the sense that they embody the principles of absolute and comparative fairness as articulated by FF ${\cal B}$ .

As such, the literature on bankruptcy problems offers little help for understanding the behaviour and properties of ${P^\unicode{x2020} }$ , or other division rules, that are defined for all Broomean problems – where claims may vary in amounts and strengths. However, an underdeveloped extension of the basic model of bankruptcy problems assigns weights to claimants. By doing so, one obtains weighted bankruptcy problems, which are formal equivalents of our Broomean problems and which we discuss in the next section.

4.2. Weighted bankruptcy problems

A weighted bankruptcy problem is a tuple ${\cal B} = \left( {E,N,a,s} \right)$ where $E \gt 0$ , $N = \left\{ {1, \ldots, n} \right\}$ , ${a_i} \ge 0$ , $\mathop \sum \nolimits_{i \in N} \,{a_i} \ge E$ and ${s_i} \gt 0$ . The interpretation of $E,N$ and $a$ carries over from the standard bankruptcy model whereas $s$ indicates a vector of weights, which …

This interpretation of the ‘weights’ $s$ is rather broad, but subsumes our interpretation of $s$ as recording claim strengths, indicating the relative strength of the reason we have for satisfying the claim of a particular individual. Indeed, a Broomean problem just is a weighted bankruptcy problem under a specific interpretation of the $a$ mounts and $s$ trengths of claims.

Whereas there is a vast literature on bankruptcy problems, their weighted versions have received considerably less attention. One of the few exceptions is Casas-Méndez et al. (Reference Casas-Méndez, Fragnelli and García-Jurado2011:161), who ‘consider a relevant topic on the subject of bankruptcy which has not to date received much attention: weighted bankruptcy problems’. The authors define, and characterize, weighted versions of the $\mathbb{P}$ , $CEA$ and $CEL$ bankruptcy rules:

$${{\mathbb{P}}^w}{({\cal B})_i} = {\rm{min}}\left\{ {\lambda {s_i}{a_i},{a_i}} \right\}{\rm{\;with\;}}\lambda \gt 0\ {\rm s.t.}\mathop \sum \limits_{i \in N} {{\mathbb{P}}^w}{({\cal B})_i} = E$$

$$CE{A^w}{({\cal {\cal B}})_i} = {\rm{min}}\left\{ {{a_i},\lambda {s_i}} \right\}{\rm{\;with\;}}\lambda \gt 0\ {\rm s.t.}\mathop \sum \limits_{i \in N} CE{A^w}{({{\cal B}})_i} = E$$

$$CE{L^w}{({\cal B})_i} = {\rm{max}}\left\{ {{a_i} - \frac{\lambda }{{{s_i}}},0} \right\}\ {\rm{with\;}}\lambda \gt 0\ {\rm s.t.}\mathop \sum \limits_{i \in N} CE{L^w}{({\cal B})_i} = E$$

Indeed, the weighted constrained proportional rule ${{\mathbb{P}}^w}$ of Casas-Méndez et al. is identical to the absolute priority rule ${P^\unicode{x2020} }$ .

Now, the main motivation that Casas-Méndez et al. (Reference Casas-Méndez, Fragnelli and García-Jurado2011: 161) give for defining and studying ${{\mathbb{P}}^w}$ , $CE{A^w}$ or $CE{L^w}$ is that ‘some of the most important bankruptcy rules have not been studied in the weighted framework’. In particular, they do little to motivate ${{\mathbb{P}}^w}$ and do not argue that ${{\mathbb{P}}^w}$ is preferable to the other two rules.

Also, Casas-Méndez et al. simply call ${{\mathbb{P}}^w} = {P^\unicode{x2020} }$ the extension of $\mathbb{P}$ to the weighted framework and do not consider $P$ or ${P^\unicode{x2021} }$ as candidates. The latter is, at least from a formal perspective, understandable. For Casas-Méndez et al. define a division rule to be efficient and claims-respecting. In fact, the latter convention is common in the economics literature. Yet, according to this convention, neither $P$ nor ${P^\unicode{x2021} }$ qualify as division rule. We think our discussion shows that it is theoretically fruitful to adopt a more liberal definition of a division rule (one that does not presuppose that a rule is efficient or claims-respecting). Indeed, by doing so we can contrast and compare ${P^\unicode{x2020} }$ with $P$ and ${P^\unicode{x2021} }$ , which fosters the study of (absolute and comparative) fairness and proportionality.

In addition to defining ${{\mathbb{P}}^w}$ , $CE{A^w}$ and $CE{L^w}$ , Casas-Méndez et al. (Reference Casas-Méndez, Fragnelli and García-Jurado2011) characterize these weighted bankruptcy rules. Translated to the framework of this paper, in which division rules are neither assumed to be efficient nor claims-respecting, they provide the following characterization of ${P^\unicode{x2020} }$ .

Thus whereas proposition 5 characterizes ${P^\unicode{x2020} }$ in terms of efficiency, fully proportionally reimbursing and satisfying partially reimbursed claims in proportion to their strength, the characterization of proposition 9 trades in our last axiom for two axioms of strategy-proofness. Roughly, a division rule is strategy-proof for amounts when no group of individuals $K$ whose claims all have the same strength $\sigma $ can benefit from aggregating their claims into a single claim with amount $\mathop \sum \nolimits_{ \in K} \,{a_j}$ and strength $\sigma $ . Conversely, a division rule is strategy-proof for strengths when no group of individuals $K$ whose claims all have the same amount $\alpha $ can benefit from aggregating their claims into a single claim with amount $\alpha $ and strength $\mathop \sum \nolimits_{j \in K} \,{s_j}$ . Formally, the two axioms of strategy-proofness are defined as follows.

Strategy-proof for amounts. Let ${\cal B} = \left( {E,N,a,s} \right)$ be a Broomean problem and let ${\cal B}{\rm{'}} = \left( {E,N{\rm{'}},a{\rm{'}},s{\rm{'}}} \right)$ be obtained from ${\cal B}$ by replacing an individual $i \in N$ with a set $K = \left\{ {{i_1}, \ldots, {i_k}} \right\}$ of individuals with claims whose amounts sum to ${a_i}$ and whose strengths are all equal to ${s_i}$ . ^{Footnote 11} A division rule $f$ is strategy-proof for amounts if for any ${\cal B}$ and ${\cal B}{\rm{'}}$ that are related as such, we have $f{({\cal B})_j} = f{({\cal B}{\rm{'}})_j}$ for all $j \in N - \left\{ i \right\}$ .

Strategy-proof for strengths. Let ${\cal B} = \left( {E,N,a,s} \right)$ be a Broomean problem and let ${\cal B}{\rm{'}} = \left( {E,N{\rm{'}},a{\rm{'}},s{\rm{'}}} \right)$ be obtained from ${\cal B}$ by replacing an individual $i \in N$ with a set $K = \left\{ {{i_1}, \ldots, {i_k}} \right\}$ of individuals with claims whose strengths sum to ${s_i}$ and whose amounts are all equal to ${a_i}$ . ^{Footnote 12} A division rule $f$ is strategy-proof for strengths if for any ${\cal B}$ and ${\cal B}{\rm{'}}$ that are related as such, we have $f{({\cal B})_j} = f{({\cal B}{\rm{'}})_j}$ for all $j \in N - \left\{ i \right\}$ .

The strategy-proof for amounts and the strategy-proof for strengths axiom generalize the strategy-proofness axiom ^{Footnote 13} for bankruptcy rules (cf. O’Neill (Reference O’Neill1982)) to the weighted framework. The proof of proposition 9 is an adaptation of a proof due to Curiel et al. (Reference Curiel, Maschler and Tijs1987), who characterize bankruptcy rule $\mathbb{P}$ in terms of efficiency, equal treatment of equals and strategy-proofness ^{Footnote 14} . Just as for their definition of ${{\mathbb{P}}^w}$ , Casas-Méndez et al. hardly motivate or justify (the properties occurring in) their characterization of ${{\mathbb{P}}^w}$ . We do not doubt that, on some occasions, it may be desirable to have a strategy-proof division rule at one’s disposal. However, desirable as they may be, it is hard to see how the strategy-proofness axioms can be justified in terms of fairness. In sharp contrast, the axioms used in the characterization given by proposition 5 can all be justified by appealing to the account of absolute and comparative fairness that we develop in our article ‘How to be absolutely fair, Part I’.

At the same time, our characterization of ${P^\unicode{x2020} }$ is less informative than that of Casas-Méndez et al. in the sense that the properties used in our characterization can be more or less ‘read off’ of the definition (4) of ${P^\unicode{x2020} }$ . In contrast, to find out that ${P^\unicode{x2020} }$ satisfies the two strategy-proofness axioms is not something that can easily be deduced from an inspection of its definition. There is a further difference between the two characterizations. We exploit three properties to characterize ${P^\unicode{x2020} }$ that are punctual, while the strategy-proofness axioms of Casas-Méndez et al. are relational, where:

For example, efficiency is a punctual axiom: it specifies that for each problem (separately) an allocation which allots all of the estate has to be selected. Similarly, fully proportionally reimbursing and satisfying partially reimbursed claims in proportion to their strength are punctual axioms. In contrast, strategy-proofness for amounts (and strengths) is a relational axiom and, as such, makes a conditional statement: if two problems ${\cal B}$ and ${\cal B}{\rm{'}}$ are related in a certain way, then the recommended allocations for ${\cal B}$ and ${\cal B}{\rm{'}}$ should also be related in a certain way – in the case of strategy-proofness the recommended allocations should be identical.