FC does not imply P
Consider the following situation.
Bankruptcy. Ari, Betty, and Claire have, respectively, deposited 800, 400 and 200 Euros on their savings account with the SAFE bank. However, SAFE goes bankrupt and its liquidation value of 900 (the ‘estate’) has to be divided amongst Ari, Betty, and Claire. Together, the money on their accounts adds up to 1400, and so a problem arises. How should we, in order to be fair, divide the estate?
Recently, Curtis (2014) has provided an important new theory of fairness that applies to such problems.Footnote 1 Curtis advocates his theory as spelling out what he maintains Broome’s (1990) well-known theory of fairness has left out: an actual method that describes ‘how to be fair’. Curtis (2014, p. 47) remarks that, to the best of his knowledge, ‘neither Broome nor anyone else had laid down a theory of precisely what one must do in order to be fair’. Yet, fair division problems like Bankruptcy have been studied extensively, both axiomatically (see Herrero and Villar 2001 and Thomson 2003 for overviews) and from the perspective of cooperative game theory (see e.g. O’Neill 1982 or Curiel et al. 1988). There is a long history to the study of such problems. For instance, the Talmud discusses a division rule, which has been studied formally by Aumann and Maschler (1985).
Classical examples of division rules are the rule of ‘constrained equalized awards’ (CEA), the ‘proportional rule’ (P), and the rule of ‘constrained equalized losses’ (CEL). There are many more rules in the axiomatic and game-theoretic literature, with some of the most important ones being introduced in this paper as we go along. Let us for now focus on the three rules just mentioned and apply them to Bankruptcy. The CEA rule equalizes the awards as much as possible, under the constraint that no agent can receive more than his or her claim.Footnote 2 In Bankruptcy, the estate is 900 and so allotting 300 to each of the agents equalizes the awards. However, doing so would give Claire more than her claim of 200. So Claire gets 200 and the remaining 700 is shared equally between Ari and Betty. Now none of the three agents gets more than his or her claim, and the resulting allocation is the one prescribed by the CEA rule, as indicated in Table 1.
Table 1 Division rules and their recommended awards for Bankruptcy
Further, P divides the estate in proportion to the size of the claims of the respective agents. Finally, the CEL rule equalizes the losses that the agents suffer with respect to their claims under the constraint that no agent can loose more than his or her claim. Suppose that we (‘temporarily’) allot Ari, Betty and Claire the totality of their claims. This would result in an infeasible allocation, as we would distribute 1,400 whereas the estate is only 900. To reach a feasible allocation, the CEL rule proposes that the agents equally share the aggregate loss of 1,400\( - 900 = 500\). Hence, the CEL rule proposes that each agent loses \(166.\bar{7}\) relative to his or her claim, resulting in the allocation that is depicted in Table 1 above.
For claims problems like Bankruptcy, Curtis (2014) endorses the proportional rule P. Indeed, P is the cornerstone of his theory of fairness. What is more, he motivates adopting it by claiming that the proportional rule is entailed by a general principle of fairness that he adopts:
The Fairness Claim (FC). In order to be fair an allocating agent must (i) do as much as she can to satisfy the claim of each receiving agent to as great a degree as possible whilst ensuring that (ii) each claim is treated equally (Curtis 2014, p. 49).
Now FC is fairly general and open to interpretation. But Curtis does not provide any. Rather, he simply states that the allocating agent ‘divides ...proportionally is demanded by part (ii) [of FC]’ and that ‘it is also clear that FC tells [the allocating agent] to follow method P’ (Curtis 2014, p. 49). In a nutshell, Curtis claims that FC implies P.
We submit that the content of FC is too vague to single out P as the only division rule that is consistent with it. For one thing, all the rules we introduced in the above example realise FC (i) and of all of them it can reasonably be said that they treat each claim equally. To be sure, P does not seem to be in conflict with FC, and it can be argued that P is a plausible way to realise FC. But to the extent to which that can be argued, the same can be said about the CEA and CEL rules. FC is simply open to many interpretations and does not lead to one division rule. In other words: FC does not imply P.
We have already mentioned that many division rules have been developed in the axiomatic literature. In Sect. 2.2, we will employ this literature to propose an explication of FC. As we will see, our FC explication does not imply P. In Sect. 2.3 we go on to argue that no plausible explication of FC in the axiomatic approach can imply P.
The following two sections, as well as Sects. 4.2 and 4.3, are more technical than the rest of the article, which we indicate by putting a star next to their title. While the general message of the article is conveyed by the non-starred sections, the starred sections demonstrate the fruitfulness of the axiomatic and game-theoretic methods for theorizing about fair divisions in detail.
Explicating FC: using the axiomatic approach\(^\star \)
The axiomatic approach analyzes cases like Bankruptcy as fair division problems with a specific structure, which is that of a so-called claims problem.
Definition 1
A claims problem
\({\mathcal {C}} := (E, N, c)\) consists of an estate \(E > 0\), a set of agents \(N = \{ 1, \ldots , n \}\) and a claims vector \(c \in {\mathbb {R}}_+^N\) specifying a claim \(c_i\) for each agent i such that \(\sum _{i \in N} c_i > E\).
A multitude of division rules for claims problems have been proposed and studied in the literature.
Definition 2
A division rule
r is a function that maps each claims problem (E, N, c) to an allocation vector
\(x \in {\mathbb {R}}_+^N\), with the property that \(x_i \le c_i\) for each agent i: no agent receives more than his claim.
On the axiomatic approach, such rules are characterized in terms of logically independent properties which facilitates a fruitful comparison of (the fairness of) distinct division rules. One of the rules considered is the proportional one, about which Thomson remarks:
The best-known rule is the proportional rule, which chooses awards proportional to claims. Proportionality is often taken as the definition of fairness [...], but we will challenge this position and start from more elementary considerations. (Thomson 2003, p. 250).
Indeed, Curtis does not want to take proportionality as the definition of fairness, but wants to start from more elementary considerations that are encapsulated in FC. And so, the axiomatic approach seems to be highly relevant for Curtis’ project.
Let us first illustrate the main tenets of the axiomatic approach by exploring the division rules already mentioned earlier. The CEA, P and CEL rule all have the Efficiency property as, in each claims problem, the rules allocate all of the available estate. Another property that is common to the three rules is that of Equal Treatment of Equals, which says that (in each claims problem) agents with the same claims receive the same amount. Efficiency and Equal Treatment of Equals are logically independent properties: a division rule may satisfy Efficiency without satisfying Equal Treatment of Equals, or vice versa. Further, Efficiency and Equal Treatment of Equals are shared by lots of division rules, and hence these properties do not (jointly) characterize a division rule, i.e. there is no unique division rule satisfying them. To characterize a division rule is to identify a rule as the only one that satisfies a certain set of (preferably logically independent) properties.
The core business of the axiomatic approach is exactly to characterize division rules in this sense. Such characterizations facilitate a fruitful comparison of (the fairness of) distinct division rules. In this section, we will use the axiomatic approach to explicate FC.
Remember that Curtis (2014, p. 49) says, ‘that [the allocating agent] must use all of the good is demanded by part (i) of FC’. In terms of the axiomatic approach, Curtis claims that FC (i) demands a division rule that satisfies Efficiency. With this, we have no qualms. However, it should be noted that at this point, Curtis’ theory differs from that of Broome. According to Broome, fairness is a strictly comparative value that only requires the proportional satisfaction of claims: Broome denies that fairness requires Efficiency. Allotting 8 Euros to Ari, 4 Euros to Betty and 2 Euros to Claire in Bankruptcy is, according to Broome, just as fair as the division proposed by P. Various authors have objected to the Broomean thought that fairness is strictly comparative. For instance, Hooker (2005, p. 341) argues that, pace Broome, ‘fairness requires the greatest possible satisfaction of claims’. Indeed, Curtis states that he departs from Broome regarding this point, for the same reasons given by Hooker. Thus, although it is clear that FC (i) expresses Efficiency, it is debatable whether fairness requires Efficiency, i.e. whether Efficiency is a fairness property. For the purpose of this paper, however, we will simply follow Curtis and Hooker and assume that Efficiency, which is expressed by FC (i), is a fairness property.
Which fairness properties— allegedly characterizing P in the presence of Efficiency—are expressed by FC (ii)? Recall that FC (ii) is fairly general and open to interpretation. But perhaps we can invoke the axiomatic approach to understand its content in a more precise way. To do so, we look for a characterization of P in terms of properties that best reflect the phrase ‘to treat equal claims equally’. For this purpose, the following characterization of P looks attractive.
Theorem 1
The proportional rule P is the only division rule that satisfies the following logically independent properties: Efficiency, Equal Treatment of Equals, Self-Duality and Composition.
Proof
See Young (1988). \(\square \)
Let us first explain these properties in turn. We already encountered Efficiency and Equal Treatment of Equals. Explaining Self-Duality requires a little more work. Given a claims problem (E, N, c) and a division rule r that satisfies Efficiency, there are two canonical manners in which r can be exploited to divide the estate amongst the agents. First, by applying r directly to the problem at hand. Second, by applying r to the loss problem
\((\sum _{i \in N} c_i - E, N,c)\) that is associated with (E, N, c), and by diminishing each agent’s claim with the loss that is allocated to him or her as such. Thus, we may exploit r either to ‘divide what there is’ or to ‘allocate what is missing’. The second manner to divide the estate may also be thought of as applying the dual rule
\(r^\star \) of r to (E, N, c):
$$\begin{aligned} r^\star \left( E,N,c\right) = c - r\left( L, N, c\right) , \quad \text {with }L = \sum _{i \in N} c_i - E\hbox { the aggregate loss}. \end{aligned}$$
Indeed, each efficient division rule comes with a dual rule. Whereas CEA is the dual rule of CEL (and vice versa), the dual rule of P is P itself, which is to say that P satisfies Self-Duality. For self-dual rules, ‘dividing what there is’ is tantamount to ‘allocating what is missing’: such rules treat awards and losses in a symmetrical manner. When a division rule satisfies Composition, it doesn’t matter whether we divide all of the estate E at once or whether we divide, first, a part \(E_1\) of the estate and, second, the remainder \(E_2\) according to the outstanding claims after the first stage. More precisely, a division rule r satisfies Composition just in case, for all \(E_1, E_2 > 0\) such that \(E_1 + E_2 = E\), we have that:
$$\begin{aligned} r(E,N,c) = r(E_1, N, c) + r(E_2, N, c - r(E_1, N, c)) \end{aligned}$$
Theorem 1 thus tells us that Efficiency, Equal Treatment of Equals, Self-Duality and Composition jointly characterize P. How then, are these properties related to FC? As we already discussed, Efficiency is expressed by FC (i) and—although debatable—we assume that fairness requires Efficiency. Equal Treatment of Equals wears its interpretation on its sleeves: (agents with) equal claims should receive the same, i.e. they should be treated equally. As FC (ii) says that each claim should be treated equally, it follows from FC (ii) that, in particular, equal claims should be treated as such. Hence, Equal Treatment of Equals is implied by FC (ii). What about Self-Duality? Although Self-Duality is not, strictly speaking, implied by FC (ii), treating awards and losses in a symmetrical manner can, being charitable to Curtis, be understood as treating claims equally: whereas CEA and CEL are biased towards equalizing awards and losses respectively, and do not satisfy Self-Duality, P is free of these biases and treats each claim equally, no matter whether we allocate what is there or what is missing.
What about the last property on our list, Composition? Here, we feel that we run out of charity. Not only do we not see any relation between FC (ii) and Composition, we think that it is hard to see any relation between fairness and Composition at all. We concur with Herrero and Villar (2001), who classify Composition as a requirement that prevents an allocation to depend on the way in which a claims problem is subdivided into partial problems. Thus, although one might interpret Composition as a generally desirable property, it is hard to qualify it as a fairness property. Moreover, like Efficiency and Equal treatment of Equals, Composition is a property that is common to CEA, P and CEL. Hence, not only is the relation between Composition and fairness opaque, Composition does not, whereas Self-Duality does, allow us to distinguish CEA and CEL from P.
Efficiency, Equal Treatment of Equals and Self-Duality do not jointly characterize P, which follows from Theorem 1 and which is illustrated more vividly by the fact that they are also satisfied by the Talmud rule (cf. Aumann and Maschler 1985), the Run to the Bank rule (cf. O’Neill 1982) and the Adjusted Proportional rule (cf. Curiel et al. 1988).Footnote 3 We discuss the Adjusted Proportional (AP) rule in some detail here, as it will play a role in the remainder of the paper.
The AP rule exploits the notion of the minimal right of an agent, which is that part of the estate that would be left when the claims of all other agents were fully satisfied. In Bankruptcy, completely satisfying the claims of Betty and Claire would leave \(900 - 400 - 200 = 300\) for Ari, which is his minimal right. The minimal rights of Betty and Claire are 0 for each. The AP rule proposes to distribute minimal rights first. Doing so in Bankruptcy leaves us with an estate of \(900 - 300 = 600\), which is called the remaining estate. Further, allotting Ari’s minimal right reduces his original claim with 300, resulting in a reduced claim of \(800 - 300 = 500\) which, as this reduced claim does not exceed the remaining estate, is also his remaining claim
Footnote 4. The remaining claims for Betty and Claire are thus equal to their original claims of, respectively, 400 and 200. The AP rule distributes minimal rights first and then divides the remaining estate according to the remaining claims, resulting in an allocation that allots \(572.\bar{7}\) to Ari, \(218.\bar{2}\) to Betty, and \(109.\bar{1}\) to Claire.
As the AP rule satisfies Efficiency, Equal Treatment of Equals and Self-Duality it cannot, in virtue of Theorem 1, satisfy Composition. However, as Composition is not even remotely connected to FC, Curtis cannot appeal to Composition to argue that P is preferable to AP according to his conception of fairness. But perhaps there are other properties—more in line with FC—that would allow Curtis to do so? Consider the property of Strict Order Preservation, which says that whenever a claims problem \({\mathcal {C}}\) is such that agent i has a claim that is strictly greater than the claim of agent j, the amount allotted to i should be strictly greater than the amount allotted to j. Strict Order Preservation seems to be not too far removed from the content of FC (ii) and certainly to be in its spirit: remember that Broome (1990, p. 95) writes that ‘stronger claims require more satisfaction than weaker ones’ which is tantamount to Strict Order Preservation. Moreover, P does satisfy Strict Order Preservation, whereas AP does not. The former is immediate from its definition, the latter is readily verified by considering a claims problem in which there are two agents that have unequal claims which both exceed the available estate. The following lemma, which will be exploited later on, attests that AP only violates Strict Order Preservation in claims problems that satisfy the aforementioned condition.
Lemma 1
Let (E, N, c) be a claims problem and let \(k, l \in N\) be such that \(c_k < c_l\). Then, \(AP_k(E, N, c) < AP_l(E, N, c)\) if and only if \(c_k < E\).
Proof
See the appendix. \(\square \)
We use Lemma 1 in the proof of the following proposition, where we present a division rule—called the Composite Proportional rule for reasons that will become apparent—that satisfies Efficiency, Equal Treatment of Equals, Self-Duality and Strict Order Preservation, but that does not coincide with P.
Proposition 1
Efficiency, Equal Treatment of Equals, Self-Duality and Strict Order Preservation do not characterize P.
Proof
With \(L = \sum _{i \in N} c_i - E\) the aggregate loss, the Composite Proportional rule CP is defined as follows.
$$\begin{aligned} CP(E, N, c) = {\left\{ \begin{array}{ll} P(E, N, c) &{} \exists \ k, l \in N : c_k < c_l \text { and } (c_k \ge E \text { or } c_k \ge L); \\ AP(E, N, c) &{} \text {otherwise} \end{array}\right. } \end{aligned}$$
That rule CP satisfies Efficiency and Equal Treatment of Equals follows immediately from the fact that P and AP do. To see that R satisfies Self-Duality observe that CP is defined in such a way that for each claims problem (E, N, c) either CP applies P to both (E, N, c) and (L, N, c) or CP applies AP to both (E, N, c) and (L, N, c). And so, as both P and AP are Self-Dual, so is CP. To see that CP satisfies Strict Order Preservation, observe that it follows from Lemma 1 that CP applies P—which satisfies Strict Order Preservation—to a claims problem when applying the AP rule would result in a violation of Strict Order Preservation. Finally, observe that R proposes to apply AP to Bankruptcy and that P and AP do not coincide on Bankruptcy. Hence CP does not coincide with P. \(\square \)
Proposition 1 shows that the explication of FC in terms of Efficiency, Equal Treatment of Equals, Self-Duality and Strict Order Preservation does not imply P. We will now go on to argue that no (satisfactory) explication of FC can imply P, since that is prevented by the type of properties that are referred to by FC.
Limits to FC explications: relational properties\(^\star \)
How might Curtis react to Proposition 1? While we do not know that, the following reaction seems natural.
Reaction Proposition 1 crucially relies on your rule CP. However, rule CP is ad hoc, as it proposes to apply P or AP, depending on whether some cooked up formal condition obtains. A minimal requirement on a fair division rule is exactly that it is not ad hoc, or more positively, that it exhibits something like ‘regularity’ or ‘predictability’. The ad hoc character of CP reveals that is not a fair division rule. P is vastly superior.
Indeed, CP has an ‘ad hoc character’ and we sympathise with the thought that this is incompatible with fairness. But then, we may ask, how can we find out whether a given rule is ‘ad hoc’ in this way? Here is a proposal. A division rule r satisfies Estate Continuity
Footnote 5 just in case small changes in the estate do not lead to large changes in the allocation vector that is proposed by r. The ad hoc character of CP can be explained by its violation of Estate Continuity, and this violation can be illustrated as follows. Suppose that Ari, Betty, and Claire have claims of 800, 400, and 200 with respect to an estate of 400.1. Inspection reveals that rule CP proposes to apply AP to this claims problem, resulting in an allocation of \(160.\bar{1}\) for Ari. However, if the estate would be slightly lowered to 400 (leaving the claims as they are), CP proposes to apply P, which results in an allocation of \(228.\bar{6}\) for Ari. Hence, a small change in the estate results in a large change in the awards vector, which shows that CP violates Estate Continuity.
By appealing to Estate Continuity, we may thus account for the ad hoc character of CP and, as P satisfies Estate Continuity, this property can be used to distinguish P from CP. Moreover, one may even go further and argue that Estate Continuity is a property that any fair division rule should have. One may, but Curtis cannot. Or so we will argue. It is not only hard to see how, intuitively, Estate Continuity is related to FC, we will also show that FC does not give us access to relational properties such as Estate Continuity; all that FC has access to are punctual properties.
The distinction between punctual and relational properties is due to Thomson (2012). To illustrate this distinction, we contrast the property of Strict Order Preservation, as discussed before, with that Strict Claim Monotonicity. When two claims problems \({\mathcal {C}}\) and \({\mathcal {C}}'\) are exactly alike, except for the fact that there is exactly one agent, say i, who has a strictly larger claim in problem \({\mathcal {C}}'\) than in \({\mathcal {C}}\), Strict Claim Monotonicity says that agent i should receive strictly more in problem \({\mathcal {C}}'\) than in problem \({\mathcal {C}}\). In some sense, both Strict Order Preservation and Strict Claim Monotonicity can be said to give substance to the thought that ‘stronger claims require more satisfaction’. In contrast to Strict Order Preservation though, Strict Claim Monotonicity allows one to distinguish P from our CP rule, as the following proposition attests.
Proposition 2
P does and CP does not satisfy Strict Claim Monotonicity.
Proof
See Appendix. \(\square \)
Now given that the substance of Strict Claim Monotonicity is not too far removed from that of Strict Order Preservation, and given that the latter property may legitimately serve in an FC explication, can’t Curtis appeal to Strict Claim Monotonicity in order to distinguish P from CP? No he cannot, for although the substance of Strict Claim Monotonicity may not be too far removed from that of Strict Order Preservation, it is a property of a different type.
Strict Order Preservation is a punctual property. In order to satisfy a punctual property, a division rule has to respect a certain condition in every claims problem. Now a division rule satisfies Strict Order Preservation just in case, in every claims problem ‘strictly higher claims receive strictly more satisfaction’. Hence, Strict Order Preservation is a punctual property. Strict Claim Monotonicity is a relational property. In order to satisfy a relational property, a division rule has to respect a certain condition pertaining to each tuple of claims problems that are related in a certain way. Now a division rule satisfies Strict Claim Monotonicity just in case, for every pair of claims problems
\({\mathcal {C}}\)
and
\({\mathcal {C}}'\)
that are related as described above, agent i receives strictly more in \({\mathcal {C}}'\) than in \({\mathcal {C}}\). Hence, Strict Claim Monotonicity is a relational property.
FC can only be explicated by punctual properties as both FC (i) and FC (ii) specify what a fair allocating agent must do in every claims problem. In particular, FC(ii) specifies that in order to be fair, an allocating agent must ‘ensure that [in every claims problem] each claim is treated equally’. As FC can only express punctual properties, we take it that it is reasonable to require, of an explication of FC, that it only appeals to punctual properties: call this the FC Explication Requirement. And so, according to the FC Explication Requirement, no explication of FC can appeal to Strict Claim Monotonicity or Estate Continuity. Thus, although these properties allow one to separate P from CP, Curtis cannot justify this separation by appealing to FC.
Just like Strict Order Preservation, Efficiency and Equal Treatment of Equals are clear instances of punctual properties. Moreover, these three properties are not only of the right type to serve in an explication of FC, but also their substance is closely related to that of FC. Hence, their occurrence in an FC explication is unproblematic. But what about Self-Duality? As discussed in Sect. 2.2, one may argue that Self-Duality can be related to FC(ii), and we considered FC explications that appealed to Self-Duality. However, the FC Explication Requirement rules out such appeal. Prima facie though, there seems to be some leeway to argue that Self-Duality is a punctual property. For don’t we say, informally, that Self-Duality requires that, in every claims problem, awards and losses are to be treated symmetrically? And don’t we say, formally, that Self-Duality requires that in every claims problem, applying a division rule r and its dual rule \(r^\star \) result in the same allocation? The problem with such informal paraphrasing of Self-Duality is that it is too imprecise to be helpful. For what does it mean to treat awards and losses symmetrically? To say that it means that r and \(r^\star \) must coincide in every claims problem serves as a legitimate answer to that question. However, that answer does not allow one to conclude that Self-Duality is a punctual property. Indeed, the application of \(r^\star \) to (E, N, c) is defined in terms of the application of r to (L, N, c), which is another claims problem that is related to (E, N, c). The following definition of Self-Duality lays bare its relational nature.
Definition 3
(Self-Duality) Let (E, N, c) and \((E',N',c')\) be claims problems that are related as follows:
$$\begin{aligned} E' = \sum _{i \in N} c_i -E, \quad N' = N, \quad c' = c \end{aligned}$$
(1)
A division rule r satisfies Self-Duality iff for any two claims problems that respect (1), we have that
$$\begin{aligned} r\left( E,N,c\right) = c - r\left( E',N',c'\right) \end{aligned}$$
(2)
Definition 3 shows that Self-Duality is a relational property: if claims problems are related as in (1)—the application condition of Self-Duality—then a division rule must, in order satisfy Self-Duality, treat them as in (2)—the treatment condition of Self-Duality. Being a relational property, the FC Explication Requirement precludes Self-Duality from figuring in an FC explication. Of course, the same goes for the other relational property that figured in Theorem 1: CompositionFootnote 6.
We investigated, by invoking the axiomatic approach, whether FC can be explicated in terms of properties of division rules and whether we could come up with an explication of FC that characterizes (or implies) P. We started out by considering a particular characterization of P due to Young (1988), viz. Theorem 1. As the substance of three of the four properties appealed to in this characterization seems not be too far removed from FC, Theorem 1 seems to be an attractive starting point for “FC explication investigations” such as ours. Although Young’s characterization of P is attractive in this sense, it must be noted that several characterizations of P have been given in the axiomatic literature. We feel that, of these characterizations, the one by Young is closest in spirit to FC. More importantly though, to the best of our knowledge, all characterizations of P that have been proposed in the literature exploit relational properties. Thus, no such characterization can, according to the FC Explication Requirement, serve as an adequate explication of FC. That is not to say that it is (logically) impossible to characterize P in terms of punctual properties only. Such is illustrated by the Trivial Proportionality property, which is defined as follows.
Definition 4
(Trivial Proportionality) A division rule r satisfies Trivial Proportionality just in case, in every claims problem (E, N, c):
$$\begin{aligned} r_i(E, N, c) = \frac{c_i}{\sum _{j \in N} c_j }\cdot E \qquad \text { for each } i \in N \end{aligned}$$
Indeed, Trivial Proportionality is a punctual property and P is the unique division rule that satisfies Trivial Proportionality. However, Trivial Proportionality can only be said to characterize P in a trivial and uninformative manner.
We think that our discussion in this section warrants the following conclusion: there cannot be a non-trivial axiomatic explication of FC that yields P in the above sense, since it will rely on axiomatic properties of a type that clearly go beyond FC.