2.1. Claim strength and claim amounts

According to the Broomean formula, claims should be satisfied in proportion to their strength. Besides a strength, each claim has an amount. Although the Broomean formula does not explicitly mention the amount of a claim, the notion is presupposed by the notion of satisfaction, which is explicitly mentioned by the formula. For the extent to which a claim is satisfied depends on the good received and the amount of the claim: amounts are needed to determine satisfaction.

As an example, consider Owing Money (cf. section 1). Romeo is the distributor in Owing Money. He owes it to Abram to repay his debt: Abram has a claim to be reimbursed by Romeo. As it takes $20$ to reimburse Abram, the claim of Abram has an amount of $20$ . If Abram receives $10$ , his claim to reimbursement is satisfied for ${{10} \over {20}} \cdot 100{\rm{\% }} = 50{\rm{\% }}$ .

Besides an amount, each claim has a strength. The strength of a claim specifies how strong the reason is, as compared with the reasons for satisfying the claims of other agents, for satisfying that particular claim. That is, claim strength is a strictly comparative notion. To illustrate this, consider Owing Money once more. Romeo has just as much reason to reimburse Abram as he has to reimburse Benvolio: both have been promised to be paid back. That is, the claims of Abram and Benvolio – to get reimbursed by Romeo – are equally strong. Thus, Abram and Benvolio have equally strong claims, with amounts of $20$ and $60$ respectively.

In Owing Money the claims are equally strong. But claims are not always equally strong, as the next example illustrates.

Investing Time. Anna and Beta have invested time in realizing a joint project. For a certain period of time, Anna has spent one day a week on the project, whereas Beta has spent three days a week on it. After some time, the value of their project is $20$ . Anna and Beta split apart and their fiduciary, Rachel, is responsible for the division of the $20$ . How, in order to be fair, should Rachel divide the $20$ ?

As Anna has contributed to realizing the joint project, Rachel the distributor in Investing Time, owes it to Anna to give her a share of its value. That is, Anna has a claim to (a share) of the project’s value. Similarly, Beta has a claim to (a share of) the project’s value. As the value of their joint project is $20$ , we say that the amount of the respective claims of Anna and Beta is $20$ . So Anna and Beta have claims with equal amounts but clearly, Beta’s claim is stronger. Indeed, (all else being equal) it is natural to say that Beta’s claim is three times as strong as Anna’s claim, in virtue of Beta’s time commitment that is three times as large. We will compactly summarize the claims-lesson of this subsection in a bullet-point and we will continue this practice in further subsections.

• A claim has both an amount and strength.

This seemingly simple observation is not only important, but ignoring or not fully analysing it has the potential to hamper fairness discussions, as we will illustrate in section 4 of this article.

2.2. Notional and absolute claims

Investing Time illustrates that claims may come in different strengths. But, when compared with Owing Money, it also illustrates a further important distinction: that between absolute and notional claims.

To illustrate the difference between what we will call absolute claims and notional claims, compare the following true statements about (1) Owing Money and (2) Investing Time respectively:

1. (1) Abram has a claim to $20$ that is equally strong as Benvolio’s claim to $60$ .

2. (2) Anna has a claim to $20$ that is ${1 \over 3}$ times as strong as Beta’s claim to $20$ .

Whereas Abram and Anna both have claims with an amount of $20$ , these claims are claims of different types. To see this, let’s discuss the claims of Abram and Anna in turn. Abram’s claim to be reimbursed by Romeo is not determined by comparisons with any other claimant’s claims or how others are treated. We call Abram’s claim absolute. As Hooker (Reference Hooker2005) puts it:

Gerard Vong (Reference Vong2018) agrees and proposes to distinguish between different types of claims:

Absolute claims require full satisfaction. When they are not completely satisfied, the claimant does not get all they should get. For instance, when Romeo realizes allocation $\left( {10,{\rm{\;}}30} \right)$ he fulfils the requirements of the Broomean formula. And one might say that he has done all for promoting fairness that he was able to, given what Owing Money describes about the situation at hand: all he had left was $40$ . However, as Abram’s absolute claim does not receive full satisfaction, something is wrong. What is wrong is that it is non-comparatively, i.e. absolutely, unfair to Abram that he remains $10$ short. Absolute claims are, as a subset of all claims, duties owed to the agent. In particular:

• Agent $A$ has an absolute claim with an amount $a$ if the distributor owes it to $A$ to allot her all of $a$ , irrespective of what claims other agents may have.

Absolute claims are a subset of all claims. Any claim that is not absolute we call a notional claim. In Investing Time, Anna contributed to the realization of the joint project, in virtue of which she has a claim to its value of $20$ . Anna’s claim is not absolute and hence notional. To see this, note that when Anna does not get the full $20$ it does not follow that something is wrong. In particular, nothing is wrong with Anna receiving $5$ which, intuitively, is her ‘fair share’.

So, Abram has an absolute claim to $20$ : he should get all of the $20$ , irrespective of which other claimants are around and irrespective of how much money Romeo has left. Anna’s claim to $20$ is notional: she should get a part of the $20$ , a part which is determined by comparing her claim with that of others. More generally:

• Agent $A$ has a notional claim with an amount $a$ if the distributor owes it to $A$ to allot her a part of $a$ , a part which is determined by comparing her claim with that of others.

So whether an agent has an absolute or notional claim with amount $a$ , the distributor never has the duty to allot more than $a$ to the agent. Our distinction between notional and absolute claims is similar to Vong (Reference Vong2018) and others (e.g. Hooker Reference Hooker2005; Saunders Reference Saunders2010; Lazenby Reference Lazenby2014; Curtis Reference Curtis2014). Indeed, it records, in precise fashion, a long-standing concern for analysing absolute fairness that is present in the Broomean fairness literature.

Next, we turn to the distinction between individual and group claims.

2.3. Individual and group claims

In Investing Time, the individuals Anna and Beta both have notional claims, with an amount of $20$ , grounded in the joint realization of their project.

However, Investing Time also involves an absolute claim. For, as Anna and Beta jointly realized the value of $20$ , Rachel owes it to Anna and Beta together to allot the $20$ , and all of the $20$ , to them. That is, the group consisting of Anna and Beta has a claim to $20$ that requires full satisfaction: the group consisting of Anna and Beta has an absolute claim with an amount of $20$ .

We concur with Broome that a claim is ‘a duty owed to the agent’ herself, as is also apparent from our account of absolute and notional claims. But we add that it is important to distinguish between individual and group claims. Individual claims are duties owed to individual (receiving) agents. Group claims are duties owed to groups of individual (receiving) agents. As we will see later, this extension will significantly increase the conceptual reach and interdisciplinary potential of Broomean fairness.

• A claim is a duty owed to an agent, which can either be an (receiving) individual or a group of (receiving) individuals.

Next we discuss a few salient notions related to claim satisfaction.

2.4. Claim satisfaction: constrained

In Owing Money, Abram has a claim with an amount of $20$ . Suppose that Abram is allotted $30$ by Romeo. Thus, Abram’s claim to $20$ is fully satisfied and he receives ‘a gift on top of that’ equal to $10$ . As a receipt of $20$ fully satisfies his claim, we say that, although $30$ amounts to $150{\rm{\% }}$ of Abram’s claim amount, by receiving $30$ the satisfaction of his claim of $20$ is (only) $100{\rm{\% }}$ . More generally, when an agent has a claim with an amount $a$ , that claim is fully satisfied when the agent receives $a$ (or more). No doubt, Abram may prefer to receive more than the amount of his claim. But whereas receipts that exceed $20$ may satisfy Abram’s preferences, they do not count towards the satisfaction of his claim. That is, we take it that, in virtue of the very meaning of claim satisfaction, claim satisfaction is constrained:

• The satisfaction of a claim is constrained by its amount $a$ : it has a maximum of $100{\rm{\% }}$ , realized by receiving any amount that is greater-than or-equal-to $a$ .

When Abram receives $10$ , in Owing Money, his claim of $20$ receives ${{10} \over {20}} \cdot 100{\rm{\% }} = 50{\rm{\% }}$ satisfaction. For short, we write: $Sat\left( {10,{\rm{\;}}20} \right) = 50{\rm{\% }}$ . Likewise when, in Investing Time, Anna and Beta receive $10$ , their claims of $20$ receive $50{\rm{\% }}$ satisfaction but in addition, the group consisting of Anna and Beta together receives $10 + 10 = 20$ so that its (absolute) group claim of $20$ receives ${{10 + 10} \over {20}} \cdot 100{\rm{\% }} = 100{\rm{\% }}$ satisfaction, or $Sat\left( {20,20} \right) = 100{\rm{\% }}$ . ^{Footnote 9}

2.5. Claim satisfaction: comparing allocations

When Abram is allotted $10$ in Owing Money, his claim with an amount of $20$ is satisfied to a larger extent than when he is allotted only $5$ : indeed, Sat $\left( {10,{\rm{\;}}20} \right)$ is strictly greater than Sat $\left( {5,{\rm{\;}}20} \right)$ . What we are interested in, however, is comparing allocations in terms of the extent to which they satisfy claims: does allocation $\left( {10,{\rm{\;}}15} \right)$ , in which Abram receives $10$ and Benvolio receives $15$ , satisfy claims to a larger extent than allocation $\left( {9,{\rm{\;}}15} \right)$ ? In order to answer that question and, more generally, to compare allocations in terms of the extent to which they satisfy claims, we will rely on the following criteria for individual and groups claims respectively.

• Allocation x satisfies the claims of the individuals to a larger extent than allocation $y$ when the satisfaction afforded by $x$ to each of the individuals is greater than or equal to that afforded by $y$ and strictly greater for at least one individual. ^{Footnote 10}

• Allocation $x$ satisfies the claim of a group of individuals to a larger extent than $y$ when the sum-total allotted to these individuals by $x$ satisfies their group claim to a larger extent than the sum-total allotted to them by $y$ . ^{Footnote 11}

Note that our criteria of comparison is rather minimal: we compare allocations by comparing claim satisfaction in an ordinal and component-wise manner. For sure, in $\left( {10,{\rm{\;}}20} \right)$ for Owing Money, claims are satisfied to a larger extent than in $\left( {9,{\rm{\;}}20} \right)$ . And, as satisfaction is constrained, in $\left( {20,{\rm{\;}}20} \right)$ claims are satisfied to a larger extent than in $\left( {25,{\rm{\;}}15} \right)$ : in both allocations Abram’s claim receives 100% satisfaction, whereas Benvolio’s claim receives more satisfaction in the first allocation than in the latter. For Investing Time, the absolute claim of the group of Anna and Beta is satisfied to a larger extent by one allocation than another just in case the total sum that is allotted by the former is larger than that of the latter.

For quite a few allocations, though, our criteria do not declare one of them as satisfying claims to a larger extent than the other. For example, in $\left( {5,{\rm{\;}}2} \right)$ Abram’s claim is satisfied to a larger extent than in $\left( {4,{\rm{\;}}30} \right)$ , but for Benvolio it is just the other way around. Hence, in $\left( {5,{\rm{\;}}2} \right)$ claims are not satisfied to a larger extent than in $\left( {4,{\rm{\;}}30} \right)$ but neither are claims satisfied to a larger extent in $\left( {4,{\rm{\;}}30} \right)$ than in $\left( {5,{\rm{\;}}2} \right)$ . Should we then say that, in $\left( {5,{\rm{\;}}2} \right)$ and $\left( {4,{\rm{\;}}30} \right)$ , claims are satisfied to the same extent? Or should we say that $\left( {5,{\rm{\;}}2} \right)$ and $\left( {4,{\rm{\;}}30} \right)$ are incomparable with respect to the extent to which they satisfy claims? For the purposes of this article, we need not and will not answer this question: it will suffice to compare allocations by the extent to which they satisfy claims on the basis of the above, ordinal, criteria only. In particular, our criteria suffice to specify the notion of an allocation which satisfies claims to as large an extent as possible.

• Allocation $x$ satisfies claims to as large an extent as possible just in case there is no allocation $y$ available which satisfies claims to a larger extent than $x$ does.

For Owing Money, e.g. allocations $\left( {0,{\rm{\;}}40} \right)$ , $\left( {10,{\rm{\;}}30} \right)$ and $\left( {20,{\rm{\;}}20} \right)$ all satisfy the absolute individual claims to as large an extent as possible. But allocation $\left( {25,{\rm{\;}}15} \right)$ does not as e.g. allocation $\left( {20,{\rm{\;}}20} \right)$ satisfies claims to a larger extent than $\left( {25,{\rm{\;}}15} \right)$ does. For Investing Time, any allocation that allots $20$ in total satisfies the absolute group claim to as large an extent as possible.

2.6. Claim satisfaction: proportional to strength

The last notion that we discuss is that of an allocation in which individual claims are satisfied in proportion to their strength. For Owing Money, individual claims are satisfied in proportion to their strength in e.g. allocation $\left( {10,{\rm{\;}}30} \right)$ . For, in this allocation, the equally strong claims of Abram and Benvolio receive equal satisfaction (both $50{\rm{\% }}$ ). But also in $\left( {5,{\rm{\;}}15} \right)$ are claims satisfied in proportion to their strength: for here, the equally strong claims also receive equal satisfaction (both $25{\rm{\% }}$ ). For Investing Time, individual claims are satisfied in proportion to their strength in e.g. $\left( {5,{\rm{\;}}15} \right)$ or $\left( {1,{\rm{\;}}5} \right)$ : in these allocations, Beta’s claim, which is three times as strong as Anna’s claim, receives three times as much satisfaction. More generally, we say that:

• In allocation $x$ individual claims are satisfied in proportion to their strength just in case, for any two individual agents $i$ and $j$ , the following holds. If $i$ ’s claim is $\sigma $ times as strong as $j$ ’s claim then, in allocation $x$ , the satisfaction of $i$ ’s claim is $\sigma $ times the satisfaction of $j$ ’s claim, i.e. Sat $\left( {{x_i},{a_i}} \right) = \sigma \cdot $ Sat $\left( {{x_j},{a_j}} \right)$ .

We have put forward a criterion which specifies whether or not an allocation satisfies claims in proportion to their strength. For the purposes of this article, we need not and will not define a measure which allows us to compare allocations in terms of the extent to which they satisfy claims in proportion to their strength.

3.1. Modelling fair division: fairness structures and functions

Consider the fair division problems Owing Money and Investing Time from the previous sections. Their description has been purely verbal. For analysing them, it is important to make explicit how the information in these descriptions is understood. This step in the analysis of fair division problems we call modelling a fair division problem. Importantly, this step is implicit in all and any fairness analysis. The following discussion makes precise how the description of a given fair division problem is used in the fairness analysis.

In a nutshell, to solve a fair division problem is to (1) model (represent) the problem as a fairness structure and to (2) recommend an allocation for the problem by applying a fairness function to that structure. The two-stage process of solving a fair division problem is summarized by Figure 1.

Figure 1. Solving a fair division problem.

A fairness structure is a formal structure which is interpreted in terms of theoretical fairness notions and on the basis of which, by applying a fairness function to it, fair divisions are obtained:

The fairness structure we use in this article are what we call Broomean problems, introduced in the next section. ^{Footnote 12} The fairness functions associated with Broomean problems we call division rules. Many division rules exist but we will motivate the use of one specific division rule.

3.2. Broomean problems

We introduce Broomean problems, which are formal representations of fair division problems. 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 $s$ . ^{Footnote 13} Fair division problems Owing Money and Investing Time can be represented as Broomean problems, augmented with information about the type of claims involved, as follows. ^{Footnote 14}

Table 1. ${\cal O}$ wing Money, ${\cal I}$ nvesting Time and their representations ${\cal O}$ and ${\cal I}$

The two representations ${\cal O}$ and ${\cal I}$ record the estate (good to be divided), the individuals involved, the respective amounts they have a claim to, and the respective strength of their claims. ^{Footnote 15} For simple fair division problems like Owing Money and Investing Time, it is straightforward (and uncontroversial) how to represent them as Broomean problems. However, as we will see, some fair division problems may be plausibly represented by more than one Broomean problem. And so, what fairness requires in a given fair division problem crucially depends on its representation as a fairness structure.

3.3. The Fairness formula

The essence of our theory of absolute and comparative fairness can be summarized by the following Fairness formula, which exploits all the notions that we defined and discussed in the previous section. The Fairness formula accommodates absolute fairness as a matter of priority over comparative fairness.

Fairness formula (FF). Fairness requires one: (i) to satisfy absolute claims (of individuals and groups) to as large an extent as possible, subject to the constraint that no one receives more than they have a claim to; (ii) to satisfy (absolute and notional) individual claims in proportion to their strength; (iii) to prioritize requirement (i) over (ii) whenever these two conflict, but in such a way that one does as much as possible to respect (ii).

The Fairness formula offers guidance for realizing fair divisions. How? In a nutshell, by modelling fair division problems as Broomean problems and applying the only division rule that is justified by the Fairness formula: the absolute priority rule. The absolute priority rule is a refinement of the so-called weighted proportional rule and it is instructive to consider the latter first, before introducing the absolute priority rule.

3.4. The weighted proportional rule

In Owing Money and Investing Time, there is not enough to go around: the sum of claim amounts exceeds the estate. Although that is typically the case in a fair division problem, it will be instructive to discuss, in section 4, cases of abundant good, i.e. cases in which the estate exceeds the sum of claims. That is why it is helpful to introduce the notion of a so-called truncated estate $\overline E$ , which cannot be larger than the sum of all claim amounts. Formally, $\overline E$ is equal to either the estate $E$ or the sum of claim amounts, whichever is smaller. So for cases such as Owing Money and Investing Time, where there is not enough to go around, the truncated estate $\overline E$ equals the estate $E$ .

The weighted proportional rule $P$ proposes to divide the truncated estate in a Broomean problem ${\cal B}$ proportional to the strength-weighted claim amounts of the individuals, so that each individual $i$ receives:

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

For ${\cal O}$ wing Money and ${\cal I}$ nvesting Time, the recommendations of the weighted priority rule, $P\left( {\cal O} \right) = \left( {10,{\rm{\;}}30} \right)$ and $P\left( {\cal I} \right) = \left( {5,{\rm{\;}}15} \right)$ , are consistent with the Fairness formula, as we will explain below.

For Owing Money, the Fairness formula demands the realization of allocation $\left( {10,30} \right)$ . For in $\left( {10,{\rm{\;}}30} \right)$ , the equally strong (individual) claims of Abram and Benvolio receive equal satisfaction (of $50{\rm{\% }}$ each) so that realizing this allocation meets the requirements of comparative fairness. But also, in $\left( {10,{\rm{\;}}30} \right)$ no individual receives more than his claim amount and there is no other allocation of the estate of $40$ in which the absolute claims of Abram and Benvolio are satisfied to a larger extent than in $\left( {10,{\rm{\;}}30} \right)$ . Hence, by realizing $\left( {10,{\rm{\;}}30} \right)$ Romeo meets the requirements of absolute fairness as well. So, by realizing $\left( {10,{\rm{\;}}30} \right)$ Romeo lives up to the requirements of both absolute and comparative fairness.

For Investing Time, the Fairness formula demands the realization of allocation $\left( {5,15} \right)$ , in which Beta’s (individual) claim, which is three times as strong as that of Anna, receives three times as much satisfaction: realizing $\left( {5,{\rm{\;}}15} \right)$ meets the requirements of comparative fairness. But also, in $\left( {5,{\rm{\;}}15} \right)$ no individual receives more than their claim amount and there is no other allocation of the estate of $20$ in which the absolute claim of the group consisting of Anna and Beta together is satisfied to a larger extent than in $\left( {5,{\rm{\;}}15} \right)$ . Hence, by realizing $\left( {5,{\rm{\;}}15} \right)$ Rachel meets the requirements of absolute fairness as well. So, by realizing $\left( {5,{\rm{\;}}15} \right)$ , Rachel lives up to the requirements of both absolute and comparative fairness.

The recommendation of the weighted proportional rule, $P\left( {\cal B} \right)$ , does not depend on whether the (individual or group) claims in the ${\cal B}$ roomean problem to which $P$ is applied are absolute or notional. However, the justification of that recommendation in terms of the Fairness formula does depend on whether we are dealing with individual claims that are notional or absolute, and whether there are group claims: for Owing Money, absolute fairness requires that the absolute individual claims receive maximal satisfaction whereas for Investing Time, it is the absolute group claim that should receive maximal satisfaction as a matter of absolute fairness. More generally, it readily follows that for any Broomean problem with claim types as in Owing Money or Investing Time, absolute fairness requires to allot as much of the (truncated) estate as possible in such a way that no individual receives more than they have a claim to. For Owing Money and Investing Time, this requirement of absolute fairness can be realized conjointly with the requirement of comparative fairness. However, as we will see in section 3.5, the joint realization of both requirements of fairness is not possible for all Broomean problems. The next proposition precisely delineates the Broomean problems for which this is possible.

Proof: See the article ‘How to be absolutely fair, Part II: philosophy meets economics’, section 3.1. □

So, the Fairness formula recommends to realize $P\left( {\cal B} \right)$ whenever $P{({\cal B})_i} \le {a_i}$ for each individual $i$ as for these cases, $P\left( {\cal B} \right)$ respects the requirements of absolute and comparative fairness. However, when $P{({\cal B})_i} \gt {a_i}$ for some individual, it is simply not possible to be both absolutely and comparatively fair. For such cases, the Fairness formula prioritizes absolute fairness. Next, we will illustrate this proposal and explain that it demands that we trade in the weighted proportional rule $P$ for the absolute priority rule ${P^\dagger }$ . Indeed, the absolute priority rule captures and operationalizes the content of the Fairness formula.

3.5. The absolute priority rule

Consider the following fair division 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 $80$ ?

Needing Owed Money is naturally represented as:

The weighted proportional rule $P$ recommends allocation $P\left( {\cal N} \right) = \left( {32,{\rm{\;}}48} \right)$ for ${\cal N}$ eeding Owed Money, which is neither absolutely nor comparatively fair. Moreover, it follows from Proposition 1 that there is no allocation for ${\cal N}$ eeding Owed Money which satisfies the requirements of both absolute and comparative fairness. What to do?

The response to Needing Owed Money seems straightforward: in order to be fair, Romeo just needs to fully reimburse Abram and Benvolio, i.e. he must realize allocation $\left( {20,{\rm{\;}}60} \right)$ . We concur that this is, indeed, what fairness requires. But note that this judgement comes down to prioritising absolute fairness over comparative fairness.

For, as Romeo’s reason for reimbursing Abram is twice as strong as it is for reimbursing Benvolio, Abram’s claim with an amount of $20$ is twice as strong as Benvolio’s claim with an amount of $60$ . Comparative fairness, which requires that individual claims are satisfied in proportion to their strength, then requires that Abram’s claim receives twice as much satisfaction as the claim of Benvolio. To ensure this, Romeo can realize, for instance, allocation $\left( {20,30} \right)$ where Abram’s claim receives $100{\rm{\% }}$ and where Benvolio’s claim receives $50{\rm{\% }}$ satisfaction. Should Romeo then realize $\left( {20,{\rm{\;}}30} \right)$ ? No, as it just seems absurd to disrespect Benvolio’s absolute claim in order to ensure that the claims of Abram and Benvolio are satisfied in proportion to their strength: fairness should not demand to level down.

In $\left( {20,{\rm{\;}}60} \right)$ , the absolute claims of Abram and Benvolio are satisfied to as large an extent as possible and neither Abram nor Benvolio receives more than they have a claim to: realizing $\left( {20,{\rm{\;}}60} \right)$ fulfils the requirements of absolute fairness. Moreover, $\left( {20,{\rm{\;}}60} \right)$ is the only allocation which fulfils the requirements of absolute fairness. However, as the claims of Abram and Benvolio receive the same satisfaction in $\left( {20,{\rm{\;}}60} \right)$ , their claims are not satisfied in proportion to their strength. Indeed, to say that fairness requires that Romeo realizes $\left( {20,{\rm{\;}}60} \right)$ is to prioritize the absolute dimension of fairness over the comparative one.

Whereas Needing Owed Money illustrates that the Fairness formula prioritizes absolute fairness, the example does not aptly illustrate what it means to do so while one does ‘as much as one can to satisfy claims in proportion to their strength’. To illustrate the latter feature, consider:

More Needed Money. Romeo owes $20$ to Abram, $60$ to Benvolio, $40$ to Capulet and has $80$ left. Abram needs his money twice as strongly as both Benvolio and Capulet do. How, in order to be fair, should Romeo divide the $80$ ?

${\cal M}$ ore Needed Money is naturally represented as:

The weighted proportional rule $P$ recommends allocation $\left( {22.86,{\rm{\;}}34.29,{\rm{\;}}22.86} \right)$ for ${\cal M}$ ore Needed Money, which is neither absolutely nor comparatively fair.

The absolute priority rule does better. We first discuss the example and then define the rule. According to the absolute priority rule ${P^\dagger }$ , we need to reimburse Abram, by allotting him $20$ , as his weighted proportional share of $22.86$ exceeds his claim amount of $20$ . After reimbursing Abram, the remaining estate is $60$ which, according to the absolute priority rule, has to be divided amongst the remaining individuals Benvolio and Capulet in accordance with the weighted proportional rule. Doing so yields $36$ for Benvolio and $24$ for Capulet so that ${P^\dagger }$ recommends $\left( {20,{\rm{\;}}36,{\rm{\;}}24} \right)$ for More Needed Money. In this allocation, Abram gets all of his claim reimbursed whereas the claims of Benvolio and Capulet are, as the reader may care to verify, satisfied in proportion to their strength.

So when there is no allocation that satisfies the requirements of both absolute and comparative fairness, the absolute priority rule ${P^\dagger }$ realizes an allocation that respects the requirements of absolute fairness while ‘it does as much as possible to promote comparative fairness’ by consecutive applications of the weighted priority rule $P$ that reimburse all individuals whose weighted proportional share exceeds their claim amounts. More generally, and precisely, for any Broomean problem ${\cal B} = \left( {E,N,a,s} \right)$ , the absolute priority rule ${P^\dagger }$ recommends an allocation for ${\cal B}$ by applying the following steps:

The absolute priority rule ${P^\dagger }$ .

(0) Let ${E_R}$ denote the remaining estate, i.e. that part of the truncated estate which has not been used for reimbursement thus far, and let ${N_R}$ denote the set of remaining individuals, i.e. the individuals who have not been reimbursed thus far.

Initially, set ${E_R} = \overline E$ and ${N_R} = N$ .

1. (1) Use the weighted proportional rule to divide the remaining estate ${E_R}$ amongst the individuals in ${N_R}$ and let $x$ denote the resulting allocation.

If there is no individual for which ${x_i} \gt {a_i}$ , allot ${x_i}$ to each $i$ in ${N_R}$ .

Otherwise, move to step (2).

1. (2) Reimburse each individual for which ${x_i} \gt {a_i}$ by allotting them their claim amount ${a_i}$ , update ${E_R}$ and ${N_R}$ accordingly, and revisit step (1).

When we apply ${P^\dagger }$ to ${\cal O}$ wing Money or ${\cal I}$ nvesting Time, we get that ${P^\dagger }\left( {\cal O} \right) = \left( {10,{\rm{\;}}30} \right)$ and ${P^\dagger }\left( {\cal I} \right) = \left( {5,{\rm{\;}}15} \right)$ without visiting step (2) of ${P^\dagger }$ ’s definition. Applying ${P^\dagger }$ to ${\cal M}$ ore Needed Money yields ${P^\dagger }\left( {\cal M} \right) = \left( {20,{\rm{\;}}36,{\rm{\;}}24} \right)$ , for which we need to visit step (2) once. To see that it may be necessary to visit step (2) more than once, consider a variant of More Needed Money which can be represented as Broomean problem ${\cal X}$ :

$${\cal X} = \left( {50,\left\{ {A,B,C} \right\},\left( {20,10,30} \right),\left( {{1 \over 2},{2 \over 6},{1 \over 6}} \right)} \right)$$

When we apply the weighted proportional rule to ${\cal X}$ , we find that $P\left( {\cal X} \right) = \left( {27.27,{\rm{\;}}9.09,{\rm{\;}}13.64} \right)$ so that only Abram’s weighted proportional share exceeds his claim amount. So we need to reimburse Abram and divide the remaining $30$ amongst Benvolio and Capulet in accordance with the weighted proportional rule. Doing so yields $12$ for Benvolio and $18$ for Capulet so that Benvolio’s weighted proportional share exceeds his claim amount. Hence, we also need to reimburse Benvolio so that, after two rounds of reimbursement, the absolute priority rule recommends $\left( {20,10,20} \right)$ for ${\cal X}$ .

4.1. The constraint of absolute fairness

Consider the following fair division problem.

Combi-Deal. Anton sells pizzas for $10$ per piece and Bernard sells ice-creams for $6$ per piece. They decide to join forces and set up a combi-deal: for $14$ customers can buy a ticket, either at Anton’s or Bernard’s shop, which gets them a pizza at Anton’s and an ice-cream at Bernard’s. Anton sells twice as many tickets for the combi-deal as Bernard. How to fairly divide the joint revenues of $14$ per combi-deal amongst Anton and Bernard?

The question of what fair division amounts to in Combi-Deal has a less clear-cut answer than in the previous problems that we considered.

What is clear-cut is that Anton and Bernard together have an absolute group claim with an amount of $14$ . For, as it is their joint combi-deal, the associated revenues of $14$ , and all of the $14$ , are owed to them. Also, it is clear that the individual claims of Anton and Bernard are notional. What is less clear, however, is how to represent the amounts and strengths of these individual claims. More than one way of doing so suggests itself, including:

The rationale of representing Combi-deal as ${{\cal C}_1}$ is as follows.

Rationale of ${{\cal C}_1}$ . Anton and Bernard have a claim to receive a part of the price of the combi-deal ticket. Thus, the amount of their (notional) claims is $14$ . The strength of their respective claims is a function of the price of their individual product and the fraction of combi-deal tickets that they sold. A plausible candidate for this function is the product function: the strength of Anton’s claim is $10 \cdot {2 \over 3}$ as the price of a pizza is $10$ and as Anton sold ${2 \over 3}$ of the tickets. Similarly, the strength of Bernard’s claim is $6 \cdot {1 \over 3}$ . Normalising the claims strength yields a strength of ${{10} \over {13}}$ for Anton and of ${3 \over {13}}$ for Bernard ^{Footnote 16} .

To be sure, we are not suggesting that one should represent Combi-Deal as ${{\cal C}_1}$ . All we suggest is that it is not, at least not prima facie, implausible to represent it as such. When Combi-Deal is represented as ${{\cal C}_1}$ , fairness requires that allocation $\left( {10.77,{\rm{\;}}3.23} \right)$ is realized. For, $\left( {10.77,{\rm{\;}}3.23} \right)$ satisfies the absolute group claim of Anton and Bernard to as large extent as possible while it does not allot more to the agents than their individual claim amount of $14$ . But also, in $\left( {10.77,{\rm{\;}}3.23} \right)$ the claims of Anton and Bernard are satisfied in proportion to their strength. Hence, $\left( {10.77,{\rm{\;}}3.23} \right)$ satisfies the requirements of both absolute and comparative fairness.

Still, one might argue that something is wrong with recommending $\left( {10.77,{\rm{\;}}3.23} \right)$ for Combi-Deal. For in this allocation, Anton receives more from selling a pizza via the combi-deal ( $10.77$ ) than from just selling a pizza ( $10$ ). And this, so one may argue, is not how it should be. For by its nature, the combi-deal yields lower joint revenues ( $14$ ) from selling a pizza plus ice-cream than from selling these products separately ( $10 + 6 = 16$ ). It seems reasonable that the lower joint revenues from selling a combi-deal translate into associated lower revenues for the pizza and ice-cream that are sold as part of the deal: any fair allocation of the $14$ should be such that Anton is allotted not more than $10$ while Bernard is allotted not more than $6$ . Recommending $\left( {10.77,{\rm{\;}}3.23} \right)$ for Combi-Deal is not fair! One may very well concur with this argument and advocate the Fairness formula at the same time. Indeed, the recommendation $\left( {10.77,{\rm{\;}}3.23} \right)$ depends as much on the Fairness formula as it does on one’s representation of Combi-Deal as ${{\cal C}_1}$ . For a proponent of the argument against $\left( {10.77,{\rm{\;}}3.23} \right)$ just given, it is natural to represent Combi-Deal as ${{\cal C}_2}$ , with the following rationale.

Rationale of ${{\cal C}_2}$ . Anton and Bernard have claims to receive a part of the regular revenues of their individual products that are part of the combi-deal. Hence, the amounts of the (notional) claims of Anton and Bernard are $10$ and $6$ respectively. The strength of their respective claim is determined by the fraction of combi-deal tickets that they sold so that Anton’s claim is $2$ times as strong as Bernard’s claim.

Applying the Fairness formula to ${{\cal C}_2}$ yields recommendation $\left( {10,{\rm{\;}}4} \right)$ , as the reader may care to verify. In $\left( {10,{\rm{\;}}4} \right)$ no agent receives more from selling their product via the combi-deal than from only selling their own product: no agent receives more than he has a claim to.

The constraint of absolute fairness that ‘no one receives more than they have a claim to’ plays a crucial role in deriving recommendation $\left( {10,{\rm{\;}}4} \right)$ on the basis of ${{\cal C}_2}$ . Let’s suppose that we delete the constraint of absolute fairness from the Fairness formula. In that case, the resulting formula would, on the basis of ${{\cal C}_2}$ , recommend allocation $\left( {11,{\rm{\;}}3} \right)$ in which Anton receives more than he has a claim to! To see this, observe that in $\left( {11,{\rm{\;}}3} \right)$ the absolute claim of $\{ $ Anton, Bernard $\} $ is satisfied to as large an extent as possible (as it is in any allocation that allots $14$ in total). But also, in $\left( {11,{\rm{\;}}3} \right)$ Anton’s claim with an amount of $10$ is fully satisfied, and so receives $100{\rm{\% }}$ satisfaction, whereas Bernard’s claim with an amount of $6$ receives $50{\rm{\% }}$ satisfaction, so that the individual claims of Anton and Bernard are satisfied in proportion to their strength. Hence, representation ${{\cal C}_2}$ of Combi-Deal illustrates the need to explicitly add the absolute constraint to the Fairness formula: without the constraint, the formula may yield recommendations that allot more to an individual than they have a claim to, which conflicts with the idea of a claim amount as an upper-bound on the receipts of the claimant.

4.2. On claim strengths and amounts

The Broomean formula only explicitly mentions claim strengths, but not amounts. Yet, as explained in section 2, the Broomean formula does mention the satisfaction of claims explicitly, which in turn depends on the good received and the amount of the claim: amounts are needed to determine satisfaction.

While there are some contributions that distinguish between claim strengths and amounts (e.g. Wintein and Heilmann Reference Wintein and Heilmann2018; Hausman Reference Hausman2023), many contributors have neglected the distinction between claims strengths and amounts altogether, such as Curtis (Reference Curtis2014). He develops a Broomean theory of fairness that identifies claims with their amounts. As a consequence, his theory is of limited scope and, in particular, cannot be used to analyse problems such as Investing Time. ^{Footnote 17} Or take Lazenby (Reference Lazenby2014), who discusses his Broomean account of fairness only in relation to problems where all individuals have claims, of varying strengths, to the same (amount of) good. ^{Footnote 18} Indeed, a problem like Owing Money cannot be dealt with by Lazenby’s account. Moreover, the fair division problems in section 3 involve agents with claims of varying amounts and strengths: these problems are outside the scope of Curtis’s, Lazenby’s and in fact any theory of fairness that we are aware of. In short, neglecting the strength-amount distinction restricts the scope of some accounts of fairness.

A second, more subtle and complex, issue is the failure to properly conceptualize the strength-amount distinction in some fairness accounts. To appreciate what is at stake, recall that the modelling of a fair division problem is a crucial step in solving it on the basis of the Fairness formula. This is illustrated by the two different representations of Combi-Deal just discussed, which lead to different recommendations. But it is also illustrated by an alternative representation of Owing Money that is sometimes given in the literature which, on our account, is flawed. This alternative representation has been worded ^{Footnote 19} most explicitly by David Morrow (Reference Morrow2017), who writes that: ^{Footnote 20}

Thus, on Morrow’s analysis, Abram and Benvolio have claims to Romeo’s money, which is $40$ , with the strength of their claims depending on the amount of money owed. In term of Broomean problems, Morrow proposes the following $alt$ ernative model of ${\cal O}$ wing Money:

$${{\cal O}^{alt}} = \left( {40,\left\{ {A,B} \right\},\left( {40,40} \right),\left( {{1 \over 4},{3 \over 4}} \right)} \right)$$

Morrow writes that ‘[fairness requires] that Romeo should give thirty ducats to Benvolio and ten to Abram’, i.e. that Romeo should realize $\left( {10,{\rm{\;}}30} \right)$ on the basis of ${{\cal O}^{alt}}$ . This recommendation coincides with that of the absolute priority rule: ${P^\dagger }\left( {{{\cal O}^{alt}}} \right) = \left( {10,{\rm{\;}}30} \right)$ .

However, Morrow does not tell us why fairness requires that Romeo should realize $\left( {10,30} \right)$ . On our account, the representation of Owing Money as ${{\cal O}^{alt}}$ is at odds with the recommendation that Romeo should realize $\left( {10,30} \right)$ . For, as we will demonstrate below, the ${{\cal O}^{alt}}$ representation fails to capture the absolute claims correctly. Without those, however, the requirements of absolute fairness, which only deals with absolute claims, cannot be invoked to argue that Romeo has to realize $\left( {10,30} \right)$ rather than, say, $\left( {5,{\rm{\;}}15} \right)$ .

The individual claims ascribed to Abram and Bevolio by ${{\cal O}^{alt}}$ are clearly notional and not absolute: Romeo owes it to Abram and Benvolio to give them a part of $40$ , and there’s nothing wrong with one of them receiving less than $40$ . Nor does it make sense to say that the group $\{ $ Abram, Benvolio $\} $ has an absolute claim to $40$ . Romeo owes $20$ to Abram and $60$ to Benvolio, for sure. But from this it does not follow that Romeo owes anything to the group consisting of Abram and Benvolio. Indeed, Abram and Benvolio may have never met, never interacted and be unaware of one another’s existence: they do not form a group in any substantive sense. In sharp contrast, Anna and Beta jointly produced the value of $20$ in Investing Time, so the group consisting of Anna and Beta has an absolute claim to receive all of the jointly produced $20$ . But in Owing Money, there is no such joint production and it does not make sense to ascribe an absolute claim to $\{ $ Abram, Benvolio $\} $ .

But then, on Morrow’s analysis, there are no absolute claims in Owing Money at all: ${{\cal O}^{alt}}$ can neither be understood as capturing absolute claims by the individuals nor by the group. Hence, absolute fairness has nothing to say about the allocation that Romeo should realize for Owing Money and, in particular, cannot explain why Romeo should realize $\left( {10,{\rm{\;}}30} \right)$ rather than $\left( {5,{\rm{\;}}15} \right)$ . Although Morrow asserts that fairness requires that Romeo realizes $\left( {10,{\rm{\;}}30} \right)$ in Owing Money, he cannot justify this assertion owing to his representation of Owing Money as ${{\cal O}^{alt}}$ . As such, Morrow’s analysis of Owing Money is flawed. Or so we argue.

We have illustrated and motivated the Fairness formula in some detail. We have done so exclusively for fair division problems in which there was not enough to go around: the amount to be divided was less than the sum of claims. However, in the literature we also find some discussions of what (Broomean) fairness requires in cases of abundant good. To further illustrate the Fairness formula, we will now apply it to such cases.

4.3. Fairness and abundant good

In Fairness between competing claims, Ben Saunders (Reference Saunders2010) criticizes Broome’s idea that fairness is strictly comparative. Saunders (Reference Saunders2010: 45) does so by considering a case where he owes ‘£20 on one friend and £10 to another’ and remarks that ‘if I have the money that I promised to pay you, then it is unfair of me either to keep it or burn it and to repay less than I can’, hinting at the absolute dimension of fairness. With respect to his example, Saunders then further remarks the following.

So, Saunders speculates about what fairness requires in cases of abundant good. In fact, he does so with respect to two different fair division problems.

Abundant Money. Marta owes £20 to Alice and £10 to Bob and has £60 in his pocket. How, in order to be fair, should Marta divide her money?

Profit. Ali and Benji have invested respectively £20 and £10 in a joint project. After some time, Ali and Benji want to split apart and the question arises how the value of their joint project, £60, should be divided. How, in order to be fair, should Ben divide the £60?

${\cal A}$ bundant Money and ${\cal P}$ rofit are represented as follows:

When applied to ${\cal A}$ , the absolute priority rule recommends $\left( {20,{\rm{\;}}10} \right)$ in which both claims receive the maximal satisfaction of $100{\rm{\% }}$ . Hence, $\left( {20,{\rm{\;}}10} \right)$ satisfies claims to as large an extent as possible and, as the claims are equally strong, in proportion to their strength. Now, any allocation in which Alice and Bob receive the full amount of their claim or more, such as $\left( {25,{\rm{\;}}15} \right)$ or $\left( {30,{\rm{\;}}30} \right)$ , satisfies claims to as large an extent as possible and in proportion to their strength. But $\left( {20,{\rm{\;}}10} \right)$ is the unique allocation which has these two features and in which no agent receives more than he has a claim to: the absolute constraint also plays a crucial role in the derivation of $\left( {20,10} \right)$ from the Fairness formula on the basis of ${\cal A}$ .

For sure, fairness does not require that Marta allots more to Alice and Bob than they have a claim to. But suppose that Marta decides to use all of her £60 in order to give Alice and Bob a ‘gift on top of repaying what [he owes]’ (Saunders Reference Saunders2010: 45). Saunders suggests that in such a situation, Marta should ‘give each an equal amount extra’, i.e. realize allocation $\left( {20,{\rm{\;}}10} \right) + \left( {15,{\rm{\;}}15} \right) = \left( {35,{\rm{\;}}25} \right)$ . We concur. However, this requirement is not grounded in fairness, but rather in what Nicholas Rescher calls subjective equity. As Rescher (Reference Rescher2002: ix) explains, fairness is based on claims so that ‘fairness is therefore something quite different from subjective equity as based on the personal evaluation of distributive goods by the claimants involved’. An important criterion of subjective equity is that of no-envy: an allocation of goods is called envy-free when no individual prefers the share of another individual over her own share. ^{Footnote 21} Now, as Alice and Bob both prefer to receive more money over less, the only division of the gift that is envy-free is the equal division. Hence, when Marta realizes $\left( {22,{\rm{\;}}38} \right)$ , or $\left( {40,{\rm{\;}}20} \right)$ for that matter, Alice can complain that the realized allocation is not envy-free. So, we concur with Saunders that when Marta decides to use her £60 to provide a gift to Alice and Bob on top what she owes them, there is a sense in which she ‘should give each an equal amount extra’, resulting in allocation $\left( {35,{\rm{\;}}25} \right)$ . However, we submit that this sense of ‘should’ is grounded in subjective equity, and not in (claims-based) fairness. When all claims are fully reimbursed, the requirements of fairness are fully met: as there are no claims on ‘gifts on top of full reimbursement’ fairness offers, in contrast to subjective equity, no guidance in how to allocate abundant good.

So, by distinguishing between fairness and subjective equity, we can provide a normative underpinning for Saunders’s intuitive suggestion for problems such as Abundant Money. Moreover, our conceptual framework allows us to account for the different recommendations for Abundant Money and Profit that Saunders alludes to. Indeed, Abundant Money gives rise to ${\cal A}$ , in which the individuals have absolute claims, whereas Profit is naturally represented as ${\cal P}$ , with individual claims being notional and invested money affecting the strength of the claims. With respect to ${\cal P}$ , the absolute priority rule recommends allocation $\left( {40,20} \right)$ which accords with Saunders’s suggestion.