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The dual serial cost-sharing rule
Mathematical Social Sciences 53 (2007) 150 – 163 www.elsevier.com/locate/econbase
The dual serial cost-sharing rule ☆ M. Josune Albizuri ⁎, José M. Zarzuelo Basque Country University, Faculty of Economics and Business Administration, Spain Received 23 December 2005; received in revised form 7 November 2006 Available online 23 January 2007
Abstract In this paper we present a cost-sharing rule for cost-sharing problems. This rule is called the dual serial cost-sharing rule because it prescribes the same allocations as the serial cost-sharing rule in the dual problem. We give two axiomatizations of this new rule, one of them by means of a consistency property. Moreover, we provide a mechanism to implement this rule. © 2006 Elsevier B.V. All rights reserved. Keywords: Cost sharing JEL classification: C7
1. Introduction In this paper we consider cost-sharing problems in which a group of n agents shares a joint process to produce a certain private good. Each one of them declares a demand q˜i of the good. The cost function is denoted C. A cost-sharing rule allocates the total production cost, i.e. C(∑iq˜i), among all the agents. In this work we study a cost-sharing rule that we have called the dual serial cost-sharing rule, because it prescribes the same allocations as the serial cost-sharing rule of Moulin and Shenker (1992) does in the dual problem. This dual rule is a particular case of the reverse serial costsharing mechanism defined by Tijs and Koster (1998). Moreover, there is a close relation between cost-sharing problems and rationing problems. Actually, in an excellent survey Moulin (2002) shows that the set of monotonic rationing methods is linearly isomorphic to that of additive costsharing rules. Then this author gives a list of rationing methods and cost-sharing rules matched by ☆
This research has been partially supported by the University of Basque Country (project 9/UPV 00031.321-15352/2003) and DGES Ministerio de Educación y Ciencia (project BEC2003-08182). ⁎Corresponding author. Lehendakari Agirre, 83, 48015 Bilbao, Spain. E-mail addresses: [email protected], [email protected] (M.J. Albizuri), [email protected] (J.M. Zarzuelo). 0165-4896/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2006.11.001
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this linear isomorphism. For instance, Moulin points out that the Uniform Gains method corresponds with the well known serial cost-sharing rule (Moulin and Shenker, 1992), meanwhile the Uniform Losses would give rise to the “dual” serial rule. The serial cost-sharing rule can be described as follows. When there are only two agents, this rule follows the standard principle: if both agents demand the same quantity they share equally the cost; and if one of them wants more, he is the only one that has to pay for it. It will be useful to give an alternative description of this procedure. That is, consider two agents i and j with q˜i ≤q˜ j. When the good production starts, each unit is equally divided between the agents, and this continues until agent i receives q˜i. Beyond that, agent i leaves the picture, and each additional unit of the good goes to agent j that is the only one that pays for it. The process stops when this agent receives his demand. So i pays C (2q˜i) / 2 and j pays the rest: C(q˜i +q˜j) −C(2q˜i) / 2. When this procedure is generalized to n agents we obtain the serial cost-sharing rule of Moulin and Shenker (1992). Alternatively, we could consider the following procedure based on a decreasing order of demands. When the good production starts every unit goes to the agent j with the highest demand, that pays accordingly. When this agent is served q˜j − q˜i units, that is when both agents are short of exactly the same quantity q˜i, then both of them are served simultaneously, and pay the same. Cð q˜ i þ q˜ j Þ−Cð q˜ i þ q˜ j −2 q˜ i Þ Cð q˜ i þ q˜ j ÞþCð q˜ i þ q˜ j −2 q˜ i Þ and j pays the rest, i.e. . Therefore agent i pays 2 2 We will say that this allocation is prescribed by the dual standard principle because as we will see with more details in Section 3, the associated rule is the dual of the serial cost-sharing rule for two agents. The generalization of this rule to all cost-sharing problems, will be called the dual serial costsharing rule. It is worthy to mention that there are other variations of the serial cost-sharing rule by allocating costs in decreasing order of demands (see, for instance, Suh, 1997; de Frutos, 1998; Leroux, 2005). In this work we will characterize axiomatically the dual serial cost-sharing rule by asking to distribute equally the cost increment when the agents ask for more by the same amount. And we also characterize the dual serial cost-sharing rule by means of a consistency property. According to this principle, if a group of agents pays their share and leaves the others in a renegotiation, then the shares of the remaining agents do not change in the resulting reduced situation. The serial cost-sharing rule was also characterized by means of consistency by Moulin and Shenker (1994), Friedman (2004), and Albizuri and Zarzuelo (2005). Finally, we propose a non-cooperative game to implement this cooperative rule when the agents demands are known to everybody. The mechanism can be described roughly as follows. The noncooperative game starts by choosing arbitrarily one of the agents with the highest demand to make a proposal. Those agents who accept the proposal pay their cost shares according to the proposal, and those who reject bargain with the proposer in ‘reduced’ bilateral negotiations where the dual standard principle is applied. If we consider the standard principle instead of the dual standard principle, then the serial cost-sharing rule arises (Albizuri and Zarzuelo, 2005). Both mechanisms are parallel to the mechanism of Dagan et al. (1997) for bankruptcy problems (see also Serrano, 1995). The paper is organized as follows. In Section 2 we give the preliminaries. In Section 3 we define and characterize the dual serial cost-sharing rule. In Section 4 we study the consistency of the serial cost-sharing rule and we characterize it by means of this principle. In Section 5 we present a mechanism to implement this rule. The last section is devoted to conclusions. 2. Preliminaries Let U denote a set of potential agents. Given a non-empty finite subset N of U, by RN we denote the |N|-dimensional euclidean space whose axes are labelled with the members of N, and
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R+N = {x ∈ RN: xi ≥ 0}. If S ⊆ N, S ≠ 0/, and x = (xi)i∈N ∈ RN, then xS denotes the projection of x onto RS, i. e., xS = (xi)i∈S ∈ RS. Finally, if x, y ∈ R we denote x+ = max {x,0} and x ∧ y = min{x,y}. A triple (N, q˜, C) is called a cost-sharing problem, if N is a non-empty finite subset of U (the set of agents involved in the problem), q˜ ∈ R+N (the demand profile of the cost-sharing problem) and C is a nondecreasing function defined on R+ such that C(0) = 0 (the cost function of the costsharing problem). Let ΓU denote the set of all cost-sharing problems with the foregoing properties. A cost-sharing rule σ on a subset Γ of ΓU associates with each (N,q˜,C ) ∈ Γ a vector σ(N,q˜, C ) ∈ R+N satisfying X
ri ðN ; q; ˜ CÞ ¼ C
iaN
X
! q˜i
ðefficiencyÞ:
iaN
Thus a cost-sharing rule must allocate total cost among the n agents. Moulin and Shenker (1992) define the serial cost-sharing rule. To give an explicit formula assume that N = {1,2,…, n} and q˜1 ≤ q˜ 2 ≤ ⋯ ≤ q˜ n. The serial cost-sharing rule of (N, q˜ , C), denoted φ, gives ˜ CÞ :¼ ui ðN ; q;
q˜ i X Cjq˜ −Cj1 j¼1
n−j þ 1
for all iaf1; N ; ng;
where C q˜0 = 0, and Cjq˜ ¼ Cððn−j þ 1Þd q˜j þ q˜j−1 þ : : : þ q˜1 Þ for all jaf1; N ; ig: Moulin and Shenker (1992) characterize the serial cost-sharing rule by means of two properties. The first one, anonymity, requires the cost shares associated with two demands to be permuted if the two demands are permuted. The other property requires agent i's cost share not to depend on demands which are higher than agent i's demand. 3. The dual serial cost-sharing rule In the Introduction we describe the serial cost-sharing rule and the dual serial cost-sharing rule when there are only two agents involved. Here we generalize these procedures to the general case. We begin by describing the serial cost-sharing rule of Moulin and Shenker (1992). Let q˜ 1 ≤ q˜ 2 ≤ … ≤ q˜ n. When the good production starts, each unit of the good is equally divided between the agents, that also share equally the incurred cost. When the quantity nq˜ 1 is produced, agent 1 with the lowest demand stops receiving the good, and leaves the picture. The production continues and each additional unit is divided equally between the remaining n − 1 agents that also share equally the cost. This in turn continues until agent 2 met his demand, i.e. the produced amount is nq˜ 1 + (n − 1)q˜ 2. At this point agent 2 stops receiving the good, and each additional unit is divided between the remaining n − 2 agents, that pay equally the corresponding cost, and so on. Alternatively, we could consider the following procedure. When the production process starts each additional unit of the good goes to agent n with the highest demand, who pays for it. He continues to do so until he receives q˜ n − q˜ n − 1. At this point agent n − 1 enters in the picture, and each additional produced unit is shared equally between agents n and n − 1, that pay the same for
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the good that is being produced. This continues until agents n and n − 1 have received their demands but q˜ n − 2. Then agent n − 2 enters in the picture and so on. We call the resulting rule the dual serial cost-sharing rule, that we denote by φ⁎. It holds ˜ ; CÞ ¼ u⁎ i ðN ; q
i b jq˜ b q˜ −C X C j1 j¼1
n−j þ 1
;
for all i ∈ {1,…,n} where b jq˜ C
! ! X X q ˜ b ¼C ¼C ð q˜ k − q˜ j Þþ for all j af1; N ; ig; and C q˜ k : 0 kaN
kaN
When comparing how agents are treated by the serial cost-sharing and the dual serial costsharing rules, it depends on the concavity or convexity of the cost function. Under convex functions agent with lower demands is better with the serial cost-sharing rule. On the contrary, the dual serial cost-sharing rule is better for these agents if the cost function is concave. The following example illustrates this fact. Example — Consider three agents 1, 2, and 3, with the demand vector q˜ = (10, 20, 30). pffiffiffi Also consider two cost functions C1 ðqÞ ¼ q, and C2(q) = q2 . The following table summarizes the allocation prescribed by the serial cost-sharing rule and the dual serial costsharing rule.
φ φ⁎
C1
C2
(1.826, 2.623, 3.298) (0.756, 1.914, 5.076)
(300, 1100, 2200) (900, 1300, 1400)
It can be easily proved the following result. Theorem 1. Let (N, q˜, C) be a cost-sharing problem. If D is a cost function that satisfies ! ! ! X X DðqÞ ¼ C q˜ k −C q˜ k −q ; kaN
kaN
then ˜ CÞ ¼ ui ðN ; q; ˜ DÞ: u⁎ i ðN ; q; The cost function D can be considered as the dual cost function of C. Then the above theorem states that the allocation prescribed by the dual serial cost-sharing rule coincides with the allocation recommended by the serial cost-sharing rule of the dual problem. Let us characterize this new rule by means of the following three axioms. Let σ be a costsharing rule and (N, q˜ , C) a cost-sharing problem. Anonymity: Let π: N → N be a one-to-one mapping and πq˜ ∈ R+N such that (πq˜)i = q˜ π− 1(i) for all i ∈ N. Then ˜ CÞ ¼ rp−1 ðiÞ ðN ; q; ˜ CÞ for all iaN : ri ðN ; p q; Independence of null demands: If q˜ i = 0 for some i ∈ N, then ˜ CÞ ¼ rj ðN ∖fig; q˜ N ∖fig ; CÞ for all jaN ∖fig: rj ðN ; q;
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Equal changes in payoff: Let r ∈ R+ and q˜ +r ∈ R+N such that q˜ i+r = q˜ i + r for all i ∈ N. Then ri ðN ; q˜ þr ; CÞ−ri ðN ; q; ˜ CÞ ¼ rj ðN ; q˜ þr ; CÞ−rj ðN ; q; ˜ CÞ for all i; jaN : The first axiom is the well known anonymity axiom. The second axiom requires the payoffs of the agents to be independent of the demands of the agents who do not demand anything. The third one requires that if all the agents increase their demand in the same amount, then the cost allocations have also to increase in the same amount. So, this axiom requires an egalitarian distribution of the costs the agents incur in when they increase their demands equally. These three axioms characterize the dual serial cost-sharing rule. Theorem 2. The dual serial cost-sharing rule is the only cost-sharing rule that satisfies anonymity, independence of null demands and equal changes in payoff. Proof. First we will prove φ⁎ satisfies the foregoing axioms. It is clear that φ⁎ satisfies anonymity and independence of null demands. Let us prove φ⁎ satisfies equal changes in payoff. First of all notice that q˜ i+r ≤ q˜ j+r if and only if q˜ i ≤ q˜ j. Therefore, the order of the demands in q˜ +r is the same as in q˜. Looking at the formula of φ⁎ notice that the term which changes when considering q˜ +r is the first one. So in φ⁎i (N,q˜ +r,C) − φ⁎i (N,q˜ ,C) and in φ⁎j (N,q˜ +r,C) − φ⁎j (N,q˜,C ) we have to consider only the first terms of the formula. Hence, ˜ CÞ u⁎ ðN ; q˜ þr ; CÞ−u⁎ ðN ; q; i
i
Cð q˜ 1 þ r þ : : : þ q˜ n þ rÞ−Cðð q˜ 1 þ r þ : : : þ q˜ n þ rÞ−nð q˜ 1 þ rÞÞ ¼ n Cð q˜ 1 þ : : : þ q˜ n Þ−Cðð q˜ 1 þ : : : þ q˜ n Þ−n q˜ 1 Þ − ¼ rj ðN ; q˜ þr ; CÞ−rj ðN ; q; ˜ CÞ n and therefore φ⁎ satisfies the required axioms. We will prove uniqueness by induction on |{i ∈ N: q˜ i ≠ 0}|. Suppose |{i ∈ N: q˜ i ≠ 0}| = 0. In this case anonymity and efficiency of the cost-sharing rule imply σi(N,q˜,C) = 0 for all i ∈ N. Suppose σ(N,q˜ ,C) is determined if |{i ∈N: q˜ i ≠ 0}| b k and let us prove it if |{i ∈N: q˜ i ≠ 0}| =k. So, let (N,q˜,C) be such that |{i ∈ N: q˜ i ≠ 0}| = k. Suppose w.l.o.g. q˜ 1 ≤ q˜ 2 ≤ … ≤ q˜ n and let j = min {i: q˜ i ≠ 0}. By independence of null demands, and taking S = {1,… j − 1}, we have ˜ CÞ ¼ ri ðN ∖S; q˜ N∖S ; CÞ for all iaN ∖S: ri ðN ; q;
ð1Þ
N ∖S such that p˜ i = q˜ i − q˜ j for all i ∈ N \ S. Since q˜ N\S = p˜ +q˜ j by equal changes in payoff Let paℝ ˜ we have
˜ CÞ ¼ ri ðN ∖S; q˜ N ∖S ; CÞ−ri ðN ∖S; p; ˜ CÞ for all iaN ∖S: rj ðN ∖S; q˜ N ∖S ; CÞ−rj ðN ∖S; p;
ð2Þ
By induction σi(N \ S,p˜, C ) is fully determined for all i ∈ N \S. If we denote α = σj(N \S,q˜ N\S,C ) − σj(N \ S,p˜,C ) by Eq. (2) we have ri ðN ∖S; q˜ N ∖S ; CÞ−ri ðN ∖S; p; ˜ CÞ ¼ a
for all iaN ∖S:
By adding all the equalities X iaN∖S
ri ðN ∖S; q˜ N ∖S ; CÞ−
X iaN ∖S
ri ðN ∖S; p; ˜ CÞ ¼ ðn−jSjÞd a:
ð3Þ
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And taking into account the efficiency of the cost-sharing rule ! ! X X C q˜ i −C p˜ i a¼
iaN ∖S
iaN ∖S
n−jSj
Therefore, by Eqs. (1) and (3), ! X C q˜ k −C ri ðN ; q; ˜ CÞ ¼
kaN∖S
:
X kaN∖S
n−jSj
! p˜ k þ ri ðN ∖S; p; ˜ CÞ for all iaN ∖S;
and by anonymity and efficiency the cost-sharing rule is determined for all i ∈ N .
□
Remark 1. The three axioms of the characterization are independent. The serial cost-sharing method satisfies the first and second ones and do not satisfy equal changes in payoff. The cost-sharing rule ξ defined by ! X C q˜ i ni ðN ; q; ˜ CÞ ¼
iaN
n
satisfies the first and third axioms but not independence of null demands. The following rule does not satisfy anonymity, but satisfies the other axioms. Without loss of generality assume that N = {1,…,n}. Let ζ the cost-sharing rule defined by ⁎ ui ðN ; q; ˜ CÞ if q˜ i ¼ minf q˜ k : kaN g; 1i ðN ; q; ˜ CÞ ¼ u⁎ ðN ; q; ˜ CÞ otherwise; pi where π is a permutation of the set A ={i: q˜ i ≠ min{q˜ k: k ∈N}}, such that jN k implies q˜ π( j) ≥q˜ π(k). That is, the first (according to the natural order) agent j such that j ∉A has to pay the allocation according to the dual serial cost-sharing rule associated to the agent with minimal demand in A. The second agent in A, the allocation associated to the agent with the second minimal demand in A, and so on. 4. Consistency In this section we characterize the dual serial cost-sharing rule by means of a consistency property. The consistency principle has played an important role in many economic allocation problems. According to this principle, if a group of agents pay their share and leave the others in a renegotiation, then the shares of the remaining agents do not change in the new reduced situation. The formal definition of consistency depends on the formal definition of the reduced situation. Several definitions of reduced cost functions have been proposed in the literature. In some of these definitions it is assumed that when computing the reduced cost of producing a given quantity q, the entire aggregate demand of the leaving group is satisfied, and their pre-payments are taken into account accordingly. These definitions can be understood as if in the production process the first units of the good go to the leaving agents (see, for instance, Thomson, 1996; Friedman, 2004). In Koster (2006), it is assumed that at some moment the leaving agent is also
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given his full demand but not necessarily at the beginning of the production. In Tijs and Koster (1998), instead, to calculate the reduced cost function of a given quantity q it is assumed that the total demands of the leaving group are only partially satisfied during the production process, (at some moment the leaving agents get their full demands). Our definition is more in this spirit, although our formulation is quite different. Indeed, assume that agents in N \ S leave, and agents in S are in charge of the production. If this production is not enough to satisfy all the demands in N, they have to ration this production. There are many ways to do this. They can distribute the production using for example the Uniform Gains rationing method (Moulin, 2002), i.e., by equalizing the ‘gains’. They may want to equalize the net losses for everybody, that is, they can use the Uniform Losses rationing method. The Uniform Gains method is applied in other paper (Albizuri and Zarzuelo, 2005) to study the serial cost-sharing rule. Here we will consider the second one. Formally, let (N, q˜, C ) be a cost-sharing problem with |N| ≥ 2, and let S ⊂ N, S ≠ ∅, and σ a cost-sharing rule. For every q ∈ R+, define ! 8 X X X > > > C q þ q ˜ r ðN ; q; ˜ CÞ if qz q˜ i ; − i > i > > > iaS iaN ∖S iaN ∖S > ! < X X C S;r ðqÞ ¼ C q þ ð q˜ i −tÞþ − q˜ i ; if qb > > > iaS iaN ∖S > X > > > > ri ðN ; ðð q˜ i −tÞþ ÞiaN ; CÞ : iaN ∖S
P where t ∈ R+ is such that iaS ð q˜ i −tÞþ ¼ q. S,σ The interpretation of this P cost function C is as follows. Suppose that agents in S produce a quantity q for them. If qz iaS q˜ i , all the demands of the members in S can be granted, and hence also agents in N \S are given their demandsP q˜ i and pay according to the rule. P That is, agents in S should have to produce a total quantity of q þ iaN ∖S q˜ i , and will receive iaN ∖S ri ðN ; q; ˜ CÞ. This is reflected in the first line of the definition above. The otherP line applies when all the demands of the agents P in S cannot be granted. In that Pcase the quantity q þ iaN ∖S ð q˜ i −tÞþ is being produced, where qb iaS q˜ i and t ∈ R+ is such that iaS ð q˜ i −tÞþ ¼ q. Agents in N \S are given (q˜ i −t)+ and they pay according to the rule taking into account the quantities they are given. Notice that if the demands cannot be granted, that is, in the second case, the original demands are reduced equally. We will say that a rule σ on ΓU satisfies consistency if for every (N,q˜,C) ∈ ΓU with |N| ≥ 2 and every S ⊂ N,S ≠ 0/ it holds 1) (S,q˜ S,CS,σ) ∈ ΓU. 2) σ(S,q˜ S,CS,σ) = σS(N,q˜ ,C). That is, σ satisfies consistency if a cost-sharing problem arises when the reduced situation is considered and the allocations prescribed by the rule for the reduced problem are the ones prescribed by the same rule for the original problem. Let us prove that the dual serial cost-sharing rule is the unique cost-sharing rule that coincides with the dual standard principle rule and satisfies consistency. Lemma 3. The dual serial cost-sharing rule satisfies consistency. ⁎ Proof. If (N,q˜ ,C) ∈ΓU and 0/ ≠S ⊂N, then (S,q˜ S,C S,φ ) ∈ΓU. Now, assume that N = {1,2,…, n} and w.l.o.g q˜ 1 ≤q˜ 2 ≤ … ≤q˜ n and S = {i1,…, is} where i1 ≤ … ≤is.
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b S;u⁎ q˜S Denote ð C Þ0 ¼ 0 ¼ i0 , then by definition we have ˜ S ; C S;u⁎ Þ ¼ u⁎ im ðS; q
S;u⁎ q˜S S;u⁎ q˜S b b m X ðC Þit−1 −ð C Þi t s−t þ 1 t¼1
m ¼ 1; N ; s:
ð4Þ
Let us calculate the differences in the summation above. S;u⁎ q˜S S;u⁎ q˜S b b ðC Þit−1 −ð C Þit X b q˜ −C b q˜ − ¼C ððu⁎ ÞS ðN ; ðð q˜ i − q˜ it−1 Þþ ÞiaN ; CÞ−ðu⁎ ÞS ðN ; ðð q˜ i − q˜ it Þþ ÞiaN ; CÞÞ: it1 it S aN ∖S
ð5Þ
If we take into account again the definition of φ\scale160%⁎, ðu⁎ ÞS ðN ; ðð q˜ i − q˜ it−1 Þþ ÞiaN ; CÞ−ðu⁎ ÞS ðN ; ðð q˜ i − q˜ it Þþ ÞiaN ; CÞ 8 if S bit−1 ; <0 ⁎ ⁎ −ðu Þ ðN ; q; ˜ CÞ þ ðu Þ ðN ; q; ˜ CÞ if it−1 b S bit ; ¼ it−1 S : ⁎ ⁎ ˜ CÞ þ ðu Þit ðN ; q; ˜ CÞ if S Nit : −ðu Þit−1 ðN ; q; Substituting in Eq. (5) we have
X S;u⁎ q˜S S;u⁎ q˜S b b b q˜ −C b q˜ − ðC Þit−1 −ð C Þit ¼ C −ðu⁎ Þit−1 ðN ; q; ˜ CÞ þ ðu⁎ ÞS ðN ; q; ˜ CÞ it1 it −
X S aN ∖S S Nit
b q˜ −C b q˜ − ¼C it1 it ¼
S aN∖S
it−1bS bit −ðu⁎ Þit−1 ðN ; q; ˜ CÞ þ ðu⁎ Þit ðN ; q; ˜ CÞ X
S X
b q˜ X b q˜ −C C r r1 − n−r þ 1 S aN ∖S S Nit
S aN ∖S r¼it−1 þ1 it−1 b S bit it X b q˜ b q˜ −C C r r1 b q˜ −C b q˜ − ððn−sÞ−r C it1 it n−r þ 1 r¼it−1 þ1 it X b q˜ b q˜ −C C r r1
S X r¼it−1 þ1
b q˜ b q˜ −C C r r1 n−r þ 1
b q˜ −C b q˜ − þ tÞ ¼ C it1 it it X
it X
r¼it−1 þ1
b q˜ −C b q˜ Þ ðC r r1
b q˜ b q˜ −C C r r1 ¼ ðs−t þ 1Þ : ðs−t þ 1Þ þ n−r þ 1 n−r þ 1 r¼it−1 þ1 r¼it−1 þ1
Substituting in Eq. (4) we obtain for m = 1,…, s m X it im b q˜ X b q˜ X b q˜ b q˜ −C C Cr1 −C ⁎ r r r1 ðu⁎ Þim ðS; q˜ S ; C S;u Þ ¼ ¼ ¼ ðu⁎ Þim ðN ; q; ˜ CÞ: n−r þ 1 n−r þ 1 r¼1 t¼1 r¼it−1 þ1 And the proof is complete. □ By DSP we denote the dual cost-sharing rule for 2 agents. Theorem 4. The dual serial cost-sharing rule is the unique cost-sharing rule that coincides with DSP on 2-person problems and satisfies consistency. Proof. By the above lemma the dual serial cost-sharing rule satisfies consistency and by definition it coincides with the dual standard principle rule when there are two agents. Now let us see that if a cost-sharing rule σ satisfies consistency and coincides with the dual standard principle rule, then it is uniquely determined.
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Suppose w.l.o.g q˜ 1 ≤ q˜ 2 ≤ … ≤ q˜ n. First notice that if q˜ i = q˜ j, then σi(N,q˜,C) = σj(N,q˜,C) since i and j are symmetric in the reduced problem ({i, j}, q˜ {i, j},C{i, j},σ) and DSP is an anonymous rule. Then σ is anonymous too. On the other hand, if q˜ i ¼ 0 then ri ðN ; q; ˜ CÞ ¼ 0:
ð6Þ
Indeed, if N = {i}, efficiency implies Eq. (6) and if N ≠ {i}, by choosing j ∈ N \{i} and applying consistency. ri ðN ; q; ˜ CÞ ¼ ri ðfi; jg; q˜ fi;jg ; C fi;jg;r Þ ¼ DSPi ðfi; jg; q˜ fi;jg ; C fi;jg;r Þ ¼ 0: Now let us prove σ(N,q˜ ,C) is determined. We will prove it by induction on K(q˜) = |{q˜i: q˜ i ≠ 0}|. If K(q˜ ) = 0 by efficiency and anonymity σi(N,q˜ ,C) = 0 for every i ∈ N. If K(q˜ ) = 1, by Eq. (6) and efficiency σ(N,q˜ ,C) is also determined. Now, let us suppose σ(N,q˜,C) is determined if K(q˜) b m(m ≥ 2) and let us prove it is so when K (q˜) = m. Let S = {i: q˜ i = 0}. By Eq. (6) ˜ CÞ ¼ 0 if iaS: ri ðN ; q;
ð7Þ
Now let i ∈ N \(S ∪{n}) and consider T = {i,n}. Consistency and the definition of DSP imply q˜T
C T ;r ð q˜ i þ q˜ n Þ− b C T ;r i ri ðN ; q˜ ; CÞ ¼ ri ðT ; q˜ T ; C T ;r Þ ¼ 2 ! q˜ T X X C q˜ j − rj ðN ; q; ˜ CÞ− b C T ;r i jaN ∖fi;ng
jaN
¼ C
X
! q˜ j −
jaN
¼
2 X
2! X C ð q˜ j − q˜ i Þþ − −
Therefore 2ri ðN ; q; ˜ CÞ þ
rj ðN ; q; ˜ CÞ
jaN ∖fi;ng
X
rj
N ; ð q˜ j − q˜ i Þþ
jaN
jaN ∖fi;ng
jaN
;C :
2 X
rj ðN ; q; ˜ CÞ ¼ C
jaN ∖fi;ng
X
! q˜ j
jaN
þ
X
X ð q˜ j − q˜ i Þþ −C
jaN
rj N ; ð q˜ j − q˜ i Þþ
jaN ∖fi;ng
And taking into account efficiency and simplifying, ! X ri ðN ; q; ˜ CÞ−rn ðN ; q; ˜ CÞ ¼ −C ð q˜ j − q˜ i Þþ þ jaN
!
X jaN ∖fi;ng
jaN
;C
rj N ; ð q˜ j − q˜ i Þþ
jaN
;C :
By induction, the right hand side of this equality is determined. Then we have n − |S| − 1 equations with n − |S| unknowns, precisely σi(N,q˜,C), i ∈ N \ S.
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By efficiency and Eq. (7) we can also consider the equation ! X X ri ðN ; q; ˜ CÞ ¼ C q˜ j : iaN ∖S
jaN
In this way we have a system with n − |S| equations and it can be easily proved that it has a unique solution. Therefore, σ(N,q˜ ,C) is fully determined. □ 5. Strategic bargaining and the dual serial cost-sharing rule Now we present a non-cooperative bargaining game whose unique subgame perfect equilibrium outcome is the allocation prescribed by the dual serial cost-sharing rule. A similar model was presented by Albizuri and Zarzuelo (2005) for the serial cost-sharing rule. Both are related with the non-cooperative model proposed by Dagan et al. (1997) for bankruptcy situations (see also Serrano, 1995). This game has subgames with two players in which the dual standard principle is taken into account to determine an allocation. Let us present the mechanism for the dual serial cost-sharing rule. Consider a cost-sharing problem (N, q˜, C) such that |N| ≥ 2. The mechanism will be referred as g(N,q˜ ,C) and it is as follows: choose arbitrarily one of the agents with the highest demand. Suppose n is P chosen. Then, agent n makes a proposal x such that 0 ≤ xi for every i ∈ N, and P ˜ i Þ. The remaining agents respond sequentially according to an arbitrary iaN xi ¼ Cð iaN q ordering. We may assume this ordering is 1,2,…, n − 1.1 If an agent accepts then he pays his component in x. If agent j rejects, then he renegotiates his cost share with the proposer n in a bilateral cost-sharing problem (defined below), by employing the dual standard principle. The proposer n pays the remainder of the total cost after all the other agents had paid their cost shares. Formally the final shares are z with zj ¼ xj forevery acceptor j; j zj ¼ DSPj f j; ng; q˜ f j;ng ; C v for every rejector j; and P b0− zj for the proposer n; zn ¼ C j pn j
where v j and Cv are defined recursively in the following way. On the one hand, the vector v j is the vector of interim cost shares that remains after renegotiation or acceptance with j − 1 agents.PThe first one is v1 = x, and the last one is vn = z. If b0− j = 2,…, n − 1, then vkj ¼ zk if kbj; vnj ¼ C vij ; and vk j = xk otherwise. i pn vj Let us turn now to define the cost function C of the reduced bilateral cost-sharing problem of agents j and n, when these two agents assume that the remaining agents pay according to the proposal v j. Notice that in order to determine zj, by definition of DSP, we only need to know the j j values Cv (q˜ n + q˜ j) and Cv (q˜ n − q˜ j). j First we define Cv (q˜n + q˜j). This is the total cost of the problem faced by j and n. If they assume that the rest of the agents pay according to the proposal v j, this total cost is: j
C v ð q˜ n þ q˜ j Þ ¼ C
X iaN
1
! q˜ i −
X i pj;n
vi j :
The same result holds if the responses are made simultaneously.
ð8Þ
160
M.J. Albizuri, J.M. Zarzuelo / Mathematical Social Sciences 53 (2007) 150–163 j
Now we turn to define Cv (q˜ n − q˜ j). Here we consider again two cases. j (a)q˜ n = q˜ j. Obviously in this case we define Cv (q˜ n − q˜ j) = 0. (b)q˜ n N q˜ j. Here assume that agents n and j are in charge of the production, and they distribute this production between all the agents according to the Uniform Losses rationing method. When these agents have the quantity q˜ n − q˜ j available for them, both of them are short of the same quantity, namely q˜ j; hence every agent i ≠ j, n will receive (q˜ i − q˜ j)+. The incurred cost of this production is Cbj. Assume that the remaining agents pay (vij − v jj)+, except those that have not received yet any quantity of the good, who pay 0. Then agents n and j to produce q˜ n − q˜ j, have to pay X j j j bj− C v ð q˜ n − q˜ j Þ :¼ C vi −vj : i pj;n q˜ i z q˜ j
þ
j
Just two exceptions to make sense: if this quantity is negative, we write Cv (q˜ n − q˜ j): = 0; and if it is j bigger than the total cost given in expression Eq. (8), we define Cv (q˜ n − q˜ j) precisely as this total cost. j Summarizing if q˜ n N q˜ j and if we denote Aj = Cv (q˜ n + q˜ j), we define 8 X j j X bj− bj− > v −v if 0b C þ b Aj ; C > j i > þ > > i pj;n i pj;n > > q˜ i z q˜ j > > Xq˜ i zq˜ j j j > < b vi −vj V0; 0 if Cj − j þ C v ð q˜ n − q˜ j Þ :¼ i pj;n > > z q ˜ q ˜ > i j > X > j j >A bj− > if C v −v zAj : j > j i > þ > i pj;n : q˜ i z q˜ j
As commented before, we have this result. Theorem 5. Let (N, q˜ , C) be a cost-sharing problem. The game g(N,q˜ ,C) has a unique subgame perfect equilibrium outcome, namely φ⁎(N,q˜,C). Proof. Let us show first uniqueness. Let s be a SPE with outcome z. We will show that z = φ ⁎(N, q˜,C) in three steps. Denote x ⁎ = φ ⁎(N,q˜ ,C). Step 1. x⁎n ≥ zn: Agent n can guarantee a cost share of x⁎n by offering xj⁎ to every j ≠ n. In this case ⁎ ⁎ ⁎ ⁎ we have C x (q˜ n + q˜ j) = Cφ ,{ j,n}(q˜ n + q˜ j) and C x (q˜ n − q˜ j) = Cφ ,{ j,n}(q˜ n − q˜ j) for every j ≠ n. Therefore, ⁎ ⁎ DSPðf j; ng; q˜ f j;ng ; C x Þ ¼ DSPðf j; ng; q˜ f j;ng ; C u ;f j;ng Þ
and taking into account φ⁎ satisfies consistency ⁎ ⁎ DSPðf j; ng; q˜ f j;ng ; C u ;f j;ng Þ ¼ ðx⁎ j ; xn Þ; and therefore substituting in Eq. (9) ⁎ ⁎ DSPðf j; ng; q˜ f j;ng ; C x Þ ¼ ðx⁎ j ; xn Þ:
ð9Þ
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Hence, the first answer j gets xj⁎ and v2 = x⁎. By an induction argument, regardless of the profile of responses, if j is a rejector then v j = x ⁎ and therefore j ⁎ DSPðf j; ng; q˜ f j;ng ; C v Þ ¼ ðx⁎ j ; xn Þ
and j gets x⁎j . As a consequence, agent n’s cost share is x⁎n Step 2. x⁎j ≥ zj for every j ≠ n: First of all we prove that for every j ≠ n X bj þ C b0 − zj : ðzi 1zj ÞV− C ip n P bj − i pj;n vij −vjj zAj then First notice that if C þ
qiz ˜ q˜j
j
ð10Þ
j
C v ð q˜ j þ q˜ n Þ ¼ C v ð q˜ n − q˜ j Þ ¼ Aj : Therefore, by definition DSP it holds zj = 0 and Eq. (10) is obviously true. P of bj − i pj;n vij −vjj bAj . Observe that in equilibrium, if x is the proposal Now, suppose C þ
qiz ˜ q˜j
made by n, it holds xi = vi1 ≥ vi2 ≥ ⋯ ≥ vin = zi for all i ≠ n, and xn = vn1 ≤ vn2 ≤ ⋯ ≤ vnn = zn. Then X jþ1 jþ1 X j j X j j X ðzi 1zj ÞV ðvi 1vj ÞV ðvi 1vj Þ þ vjjþ1 ¼ ðvi 1vj Þ þ zj i pn
¼
P i pj;n
i pn
i pj;n
i pj;n
X X j j j b0 −ðvnj þ vjj Þ− vi − ðvij −vjj Þþ þ zj ¼ C ðvi −vj Þþ i pn
i pn
b0 −ðvnj þ vjj Þ−2zj þ ðvnj þ vjj Þ− C bj þ zj ¼ C b0 − C bj −zj : þ zj V C and Eq. (10) is proved. Now by contradiction assume T = { j ∈ N: xj⁎ b zj, j ≠ n} is not empty. Let j ∈ T such that zj ≥ zi for every i ∈ T. Then X X X X ⁎ b0 −zn ¼ b0 − C bj −zj þ C zi ¼ ðzi 1zj Þ þ ðzi zj Þþ b C ðx⁎ i −xj Þþ i pn
i pn
b0 − C bj −zj þ ¼ C
i pn
X i pn
X ⁎ x⁎ ðx⁎ i − i 1xj Þ
i pn ⁎ x⁎ i Nxj
i pn
b0 − C bj −zj þ C b0 −x⁎ b b b ⁎ ¼ C n − C 0 þ C j þ zj b C 0 −xn : The first inequality follows from Eq. (10); and from the fact that zj ≥ zi implies that if i ∉ T then xi⁎ b zi and therefore xi⁎ − xj⁎ > zi − zj. In the fourth equality we take into account that P ⁎ ⁎ b b i pn ðxi 1xj Þ ¼ − C 0 þ C j þ zj : Hence zn N x⁎n which contradicts Step 1. Step 3: x⁎ =z follows from Steps 1 and 2 taking also into account that x ⁎ and z are efficient. As for the existence the following strategy profile forms a subgame perfect equilibrium: Agent n proposes x ⁎ =φ ⁎ (N,C). Agent j accepts a proposal x if and only if xj ≤ DSPj ({ j,n},q˜ { j, n}, j Cv ). □ We end this section by showing that for all cost-sharing problems (N,q˜ ,C),all the subgames perfect equilibria of g(N,q˜ ,C) are coalition-proof Nash equilibria (Bernheim et al., 1987) of
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the game in strategic form G(N,q˜ ,C) associated with g(N,q˜ ,C). This can be done in the same way as for the serial cost-sharing rule and the games associated with it. The definition of coalition-proof Nash equilibrium is as follows. Let (N,(Si)i ∈ N, (ui)i ∈ N) be a game in normal form, where N is the players set, and Si and ui are the strategy set and the payoff function respectively of player i. Let 0/ ≠ M ⊆ N. Denote SM = Πi ∈ M Si and if s ∈ Πi ∈ N Si and let sM denote its restriction to M. If t ∈ Πi ∈ M Si and s ∈ Πi ∈ M Si, t is an internally consistent upon s if 1) ui(tI,s− i)Nui(s)if M = {i}for some i ∈ N. 2) Otherwise, i) ui (tM,sN\M) N ui(s) for i ∈ M, and ii) there is not any 0/ ≠ L ⊂ M such that L has an internally consistent improvement upon (tM,sN\M). According to the definition of Bernheim et al. (1987) s is a coalition-proof Nash equilibrium, if there is not any 0/ ≠ M ⊆ N such that M has an internally consistent improvement upon s. Theorem 6. For all cost-sharing problems (N,q˜ ,C ), all the perfect equilibria of g(N,q˜ ,C) are coalition-proof Nash equilibria of the game G(N,q˜,C). Proof. Let s be a subgame perfect equilibrium and suppose there is a coalition M that has an internally consistent improvement upon s. There are two possible situations. First, suppose n ∉ M. Consider k be the first respondent of the deviating coalition. Since his payoff is independent of the strategies of the agents who follow him in the ordering, if he improves his payoff, the strategy profile s would not be an equilibrium. Therefore, the second situation holds: n ∈ M. Let x be the proposal made in the deviation, {v j } be the sequence of interim payoffs determined by the joint deviation, and let {w j } be the sequence associated with the case in which all responders follow their equilibrium strategies after x has been proposed. We will show by induction that vnj = wnjfor all j. By definition v1n = wn1 = xn. And suppose vnj′ = wnj′ for all j′ b j. If j is not an agent belonging to the deviating coalition, then vnj = wnj because vnj−1 = wnj−1 and he does not deviate. If j belongs to the deviating coalition, then agent j cannot benefit if he deviates from the joint deviation (due to the fact that the deviation is internally consistent). Therefore vnj ≥ wnj. Moreover, taking into account that vnj−1= wnj−1 and s is a subgame perfect equilibrium, it must be vnj ≤wnj. And as a result vnj = wnj . But this contradicts the fact that the proposer belongs to a coalition that has an internally consistent improvement upon s. Therefore, there is not such a coalition. □ References Albizuri, M.J., Zarzuelo, J.M., 2005. “Consistency and strategic bargaining in cost sharing problems of excludable goods”, mimeo, Basque Country University. Bernheim, B.D., Peleg, B., Whinston, M.D., 1987. Coalition-proof Nash equilibria I Concepts. Journal of Economic Theory 43, 1–12. Dagan, N., Serrano, R., Volij, O., 1997. A noncooperative view of consistent bankruptcy rules. Games and Economic Behavior 18, 55–72. de Frutos, M.A., 1998. Decreasing serial cost sharing under economies of scale. Journal of Economic Theory 79, 245–275. Friedman, E.J., 2004. Paths and consistency in additive cost sharing. International Journal of Game Theory 32, 501–518. Koster, M., 2006. Consistent Cost Sharing and RationingAvailable at SSRN: http://ssrn.com/abstract=903912. Leroux, J., 2005. Strategy Proof Profit Sharing in Partnerships: Improving Upon Autarky. Working Paper 2005–06. Rice University.
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Moulin, H., 2002. Axiomatic cost and surplus sharing. In: Arrow, K.J., Sen, A., Suzumura, K. (Eds.), Handbook in Social Choice and Welfare, vol. 1. Elsevier Science. Moulin, H., Shenker, S., 1992. Serial cost sharing. Econometrica 60, 1009–1037. Moulin, H., Shenker, S., 1994. Average cost pricing versus serial cost-sharing: an axiomatic comparison. Journal of Economic Theory 64, 178–201. Serrano, R., 1995. Strategic bargaining, surplus sharing problems and the nucleolus. Journal of Mathematical Economics 24, 319–329. Suh, S.C., 1997. “Two serial mechanisms in a surplus sharing problem”, mimeo, University of Windsor (Canada). 319–329. Thomson, W., 1996. “Consistent allocation rules”, University of Rochester, mimeo. Tijs, S., Koster, M., 1998. General aggregation of demand and cost sharing methods. Annals of Operation Research 84, 137–164.
The dual serial cost-sharing rule ☆ M. Josune Albizuri ⁎, José M. Zarzuelo Basque Country University, Faculty of Economics and Business Administration, Spain Received 23 December 2005; received in revised form 7 November 2006 Available online 23 January 2007
Abstract In this paper we present a cost-sharing rule for cost-sharing problems. This rule is called the dual serial cost-sharing rule because it prescribes the same allocations as the serial cost-sharing rule in the dual problem. We give two axiomatizations of this new rule, one of them by means of a consistency property. Moreover, we provide a mechanism to implement this rule. © 2006 Elsevier B.V. All rights reserved. Keywords: Cost sharing JEL classification: C7
1. Introduction In this paper we consider cost-sharing problems in which a group of n agents shares a joint process to produce a certain private good. Each one of them declares a demand q˜i of the good. The cost function is denoted C. A cost-sharing rule allocates the total production cost, i.e. C(∑iq˜i), among all the agents. In this work we study a cost-sharing rule that we have called the dual serial cost-sharing rule, because it prescribes the same allocations as the serial cost-sharing rule of Moulin and Shenker (1992) does in the dual problem. This dual rule is a particular case of the reverse serial costsharing mechanism defined by Tijs and Koster (1998). Moreover, there is a close relation between cost-sharing problems and rationing problems. Actually, in an excellent survey Moulin (2002) shows that the set of monotonic rationing methods is linearly isomorphic to that of additive costsharing rules. Then this author gives a list of rationing methods and cost-sharing rules matched by ☆
This research has been partially supported by the University of Basque Country (project 9/UPV 00031.321-15352/2003) and DGES Ministerio de Educación y Ciencia (project BEC2003-08182). ⁎Corresponding author. Lehendakari Agirre, 83, 48015 Bilbao, Spain. E-mail addresses: [email protected], [email protected] (M.J. Albizuri), [email protected] (J.M. Zarzuelo). 0165-4896/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2006.11.001
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this linear isomorphism. For instance, Moulin points out that the Uniform Gains method corresponds with the well known serial cost-sharing rule (Moulin and Shenker, 1992), meanwhile the Uniform Losses would give rise to the “dual” serial rule. The serial cost-sharing rule can be described as follows. When there are only two agents, this rule follows the standard principle: if both agents demand the same quantity they share equally the cost; and if one of them wants more, he is the only one that has to pay for it. It will be useful to give an alternative description of this procedure. That is, consider two agents i and j with q˜i ≤q˜ j. When the good production starts, each unit is equally divided between the agents, and this continues until agent i receives q˜i. Beyond that, agent i leaves the picture, and each additional unit of the good goes to agent j that is the only one that pays for it. The process stops when this agent receives his demand. So i pays C (2q˜i) / 2 and j pays the rest: C(q˜i +q˜j) −C(2q˜i) / 2. When this procedure is generalized to n agents we obtain the serial cost-sharing rule of Moulin and Shenker (1992). Alternatively, we could consider the following procedure based on a decreasing order of demands. When the good production starts every unit goes to the agent j with the highest demand, that pays accordingly. When this agent is served q˜j − q˜i units, that is when both agents are short of exactly the same quantity q˜i, then both of them are served simultaneously, and pay the same. Cð q˜ i þ q˜ j Þ−Cð q˜ i þ q˜ j −2 q˜ i Þ Cð q˜ i þ q˜ j ÞþCð q˜ i þ q˜ j −2 q˜ i Þ and j pays the rest, i.e. . Therefore agent i pays 2 2 We will say that this allocation is prescribed by the dual standard principle because as we will see with more details in Section 3, the associated rule is the dual of the serial cost-sharing rule for two agents. The generalization of this rule to all cost-sharing problems, will be called the dual serial costsharing rule. It is worthy to mention that there are other variations of the serial cost-sharing rule by allocating costs in decreasing order of demands (see, for instance, Suh, 1997; de Frutos, 1998; Leroux, 2005). In this work we will characterize axiomatically the dual serial cost-sharing rule by asking to distribute equally the cost increment when the agents ask for more by the same amount. And we also characterize the dual serial cost-sharing rule by means of a consistency property. According to this principle, if a group of agents pays their share and leaves the others in a renegotiation, then the shares of the remaining agents do not change in the resulting reduced situation. The serial cost-sharing rule was also characterized by means of consistency by Moulin and Shenker (1994), Friedman (2004), and Albizuri and Zarzuelo (2005). Finally, we propose a non-cooperative game to implement this cooperative rule when the agents demands are known to everybody. The mechanism can be described roughly as follows. The noncooperative game starts by choosing arbitrarily one of the agents with the highest demand to make a proposal. Those agents who accept the proposal pay their cost shares according to the proposal, and those who reject bargain with the proposer in ‘reduced’ bilateral negotiations where the dual standard principle is applied. If we consider the standard principle instead of the dual standard principle, then the serial cost-sharing rule arises (Albizuri and Zarzuelo, 2005). Both mechanisms are parallel to the mechanism of Dagan et al. (1997) for bankruptcy problems (see also Serrano, 1995). The paper is organized as follows. In Section 2 we give the preliminaries. In Section 3 we define and characterize the dual serial cost-sharing rule. In Section 4 we study the consistency of the serial cost-sharing rule and we characterize it by means of this principle. In Section 5 we present a mechanism to implement this rule. The last section is devoted to conclusions. 2. Preliminaries Let U denote a set of potential agents. Given a non-empty finite subset N of U, by RN we denote the |N|-dimensional euclidean space whose axes are labelled with the members of N, and
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R+N = {x ∈ RN: xi ≥ 0}. If S ⊆ N, S ≠ 0/, and x = (xi)i∈N ∈ RN, then xS denotes the projection of x onto RS, i. e., xS = (xi)i∈S ∈ RS. Finally, if x, y ∈ R we denote x+ = max {x,0} and x ∧ y = min{x,y}. A triple (N, q˜, C) is called a cost-sharing problem, if N is a non-empty finite subset of U (the set of agents involved in the problem), q˜ ∈ R+N (the demand profile of the cost-sharing problem) and C is a nondecreasing function defined on R+ such that C(0) = 0 (the cost function of the costsharing problem). Let ΓU denote the set of all cost-sharing problems with the foregoing properties. A cost-sharing rule σ on a subset Γ of ΓU associates with each (N,q˜,C ) ∈ Γ a vector σ(N,q˜, C ) ∈ R+N satisfying X
ri ðN ; q; ˜ CÞ ¼ C
iaN
X
! q˜i
ðefficiencyÞ:
iaN
Thus a cost-sharing rule must allocate total cost among the n agents. Moulin and Shenker (1992) define the serial cost-sharing rule. To give an explicit formula assume that N = {1,2,…, n} and q˜1 ≤ q˜ 2 ≤ ⋯ ≤ q˜ n. The serial cost-sharing rule of (N, q˜ , C), denoted φ, gives ˜ CÞ :¼ ui ðN ; q;
q˜ i X Cjq˜ −Cj1 j¼1
n−j þ 1
for all iaf1; N ; ng;
where C q˜0 = 0, and Cjq˜ ¼ Cððn−j þ 1Þd q˜j þ q˜j−1 þ : : : þ q˜1 Þ for all jaf1; N ; ig: Moulin and Shenker (1992) characterize the serial cost-sharing rule by means of two properties. The first one, anonymity, requires the cost shares associated with two demands to be permuted if the two demands are permuted. The other property requires agent i's cost share not to depend on demands which are higher than agent i's demand. 3. The dual serial cost-sharing rule In the Introduction we describe the serial cost-sharing rule and the dual serial cost-sharing rule when there are only two agents involved. Here we generalize these procedures to the general case. We begin by describing the serial cost-sharing rule of Moulin and Shenker (1992). Let q˜ 1 ≤ q˜ 2 ≤ … ≤ q˜ n. When the good production starts, each unit of the good is equally divided between the agents, that also share equally the incurred cost. When the quantity nq˜ 1 is produced, agent 1 with the lowest demand stops receiving the good, and leaves the picture. The production continues and each additional unit is divided equally between the remaining n − 1 agents that also share equally the cost. This in turn continues until agent 2 met his demand, i.e. the produced amount is nq˜ 1 + (n − 1)q˜ 2. At this point agent 2 stops receiving the good, and each additional unit is divided between the remaining n − 2 agents, that pay equally the corresponding cost, and so on. Alternatively, we could consider the following procedure. When the production process starts each additional unit of the good goes to agent n with the highest demand, who pays for it. He continues to do so until he receives q˜ n − q˜ n − 1. At this point agent n − 1 enters in the picture, and each additional produced unit is shared equally between agents n and n − 1, that pay the same for
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the good that is being produced. This continues until agents n and n − 1 have received their demands but q˜ n − 2. Then agent n − 2 enters in the picture and so on. We call the resulting rule the dual serial cost-sharing rule, that we denote by φ⁎. It holds ˜ ; CÞ ¼ u⁎ i ðN ; q
i b jq˜ b q˜ −C X C j1 j¼1
n−j þ 1
;
for all i ∈ {1,…,n} where b jq˜ C
! ! X X q ˜ b ¼C ¼C ð q˜ k − q˜ j Þþ for all j af1; N ; ig; and C q˜ k : 0 kaN
kaN
When comparing how agents are treated by the serial cost-sharing and the dual serial costsharing rules, it depends on the concavity or convexity of the cost function. Under convex functions agent with lower demands is better with the serial cost-sharing rule. On the contrary, the dual serial cost-sharing rule is better for these agents if the cost function is concave. The following example illustrates this fact. Example — Consider three agents 1, 2, and 3, with the demand vector q˜ = (10, 20, 30). pffiffiffi Also consider two cost functions C1 ðqÞ ¼ q, and C2(q) = q2 . The following table summarizes the allocation prescribed by the serial cost-sharing rule and the dual serial costsharing rule.
φ φ⁎
C1
C2
(1.826, 2.623, 3.298) (0.756, 1.914, 5.076)
(300, 1100, 2200) (900, 1300, 1400)
It can be easily proved the following result. Theorem 1. Let (N, q˜, C) be a cost-sharing problem. If D is a cost function that satisfies ! ! ! X X DðqÞ ¼ C q˜ k −C q˜ k −q ; kaN
kaN
then ˜ CÞ ¼ ui ðN ; q; ˜ DÞ: u⁎ i ðN ; q; The cost function D can be considered as the dual cost function of C. Then the above theorem states that the allocation prescribed by the dual serial cost-sharing rule coincides with the allocation recommended by the serial cost-sharing rule of the dual problem. Let us characterize this new rule by means of the following three axioms. Let σ be a costsharing rule and (N, q˜ , C) a cost-sharing problem. Anonymity: Let π: N → N be a one-to-one mapping and πq˜ ∈ R+N such that (πq˜)i = q˜ π− 1(i) for all i ∈ N. Then ˜ CÞ ¼ rp−1 ðiÞ ðN ; q; ˜ CÞ for all iaN : ri ðN ; p q; Independence of null demands: If q˜ i = 0 for some i ∈ N, then ˜ CÞ ¼ rj ðN ∖fig; q˜ N ∖fig ; CÞ for all jaN ∖fig: rj ðN ; q;
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Equal changes in payoff: Let r ∈ R+ and q˜ +r ∈ R+N such that q˜ i+r = q˜ i + r for all i ∈ N. Then ri ðN ; q˜ þr ; CÞ−ri ðN ; q; ˜ CÞ ¼ rj ðN ; q˜ þr ; CÞ−rj ðN ; q; ˜ CÞ for all i; jaN : The first axiom is the well known anonymity axiom. The second axiom requires the payoffs of the agents to be independent of the demands of the agents who do not demand anything. The third one requires that if all the agents increase their demand in the same amount, then the cost allocations have also to increase in the same amount. So, this axiom requires an egalitarian distribution of the costs the agents incur in when they increase their demands equally. These three axioms characterize the dual serial cost-sharing rule. Theorem 2. The dual serial cost-sharing rule is the only cost-sharing rule that satisfies anonymity, independence of null demands and equal changes in payoff. Proof. First we will prove φ⁎ satisfies the foregoing axioms. It is clear that φ⁎ satisfies anonymity and independence of null demands. Let us prove φ⁎ satisfies equal changes in payoff. First of all notice that q˜ i+r ≤ q˜ j+r if and only if q˜ i ≤ q˜ j. Therefore, the order of the demands in q˜ +r is the same as in q˜. Looking at the formula of φ⁎ notice that the term which changes when considering q˜ +r is the first one. So in φ⁎i (N,q˜ +r,C) − φ⁎i (N,q˜ ,C) and in φ⁎j (N,q˜ +r,C) − φ⁎j (N,q˜,C ) we have to consider only the first terms of the formula. Hence, ˜ CÞ u⁎ ðN ; q˜ þr ; CÞ−u⁎ ðN ; q; i
i
Cð q˜ 1 þ r þ : : : þ q˜ n þ rÞ−Cðð q˜ 1 þ r þ : : : þ q˜ n þ rÞ−nð q˜ 1 þ rÞÞ ¼ n Cð q˜ 1 þ : : : þ q˜ n Þ−Cðð q˜ 1 þ : : : þ q˜ n Þ−n q˜ 1 Þ − ¼ rj ðN ; q˜ þr ; CÞ−rj ðN ; q; ˜ CÞ n and therefore φ⁎ satisfies the required axioms. We will prove uniqueness by induction on |{i ∈ N: q˜ i ≠ 0}|. Suppose |{i ∈ N: q˜ i ≠ 0}| = 0. In this case anonymity and efficiency of the cost-sharing rule imply σi(N,q˜,C) = 0 for all i ∈ N. Suppose σ(N,q˜ ,C) is determined if |{i ∈N: q˜ i ≠ 0}| b k and let us prove it if |{i ∈N: q˜ i ≠ 0}| =k. So, let (N,q˜,C) be such that |{i ∈ N: q˜ i ≠ 0}| = k. Suppose w.l.o.g. q˜ 1 ≤ q˜ 2 ≤ … ≤ q˜ n and let j = min {i: q˜ i ≠ 0}. By independence of null demands, and taking S = {1,… j − 1}, we have ˜ CÞ ¼ ri ðN ∖S; q˜ N∖S ; CÞ for all iaN ∖S: ri ðN ; q;
ð1Þ
N ∖S such that p˜ i = q˜ i − q˜ j for all i ∈ N \ S. Since q˜ N\S = p˜ +q˜ j by equal changes in payoff Let paℝ ˜ we have
˜ CÞ ¼ ri ðN ∖S; q˜ N ∖S ; CÞ−ri ðN ∖S; p; ˜ CÞ for all iaN ∖S: rj ðN ∖S; q˜ N ∖S ; CÞ−rj ðN ∖S; p;
ð2Þ
By induction σi(N \ S,p˜, C ) is fully determined for all i ∈ N \S. If we denote α = σj(N \S,q˜ N\S,C ) − σj(N \ S,p˜,C ) by Eq. (2) we have ri ðN ∖S; q˜ N ∖S ; CÞ−ri ðN ∖S; p; ˜ CÞ ¼ a
for all iaN ∖S:
By adding all the equalities X iaN∖S
ri ðN ∖S; q˜ N ∖S ; CÞ−
X iaN ∖S
ri ðN ∖S; p; ˜ CÞ ¼ ðn−jSjÞd a:
ð3Þ
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And taking into account the efficiency of the cost-sharing rule ! ! X X C q˜ i −C p˜ i a¼
iaN ∖S
iaN ∖S
n−jSj
Therefore, by Eqs. (1) and (3), ! X C q˜ k −C ri ðN ; q; ˜ CÞ ¼
kaN∖S
:
X kaN∖S
n−jSj
! p˜ k þ ri ðN ∖S; p; ˜ CÞ for all iaN ∖S;
and by anonymity and efficiency the cost-sharing rule is determined for all i ∈ N .
□
Remark 1. The three axioms of the characterization are independent. The serial cost-sharing method satisfies the first and second ones and do not satisfy equal changes in payoff. The cost-sharing rule ξ defined by ! X C q˜ i ni ðN ; q; ˜ CÞ ¼
iaN
n
satisfies the first and third axioms but not independence of null demands. The following rule does not satisfy anonymity, but satisfies the other axioms. Without loss of generality assume that N = {1,…,n}. Let ζ the cost-sharing rule defined by ⁎ ui ðN ; q; ˜ CÞ if q˜ i ¼ minf q˜ k : kaN g; 1i ðN ; q; ˜ CÞ ¼ u⁎ ðN ; q; ˜ CÞ otherwise; pi where π is a permutation of the set A ={i: q˜ i ≠ min{q˜ k: k ∈N}}, such that jN k implies q˜ π( j) ≥q˜ π(k). That is, the first (according to the natural order) agent j such that j ∉A has to pay the allocation according to the dual serial cost-sharing rule associated to the agent with minimal demand in A. The second agent in A, the allocation associated to the agent with the second minimal demand in A, and so on. 4. Consistency In this section we characterize the dual serial cost-sharing rule by means of a consistency property. The consistency principle has played an important role in many economic allocation problems. According to this principle, if a group of agents pay their share and leave the others in a renegotiation, then the shares of the remaining agents do not change in the new reduced situation. The formal definition of consistency depends on the formal definition of the reduced situation. Several definitions of reduced cost functions have been proposed in the literature. In some of these definitions it is assumed that when computing the reduced cost of producing a given quantity q, the entire aggregate demand of the leaving group is satisfied, and their pre-payments are taken into account accordingly. These definitions can be understood as if in the production process the first units of the good go to the leaving agents (see, for instance, Thomson, 1996; Friedman, 2004). In Koster (2006), it is assumed that at some moment the leaving agent is also
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given his full demand but not necessarily at the beginning of the production. In Tijs and Koster (1998), instead, to calculate the reduced cost function of a given quantity q it is assumed that the total demands of the leaving group are only partially satisfied during the production process, (at some moment the leaving agents get their full demands). Our definition is more in this spirit, although our formulation is quite different. Indeed, assume that agents in N \ S leave, and agents in S are in charge of the production. If this production is not enough to satisfy all the demands in N, they have to ration this production. There are many ways to do this. They can distribute the production using for example the Uniform Gains rationing method (Moulin, 2002), i.e., by equalizing the ‘gains’. They may want to equalize the net losses for everybody, that is, they can use the Uniform Losses rationing method. The Uniform Gains method is applied in other paper (Albizuri and Zarzuelo, 2005) to study the serial cost-sharing rule. Here we will consider the second one. Formally, let (N, q˜, C ) be a cost-sharing problem with |N| ≥ 2, and let S ⊂ N, S ≠ ∅, and σ a cost-sharing rule. For every q ∈ R+, define ! 8 X X X > > > C q þ q ˜ r ðN ; q; ˜ CÞ if qz q˜ i ; − i > i > > > iaS iaN ∖S iaN ∖S > ! < X X C S;r ðqÞ ¼ C q þ ð q˜ i −tÞþ − q˜ i ; if qb > > > iaS iaN ∖S > X > > > > ri ðN ; ðð q˜ i −tÞþ ÞiaN ; CÞ : iaN ∖S
P where t ∈ R+ is such that iaS ð q˜ i −tÞþ ¼ q. S,σ The interpretation of this P cost function C is as follows. Suppose that agents in S produce a quantity q for them. If qz iaS q˜ i , all the demands of the members in S can be granted, and hence also agents in N \S are given their demandsP q˜ i and pay according to the rule. P That is, agents in S should have to produce a total quantity of q þ iaN ∖S q˜ i , and will receive iaN ∖S ri ðN ; q; ˜ CÞ. This is reflected in the first line of the definition above. The otherP line applies when all the demands of the agents P in S cannot be granted. In that Pcase the quantity q þ iaN ∖S ð q˜ i −tÞþ is being produced, where qb iaS q˜ i and t ∈ R+ is such that iaS ð q˜ i −tÞþ ¼ q. Agents in N \S are given (q˜ i −t)+ and they pay according to the rule taking into account the quantities they are given. Notice that if the demands cannot be granted, that is, in the second case, the original demands are reduced equally. We will say that a rule σ on ΓU satisfies consistency if for every (N,q˜,C) ∈ ΓU with |N| ≥ 2 and every S ⊂ N,S ≠ 0/ it holds 1) (S,q˜ S,CS,σ) ∈ ΓU. 2) σ(S,q˜ S,CS,σ) = σS(N,q˜ ,C). That is, σ satisfies consistency if a cost-sharing problem arises when the reduced situation is considered and the allocations prescribed by the rule for the reduced problem are the ones prescribed by the same rule for the original problem. Let us prove that the dual serial cost-sharing rule is the unique cost-sharing rule that coincides with the dual standard principle rule and satisfies consistency. Lemma 3. The dual serial cost-sharing rule satisfies consistency. ⁎ Proof. If (N,q˜ ,C) ∈ΓU and 0/ ≠S ⊂N, then (S,q˜ S,C S,φ ) ∈ΓU. Now, assume that N = {1,2,…, n} and w.l.o.g q˜ 1 ≤q˜ 2 ≤ … ≤q˜ n and S = {i1,…, is} where i1 ≤ … ≤is.
M.J. Albizuri, J.M. Zarzuelo / Mathematical Social Sciences 53 (2007) 150–163
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b S;u⁎ q˜S Denote ð C Þ0 ¼ 0 ¼ i0 , then by definition we have ˜ S ; C S;u⁎ Þ ¼ u⁎ im ðS; q
S;u⁎ q˜S S;u⁎ q˜S b b m X ðC Þit−1 −ð C Þi t s−t þ 1 t¼1
m ¼ 1; N ; s:
ð4Þ
Let us calculate the differences in the summation above. S;u⁎ q˜S S;u⁎ q˜S b b ðC Þit−1 −ð C Þit X b q˜ −C b q˜ − ¼C ððu⁎ ÞS ðN ; ðð q˜ i − q˜ it−1 Þþ ÞiaN ; CÞ−ðu⁎ ÞS ðN ; ðð q˜ i − q˜ it Þþ ÞiaN ; CÞÞ: it1 it S aN ∖S
ð5Þ
If we take into account again the definition of φ\scale160%⁎, ðu⁎ ÞS ðN ; ðð q˜ i − q˜ it−1 Þþ ÞiaN ; CÞ−ðu⁎ ÞS ðN ; ðð q˜ i − q˜ it Þþ ÞiaN ; CÞ 8 if S bit−1 ; <0 ⁎ ⁎ −ðu Þ ðN ; q; ˜ CÞ þ ðu Þ ðN ; q; ˜ CÞ if it−1 b S bit ; ¼ it−1 S : ⁎ ⁎ ˜ CÞ þ ðu Þit ðN ; q; ˜ CÞ if S Nit : −ðu Þit−1 ðN ; q; Substituting in Eq. (5) we have
X S;u⁎ q˜S S;u⁎ q˜S b b b q˜ −C b q˜ − ðC Þit−1 −ð C Þit ¼ C −ðu⁎ Þit−1 ðN ; q; ˜ CÞ þ ðu⁎ ÞS ðN ; q; ˜ CÞ it1 it −
X S aN ∖S S Nit
b q˜ −C b q˜ − ¼C it1 it ¼
S aN∖S
it−1bS bit −ðu⁎ Þit−1 ðN ; q; ˜ CÞ þ ðu⁎ Þit ðN ; q; ˜ CÞ X
S X
b q˜ X b q˜ −C C r r1 − n−r þ 1 S aN ∖S S Nit
S aN ∖S r¼it−1 þ1 it−1 b S bit it X b q˜ b q˜ −C C r r1 b q˜ −C b q˜ − ððn−sÞ−r C it1 it n−r þ 1 r¼it−1 þ1 it X b q˜ b q˜ −C C r r1
S X r¼it−1 þ1
b q˜ b q˜ −C C r r1 n−r þ 1
b q˜ −C b q˜ − þ tÞ ¼ C it1 it it X
it X
r¼it−1 þ1
b q˜ −C b q˜ Þ ðC r r1
b q˜ b q˜ −C C r r1 ¼ ðs−t þ 1Þ : ðs−t þ 1Þ þ n−r þ 1 n−r þ 1 r¼it−1 þ1 r¼it−1 þ1
Substituting in Eq. (4) we obtain for m = 1,…, s m X it im b q˜ X b q˜ X b q˜ b q˜ −C C Cr1 −C ⁎ r r r1 ðu⁎ Þim ðS; q˜ S ; C S;u Þ ¼ ¼ ¼ ðu⁎ Þim ðN ; q; ˜ CÞ: n−r þ 1 n−r þ 1 r¼1 t¼1 r¼it−1 þ1 And the proof is complete. □ By DSP we denote the dual cost-sharing rule for 2 agents. Theorem 4. The dual serial cost-sharing rule is the unique cost-sharing rule that coincides with DSP on 2-person problems and satisfies consistency. Proof. By the above lemma the dual serial cost-sharing rule satisfies consistency and by definition it coincides with the dual standard principle rule when there are two agents. Now let us see that if a cost-sharing rule σ satisfies consistency and coincides with the dual standard principle rule, then it is uniquely determined.
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Suppose w.l.o.g q˜ 1 ≤ q˜ 2 ≤ … ≤ q˜ n. First notice that if q˜ i = q˜ j, then σi(N,q˜,C) = σj(N,q˜,C) since i and j are symmetric in the reduced problem ({i, j}, q˜ {i, j},C{i, j},σ) and DSP is an anonymous rule. Then σ is anonymous too. On the other hand, if q˜ i ¼ 0 then ri ðN ; q; ˜ CÞ ¼ 0:
ð6Þ
Indeed, if N = {i}, efficiency implies Eq. (6) and if N ≠ {i}, by choosing j ∈ N \{i} and applying consistency. ri ðN ; q; ˜ CÞ ¼ ri ðfi; jg; q˜ fi;jg ; C fi;jg;r Þ ¼ DSPi ðfi; jg; q˜ fi;jg ; C fi;jg;r Þ ¼ 0: Now let us prove σ(N,q˜ ,C) is determined. We will prove it by induction on K(q˜) = |{q˜i: q˜ i ≠ 0}|. If K(q˜ ) = 0 by efficiency and anonymity σi(N,q˜ ,C) = 0 for every i ∈ N. If K(q˜ ) = 1, by Eq. (6) and efficiency σ(N,q˜ ,C) is also determined. Now, let us suppose σ(N,q˜,C) is determined if K(q˜) b m(m ≥ 2) and let us prove it is so when K (q˜) = m. Let S = {i: q˜ i = 0}. By Eq. (6) ˜ CÞ ¼ 0 if iaS: ri ðN ; q;
ð7Þ
Now let i ∈ N \(S ∪{n}) and consider T = {i,n}. Consistency and the definition of DSP imply q˜T
C T ;r ð q˜ i þ q˜ n Þ− b C T ;r i ri ðN ; q˜ ; CÞ ¼ ri ðT ; q˜ T ; C T ;r Þ ¼ 2 ! q˜ T X X C q˜ j − rj ðN ; q; ˜ CÞ− b C T ;r i jaN ∖fi;ng
jaN
¼ C
X
! q˜ j −
jaN
¼
2 X
2! X C ð q˜ j − q˜ i Þþ − −
Therefore 2ri ðN ; q; ˜ CÞ þ
rj ðN ; q; ˜ CÞ
jaN ∖fi;ng
X
rj
N ; ð q˜ j − q˜ i Þþ
jaN
jaN ∖fi;ng
jaN
;C :
2 X
rj ðN ; q; ˜ CÞ ¼ C
jaN ∖fi;ng
X
! q˜ j
jaN
þ
X
X ð q˜ j − q˜ i Þþ −C
jaN
rj N ; ð q˜ j − q˜ i Þþ
jaN ∖fi;ng
And taking into account efficiency and simplifying, ! X ri ðN ; q; ˜ CÞ−rn ðN ; q; ˜ CÞ ¼ −C ð q˜ j − q˜ i Þþ þ jaN
!
X jaN ∖fi;ng
jaN
;C
rj N ; ð q˜ j − q˜ i Þþ
jaN
;C :
By induction, the right hand side of this equality is determined. Then we have n − |S| − 1 equations with n − |S| unknowns, precisely σi(N,q˜,C), i ∈ N \ S.
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159
By efficiency and Eq. (7) we can also consider the equation ! X X ri ðN ; q; ˜ CÞ ¼ C q˜ j : iaN ∖S
jaN
In this way we have a system with n − |S| equations and it can be easily proved that it has a unique solution. Therefore, σ(N,q˜ ,C) is fully determined. □ 5. Strategic bargaining and the dual serial cost-sharing rule Now we present a non-cooperative bargaining game whose unique subgame perfect equilibrium outcome is the allocation prescribed by the dual serial cost-sharing rule. A similar model was presented by Albizuri and Zarzuelo (2005) for the serial cost-sharing rule. Both are related with the non-cooperative model proposed by Dagan et al. (1997) for bankruptcy situations (see also Serrano, 1995). This game has subgames with two players in which the dual standard principle is taken into account to determine an allocation. Let us present the mechanism for the dual serial cost-sharing rule. Consider a cost-sharing problem (N, q˜, C) such that |N| ≥ 2. The mechanism will be referred as g(N,q˜ ,C) and it is as follows: choose arbitrarily one of the agents with the highest demand. Suppose n is P chosen. Then, agent n makes a proposal x such that 0 ≤ xi for every i ∈ N, and P ˜ i Þ. The remaining agents respond sequentially according to an arbitrary iaN xi ¼ Cð iaN q ordering. We may assume this ordering is 1,2,…, n − 1.1 If an agent accepts then he pays his component in x. If agent j rejects, then he renegotiates his cost share with the proposer n in a bilateral cost-sharing problem (defined below), by employing the dual standard principle. The proposer n pays the remainder of the total cost after all the other agents had paid their cost shares. Formally the final shares are z with zj ¼ xj forevery acceptor j; j zj ¼ DSPj f j; ng; q˜ f j;ng ; C v for every rejector j; and P b0− zj for the proposer n; zn ¼ C j pn j
where v j and Cv are defined recursively in the following way. On the one hand, the vector v j is the vector of interim cost shares that remains after renegotiation or acceptance with j − 1 agents.PThe first one is v1 = x, and the last one is vn = z. If b0− j = 2,…, n − 1, then vkj ¼ zk if kbj; vnj ¼ C vij ; and vk j = xk otherwise. i pn vj Let us turn now to define the cost function C of the reduced bilateral cost-sharing problem of agents j and n, when these two agents assume that the remaining agents pay according to the proposal v j. Notice that in order to determine zj, by definition of DSP, we only need to know the j j values Cv (q˜ n + q˜ j) and Cv (q˜ n − q˜ j). j First we define Cv (q˜n + q˜j). This is the total cost of the problem faced by j and n. If they assume that the rest of the agents pay according to the proposal v j, this total cost is: j
C v ð q˜ n þ q˜ j Þ ¼ C
X iaN
1
! q˜ i −
X i pj;n
vi j :
The same result holds if the responses are made simultaneously.
ð8Þ
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Now we turn to define Cv (q˜ n − q˜ j). Here we consider again two cases. j (a)q˜ n = q˜ j. Obviously in this case we define Cv (q˜ n − q˜ j) = 0. (b)q˜ n N q˜ j. Here assume that agents n and j are in charge of the production, and they distribute this production between all the agents according to the Uniform Losses rationing method. When these agents have the quantity q˜ n − q˜ j available for them, both of them are short of the same quantity, namely q˜ j; hence every agent i ≠ j, n will receive (q˜ i − q˜ j)+. The incurred cost of this production is Cbj. Assume that the remaining agents pay (vij − v jj)+, except those that have not received yet any quantity of the good, who pay 0. Then agents n and j to produce q˜ n − q˜ j, have to pay X j j j bj− C v ð q˜ n − q˜ j Þ :¼ C vi −vj : i pj;n q˜ i z q˜ j
þ
j
Just two exceptions to make sense: if this quantity is negative, we write Cv (q˜ n − q˜ j): = 0; and if it is j bigger than the total cost given in expression Eq. (8), we define Cv (q˜ n − q˜ j) precisely as this total cost. j Summarizing if q˜ n N q˜ j and if we denote Aj = Cv (q˜ n + q˜ j), we define 8 X j j X bj− bj− > v −v if 0b C þ b Aj ; C > j i > þ > > i pj;n i pj;n > > q˜ i z q˜ j > > Xq˜ i zq˜ j j j > < b vi −vj V0; 0 if Cj − j þ C v ð q˜ n − q˜ j Þ :¼ i pj;n > > z q ˜ q ˜ > i j > X > j j >A bj− > if C v −v zAj : j > j i > þ > i pj;n : q˜ i z q˜ j
As commented before, we have this result. Theorem 5. Let (N, q˜ , C) be a cost-sharing problem. The game g(N,q˜ ,C) has a unique subgame perfect equilibrium outcome, namely φ⁎(N,q˜,C). Proof. Let us show first uniqueness. Let s be a SPE with outcome z. We will show that z = φ ⁎(N, q˜,C) in three steps. Denote x ⁎ = φ ⁎(N,q˜ ,C). Step 1. x⁎n ≥ zn: Agent n can guarantee a cost share of x⁎n by offering xj⁎ to every j ≠ n. In this case ⁎ ⁎ ⁎ ⁎ we have C x (q˜ n + q˜ j) = Cφ ,{ j,n}(q˜ n + q˜ j) and C x (q˜ n − q˜ j) = Cφ ,{ j,n}(q˜ n − q˜ j) for every j ≠ n. Therefore, ⁎ ⁎ DSPðf j; ng; q˜ f j;ng ; C x Þ ¼ DSPðf j; ng; q˜ f j;ng ; C u ;f j;ng Þ
and taking into account φ⁎ satisfies consistency ⁎ ⁎ DSPðf j; ng; q˜ f j;ng ; C u ;f j;ng Þ ¼ ðx⁎ j ; xn Þ; and therefore substituting in Eq. (9) ⁎ ⁎ DSPðf j; ng; q˜ f j;ng ; C x Þ ¼ ðx⁎ j ; xn Þ:
ð9Þ
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161
Hence, the first answer j gets xj⁎ and v2 = x⁎. By an induction argument, regardless of the profile of responses, if j is a rejector then v j = x ⁎ and therefore j ⁎ DSPðf j; ng; q˜ f j;ng ; C v Þ ¼ ðx⁎ j ; xn Þ
and j gets x⁎j . As a consequence, agent n’s cost share is x⁎n Step 2. x⁎j ≥ zj for every j ≠ n: First of all we prove that for every j ≠ n X bj þ C b0 − zj : ðzi 1zj ÞV− C ip n P bj − i pj;n vij −vjj zAj then First notice that if C þ
qiz ˜ q˜j
j
ð10Þ
j
C v ð q˜ j þ q˜ n Þ ¼ C v ð q˜ n − q˜ j Þ ¼ Aj : Therefore, by definition DSP it holds zj = 0 and Eq. (10) is obviously true. P of bj − i pj;n vij −vjj bAj . Observe that in equilibrium, if x is the proposal Now, suppose C þ
qiz ˜ q˜j
made by n, it holds xi = vi1 ≥ vi2 ≥ ⋯ ≥ vin = zi for all i ≠ n, and xn = vn1 ≤ vn2 ≤ ⋯ ≤ vnn = zn. Then X jþ1 jþ1 X j j X j j X ðzi 1zj ÞV ðvi 1vj ÞV ðvi 1vj Þ þ vjjþ1 ¼ ðvi 1vj Þ þ zj i pn
¼
P i pj;n
i pn
i pj;n
i pj;n
X X j j j b0 −ðvnj þ vjj Þ− vi − ðvij −vjj Þþ þ zj ¼ C ðvi −vj Þþ i pn
i pn
b0 −ðvnj þ vjj Þ−2zj þ ðvnj þ vjj Þ− C bj þ zj ¼ C b0 − C bj −zj : þ zj V C and Eq. (10) is proved. Now by contradiction assume T = { j ∈ N: xj⁎ b zj, j ≠ n} is not empty. Let j ∈ T such that zj ≥ zi for every i ∈ T. Then X X X X ⁎ b0 −zn ¼ b0 − C bj −zj þ C zi ¼ ðzi 1zj Þ þ ðzi zj Þþ b C ðx⁎ i −xj Þþ i pn
i pn
b0 − C bj −zj þ ¼ C
i pn
X i pn
X ⁎ x⁎ ðx⁎ i − i 1xj Þ
i pn ⁎ x⁎ i Nxj
i pn
b0 − C bj −zj þ C b0 −x⁎ b b b ⁎ ¼ C n − C 0 þ C j þ zj b C 0 −xn : The first inequality follows from Eq. (10); and from the fact that zj ≥ zi implies that if i ∉ T then xi⁎ b zi and therefore xi⁎ − xj⁎ > zi − zj. In the fourth equality we take into account that P ⁎ ⁎ b b i pn ðxi 1xj Þ ¼ − C 0 þ C j þ zj : Hence zn N x⁎n which contradicts Step 1. Step 3: x⁎ =z follows from Steps 1 and 2 taking also into account that x ⁎ and z are efficient. As for the existence the following strategy profile forms a subgame perfect equilibrium: Agent n proposes x ⁎ =φ ⁎ (N,C). Agent j accepts a proposal x if and only if xj ≤ DSPj ({ j,n},q˜ { j, n}, j Cv ). □ We end this section by showing that for all cost-sharing problems (N,q˜ ,C),all the subgames perfect equilibria of g(N,q˜ ,C) are coalition-proof Nash equilibria (Bernheim et al., 1987) of
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the game in strategic form G(N,q˜ ,C) associated with g(N,q˜ ,C). This can be done in the same way as for the serial cost-sharing rule and the games associated with it. The definition of coalition-proof Nash equilibrium is as follows. Let (N,(Si)i ∈ N, (ui)i ∈ N) be a game in normal form, where N is the players set, and Si and ui are the strategy set and the payoff function respectively of player i. Let 0/ ≠ M ⊆ N. Denote SM = Πi ∈ M Si and if s ∈ Πi ∈ N Si and let sM denote its restriction to M. If t ∈ Πi ∈ M Si and s ∈ Πi ∈ M Si, t is an internally consistent upon s if 1) ui(tI,s− i)Nui(s)if M = {i}for some i ∈ N. 2) Otherwise, i) ui (tM,sN\M) N ui(s) for i ∈ M, and ii) there is not any 0/ ≠ L ⊂ M such that L has an internally consistent improvement upon (tM,sN\M). According to the definition of Bernheim et al. (1987) s is a coalition-proof Nash equilibrium, if there is not any 0/ ≠ M ⊆ N such that M has an internally consistent improvement upon s. Theorem 6. For all cost-sharing problems (N,q˜ ,C ), all the perfect equilibria of g(N,q˜ ,C) are coalition-proof Nash equilibria of the game G(N,q˜,C). Proof. Let s be a subgame perfect equilibrium and suppose there is a coalition M that has an internally consistent improvement upon s. There are two possible situations. First, suppose n ∉ M. Consider k be the first respondent of the deviating coalition. Since his payoff is independent of the strategies of the agents who follow him in the ordering, if he improves his payoff, the strategy profile s would not be an equilibrium. Therefore, the second situation holds: n ∈ M. Let x be the proposal made in the deviation, {v j } be the sequence of interim payoffs determined by the joint deviation, and let {w j } be the sequence associated with the case in which all responders follow their equilibrium strategies after x has been proposed. We will show by induction that vnj = wnjfor all j. By definition v1n = wn1 = xn. And suppose vnj′ = wnj′ for all j′ b j. If j is not an agent belonging to the deviating coalition, then vnj = wnj because vnj−1 = wnj−1 and he does not deviate. If j belongs to the deviating coalition, then agent j cannot benefit if he deviates from the joint deviation (due to the fact that the deviation is internally consistent). Therefore vnj ≥ wnj. Moreover, taking into account that vnj−1= wnj−1 and s is a subgame perfect equilibrium, it must be vnj ≤wnj. And as a result vnj = wnj . But this contradicts the fact that the proposer belongs to a coalition that has an internally consistent improvement upon s. Therefore, there is not such a coalition. □ References Albizuri, M.J., Zarzuelo, J.M., 2005. “Consistency and strategic bargaining in cost sharing problems of excludable goods”, mimeo, Basque Country University. Bernheim, B.D., Peleg, B., Whinston, M.D., 1987. Coalition-proof Nash equilibria I Concepts. Journal of Economic Theory 43, 1–12. Dagan, N., Serrano, R., Volij, O., 1997. A noncooperative view of consistent bankruptcy rules. Games and Economic Behavior 18, 55–72. de Frutos, M.A., 1998. Decreasing serial cost sharing under economies of scale. Journal of Economic Theory 79, 245–275. Friedman, E.J., 2004. Paths and consistency in additive cost sharing. International Journal of Game Theory 32, 501–518. Koster, M., 2006. Consistent Cost Sharing and RationingAvailable at SSRN: http://ssrn.com/abstract=903912. Leroux, J., 2005. Strategy Proof Profit Sharing in Partnerships: Improving Upon Autarky. Working Paper 2005–06. Rice University.
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