The constrained equal loss rule in problems with

The constrained equal loss rule in problems with constraints and claims∗ Leticia Lorenzo† Facultade de CCEE e Empresariais. Departamento de Estat´ıstica. Universidade de Vigo. Lagoas - Marcosende s/n, 36310 - Vigo, Pontevedra, Spain.

Abstract In this paper we study how to distribute a resource among different agents, claiming on it, when there are some constraints in the problem according to the equal loss principle. We introduce one single-valued rule. Some properties of this rule are given as well as an axiomatic characterization.

∗ Financial

support from the Ministerio de Ciencia y Tecnologia and FEDER, and Xunta de Galicia through

grants BEC2002-04102-C02-01 and PGIDIT03PXIC30002PN are gratefully acknowledged. † E-mail

address: [email protected]; phone: +34 986 812443; fax: +34 986 812401.

1

1

Introduction

When a firm goes bankrupt, how should its liquidation value be divided among its creditors? How should an extra time be divided among the activities involved in a project without delaying it? Both problems have been studied in the literature. The first one is answered in the field of bankruptcy problems. A bankruptcy problem is a pure distribution problem in which the amount to divide is insufficient to cover the agents´ claims. They have been introduced and studied by O´Neill (1982) and Aumann and Maschler (1985) among others. The reader is referred to Thomson (2003) and Moulin (2002) for a complete survey. The second question arises from PERT problems, introduced by Berganti˜ nos and S´anchez (2002B). These problems are inspired by the Project Evaluation Review Technique, an operational research tool described in detail in Moder and Philips (1970). This tool is designed to schedule and coordinate the activities involved in a project. Some of these activities must be performed sequentially and others can be performed in parallel with other activities and this collection of serial and parallel tasks can be modeled as a network. Once the schedule is designed and the PERT time (the time required to finish the project) is computed, the coordinator identifies the activities that can be allowed more time if required without delaying the project and estimates the slack associated with each of them. At this point the PERT problem focuses on how to allocate the extra time among these activities without delaying the project in a ”well-behaved” manner. Berganti˜ nos and S´ anchez (2002A) introduce a general class of problems that contains as subsets bankruptcy and PERT problems and call them problems with constraints and claims (PCC). They study how the proportionality principle works on this new class of problems by proposing two different rules based on it: the proportional rule (weak Pareto optimal) and the extended proportional rule (Pareto optimal). They also characterize the extended rule. Later, Berganti˜ nos and Lorenzo (2004) center their work on defining allocation rules to solve any problem with constraints and claims following the equal award principle. In this paper we focus our study on equality but according to losses, regarding that no

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agent receives a negative amount. We define two single-valued rules: the constrained equal loss rule, a weakly Pareto optimal rule, and the extended constrained equal loss rule, a Pareto optimal rule. The object of this paper is to characterize the extended rule. There are several characterizations of the constrained equal loss rule in the bankruptcy literature: Herrero (2003) proved that the constrained equal loss rule is the only efficient rule satisfying symmetry, consistency, minimal rights first and composition down. Herrero and Villar (2001) characterized it by means of consistency, exclusion and composition down. Later, Herrero and Villar (2002) characterized the constrained equal loss rule in terms of independence of residual claims and composition down. Our aim was to extend these characterizations to problems with constraints and claims. For it, we adapt the definitions of the properties to the new model. Berganti˜ nos and S´anchez (2002A) show that the relations among properties in PCC are completely different from the relations of their counterparts in bankruptcy problems. For instance, even though, in bankruptcy problems, Pareto optimality is compatible with composition down, in the new model Pareto optimality becomes incompatible with this property. Thus we define a weaker property called limited composition down which is compatible with Pareto optimality. We find out that the two characterizations by Herrero and Villar can not be extended since the extended constrained equal loss rule fails to satisfy both exclusion and independence of residual claims. Finally, we extend the characterization by Herrero (2003) by means of Pareto optimality, symmetry, consistency, minimal rights first, limited composition down and we need to add a new property specially defined for PCC, lower bound requirement over subsets. Thus no characterization of the constrained equal loss rule can be extended to our model just by using the same properties as in bankruptcy. The paper is organized as follows. Section 2 introduces and motivates problems with constraints and claims. Section 3 defines different appealing properties of the allocation rules and finally section 4 is devoted to the rules following the equal loss principle centering the study on the extended constrained equal loss rule and its characterization.

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2

Problems with constraints and claims

Firstly we introduce some notation. Given a finite set N we denote by n the cardinality of N . Given x = (xi )i∈N , y = (yi )i∈N ∈ RN , x ≥ y means xi ≥ yi for all i ∈ N ; x > y means xi > yi for all i ∈ N and x + y = (xi + yi )i∈N . Given S ⊂ N , xS = (xi )i∈S ∈ RS , particularly x−i = xN \{i} . We S  p denote by P = {π1 , . . . , πp } a collection of subsets of N that covers N πi = N with i=1

cardinality p. Given M ⊂ N with M 6= ∅ we denote by PM the collection of subsets induced by P in M , i.e. PM = {π ∩ M : π ∈ P, π ∩ M 6= ∅}. Given Y ∈ RN we define the Pareto Boundary of Y as P B(Y ) = {y ∈ Y : if x ∈ RN , x ≥ y and x 6= y then x ∈ / Y} and the Weak Pareto Boundary as W P B(Y ) = {y ∈ Y : if x ∈ RN , x > y then x ∈ / Y} In this kind of problems the issue does not differ too much from bankruptcy (taxation) problems. We have several agents claiming on an homogeneous and infinitely indivisible resource. But now we assume that there are several subsets of agents, not necessarily disjoint, given by P and each subset can not be awarded more than a part of the resource available, as it is showed in the next definitions. Definition 1 A problem with constraints and claims (briefly PCC) is a 4-tuple (N, P, c, E) S where N = π is the set of agents, P is a collection of subsets that covers N , 0 ≤ c ∈ RN π∈P

is the vector that holds the claims of the agents and E = (Eπ )π∈P ∈ Rp , where Eπ represents an upper bound of a part of the resource available for the subset π. We will often write (c, E) instead of (N, P, c, E). We denote by G(N ) the class of PCC with set of agents N , and by G the class of all PCC. Definition 2 Given (c, E) ∈ G(N ) we define the set of feasible allocations of (c, E) by:     0 ≤ xi ≤ ci , for every i ∈ N   F (c, E) = x ∈ RN P    xi ≤ Eπ , for every π ∈ P  i∈π

4

and a rule is a map R that associates with each problem (c, E) ∈ G(N ) an allocation R(c, E) ∈ RN and where each Ri (c, E) represents the award received by agent i. O´Neill (1982) defines a bankruptcy problem as a 3-tuple (N, c, E) where N is the set of claimants, c denotes the vector of claims and E is the resource available. Berganti˜ nos and S´ anchez (2002B) define a PERT problem as a 3-tuple (G, (asi )i∈N , (psπ )π∈P ) where G is the directed graph that models the project, N holds the set of activities involved in it, P denotes the set of paths from the beginning till the end of the project and psπ (asi ) is the slack associated with path π (activity i). Both problems can be seen as particular cases of a problem with constraints and claims as the following table shows:

Bankruptcy problems

PERT problems

(N, c, E)

(G, (asi )i∈N , (psπ )π∈P ) N = {activities}

PCC

P = {N }

P = {paths}

(N, P, c, E)

c = (asi )i∈N E = (psπ )π∈N

3

Properties of the allocation rules

We devote this section to define some desirable properties of the rules. Most of them are well-known properties of the bankruptcy literature but adapted to this general framework. Although there is one property specifically defined for PCC. Consider an allocation rule R. Two properties about optimality are defined. Pareto Optimality (PO): for all (c, E) ∈ G(N ) we have that R(c, E) ∈ P B(F (c, E)) and Weak Pareto Optimality (WPO): for all (c, E) ∈ G(N ) it is verified that R(c, E) ∈ W P B(F (c, E)).

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It is logical to require that agents with equal claims, under the same constraints, receive equal awards. This property is called Symmetry (SYM): for all (c, E) ∈ G(N ), given i, j ∈ N such that ci = cj and i ∈ π ⇔ j ∈ π, then Ri (c, E) = Rj (c, E). A PCC (c, E) can be interpreted as solving many subproblems as subsets in P . And it would be desirable for an agent not to receive less than the minimum amount he is awarded in every subproblem in which he is involved. This is stated by the next property. Lower Bound Requirement over Subsets (LS): for all (c, E) ∈ G(N ) and every i ∈ N , it is verified that: Ri (c, E) ≥ min Ri (cπ , Eπ ) i∈π∈P

This property is trivially satisfied in bankruptcy problems (it states that R(c, E) ≥ R(c, E)), although in PCC there are rules which do not satisfy LS. We define the minimal right of an agent i as the minimum of the difference between the amount available, in every state in which agent i participates, and the sum of the claims of the other claimants joining that state when this difference is non-negative, and 0 otherwise. ( ( )) X mi (c, E) = max 0, min ci , Eπ − cj i∈π∈P

j∈π\{i}

It is interesting to require that the awards vector should be equivalently obtainable directly or by first assigning to each agent his minimal right, adjusting claims and states down by m(c, E), and finally applying the rule to divide the remainder.

We say that

a rule satisfies Minimal Rights First (MRF) if for all (c, E) ∈ G(N ) it is verified that P R(c, E) = m(c, E) + R(c0 , E 0 ) where c0 = c − m(c, E) and Eπ0 = Eπ − mi (c, E) for i∈π

all π ∈ P . Composition Down (CD), also known as Path Independence, states that if the estate value is re-evaluated and found to be worth less than initially thought, if we cancel the initial division and apply the rule to the revised problem or if we consider the initial awards as claims on the revised value and apply the rule to the problem so defined, the awards vectors obtained in both ways of proceeding should coincide. Given (c, E) ∈ G(N ) and E 0 ∈ Rp such that 0 ≤ E 0 < E then R(c, E 0 ) = R(R(c, E), E 0 ). 6

We will denote by Limited Composition Down (LCD) the property composition down restricted to problems with P = {N }. Consistency says that if some agents leave with their awards, and the PCC is reevaluated from the point of view of the remaining agents, the rule should allocate to these agents the same awards as those obtained in the initial problem. Consistency (CONS): for all ∗ ∗ (N, P, c, E) ∈ G(N ), given M ⊂ N , M 6= ∅ and (M, PM , cM , EM ) ∈ G(M ) where Eπ∩M = P Eπ − Ri (c, E) for all π ∩ M ∈ PM then i∈π∩(N \M ) ∗ Ri (M, PM , cM , EM ) = Ri (N, P, c, E) for all i ∈ M.

Remark 1 Berganti˜ nos and S´ anchez (2002A) show that the relation between these properties changes completely when defined in this general framework. In particular they prove that Pareto optimality is incompatible with composition up. Following a similar argument it can be proved that Pareto optimality is also incompatible with composition down. Thus PCC are not an straight forward extension of bankruptcy problems.

4

Rules according to the equal loss principle

In this section we introduce two single-valued rules based on the constrained equal loss principle. This principle states that all agents suffer the same loss, subject to the condition that no creditor ends up with a negative award. The definition of the constrained equal loss rule is very similar to its definition in bankruptcy problems and in order to obtain the extended constrained equal loss rule we apply the equal loss principle repeatedly until no agent can improve his award. The constrained equal loss rule (briefly CEL) divides equally the losses, provided no agent receives a negative transfer: CELi (c, E) = ci − min{ci , λ} = max{0, ci − λ} where λ = min {λ0 : (ci − min{ci , λ0 })i∈N ∈ F (c, E)}. It is easy to prove that when c ∈ F (c, E), λ = 0. Otherwise λ can be computed in the following way: λ = max {λπ : π ∈ P } 7

where λπ is such that

P

(ci − min{λπ , ci }) = Eπ .

i∈π

This rule was defined in S´ anchez (1999). She proved that CEL satisfies WPO and there always exists a part of the resource available which is completely allocated between P its claimants, i.e., there exists π0 ∈ P such that CELi (c, E) = Eπ0 . CEL also satisfies i∈π0

SYM and CD. Nevertheless it fails MRF, CONS and PO. So we will extend this rule such that it satisfies PO, just by applying it repeatedly, according to the following process, until no agent can improve his award. • step 1. (N 1 , P 1 , c1 , E 1 ) = (N, P, c, E). For any i ∈ N we compute CELi (c1 , E 1 ) = c1i − min{c1i , λ1 } where λ1 = max{λπ : π ∈ P }. Assume that we have already calculated (N s , P s , cs , E s ) and CEL(cs , E s ) for any s ≤ t. • step t+1. Let (N t+1 , P t+1 , ct+1 , E t+1 ) be defined by:    ∃ ε > 0 such that t t+1 – N = i∈N   (CELi (ct , E t ) + ε, CEL−i (ct , E t )) ∈ F (ct , E t )  – P t+1 = π t ∩ N t+1 : π t ∈ P t , π t ∩ N t+1 6= ∅ = PNt t+1 ,

   ,  

– ct+1 = (cti − CELi (ct , E t ))i∈N t+1 = (min {cti , λt })i∈N t+1 and t – Eπt+1 t+1 = Eπ t −

CELi (ct , E t ) for any π t+1 = π t ∩ N t+1 ∈ P t+1 .

P i∈π t

 For any i ∈ N t+1 we compute CELi (ct+1 , E t+1 ) = ct+1 − min ct+1 , λt+1 . i i This process ends when there exists an stage T where N T 6= ∅ but N T +1 = ∅. Since CEL satisfies WPO, it is always verified that N t+1

N t and hence the process described

above will end in a finite number of steps T . For every i ∈ N there also exists an stage Ti / N Ti +1 . Then for each i ∈ N we define the extended constrained such that i ∈ N Ti but i ∈ equal loss rule ECEL as ECELi (c, E) =

Ti X t=1

8

CELi (ct , E t )

Remark 2 Notice that if during the process above there is an stage t where λt = 0 then the process ends at stage t, ct ∈ F (ct , E t ) and every agent involved in that stage is awarded all his claim. Proposition 1 Given (c, E) ∈ G(N ) it is verified that: 1. λt+1 < λt for every t = 1, . . . , T − 1.  2. ECELi (c, E) = ci − min ci , λTi 3. pt+1 < pt for every t = 1, . . . , T − 1. 4. Given i ∈ N with ECELi (c, E) < ci there exists π ∈ P verifying i ∈ π, π ∩ N Ti 6= ∅, P π ∩ N Ti +1 = ∅ and ECELj (c, E) = Eπ . j∈π

Proof. Given (c, E) ∈ G(N ). 1. Let us suppose that λt ≤ λt+1 . Given the problem (ct+1 , E t+1 ) we know by the definition of CEL that λt+1 is the minimum feasible loss. We will show that λt is also feasible. Since ct+1 = min {cti , λt } i we have that P j∈π t+1

P

t (ct+1 − min{ct+1 j j , λ }) =

(min{ctj , λt } − min{min{ctj , λt }, λt }) = 0 ≤ Eπt+1 for all π t+1 ∈ P t+1

j∈π t+1

Thus λt ≥ λt+1 . Therefore λt = λt+1 and X


t+1 ct+1 − min ct+1 = 0 for all π t+1 ∈ P t+1 . j j ,λ

j∈π t+1 t+1 Since Eπt+1 ∈ P t+1 we can find a feasible loss λ < λt+1 and we have t+1 > 0 for all π

a contradiction with the fact that λt+1 is the minimum feasible loss in the problem (ct+1 , E t+1 ). Hence λt > λt+1 for every t = 1, . . . , T − 1. 2. Consider i ∈ N such that i ∈ N Ti but i ∈ / N Ti +1 . Thus ECELi (c, E) =

Ti X t=1

9


cti − min cti , λt

 We know that cti = min ct−1 , λt−1 . Following on with this argument we obtain that i n  cti = min ct−2 , λt−2 , λt−1 = . . . = min ci , i

min



k=1,...,t−1

λk

o

 and by proposition 1.1 cti = min ci , λt−1 , where we take λ0 = ci by the sake of convenience. Given this new definition of cti , we can rewrite the extended equal loss rule as Ti 
P min ci , λt−1 − min {ci , λt } = ECELi (c, E) =  t=1   min ci , λ0 − min ci , λTi = ci − min ci , λTi

3. We will prove that for any t = 1, . . . , T − 1 there always exists a subset π ∈ P such that π ∩ N t 6= ∅ but π ∩ N t+1 = ∅. We study what happens for t = 1 as the other cases are similar. By the definition of CEL there always exists π0 ∈ P such that

P

CELi (c1 , E 1 ) =

i∈π0

Eπ0 . Thus π0 ∩ N 2 = ∅. 4. Given i ∈ N with ECELi (c, E) < ci , suppose that for every π ∈ P such that i ∈ π, P π ∩ N Ti 6= ∅ and π ∩ N Ti +1 = ∅ it is verified that ECELj (c, E) < Eπ . Let us define j∈π

ε = min

  P    Eπ − ECELj (c, E) for every π ∈ P such        j∈π   that i ∈ π, π ∩ N Ti 6= ∅ and π ∩ N Ti +1 = ∅,       ci − ECELi (c, E)

>0

     

Due to the definition of ε we have that

CELi cTi , E Ti + ε, CEL−i (cTi , E Ti ) ∈ F (cTi , E Ti ) and this is a contradiction.

Proposition 2 The extended constrained equal loss rule satisfies WPO, PO, LCD, SYM, LS, CONS and MRF. 10

Proof. Of course it satisfies WPO and PO because of the definition. It also satisfies LCD because when P = {N }, ECEL(c, E) = CEL(c, E) and the constrained equal loss rule satisfies composition down (S´ anchez (1999)).

• ECEL satisfies SYM. Given agents i, j ∈ N such that ci = cj and i ∈ π ⇔ j ∈ π, we have to prove that ECELi (c, E) = ECELj (c, E). Since CEL satisfies symmetry, it is verified that CELi (c1 , E 1 ) = CELj (c1 , E 1 ).   Consider (N 2 , P 2 , c2 , E 2 ). We have that c2i = min c1i , λ1 = min c1j , λ1 = c2j . Moreover i ∈ π 2 ⇔ j ∈ π 2 . Thus we are again under symmetry conditions and therefore CELi (c2 , E 2 ) = CELj (c2 , E 2 ). Furthermore CELi (ct , E t ) = CELj (ct , E t ) for every t ≤ min{Ti , Tj }. There is only rest to prove that Ti = Tj . Suppose Ti < Tj ≤ T . Then ECELi (c, E) < ci . By proposition 1.4 there exists π ∈ P P ECELk (c, E) = Eπ . Since such that i ∈ π with π ∩ N Ti 6= ∅, π ∩ N Ti +1 = ∅ and k∈π

j ∈ π, Tj = Ti which is a contradiction. Assuming Tj < Ti ≤ T we will reach another contradiction. Thus Ti = Tj and hence ECELi (c, E) = ECELj(c, E). • ECEL satisfies LS. Given an agent i ∈ N we will prove that n o ECELi (c, E) = ci − min{ci , λTi } ≥ min ECELi (cα , Eα ) = ci − min ci , max λα i∈α∈P

i∈α∈P

Given an agent i ∈ N with Ti = 1, we know that ECELi (c, E) = ci − min{ci , λ1 }. If λ1 = 0, by remark 2, c ∈ F (c, E), ECELi (c, E) = ci and LS holds trivially. So we will assume that λ1 > 0. Then ECELi (c, E) < ci . By proposition 1.4 there P exists π ∈ P with i ∈ π such that π ∩ N 1 6= ∅, π ∩ N 2 = ∅ and ECELj (c, E) = j∈π
P cj − min{cj , λ1 } = Eπ where λ1 = λπ = max{λα }. Thus α∈P

j∈π

n o  ci − min ci , λ1 = ci − min ci , max λα . i∈α∈P

Given an agent i ∈ N with Ti = 2. By proposition 1.2 ECELi (c, E) = ci − min{ci , λ2 }. Again if λ2 = 0, ECELi (c, E) = ci and LS holds trivially. Thus we will assume λ2 > 0 11

and hence ECELi (c, E) < ci . By proposition 1.4 there exists π ∈ P with i ∈ π such P that π ∩ N 2 6= ∅, π ∩ N 3 = ∅ and ECELj (c, E) = Eπ . Therefore j∈π

X

X


cj − min{cj , λ1 } +


cj − min cj , λ2 = Eπ

j∈π∩N 2

j∈π∩(N 1 \N 2 )

Taking into account the equation above and the following X

(cj − min {cj , λπ }) +

X

(cj − min {cj , λπ }) = Eπ

j∈π∩N 2

j∈π∩(N 1 \N 2 )

 we know that cj − min cj , λ1 ≤ cj − min{cj , λπ } for all j ∈ π ∩ (N 1 \N 2 ) because λ1 = max{λα } and therefore α∈P

X


cj − min cj , λ2 ≥

j∈π∩N 2

X

(cj − min {cj , λπ }) .

j∈π∩N 2

n o  Hence ci − min ci , λ2 ≥ ci − min {ci , λπ } ≥ ci − min ci , max λα . i∈α∈P

Using similar arguments to those used before we can conclude that for any agent i ∈ N with Ti = 3, . . . , T , ECELi (c, E) ≥ min ECELi (cα , Eα ). Hence ECEL satisfies LS. i∈α∈P

• ECEL satisfies CONS : Given (c, E) and M ⊂ N we have to prove that ∗ ECELi (N, P, c, E) = ECELi (M, PM , cM , EM ) for every i ∈ M ∗ where Eπ∩M = Eπ −

P

ECELj (c, E) for every π ∩ M ∈ PM .

j∈π∩(N \M )

When c ∈ F (c, E) the property holds trivially, thus we will restrict the proof to the case when c ∈ / F (c, E). We introduce some sets and notation that will be used during the proof: – M0 = {i ∈ M : ∃π ∩ M, i ∈ π and Eπ∩M = 0}. Notice that for every agent in M0 it is verified that ECELi (c, E) = 0. 1 – TM = min {Ti : i ∈ M \M0 }.

 1 – M ∗ = i ∈ M0 : Ti < TM .  ∗ – Consider CELi (cM , EM ) = ci − min ci , λ1M . 12

 1 We will prove that λ1M = max∗ λTM , ci . i∈M

 1 1 If max∗ λTM , ci = 0 it is verified that ci = 0 for all i ∈ M ∗ and λTM = λT = 0. i∈M

By remark 2 we know that cT ∈ F (cT , E T ) and ECELi (c, E) = ci for all i ∈ N t . It is trivial to prove that cM ∈ F (cM , EM ) and therefore ECELi (cM , EM ) = ci = ECELi (c, E) for all i ∈ M . Thus in order to avoid trivial situations we will assume  1 that max∗ λTM , ci > 0. i∈M

 1 First we will prove that max∗ λTM , ci is a feasible loss, i.e. i∈M

X 

 n  1 o ∗ cj − min cj , max∗ λTM , ci ≤ Eπ∩M i∈M

j∈π∩M

1

1 . By proposition 1.1 λTi ≤ λTM . Moreover, for Notice that for all i ∈ M \M ∗ , Ti ≥ TM n  1 o every agent i ∈ M ∗ we know that ECELi (c, E) = 0 = ci − min ci , max∗ λTM , ci . i∈M

Thus  P j∈π∩M

 n  1 o P cj − min cj , max∗ λTM , ci ≤ ECELi (c, E) ≤ i∈M j∈π∩M P ∗ Eπ − ECELi (c, E) = Eπ∩M j∈π∩(N \M )

 1 As long as λ1M is the minimum feasible loss, λ1M ≤ max∗ λTM , ci . i∈M

Reached this point we need to consider two different cases:  1 1 1 – max∗ λTM , ci = λTM > 0. Consider i ∈ M \ M0 such that Ti = TM : i∈M

1

ECELi (c, E) = ci − min{ci , λTM } < ci 1

By proposition 1.4 there exists π ∈ P with i ∈ π such that π ∩ N TM 6= ∅, 1

π ∩ N TM +1 = ∅ and X

ECELj (c, E) = Eπ ⇒

j∈π

X

∗ ECELj (c, E) = Eπ∩M .

j∈π∩M

Given j ∈ π ∩ (M \ M0 ): j ∈ π ⇒ Tj ≤

1 TM

  

 1  j ∈ M \ M0 ⇒ T j ≥ T M Given j ∈ π ∩ M0 , we know that ECELj (c, E) = 0 13

1

⇒ ECELj (c, E) = cj − min{cj , λTM }

1

1

1 ∗ If Tj ≥ TM ⇒ λTj ≤ λTM and cj − min{cj , λTM } = 0 1

1

1 ∗ If Tj < TM ⇒ j ∈ M ∗ and cj − min{cj , λTM } = 0 because λTM ≥ max∗ {ci }. i∈M

Thus

P

ECELj (c, E) =

j∈π∩M



P

 1 cj − min cj , λTM

j∈π∩M



∗ = Eπ∩M .

1

1 Hence λ1M = λTM , M 1 \ M 2 = {i ∈ M : Ti = TM } ∪ M0 and

ECELi (cM , EM ) = ECELi (c, E) for all i ∈ M 1 \ M 2  1 – max∗ λTM , ci = ci∗ > 0. We have just proved that ci∗ ≥ λ1M . Let us suppose i∈M

that ci∗ > λ1M . If we compute the award received by agent i∗ we obtain that ∗ CELi∗ (cM , EM ) = ci∗ − min{ci∗ , λ1M } > 0. While there exist π ∩ M ∈ PM such ∗ = 0 which is a contradiction. Thus ci∗ ≤ λ1M and that i ∈ π ∩ M and Eπ∩M

therefore ci∗ = λ1M . It is to notice that in this case M 1 \ M 2 = M0 and again ∗ ) = 0 = ECELi (c, E) for all i ∈ M 1 \ M 2 ECELi (cM , EM

2 2 ) where: , c2M , EM From now on, we consider the problem (M 2 , PM  1   M \ ({i ∈ M : Ti = T 1 } ∪ M0 ) when λ1 = λTM M M 2 – M =   M \ M0 when λ1 = ci∗ M

2 – PM = PM 2 ,

– c2i = ci − CELi (cM , EM ) = min{ci , λ1M } and 2 ∗ – Eπ∩M 2 = Eπ∩M −

P

CELj (cM , EM )

j∈π∩M

2 2 2 It is remarkable that in this problem Eπ∩M ∈ PM . 2 > 0 for all π ∩ M

 2 Consider CEL(c2M , EM ) = c2i − min c2i , λ2M .  2 2 2 We will prove that λ2M = λTM where TM = min Ti : i ∈ M 2 . If λTM = 0 by remark 2 2

2

2

2

we know that cTM ∈ F (cTM , E TM ) and ECELi (c, E) = ci for all i ∈ N TM . It is easy 2 to prove that in this case c2M ∈ F (c2M , EM ) and therefore ECELi (cM , EM ) = ci = 2

ECELi (c, E) for all i ∈ M 2 . Thus we will assume that λTM > 0.

14

2

First we will prove that λTM is a feasible loss i.e. 

X

 2 2 2 2 ∈ PM . c2j − min{c2j , λTM } ≤ Eπ∩M 2 for all π ∩ M

j∈π∩M 2

We can rewrite this expression taking into account the definitions above 

 2 min{cj , λ1M } − min{min{cj , λ1M }, λTM } ≤ j∈π∩M 2
P P ∗ − CELj (cM , EM ) = Eπ∩M − cj − min{cj , λ1M } . P

∗ Eπ∩M

j∈π∩M

j∈π∩M

1

2

By proposition 1.1 λ1M ≥ λTM > λTM , so the inequality above is equivalent to 

 2 min{cj , λ1M } − min{cj , λTM } ≤ j∈π∩M 2

P P cj − min{cj , λ1M } − cj − min{cj , λ1M } . P

∗ Eπ∩M −

j∈π∩M 2

j∈π∩(M \M 2 )

For all j ∈ π ∩ (M \ M 2 ) we have just proved that cj − min{cj , λ1M } = ECELj (c, E) thus 
P 2 min{cj , λ1M } − min{cj , λTM } + cj − min{cj , λ1M } = 2 2 j∈π∩M j∈π∩M  P  P 2 ∗ TM ∗ ECELj (c, E) = Eπ∩M cj − min{cj , λ } ≤ Eπ∩M − 2. P



j∈π∩M 2

j∈π∩(M \M 2 )

2

So proving the feasibility of λTM is equivalent to prove that P j∈π∩M 2



 2 ∗ cj − min{cj , λTM } ≤ Eπ∩M 2

2 for all π ∩ M 2 ∈ PM . 2

2 2 Given j ∈ π∩M 2 , due to the definition of TM , Tj ≥ TM . By proposition 1.1 λTj ≤ λTM . 2

Therefore ECELj (c, E) ≥ cj − min{cj , λTM } and  P 2 cj − min{cj , λTM } ≤ ECELj (c, E) ≤ j∈π∩M 2 j∈π∩M 2 P ∗ Eπ − ECELj (c, E) = Eπ∩M 2 P



j∈π∩(N \M 2 )

2

Since λ2M is the minimum feasible loss it is verified that λ2M ≤ λTM .

15

2

2 Consider an agent i ∈ M 2 such that Ti = TM then ECELi (c, E) = ci −min{ci , λTM } < 2

2

ci . By proposition 1.4 there exists π ∈ P such that i ∈ π, π ∩ N TM 6= ∅, π ∩ N TM +1 = ∅ P P ∗ and ECELj (c, E) = Eπ . In particular ECELj (c, E) = Eπ∩M 2. j∈π∩M 2

j∈π

Given j ∈ π ∩ M 2 2 j ∈ π ⇒ Tj ≤ TM

  

2 j ∈ M 2 ⇒ T j ≥ TM

 

X

Since

P

X

ECELj (c, E) =

j∈π∩M 2



2

⇒ ECELj (c, E) = cj − min{cj , λTM } and



 2 ∗ cj − min{cj , λTM } = Eπ∩M 2

j∈π∩M 2

 2 ∗ 2 cj − min{cj , λ2M } ≤ Eπ∩M we have that λTM ≤ λ2M . 2 for all π ∩ M

j∈π∩M 2 2 Hence λ2M = λTM

and ECELi (cM , EM ) = ECELi (c, E) ∀ i ∈ M 2 \ M 3 .

Applying the same argument consecutively, we can conclude that ECEL satisfies CONS. • It satisfies MRF. Given (c, E) ∈ G(N ) we have to prove that ECEL(c, E) = m(c, E) + ECEL(c0 , E 0 )  n o P where mi (c, E) = max 0, min ci , Eπ − cj is the minimal right for agent i∈π∈P j∈π:j6=i P i, c0i = ci − mi (c, E) and Eπ0 = Eπ − mj (c, E). j∈π

First we will prove that ECELi (c, E) ≥ mi (c, E) for all i ∈ N . It is trivial when ECELi (c, E) = ci or when mi (c, E) = 0. Moreover, when mi (c, E) = ci this implies that cπ ∈ F (cπ , Eπ ) for all π ∈ P : i ∈ π and since ECEL satisfies PO we have that ECELi (c, E) = ci = mi (c, E). Thus we will prove that ECELi (c, E) ≥ mi (c, E) when ECELi (c, E) < ci and 0 < mi (c, E) < ci . Notice that the last inequality implies that n o P mi (c, E) = min Eπ − cj . i∈π∈P

j∈π:j6=i

Since ECELi (c, E) < ci , by proposition 1.4, there exists π ∈ P such that i ∈ π, P π ∩ N Ti 6= ∅, π ∩ N Ti +1 = ∅ and ECELj (c, E) = Eπ . j∈π

16

P P ECELi (c, E) = Eπ − ECELj ≥ Eπ − cj ≥ j∈π:j6=i j∈π:j6=i n o P min Eα − cj = mi (c, E) i∈α∈P

j∈α:j6=i

Hence min{ci , λTi } ≤ ci − mi (c, E) = c0i . Moreover, it is verified that min{c0i , λTi } = min{ci , λTi } for all i ∈ N : – If min{ci , λTi } = ci ≤ c0i then mi (c, E) = 0 and the equality holds. – If min{ci , λTi } = λTi ≤ c0i then min{c0i , λTi } = min{ci , λTi }.  Consider an agent i ∈ N such that Ti = 1. Thus ECELi (c, E) = ci − min ci , λ1 . If λ1 = 0, by remark 2, c ∈ F (c, E), ECEL(c, E) = c = m(c, E) and MRF holds trivially. So we will assume λ1 > 0.  Given the problem (c0 , E 0 ), consider CELj (c0 , E 0 ) = c0j − min c0j , λ01 . We will prove that λ01 = λ1 . Firstly we will show that λ1 is a feasible loss: 

P cj − min c0j , λ1 ≤ cj − min c0j , λTj = j∈α
j∈αP  P ECELj (c, E) ≤ Eα cj − min cj , λTj = P

j∈α

j∈α

and therefore P j∈α



P P c0j − min c0j , λ1 = cj − min c0j , λ1 − mj (c, E) ≤ j∈α j∈α P Eα − mj (c, E) = Eα0 j∈α

thus λ1 ≥ λ01 . Since λ1 > 0 we have that ECELi (c, E) = ci − {ci , λ1 } < ci . By proposition 1.4 there exists π ∈ P such that i ∈ π with π ∩ N 1 6= ∅, π ∩ N 2 = ∅ and P j∈π

P j∈π

ECELj (c, E) =

P j∈π



 P cj − min cj , λ1 = cj − min c0j , λ1 = Eπ . j∈π



P P c0j − min c0j , λ1 = cj − mj (c, E) − min c0j , λ1 = Eπ − mj (c, E) = Eπ0 j∈π

j∈π

17

Hence λ1 ≤ λ01 and therefore λ1 = λ01 . Thus all the agents in N 1 \ N 2 also leave the process in the first stage, in the problem   (c0 , E 0 ), and are awarded c0j − min c0j , λ1 = cj − mj (c, E) − min cj , λ1 . In particular for agent i we obtain: mi (c, E) + ECELi (c0 , E 0 ) = mi (c, E) + ci − mi (c, E)−   min ci , λ1 = ci − min ci , λ1 = ECELi (c, E). Consider an agent i ∈ N such that Ti = 2. Given M = N 2 , we define the problem (M, PM , cM , EM ) in the usual way. Since ECEL satisfies CONS, in order to prove that ECEL satisfies MRF we will prove that 0 ECELi (c, E) = mi (c, E) + ECEL(c0M , EM )

0 where c0M = (c0i )i∈M and Eπ∩M = Eπ0 −

P

ECELj (c0 , E 0 ).

j∈π∩(N \M )

We have just proved that ECELj (c, E) = mj (c, E) + ECELj (c0 , E 0 ) for every j ∈ N 1 \ N 2 = N \ M and therefore P 0 (ECELj (c, E) − mj (c, E)) = = Eπ0 − Eπ∩M j∈π∩(N P P \M ) P Eπ − mj (c, E) − ECELj (c, E) + mj (c, E) = j∈π j∈π∩(N \M ) P j∈π∩(N \M ) P P ∗ ECELj (c, E) − mj (c, E) = Eπ∩M − mj (c, E). Eπ − j∈π∩(N \M )

j∈π∩M

j∈π∩M

0 ) = c0i − min {c0i , λ02 }. Let us prove that λ02 = λ2 . Firstly we will Consider CEL(c0M , EM

prove that λ2 is a feasible loss. 

P cj − min c0j , λ2 ≤ cj − min c0j , λTj = j∈π∩M 
j∈π∩M P P ∗ cj − min cj , λTj = ECELj (c, E) ≤ Eπ∩M , P

j∈π∩M

j∈π∩M

hence P j∈π∩M



P c0j − min c0j , λ2 = cj − mj (c, E) − min c0j , λ2 ≤ j∈π∩M P ∗ 0 Eπ∩M − mj (c, E) = Eπ∩M . j∈π∩M

18

Thus λ2 ≥ λ02 . In order to avoid trivial situations we assume λ2 > 0. Therefore ECELi (c, E) = ci −min{ci , λ2 } < ci . By proposition 1.4 there exists π ∈ P such that i ∈ π, π∩N 2 6= ∅, P P π ∩ N 3 = ∅ and ECELj (c, E) = Eπ which implies that ECELj (c, E) = j∈π

j∈π∩M

∗ Eπ∩M . Moreover, since M = N 2 and π ∩ N 3 = ∅ we have that


P ECELj (c, E) = cj − min cj , λ2 = j∈π∩M j∈π∩M 
P ∗ cj − min c0j , λ2 = Eπ∩M P

j∈π∩M

And therefore P j∈π∩M


∗ c0j − min c0j , λ2 = Eπ∩M −

P j∈π∩M

0 mj (c, E) = Eπ∩M

Hence λ2 ≤ λ02 and λ2 = λ02 .   0 ) = c0i − min c0i , λ2 = c0i − min ci , λ2 and Thus ECELi (c0M , EM 0 )= mi (c, E) + ECELi (c0 , E 0 ) = mi (c, E) + ECELi (c0M , EM   = mi (c, E) + c0i − min ci , λ2 = ci − min ci , λ2 = ECELi (c, E)

Following the same argument for every stage t = 3, . . . , T we can conclude that ECEL satisfies MRF.

Theorem 1 The extended constrained equal loss rule is the only rule satisfying PO, LS, SYM, MRF, CONS and LCD. Proof. By proposition 2 we know that ECEL satisfies PO, LS, SYM, MRF, CONS and LCD. Thus to prove the uniqueness let us suppose that there exists an allocation rule R satisfying PO, LS, SYM, MRF, CONS and LCD. When c ∈ F (c, E), by PO R(c, E) = c = ECEL(c, E). So we will assume c ∈ / F (c, E). We will prove the uniqueness by induction in the number of estates p = |P |. 19

• When p = 1, P = {N }: In this case ECELi (c, E) = CELi (c, E). We know, from Herrero (2003), that the constrained equal loss rule is the only efficient rule satisfying MRF, CD and SYM. As R satisfies PO, MRF, LCD and SYM we have that R(c, E) = CEL(c, E) = ECEL(c, E). • Let us assume that R(c, E) = ECEL(c, E) when p ≤ k − 1. • Given p = k, consider CELi (c, E) = ci − min{λ1 , ci } By LS, Ri (c, E) ≥ min Ri (cπ , Eπ ). We have just proved that when there is only i∈π∈P

one estate, R coincides with ECEL which also coincides with CEL. Thus Ri (c, E) ≥ n o min CELi (cπ , Eπ ) = min {ci − min{ci , λπ }} = ci − min ci , max λπ .

i∈π∈P

i∈π∈P

i∈π∈P



Since CELi (c, E) = ci − min ci , λ

1

where λ1 = max{λπ }. We have that π∈P

Ri (c, E) ≥ CELi (c, E) for every i ∈ N . By definition of CEL we know that there always exists π0 ∈ P such that P i∈π0

CELi (c, E) =

P


ci − min ci , λ1 = Eπ0

i∈π0

Therefore π0 ⊂ N 1 \ N 2 and CELi (c, E) = ECELi (c, E) for every i ∈ π0 . As Ri (c, E) ≥ CELi (c, E) for all i ∈ N and both rules satisfy PO, we can conclude that Ri (c, E) = CELi (c, E) = ECELi (c, E) for all i ∈ π0 . ∗ Consider M = N \{π0 } and the following problem (M, PM , cM , EM ). Since |PM | = k−1

by the induction supposition and taking into account that both rules satisfy CONS we have that Ri (c, E) = Ri (cM , EM ) = ECELi (cM , EM ) = ECELi (c, E) for every i ∈ M . Since Ri (c, E) = ECELi (c, E) for every i ∈ π0 we have that R(c, E) = ECEL(c, E).

20

Remark 3 Not every characterization of the constrained equal loss in bankruptcy problems can be extended to problems with constraints and claims by the aid of consistency and lower bound requirement over subsets. Herrero and Villar (2001) characterize the constrained equal loss rule with CONS, CD and Exclusion ( EXC). This property states that if an agent’s claim is smaller than the loss per capita, he should be awarded nothing. We can adapt this property to this new framework in the following way: we say that a rule R satisfies Lπ EXC if given an agent whose claim verifies that ci ≤ for all π ∈ P with i ∈ π, where |π| n P o Lπ = max 0, cj − Eπ . Then Ri (c, E) = 0. j∈π

But ECEL does not satisfy EXC: Consider N = {1, 2, 3, 4, 5}, P = {π1 , π2 , π3 } with π1 = {1, 2, 5}, π2 = {1, 3, 5}, π3 = {4, 5}, c = (1, 5, 5, 4, 9) and E = (11, 11, 5). The vector of losses per capita is L = (4, 4, 8) and for every π ∈ P with 1 ∈ π it is verified that 4 Lπ = . Although ECEL1 (c, E) = 1. Notice that not only he receives a positive c1 = 1 < |π| 3 award but he is awarded all his claim. Furthermore, Herrero and Villar (2002) again characterize the constrained equal loss rule in terms of CD and Independence of residual claims (IRC). This property states that if an agent’s claim is residual, i.e, the aggregate claim that results from deducting ci from all agents whose claims are higher than ci exceeds the worth of the available estate, this agent should be P awarded nothing. In PCC this property says that if Eπ ≤ max{0, cj − ci } for all π ∈ P j∈π

such that i ∈ π then Ri (c, E) = 0. But again ECEL fails to satisfy IRC. If we consider the example above, the claim of agent 1 is residual, although ECEL1 (c, E) = 1. This shows that in the new model the extended constrained equal loss rule is less detrimental with agents with small claims than it is in bankruptcy problems. To end this section we proof that the properties that characterize the extended constrained equal loss rule are independent but LCD. We do not know what happens with LCD. • Given (c, E) ∈ G(N ). We define the following rule R(c, E) = c. It clearly satisfies all properties but PO. • Given (c, E) ∈ G(N ) consider the extended proportional rule EP RO (Berganti˜ nos and 21

S´ anchez (2002A)). Of course it satisfies PO, LS, CONS, SYM and LCD although it does not satisfy MRF. Consider the problem (N, P, c, E) with N = {1, 2, 3}, P = {N }, c = (6, 6, 3) and E = 12. We have that m(c, E) + EP RO(c0 , E 0 ) = (3, 3, 0) + (2, 2, 2) = (5, 5, 2), although EP RO(c, E) = (4.8, 4.8, 2.4). • Given (c, E) ∈ G(N ) let us define a priority rule R(c, E) where agents get their award according to the order 1, 2, . . . , n.  n o X Ri (c, E) = max 0, min ci , Eπ − Rj (c, E) i∈π∈P

j∈π:j