Additive adjudication of conflicting claims

In a “claims problem” (O’Neill 1982), a group of individuals have claims on a resource but its endowment is not sufficient to honour all of the claims. We examine the following question: If a claims problem can be decomposed into smaller claims problems, can the solutions of these smaller problems be added to obtain the solution of the original problem? A natural condition for this decomposition is that the solution to each of the smaller problems is non-degenerate, assigning positive awards to each claimant. We identify the only consistent and endowment monotonic adjudication rules satisfying this property; they are generalizations of the canonical “constrained equal losses rule” sorting claimants into priority classes and distributing the amount available to each class using a weighted constrained equal losses rule. The constrained equal losses rule is the only symmetric rule in this family of rules.


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Additive adjudication of conflicting claims ∑ N z i = E and, for each i ∈ N , z i ≤ c i . We refer to z i as the award of claimant i and to c i − z i as her loss. Let Z(c, E) denote the collection of all allocations for (c, E). An adjudication rule is a function f recommending an allocation for each possible claims problem: for each N ∈ N and each (c, E) ∈ C N , f (c, E) ∈ Z (c, E).
For each rule f, its dual (Aumann and Maschler 1985) is the rule g defined by setting for each (c, E) ∈ C N ,

Rules
In this paper, we examine the relationship between additivity properties and two canonical rules attributed to medieval philosopher Maimonides (Aumann and Maschler 1985): the "constrained equal losses" and the "constrained equal awards" rules.
The constrained equal losses rule, denoted by CEL, equalizes the losses imposed on claimants subject to the constraint that no claimant receives a negative award: for each (c, E) ∈ C N and each i ∈ N, where λ ∈ ℝ + is chosen so as to satisfy ∑ j∈N max{0, c j − λ} = E. The constrained equal losses rule can be extended to allow for asymmetric treatment, by equalizing the weighted losses imposed on claimants (Moulin 2000). 3 The weighted constrained equal losses rule corresponding to a weights profile w ∈ ℝ A ++ , denoted by CEL w , is such that for each (c, E) ∈ C N and each i ∈ N where λ ∈ ℝ + is chosen so as to satisfy ∑ N max{0, c j − w j λ} = E. A weighted constrained equal losses rule can be further extended to include priority classes, whereby claimants in lower priority classes receive awards conditional on full compensation among those in higher classes (Moulin 2000). We refer to such a rule as priority-augmented weighted constrained equal losses rule, or PWCEL rule. Formally, a rule f is a PWCEL rule, if there is a partition of A into n ≤ |A| non-empty priority classes A 1 , … , A n and a weights profile w ∈ ℝ A ++ such that, for each N ∈ N and each (c, E) ∈ C N , f(c, E) can be computed sequentially as follows: In the special case where each of the sets A 1 , … , A n is a singleton, we refer to the resulting rule as a priority rule.
The constrained equal awards rule, denoted by CEA, equalizes awards subject to the constraint that no claimant receives more than her claim: for each (c, E) ∈ C N and each i ∈ N, where λ ∈ ℝ + is chosen so as to satisfy ∑ j∈N min{c j , λ} = E. Like the constrained equal losses rule, this rule can also be generalized to allow for weights and priority classes (Moulin 2000).

Axioms
We start recalling the two classical properties of adjudication rules, consistency and endowment monotonicity. Consistency is a basic property in the theory of distributive justice; it requires that if an allocation is considered desirable for a group of individuals, then it should remain so when restricted to each sub-group (Young 1987). More precisely, suppose that a rule is applied to settle a claims problem and a group of claimants is withdrawn along with their awards. If the situation is re-evaluated from the viewpoint of those who remain, in distributing the remaining endowment, a consistent rule assigns the same awards it did initially. Consistency: . Whereas consistency allows us to deduce that an allocation is desirable for each pair of individuals from its overall desirability, its converse allows us to deduce the desirability of an overall allocation from its desirability for each pair of individuals.
Converse consistency: The following two properties are standard: Endowment monotonicity: For each (c, E) ∈ C N and each E � ∈ [0, E],

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Additive adjudication of conflicting claims Endowment monotonicity implies endowment continuity. We say that a rule satisfies bilateral endowment monotonicity if it satisfies endowment monotonicity for the two-claimant case. 4 We now examine additivity in claims problems. 5 Additivity requires the overall allocation is invariant to whether two claims problems are solved independently or jointly, by aggregating each individual's claims on the aggregate endowment. Additivity: Despite appearing natural at first sight, the condition is demanding. Consider the example in the introduction stated formally: Then, the definition of an allocation and additivity imply that, for each rule f, f i (c * , E * ) = 1 and f j (c * , E * ) = 0. Bergantiños and Méndez-Naya (2001) and Bergantiños and Vidal-Puga (2004) use similar examples to show that, in fact, no rule satisfies additivity. 6 Another difficulty is that it requires ignoring an individual's claim by transferring it into a subproblem with a null-endowment.
To rule out pathological cases featuring arbitrary transfers of claims and endowments across the subproblems, we consider a weaker property: additivity holds conditional on all individuals receiving positive awards in each of the smaller problems.
Positive-awards-conditional additivity: For brevity, we refer to the above property as PAC additivity.

Results
The following lemma establishes that each PWCEL rule satisfies all of the properties that we will invoke. A proof can be found in the Appendix.
We can now state our result: Theorem 1 A rule satisfies consistency, endowment continuity, and PAC additivity if and only if it is a PWCEL rule.
The axioms in Theorem 1 are logically independent (see Table 1 in the Appendix). Each PWCEL rule is not only endowment continuous, but continuous in both the claims and the endowment (a standard property used by Young 1987). Moreover, since each PWCEL rule is endowment monotonic, and hence endowment continuous, our main result is a corollary of Theorem 1:

Corollary 1 A rule satisfies consistency, endowment monotonicity, and PAC additivity if and only if it is a PWCEL rule.
The basic equity condition in claims problems specifies that claimants with equal claims receive equal awards: Equal treatment of equals: For each (c, E) ∈ C N and each pair i, j ∈ N such that Since the only PWCEL rule satisfying equal treatment of equals is the constrained equal losses rule, the following is a corollary of Theorem 1 as well: Corollary 2 A rule satisfies consistency, endowment continuity, PAC additivity, and equal treatment of equals if and only if it is the constrained equal losses rule.
The axioms in Corollaries 1 and 2 are also logically independent (see the Appendix).

Duality
The following axiom is linked through duality to PAC additivity and is used in the axiomatic derivation of the PWCEL rules.
Positive-losses-conditional additivity: For brevity, we refer to the above property as PLC additivity.
We say that two properties are dual if whenever a rule satisfies one of them, its dual satisfies the other (Thomson and Yeh 2008).

Lemma 2 PAC additivity and PLC additivity are dual.
Proof Let f denote a rule satisfying PAC additivity and let g denote its dual rule. Let (c, E), (c � , E � ) ∈ C N be such that g(c, E) < c and g(c � , E � ) < c � . Thus, letting 1 3 Additive adjudication of conflicting claims Conversely, if a rules satisfies PLC additivity, then its dual satisfies PAC additivity.
Let g denote a rule satisfying PLC additivity and let f denote its dual rule.
Since g satisfies PLC additivity, The next lemma shows that PLC additivity implies that increasing the claims of a claimant does not change the outcome whenever no claimant achieved her claim.
Lemma 3 Let f denote a rule satisfying PLC additivity. Then, for each (c, E) ∈ C N and each A dual analysis of the results above concludes that (relying on Lemma 2), if we replace PAC additivity by PLC additivity, then we obtain the dual results of Theorem 1 and Corollaries 1 and 2.

Corollary 3 (Dual to Theorem 1). A rule is consistent, endowment continuous, and satisfies PLC additivity if and only if it is a PWCEA rule.
Analogously,

And,
Corollary 5 (Dual to Corollary 2). A rule satisfies consistency, endowment continuity, PLC additivity, and equal treatment of equals if and only if it is the constrained equal awards rule.
The axioms in Corollaries 3, 4, and 5 are logically independent. Their independence follows from the independence of the axioms in the original axiomatization.

Proofs
To prove Theorem 1, we proceed in two steps, corresponding to the following subsections. In the first step, we consider the implications of PAC-additivity and endowment continuity in two-claimant problems (Lemmata 4 and 5 below). Here we establish that, if a rule is endowment continuous and its dual satisfies PAC additivity, then it is either a weighted constrained equal awards or a priority rule (Lemma 6). The dual of a weighted constrained equal awards or a priority rule is a weighted constrained equal losses rule or a priority rule (Lemma 7).
The second step of the proof uses consistency to extend the two-claimant result to general claims problems. It relies on the fact that the PWCEL rules satisfy the property of "converse" consistency.

Two-claimant problems
The lemma below states that, for problems with two claimants, the additivity axioms jointly with endowment continuity imply endowment monotonicity.

Lemma 4 If a rule satisfies endowment continuity and either PAC or PLC additivity, then it is bilaterally endowment monotonic.
Proof Let g denote a rule satisfying endowment continuity and PLC additivity. Let i, j ∈ A and c ∈ ℝ {i,j} + . We first prove that, for each pair E, E � ∈ [0, c i + c j ], Let E and E ′ be as specified in (3) and let c � ∈ ℝ {i,j} Since g is endowment continuous, E i and E j are well defined. Without loss of generality, suppose that E j ≤ E i . We will prove that By way of contradiction, suppose that there is } is a continuous path in ℝ {i,j} + connecting g(c, E j ) and c, and containing g(c, E * ) , as illustrated by the thick curve in Fig. 1 Fig. 1. This contradicts (3). This contradiction establishes (4).
To conclude the proof, note that by (3), g i (c, ⋅) and g j (c, ⋅) are non-decreasing on Recall that the claimants i, j and the claims profile c were chosen arbitrarily, and thus g is bilaterally endowment monotonic.
Let f denote a rule satisfying endowment continuity and PAC additivity. Then, immediately, the dual of f satisfies endowment continuity and PLC additivity. By the argument above, the dual of f is bilaterally endowment monotonic. This immediately implies that f is bilaterally endowment monotonic. ◻ The next lemma shows that PLC additivity jointly with endowment monotonicity implies a linearity-type property (endowment linearity) whenever no claimant achieved her claim.
Lemma 5 Let g denote a rule satisfying bilateral endowment monotonicity and PLC additivity. Then, for each Proof Let g denote a rule satisfying bilateral endowment monotonicity and PLC additivity. By PLC additivity, for each k ∈ ℕ and each (c, By PLC additivity and (5), we have that, It follows that, as desired.
Case 2: ∈ [0, 1] ⧵ ℚ . Take an increasing sequence of rational numbers { t } and a decreasing sequence of rational numbers { t } , both converging to . By bilateral endowment monotonicity, for each t, The lemma below shows that if a rule satisfies bilateral endowment monotonicity and PLC additivity, then we obtain either a weighted CEA rule or a priority rule.

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Additive adjudication of conflicting claims Lemma 6 Let g denote a rule satisfying bilateral endowment monotonicity and PLC additivity. Then, for each pair i, j ∈ A , one and only one of the following statements is true: where λ ∈ ℝ + is chosen so as to satisfy Proof Let g denote a rule satisfying bilateral endowment monotonicity and PLC additivity and let i, j ∈ A . The proof consists of three claims.
Let (c, E) ∈ C {i,j} be such that 0 < g(c, E) < c and define = g j (c,E) g i (c,E) . By Lemma 5, for each ∈ (0, 1], } is a line with a constant slope of connecting g(c, 0) to g(c, E), as illustrated in Fig. 2

. Let
Since g is bilaterally endowment monotonic, g(c, ⋅) is continuous and thus, E i and E j are well defined. Without loss of generality, suppose that E j ≤ E i . By definition, for each E ′ < E j , g(c, E � ) < c . Thus, by Lemma 5, for each ∈ (0, 1), This implies that the set of points {x ∈ ℝ {i,j} + ∶ x = g(c, e), e ∈ [0, E j ]} is a line connecting g(c, 0) to g(c, E j ) . Since there is ∈ (0, 1) such that E = E j , the slope of this line is , as illustrated in Fig. 2. Thus, for each E � ∈ [0, E j ], By bilateral endowment monotonicity, the same is true for each This follows immediately from the assumption that there is no (c, E) ∈ C {i,j} such that 0 < g(c, E) < c and the fact that, since g is bilaterally endowment monotonic, g(c, ⋅) is continuous.
Claim 3 Let (c, E) ∈ C {i,j} . We have: We prove statement (i); the proofs of (ii) and (iii) are analogous. Let (c, E) ∈ C {i,j} and suppose that g i (c, E) = min{λ, c i } and g j (c, E) = min{ λ, c j } where . By Claim 1, Therefore, g recommends allocations for both claims profiles c and c ∧ c � following the same ray starting from the origin. Similarly, by Lemma 3, c ∧ c � ≤ c � implies g(c � , E �� ) = (c ∧ c � , E �� ) . By Claim 1, Combining the above claims establishes the Lemma.
◻ The lemma below shows that if a rule satisfies endowment continuity and PAC additivity, then we obtain either a weighted CEL rule or a priority rule.
Lemma 7 Let f denote a rule satisfying endowment continuity and PAC additivity. Then, for each pair i, j ∈ A , one and only one of the following statements is true: where λ ∈ ℝ + is chosen so as to satisfy max{0, c i − λ} + max{0, c j − λ} = E. (ii) There is k ∈ {i, j} such that, for each (c, E) ∈ C {i,j} , f k (c, E) = min{c k , E}.
A proof can be found in the Appendix.

Proof of Theorem 1
By Lemma 1, each PWCEL rule satisfies the axioms in Theorem 1. Conversely, let f denote a rule satisfying the axioms in Theorem 1. We will prove that f is a PWCEL rule. For

By Lemma 7, for each pair i, j ∈ A,
Throughout the rest of the proof, for each pair i, j ∈ A such that i ∼ j , we will use the notation ij for the corresponding parameter in (6). Completeness and transitivity of ≿ : Completeness follows immediately from Lemma 7. To establish transitivity, let i, j, k ∈ A be such that i ≿ j ≿ k . We need to show that i ≿ k . By way of contradiction, suppose that this is not true, so that k ≻ i . Let (c, E) ∈ C {i,j,k} be such that c i = 1∕ ij and c j = c k = E = 1 . Let x = f (c, E) . By consistency,

Thus 78
and, by the definition of ≻, We consider four cases: (a) i ≻ j ≻ k . By the definition of ≻ and (7), i ≻ j implies x j = 0 and j ≻ k implies x k = 0 . Since x i + x j + x k = 1 , x i = 1 , contradicting (10). Thus, i ≻ j ≻ k is not possible. (b) i ∼ j ∼ k . By the (9), x i = 0 . Thus, x j + x k = 1 . By (9), x j > 0 . By (8), x i > 0 , a contradiction. Thus, i ∼ j ∼ k is not possible. (c) i ∼ j ≻ k . By the definition of ≻ and (7), j ≻ k implies x k = 0 . Since By the definition of ≻ and (13), i ≻ j implies x j = 0 . Since x i + x j + x k = 1 , by (10), x k = 1 . By (9), x j > 0 , a contradiction. Thus, i ≻ j ∼ k is not possible.
Thus, indeed, i ≿ k , and ≿ is transitive.

Construction of the priority classes: Recursively define the subsets of A:
Since A is finite and ≿ is complete and transitive, there is n ≤ |A| such that A is partitioned by the sets A 1 , … , A n .

Construction of the weights:
Let m ∈ {1, … , n} and (c, E) ∈ C A m be such that, for each i ∈ A m , c i = 1 and E = 1 . Let r m = c − f (c, E) . Note that r m ≥ 0 . We next prove that Suppose, instead, that there is i ∈ A m such that r m i = 0 and let j ∈ A m ⧵ {i} . Then, f i (c, E) = 1 and f j (c, E) = 0 . By consistency, f i (c {i,j} , 1 + 0) = 1 and f j (c {i,j} , 1 + 0) = 0 . Thus, (ii) in Lemma 7 holds, implying that i ≻ j . However, by definition, for each pair i, j ∈ A m , i ∼ j . This contradiction establishes (11).
By (11), we can define a weights profile w ∈ ℝ A ++ such that, for each m ∈ {1, … , n} and each i ∈ A m , w i = r m i . We make an observation regarding the relationship between the coordinates of w corresponding to each agent. Suppose again that |A m | ≥ 2 and let i, j ∈ A m . Thus, i ∼ j . Thus, by (6), Concluding the proof: Let F denote the PWCEL rule specified by the partition A 1 , … , A n and weights profile w constructed above. We prove that f = F. Let N ∈ N , (c, E) ∈ C N , and x = f (c, E) . By consistency, Suppose that i, j ∈ N are such that i ∼ j . Then, by (6), By the definition of F, is chosen so as to satisfy max{0, c i − λ} + max{0, c j − ij λ} = x i + x j .

Thus, by (12),
Suppose that i, j ∈ N are such that i ≻ j . Then, by (ii) in Lemma 7, By the definition of F, F i (c {i,j} , x i + x j ) = min{c i , x i + x j } as well. Thus, By Lemma 1, F is conversely consistent. Thus, by (14) and (15) and since ≿ is complete, F(c, E) = x = f (c, E) . Since N ∈ N and (c, E) ∈ C N were chosen arbitrarily, F = f . ◻ Note that, as established in the Appendix, the axioms in Theorem 1 are logically independent.

Concluding remarks
We discuss here some axiomatizations that are related to other results in the literature. For instance, Theorem 3* in Herrero and Villar (2001) states that the constrained equal losses rule is the only one satisfying equal treatment of equals, composition down, and minimal rights first. The result in our Corollary 2 adds consistency and replaces composition down and minimal rights first by endowment continuity and PAC additivity. Similarly, Flores-Szwagrzak (2015) says that a rule satisfies consistency, composition down, and minimal rights first if and only if is a PWCEL rule. Our main result in Theorem 1 replaces once again composition down and minimal rights first by endowment continuity and PAC additivity in order to axiomatize the PWCEL rules.
Regarding the constrained equal awards, we can invoke duality to conjecture similar conclusions. For instance, a rule satisfies equal treatment of equals, composition up (dual to composition down), and claims truncation invariance (dual to minimal rights first) if and only if is a constrained equal awards (Dagan 1996). Similarly, Flores-Szwagrzak (2015) states that a rule satisfies consistency, composition up, and claims truncation invariance if and only if is a PWCEA rule. The result dual to Theorem 1 replaces composition up and claims truncation invariance by endowment continuity and PLC additivity in order to axiomatize the PWCEA rules. Finally, if we replace endowment continuity by endowment monotonicity in all our results mentioned above, we obtain the very same conclusions.
Interestingly, Flores-Szwagrzak et al. (2020) study a version of additivity where a problem can be decomposed into two smaller problems whenever i) all individuals receive positive awards in each of the smaller problems, and ii) these smaller problems keep the same claims vector.
Restricted additivity: For each c ∈ ℝ N + , and each pair E, E � ∈ [0, In contrast, PAC additivity does not impose the restriction of keeping the same claim vectors in the decomposed problems. Note that if we replace PAC additivity by restricted additivity, the main result in Theorem 1 does not hold anymore since solutions like the priority-augmented proportional rules would also satisfy consistency, endowment continuity, and restricted additivity.
We finally discuss a conjecture suggested to us by Youngsub Chun, Hervé Moulin, and José Zarzuelo. Consider the following property: for each pair (c, E), Clearly, this property is implied by both PAC and PLC additivity. Is the family of consistent and endowment continuous rules satisfying this property the union of the PWCEA and the PWCEL rules? The answer is no. For a counterexample, see the subsection Counterexample 1 in the Appendix.
The following rule satisfies consistency, PAC additivity, and equal treatment of equals but does not satisfy endowment continuity. Let |N| = n and ĉ = (9, 7, 1, … , 1) ∈ ℝ n . Let G such that for each N ∈ N and each (c, E) ∈ C N , When |N| = 2 7) and E = 1 . The rule coincides with the CEL otherwise. • If N ≠ {1, 2} , the rule G coincides with the CEL.
To see that G is not endowment continuous consider the following claims c = (9, 7, 1) and The rule H defined below satisfies endowment continuity, PAC additivity, and equal treatment of equals but does not satisfy consistency. Let H such that for each N ∈ N and each (c, E) ∈ C N .
For instance, let c = (1, 8, 9) and E = 3 . Thus, H ((1, 8, 9), 3) = (0, 0, 3) . By consistency, if the claimant number one leaves with her outcome, i.e., with 0, the rest of the claimants remain unaffected, but If we replace PAC additivity by PLC additivity, results dual to Theorem 1, and Corollaries 1 and 2 mentioned above hold too. Analogously, we show the independence of the axioms of these dual results below (see a summary in Table 2).
The PWCEA rule with non trivial priorities satisfies consistency, endowment continuity, and PLC additivity but does not satisfy equal treatment of equals. For instance, The constrained equal losses rule satisfies consistency, endowment continuity, and equal treatment of equals but does not satisfy PLC additivity. It is well-known that the CEL rule satisfies equal treatment of equals. We provide a numerical example showing that the CEL rule fails PLC additivity.
The rule H * satisfies endowment continuity, PAC additivity, and equal treatment of equals but does not satisfy consistency. For instance, let c = (1,8,9) and E = 6 .

Proof of Lemma 1
The PWCEL rules belongs to the wider family of consistent, continuous, and endowment monotonic rules characterized by Moulin (2000). An endowment monotonic and consistent rule is conversely consistent (Chun 1999); thus, each PWCEL rule is conversely consistent. We now prove that these rules satisfy PAC additivity.
Let f denote a PWCEL rule associated with the partition of A into n ≤ |A| priority classes A 1 , … , A n and the weights profile w ∈ ℝ A Let B 1 , … , B m denote non-empty and distinct elements of {N ∩ A 1 , … , N ∩ A n } such that B 1 corresponds to the N ∩ A t with the smallest index t, B 2 corresponds to the N ∩ A t with the second smallest index t, and so forth; moreover m is chosen so that the union of B 1 , … , B m is a partition of N. Thus, B 1 consists of the claimants in N with the highest priority, B 2 consists of the claimants in N with second highest priority, and so forth. Let B ≡ ⋃ m−1 l=1 B l . By the definition of a PWCEL rule, for each i ∈ B , H((8, 9), 6 − 1) = ( 5 2 , 5 2 ) ≠ (1, 4)

Proof of Lemma 7
Let f denote a rule satisfying PAC additivity and endowment continuity, let g denote its dual, and let i, j ∈ A . By Lemma 2, g satisfies PLC additivity. By Lemma 4, g is bilaterally endowment monotonic. Thus, by Lemma 6, one and only one of the following statements is true: (a) There is > 0 such that, for each (c, E) ∈ C {i,j} , g i (c, E) = min{c i , λ} and g j (c, E) = min{c j , λ} where λ ∈ ℝ + is chosen so as to satisfy min{c i , λ} + min{c j , λ} = E. (b) There is k ∈ {i, j} such that, for each (c, E) ∈ C {i,j} , g k (c, E) = min{c k , E}.
Suppose (a) is true. Let (c, E) ∈ C {i,j} . Then, by (a) and since g is the dual of f, where λ ∈ ℝ + is chosen so as to satisfy min{c i , λ} + min{c j , λ} = c i + c j − E . Rearranging, Thus, if (a) is true, (i) in Lemma 7 is true. Similarly, if the alternative and mutually exclusive statement (b) is true, (ii) in Lemma 7 is true. ◻

Counter example 1
There are rules that are consistent, endowment continuous, and satisfy property (16) that are neither a PWCEL nor a PWCEA rule. For example, let i ∈ A and define rule F as follows: Clearly, F satisfies consistency and endowment continuity. It remains to show that it satisfies property (16). Since the constrained equal awards rule satisfies the property and F coincides with it when claimant i is not present in the claims problem, there is nothing to show unless i is present. Let N ∈ N be such that i ∈ N and (c, E), (c � , E � ) ∈ C N be such that c > F(c, E) > 0 and c � > F(c � , E � ) > 0 . These inequalities imply that E > 0.5c i and E ′ > 0.5c ′ i . Thus, Since c > F(c, E) and c � > F(c � , E � ), Thus, since the constrained equal awards rule satisfies PLC-additivity, equals f i (c, E) = c i − min{c i , λ} = max{0, c i − λ} f j (c, E) = c j − min{c j , λ} = max{0, c j − λ} max{0, c i − λ} + max{0, c j − λ} = E.
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