The Claims Approach and Aggregativity
In analysing fair division problems such as Problem I and Problem II, Paseau and Saunders adopt what we will call the claims approach to fair division. In the philosophical literature, this approach has arguably started with John Broome’s seminal paper on fairness (Broome 1990). Broomean theories of fairness focus on those fair division problems in which some agents have claims to a good that is to be distributed. Roughly, a claim is a specific type of reason, owed to the agent herself, as to why she should have some of the good that is to be divided. A thorough analysis of what is a claim is not to be found in the literature, but need, desert, and promises are typically taken to induce claims. In the introduction, we also (implicitly) analysed Problem I and Problem II as claims problems, the official definition of which is as follows.
Claims problems A claims problem\({\mathcal {C}} := (E, N, c)\) consists of an amount of good \(E \ge 0\), also called the estate, a set of receiving agents N and a claims vector c specifying the amount of the estate that agent i has a claim to (\(c_i \ge 0\)), and which is such that together the claims exceed the amount of the good available (\(\sum _{i \in N} c_i \ge E\)).
Hence, on the claims approach, Problem I and Problem II are analysed as claims problems \({\mathcal {C}}^I\) and \({\mathcal {C}}^{II}\), respectively, where:
$$\begin{aligned} {\mathcal {C}}^I = (60, \{A, B\}, (80, 40)) \qquad {\mathcal {C}}^{II} = (90, \{A, B\}, (40, 80)) \end{aligned}$$
We already discussed the proportional rule P, which is an example of a division rule. More generally, a division rule is defined as follows.
Division rules A division ruler is a function that maps each claims problem (E, N, c) to an allocation\(x \in {\mathbb {R}}^N\), with the property that each agent receives a non-negative amount that does not exceed his or her claim (\(0 \le x_i \le c_i\)), and the sum of what is allocated does not exceed the estate (\(\sum _{i \in N} x_i \le E\)).
This definition permits a multitude of different division rules, as it does make very little specific demands on what the allocation should look like. Notably, this general definition of a division rule does not even prescribe that such a rule is efficient, which means it is not required that a division rule proposes to allocate all of the estate. Indeed, the rule which allots 0 to each agent in each claims problem, call this the trivial rule, respects the definition of a division rule. Clearly, the trivial rule is, in sharp contrast to the proportional rule, a rather uninteresting division rule.
In the economic literature on fair division, claims problems and division rules are studied extensively: see Thomson (2003) for an overview. Here, we just mention one further example of a non-trivial division rule, the so-called constrained equal losses rule (CEL rule). Given a claims problem (E, N, c), the CEL rule proposes an efficient division of the estate E in such a way that each agent loses an equal amount with respect to her claim, subject to the constraint that no agent loses more than her claim.Footnote 2 Applying the CEL rule to claims problem \({\mathcal {C}}^I\) results in the allocation (50, 10), so that both A and B lose an equal amount of 30 with respect to their claims of 80 and 40, respectively. Also, applying the CEL rule to claims problem \({\mathcal {C}}^{II}\) results in the allocation (25, 65) so that both agents lose 15 with respect to their respective claims.
Given any two claims problems \({\mathcal {C}} = (E,N,c)\) and \({\mathcal {C}}' = (E', N, c')\) that involve the same set of agents N, by aggregating \({\mathcal {C}}\) and \({\mathcal {C}}'\), i.e. by adding the estates and claims vectors of \({\mathcal {C}}\) and \({\mathcal {C}}'\), one obtains a further claims problem \({\mathcal {C}} + {\mathcal {C}}' := (E + E', N, c + c')\), which is then called the aggregated problem of \({\mathcal {C}}\) and \({\mathcal {C}}'\). For example, by aggregating claims problems \({\mathcal {C}}^I\) and \({\mathcal {C}}^{II}\) one obtains the aggregated problem \({\mathcal {C}}^{I} + {\mathcal {C}}^{II}\) which we will also denote as \({\mathcal {C}}^{I+II}\):
$$\begin{aligned} {\mathcal {C}}^{I+II} = (150, \{A, B\}, (120, 120)) \end{aligned}$$
In the introduction, we informally demonstrated that the proportional rule P is not aggregative with respect to\({\mathcal {C}}^{I}\)and\({\mathcal {C}}^{II}\), meaning that:
$$\begin{aligned} P({\mathcal {C}}^{I}) + P({\mathcal {C}}^{II}) \not = P({\mathcal {C}}^{I +II}) \end{aligned}$$
In contrast, the CEL rule is aggregative with respect to \({\mathcal {C}}^{I}\) and \({\mathcal {C}}^{II}\), as
$$\begin{aligned} CEL({\mathcal {C}}^{I}) + CEL({\mathcal {C}}^{II}) = CEL({\mathcal {C}}^{I +II}) \end{aligned}$$
In order for a division rule r to be aggregative, it has to be aggregative with respect to any two claims problems \({\mathcal {C}} = (E,N,c)\) and \({\mathcal {C}}' = (E', N, c')\) that involve the same set of agents. That is, r is aggregative just in case for any such \({\mathcal {C}}\) and \({\mathcal {C}}'\) we have:
$$\begin{aligned} r({\mathcal {C}}) + r({\mathcal {C}}') = r({\mathcal {C}} + {\mathcal {C}}') \end{aligned}$$
(1)
The trivial rule that was discussed above is clearly an aggregative, albeit an uninteresting, division rule. Claims problems \({\mathcal {C}}^{I}\) and \({\mathcal {C}}^{II}\) testify that the proportional rule is not aggregative. But what about the CEL rule, is it aggregative? It follows from Theorem 1 below that the answer is ‘no’, as there are no non-trivial aggregative division rules.
Theorem 1
(There are no non-trivial aggregative division rules)
Proof
See “Appendix”. \(\square \)
Paseau and Saunders (2015) prove Theorem 1 themselves but remark that ‘there is an extensive and sophisticated economics literature in this area which appears to imply [Theorem 1]’. Their presumption is definitely correct as a proof of Theorem 1 can also be found in Bergantiños and Méndez-Naya (2001: 227).Footnote 3 Bergantiños and Méndez-Naya’s result has led to some interesting studies of ‘aggregativity in claims problems’ in the economic literatureFootnote 4, which is not directly relevant for our purposes here (but see Sect. 4).
What is directly relevant for our purposes here is the philosophical upshot of Theorem 1. According to Paseau and Saunders, Theorem 1 implies NAT.
NAT There is no non-trivial aggregative theory of fairness.
The reason that Paseau and Saunders take Theorem 1 to imply NAT is simply that they equate theories of fairness with division rules. Although it makes sense to do so on the claims approach to fair division, that is not the only approach. In the next section, we will discuss another framework for fair division, which we call the games approach, that can also be used to analyse fair division problems such as Problem I and Problem II. In order to analyse such problems, the games approach does not model them as claims problems, but rather as cooperative games. On the claims approach, theories of fairness are division rules that act on claims problem but, as we will see, on the games approach theories of fairness are solution values that act on cooperative games: theories of fairness on the claims and games approach act on different fairness structures. On the claims approach, there are no (non-trivial) aggregative theories of fairness, but on the games approach there are such theories. Indeed, as a categorical statement, NAT is simply false, as we will explain in the remainder of this section.
A First Look at the Games Approach
Let us consider Problem I once more and observe the following. If John would fully repay Bob there is still \(60 - 40 = \pounds 20\) left for Ann, which is to say that Ann can guarantee herself £20. When John does his utmost to fully reimburse Ann, he has to give her £60 leaving nothing for Bob, which is to say that Bob can guarantee himself £0. Now consider the group consisting of Ann and Bob. If all receiving agents other than Ann and Bob (of which there are none) are fully reimbursed, there is £60 left for Ann and Bob together: the group consisting of Ann and Bob can guarantee itself £60. We have now implicitly analysed Problem I as a cooperative game, the definition of which is as follows.
Cooperative games A cooperative game is a pair (N, v), with N a set of agents and with \(v : \mathcal {P}(N) \rightarrow {\mathbb {R}}_+\), \(v(\emptyset ) = 0\) the characteristic function of the game which specifies the value that each group of agents (or coalition) can guarantee itself.Footnote 5 In particular, v(N) represents the value that the grand coalitionN can guarantee itself.
Thus, the cooperative games that are associated with Problem I and Problem II are given by \((\{A, B \}, v^I)\) and \((\{A, B \}, v^{II})\) respectively, where:
$$\begin{aligned}&v^I(\emptyset ) = 0\quad v^I(\{A \}) = 20 \quad v^I(\{B \}) = 0 \quad v^I(\{A, B \}) = 60\\&v^{II}(\emptyset ) = 0 \quad v^{II}(\{A \}) = 10 \quad v^{II}(\{B \}) = 50 \quad v^{II}(\{A, B \}) = 90 \end{aligned}$$
One central question that is studied by cooperative game theory is the following.
$$\begin{aligned} \text{ Given } \text{ a } \text{ game } (N,v), \hbox { how to divide } v(N) \hbox { amongst the agents in } N? \end{aligned}$$
(2)
Note that the money that is to be divided in Problem I and Problem II is, per definition, the value of the ‘grand coalition’ (the group of all agents, i.e. in this case, Ann and Bob together) in the associated games \(v^I\) and \(v^{II}\), respectively. Hence, an answer to question (2) can resolve fair division problems such as Problem I and Problem II, as discussed in more detail in Sect. 2.3. In the literature on cooperative game theory, question (2) is answered by specifying a solution value.
Solution values A solution value\(\varphi \) is a function that maps each cooperative game (N, v) to an allocation\(x \in {\mathbb {R}}^N\) with the property that \(\sum _{i \in N} x_i \le v(N)\).
There is an extensive literature in cooperative game theory that proposes and compares different solution values and all of them can be—and typically are—understood as proposals to divide the value of the grand coalition fairly. Prominent solution values that have been proposed in the literature are the Shapley value (cf. Shapley 1953), the nucleolus (cf. Schmeidler 1969), and the \(\tau \)-value (cf. Tijs 1981). We will revisit the Shapley value in some detail later on and apply it to \(v^I\) and \(v^{II}\). However, it will be instructive to first spend a few words on the notions of a theory of fairness and a fairness structure.
Fairness: Theories, Structures, and Aggregation
Let us revisit Problem I once more, in which John owes £80 to Ann, £40 to Bob but has only £60 left. We have seen that there are two different ways to divide the £60 in this fair division problem. One way is to analyse Problem I as claims problem \({\mathcal {C}}^I\) and then to apply a division rule to \({\mathcal {C}}^I\) in order divide the £60. The other way is to analyse Problem I as cooperative game \(v^I\) and then to apply a solution value to \(v^I\) in order to do so. To be sure, the availability of these two approaches is not confined to Problem I. Indeed, any fair division problem that can be analysed as a claims problem\({\mathcal {C}} = (E, N, c)\)can also be analysed as a cooperative gameFootnote 6\((N, v^{\mathcal {C}})\), where
$$\begin{aligned} v^{\mathcal {C}}(S) = max\{0, E - \sum _{i \not \in S} c_i \} \qquad {\text {for}}\, {\text {each}}\,S \subseteq N. \end{aligned}$$
(⋆)
Note that the estate E in claims problem \({\mathcal {C}}\) coincides with the value of the grand coalition \(v^{\mathcal {C}}(N)\) in the game that is associated with \({\mathcal {C}}\) via (\(\star \)). Hence, division rules and solution values provide two different ways to divide \(E = v^{\mathcal {C}}(N)\).
Paseau and Saunders equate theories of fairness with division rules, which makes sense from the perspective of the claims approach. However, from the perspective of the games approach, it makes just as much sense to equate a theory of fairness with a solution value. By a theory of fairness, we mean a function that assigns an allocation of the good-to-be-divided for each fairness structure that is within its domain. A fairness structure is obtained by modelling a fair division problem, that is by extracting the characteristics of the problem on the basis of which, according to the model, fair division should proceed. Thus, the fairness structures associated with the claims approach are claims problems whereas the fairness structures associated with the games approach are cooperative games. Both division rules and solution values are theories of fairness, albeit theories that take different fairness structures as their input.
Although our notions of theory of fairness and of fairness structure are abstract, they are theoretically fruitful. For one thing, they allow us to spell out the notion of an aggregative theory of fairness in a way that does not privilege the claims or games approach to fairness.
Aggregative Theories of Fairness A theory of fairness ToF is called aggregative when the following holds. Given any two fairness structures \(S_1\) and \(S_2\) that involve the same set of receiving agents, the sum of the allocations that result from applying ToF to \(S_1\) and \(S_2\) is equal to the allocation that results from applying ToF to the fairness structure that results from aggregating \(S_1\) and \(S_2\).
We have seen how aggregation works on the claim approach in Sect. 2.1. To aggregate claims problems \({\mathcal {C}} = (E,N,c)\) and \({\mathcal {C}}' = (E', N, c')\) one adds the estates and claims vectors of both problems and thus obtains the aggregated claims problem \({\mathcal {C}} + {\mathcal {C}}' = (E + E', N, c + c')\). When we abstract away from the particular fairness structures that are exploited by the claims approach, we may say that to aggregate two fairness structures (that involve the same set of receiving agents) one adds—component wise—all information of the two structures. From the more abstract notion of aggregation thus arrived at, it readily follows how to aggregate fairness structures on the games approach: to aggregate games (N, v) and \((N,v')\) one adds, for each coalition \(S \subseteq N\), its value in both problems and thus obtains the aggregated game \((N, v+v')\). As an example, by aggregating games \((N, v^I)\) and \((N, v^{II})\) that are associated with Problem I and II, respectively, we obtainFootnote 7 the aggregated game \((N, v^{I+II})\), where:
$$\begin{aligned} v^{I+II}(\emptyset ) = 0 \quad v^{I+II}(\{A \}) = 30 \quad v^{I+II}(\{B \}) = 50 \quad v^{I+II}(\{A, B \}) = 150 \end{aligned}$$
The definition of an aggregative solution value now readily follows from the general definition of an aggregative theory of fairness. A solution value \(\varphi \) is said to be aggregative with respect to games (N, v) and \((N,v')\) when:
$$\begin{aligned} \varphi (N, v) + \varphi (N, v') = \varphi (N, v + v') \end{aligned}$$
(3)
A solution value \(\varphi \) is aggregative when \(\varphi \) is aggregative with respect to any pair of cooperative games that involve the same set of agents N, i.e. \(\varphi \) is aggregative when (3) holds for any games (N, v) and \((N,v')\).
The trivial solution value, i.e. the solution value that assigns 0 to each agent in each cooperative game, is a rather uninteresting example of an aggregative solution value. The question arises whether there are also non-trivial aggregative solution values, i.e. whether the games approach harbours non-trivial aggregative theories of fairness. As we will explain in the next section, the Shapley value testifies that the answer to that question is ‘yes’.
The Shapley Value and the Failure of NAT
Let us illustrate the Shapley value by showing how the Shapley values for \(v^I\) and \(v^{II}\), the games associated with Problem I and Problem II as described in Sect. 2.2, are obtained. For the sake of convenience, let us first display these games once more.
$$\begin{aligned} v^I(\emptyset )&= 0 \quad v^I(\{A \}) = 20 \quad v^I(\{B \}) = 0 \quad v^I(\{A, B \}) = 60\\ v^{II}(\emptyset )&= 0 \quad v^{II}(\{A \}) = 10 \quad v^{II}(\{B \}) = 50 \quad v^{II}(\{A, B \}) = 90 \end{aligned}$$
Given a cooperative game (N, v), the Shapley value considers all orders of the agents in N. As \(v^I\) and \(v^{II}\) only involve two agents, A and B, there are just two such orders in these games: \(\langle A, B \rangle \) and \(\langle B, A \rangle \) respectively. Agent orders may be thought of as possible manners in which the grand coalition, consisting of all agents, can be formed. So in the order \(\langle A, B \rangle \), agent A is the first to arrive and thereby realises the singleton coalition \(\{A\}\). Next B arrives and he joins A to form the grand coalition \(\{A, B\}\). The Shapley value records, for each order, the marginal contributions that the agents make with respect to the coalitions that are formed upon their arrival. Consider the order \(\langle A, B \rangle \) for \(v^I\). When A arrives, she realises a marginal contribution of \(v^I(\{A \}) - v^I(\emptyset ) = 20 - 0 = 20\) with respect to the empty coalition. Then B arrives and he realises a marginal contribution of \(v^I(\{A, B \}) - v^I(\{A \}) = 60 - 20 = 40\) with respect to coalition \(\{A \}\). So, in \(v^I\), the marginal contributions of A and B induced by \(\langle A, B \rangle \) are 20 and 40, respectively. According to the Shapley value, agents receive their average marginal contribution over all agent orders. As the reader may care to verify, the order \(\langle B, A \rangle \) in \(v^{I}\) induces marginal contributions for A and B of 60 and 0, respectively. Hence the Shapley value for \(v^{I}\) allots \(\frac{20 + 60}{2} = 40\) to A and \(\frac{40 + 0}{2} = 20\) to B: \(\mathsf{Sh}(v^I) = (40, 20)\). Table 1 below conveniently summarises the computation of the Shapley value for \(v^{I}\) and also presents this computation for \(v^{II}\).
Table 1 Shapley values for \(v^I\) and \(v^{II}\) Given an arbitrary cooperative game (N, v), let us write \(\Pi (N)\) to denote the set of all orders of the agents in N and, given such an order \(\pi \), let \(MC^v(\pi )\) denote the vector that records the marginal contributions that the agents realize in order \(\pi \) on the basis of v. The general definition of the Shapley value can then be stated as follows.Footnote 8
$$\begin{aligned} \mathsf{Sh}(N,v) = \frac{\sum _{\pi \in \Pi (N)} MC^v(\pi )}{\mid N\mid !} \end{aligned}$$
(4)
From this definition of the Shapley value, it can readily be shown that the Shapley value is aggregative, as recorded by the following theorem.
Theorem 2
Sh is aggregative: \(\mathsf{Sh}(N, v) + \mathsf{Sh}(N, v') = \mathsf{Sh}(N, v + v')\)
Proof
See appendix. \(\square \)
Theorem 2 testifies that NAT, when taken as a categorical statement, is simply false. Indeed, the Shapley value is a (non-trivial) theory of fairness that is aggregative.
Aggregativity After the Failure of NAT
In demonstrating that NAT is false, pacePaseau and Saunders (2015), we have achieved a narrow, but important, argumentative goal. In the remainder of the paper, we turn to discussing questions of broader significance that are implied by what we have demonstrated. These questions are, on the one hand, raised by the strategy we adopted to show that NAT is false. On the other hand, the very fact that NAT is false needs evaluation in terms of its implications for Paseau and Saunders (2015) and beyond. There are two issues.
Firstly, we have introduced the claims and the games approach to modelling and analysing fair division. These two approaches clearly differ in a number of important respects, not least that there are aggregative theories of fairness on the games approach, but not on the claims approach. Yet, we have also seen that the two approaches are closely related in some other respects. Recall the equation (\(\star \)) which shows how any fair division problem that can be modelled as a claims problem can also be modelled as a game. This suggests that there could be more similarities between the two approaches. Indeed, as we will see, there exist division rules and solution values that give the very same recommendations. We will explore the potential tension between these kinds of similarities and differences between the claims and games approach. This, in turn, allows us to make the nature and scope of the aggregativity condition much more precise. We will do so in Sect. 3.
Secondly, we will investigate what kind of role aggregativity should play in theorising about fair division. We will argue that aggregativity is not a property of fairness, and that non-aggregative theories of fairness are not problematic. That is, the proponent of the claims approach might face the choice between giving up aggregativity or the claims approach in favour of the games approach. Contrary to what Paseau and Saunders (2015) say, non-aggregativity should thus not be viewed as a problem, or so we argue in Sect. 4.
Together, exploring these issues will yield a comprehensive treatment of aggregativity in the claims approach and in the games approach.