- Email: [email protected]
Monotonicity and the Aumann–Shapley cost-sharing method in the discrete case
European Journal of Operational Research xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Decision Support
Monotonicity and the Aumann–Shapley cost-sharing method in the discrete case M.J. Albizuri ⇑, H. Díez, A. Sarachu University of the Basque Country, Faculty of Economics and Business, Department of Applied Economics IV, Bilbao, Spain
a r t i c l e
i n f o
Article history: Received 4 August 2013 Accepted 27 March 2014 Available online xxxx Keywords: Cost sharing Aumann–Shapley method Monotonicity
a b s t r a c t We give an axiomatization of the Aumann–Shapley cost-sharing method in the discrete case by means of monotonicity and no merging or splitting (Sprumont, 2005). Monotonicity has not yet been employed to characterize this method in such a case, by contrast with the case in which goods are perfectly divisible, for which Monderer and Neyman (1988) and Young (1985b) characterize the Aumann–Shapley price mechanism. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction In this paper we provide an axiomatization for the Aumann– Shapley cost-sharing method in the discrete case by employing a monotonicity property. Aumann–Shapley pricing is a method for solving cost-sharing problems in which agents consume a quantity of possibly different goods. Assuming that a finite set N ¼ f1; . . . ; ng denotes the set of agents i; qi is the quantity of good i demanded by agent i and q is the corresponding consumption profile, we want to split CðqÞ among the agents. Goods may be priced instead of sharing the cost CðqÞ. The two approaches are equivalent, and we have chosen the latter, following Sprumont (2005). In the discrete case agents can consume several units of the good. The consumption qi of agent i is a non-negative integer which indicates how many units of good i are consumed by agent i. A simple cost-sharing method for solving such problems is the Shapley– Shubik method introduced by Shubik (1962). Given a coalition S # N of agents, Shubik considers the cost of the consumptions of those agents: v ðSÞ ¼ CðqS ; 0NnS Þ; and calculates the Shapley value (1953) of the transferable utility game v with N as the set of players. This Shapley value gives precisely the assignment by the Shapley– Shubik method. The Aumann–Shapley method is also defined by means of a transferable utility game, but now more information about cost is taken into account. Indeed, costs of intermediate consumptions of agents are considered and not just those associated with consumptions qi . Given an agent i, each unit consumed by i is viewed as a different player, so there is a set N i of jN i j ¼ qi players ⇑ Corresponding author. Tel.: +34 94 601 3805. E-mail addresses: [email protected] (M.J. (H. Díez), [email protected] (A. Sarachu).
Albizuri),
[email protected]
S and the grand set N q ¼ i2N N i . Therefore there are coalitions of players, which are subsets of N q . Given S # N q , dðSÞ ¼ ðjS \ N i jÞi2N represents the units demanded by agents in S and the cost associated with those demands is v q ðSÞ ¼ CðdðSÞÞ. The Shapley value of v q gives allocations for players in Nq . Since each agent i is represented by the elements in N i , the sum of the allocations of these elements is by definition the allocation for agent i provided by the Aumann–Shapley method. The Aumann–Shapley pricing originates in the work of Billera, Heath, and Raanan (1978), when these authors used Aumann and Shapley’s (1974) theory of values for nonatomic games to set the telephone billing rates in order to share the cost of service among the users. Subsequently, Billera and Heath (1982) and Mirman and Tauman (1982) transfer the axiomatic approach introduced by Aumann and Shapley (1974) to characterize this pricing mechanism for cost-sharing problems in which goods are perfectly divisible. Samet and Tauman (1982) offer an axiomatic characterization for the Aumann–Shapley price mechanism based on the one given by Dubey, Neyman, and Weber (1981) for the semivalues; and latterly, McLean, Pazgal, and Sharkey (2004) propose another one using the potential and consistency. We also find many applications of Aumann–Shapley pricing, notably from the pioneering works as the above by Billera et al. (1978) and Samet, Tauman, and Zang (1984) to the contributions made by Castaño-Pardo and García-Díaz (1995), Krus´ and Bronisz (2000) and Haviv (2001), or the more recent works by Faria, Barroso, Kelman, Granville, and Pereira (2009), Pierru (2010) and Chen, Iyengar, and Moallemi (2013). These applications range from the pricing of transportation, water, phone, electricity or highways, to the allocation of firmenergy rights, the allocation of the CO2 emissions, the analysis of
http://dx.doi.org/10.1016/j.ejor.2014.03.044 0377-2217/Ó 2014 Elsevier B.V. All rights reserved.
Please cite this article in press as: Albizuri, M. J., et al. Monotonicity and the Aumann–Shapley cost-sharing method in the discrete case. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.03.044
2
M.J. Albizuri et al. / European Journal of Operational Research xxx (2014) xxx–xxx
systemic risk or the study of multiple commodity situations. Another related work is Hougaard and Tind (2009), where parametric programming is employed to study the Aumann–Shapley pricing. Only more recently the discrete case has been given attention. The Aumann–Shapley method is characterized in two papers. In the first, by Calvo and Santos (2000), the method is characterized by means of Balanced Contributions, as Myerson (1980) does for the Shapley value, and requiring agents who consume nothing to pay zero. In the second Sprumont (2005) employs Additivity, Dummy and No Merging or Splitting. Dummy is the natural extension for cost-sharing problems of the Dummy axiom for transferable utility games. It requires that dummies get zero. An agent i is a dummy if there are no consumption profile whose cost changes if i consumes one more unit. No Merging or Splitting, which is described in detail in Section 3, prevents splitting manipulations. Accordingly, if an agent i splits into several agents (and therefore splits his/her consumption into several consumptions), the split agents have to pay in sum the same quantity as agent i had to. The characterization by Sprumont (2005) can be seen as formed by basic axioms, like the one given by Mirman and Tauman (1982) for the Aumann–Shapley price mechanism with perfectly divisible goods and Shapley’s (1953) characterization of his value. A look at other characterizations for the Aumann–Shapley price mechanism with perfectly divisible goods reveals that Monotonicity has also been employed, as Young (1985a) does when he characterizes the Shapley value. Monotonicity for cost-sharing problems states that if the marginal costs associated with an agent at all consumption levels decrease, then the cost share corresponding to that agent cannot increase. Young (1985b) and Monderer and Neyman (1988) obtain two characterizations for the Aumann–Shapley price mechanism using this axiom. In this paper we provide precisely an axiomatization for the Aumann–Shapley method with Monotonicity. We prove that this method is the only one that satisfies Monotonicity and No Merging or Splitting. Thus, we prove that Additivity and Dummy can be replaced by Monotonicity in the characterization of Sprumont (2005). Observe that the same happens when Young (1985a) characterizes the Shapley value (1953) for transferable utility games with Symmetry and Monotonicity. Young (1985a) proves that Additivity and Dummy can be replaced by Monotonicity in Shapley’s (1953) characterization. Moreover, an examination of the characterizations of the Aumann–Shapley price mechanism given by Young (1985b) and by Monderer and Neyman (1988) reveals that they employ stronger axioms than No Merging or Splitting. Wang (1999) characterizes the methods satisfying Additivity and Dummy in the discrete case. Clearly, all such methods satisfy Monotonicity. In this work we prove that in the presence of No Merging or Splitting, Monotonicity implies Additivity and Dummy. The presence of No Merging or Splitting is crucial for the result. For example, Bahel and Trudeau (2013) study a more general discrete cost sharing model where any two properties among Additivity, Dummy and Monotonicity fail to imply the third. It must be said that, like Sprumont (2005), we deal with nondecreasing cost functions, unlike Young (1985b) and Monderer and Neyman (1988), who use cost functions which are not necessarily non-decreasing. In addition, as Sprumont (2005) does, we assume that cost shares are non-negative. If we considered any cost function we would characterize the Aumann–Shapley method just requiring cost shares to be non-negative if the cost function is nondecreasing. The rest of the paper is structured as follows. Section 2 gives preliminaries, Section 3 presents the axiomatization and the paper ends with conclusions in Section 4 and references.
2. Preliminaries First some notation must be given. Denote by N the set of nonnegative integers, and by N the set of non-empty finite subsets of N. If N 2 N , denote by jNj the cardinality of N, and let NN be the Cartesian product of N with itself jNj times. Coordinates of elements in NN are indexed by the elements of N. If x ¼ ðxi Þi2N 2 NN P M and M # N we write xðMÞ for denotes the i2M xi , and xM 2 N restriction of x to M. If x; y 2 NN we write x 6 y to denote xi 6 yi for all i 2 N. Some distinguished vectors in NN are 0 ¼ ð0; . . . ; 0Þ and for every M # N its indicator 1M , defined by 1M i ¼ 1 if i 2 M i and 1M i ¼ 0 otherwise. We write 1 instead of the more cumbersome 1fig and N n i instead of N n fig. A pair ðN; qÞ defines a consumption profile, where N 2 N is the set of agents and q 2 NN is the list of their consumptions. If q; q0 2 NN , write ½q; q0 for the set fx 2 NN : q 6 x 6 q0 g. For each consumption profile ðN; qÞ, denote by CðN; qÞ the set of cost functions for ðN; qÞ, that is non-decreasing functions C : ½0; q NN ! Rþ such that Cð0Þ ¼ 0.1 A cost-sharing problem is a triple ðN; q; CÞ, where ðN; qÞ is a consumption profile and C 2 CðN; qÞ. The set of all problems is denoted by P. Then a cost-sharing method is a mapping / that assigns to each cost-sharing problem ðN; q; CÞ 2 P a vector /ðN; q; CÞ 2 RNþ P that satisfies the Budget Balance condition: i2N /i ðN; q; CÞ ¼ CðqÞ, that is, that exactly shares all the costs. We give some more notation. Let p : N ! N be a bijection. If pN N 2 N , write pN ¼ fpðiÞ : i 2 Ng. If x 2 RNþ , define px 2 Rþ by ðpxÞpðiÞ ¼ xi for all i 2 N. Moreover if ðN; qÞ is a consumption profile and C 2 CðN; qÞ, define pC 2 CðpN; pqÞ by pCðpxÞ ¼ CðxÞ. Write pðN; q; CÞ ¼ ðpN; pq; pCÞ. Given ðN; q; CÞ 2 P two agents i; j 2 N are said to be symmetric if for any bijection p : N ! N such that pðiÞ ¼ j; pðjÞ ¼ i, and pðkÞ ¼ k for every k 2 N n fi; jg it holds that pðN; q; CÞ ¼ ðN; q; CÞ. Notice that the consumption levels corresponding to i and j must be equal, and the cost function C must be symmetric in their demands. The next step is to formalize the Aumann–Shapley method. Given ðN; q; CÞ 2 P, let fN i gi2N be a family of pairwise disjoint sets S such that jN i j ¼ qi , and let N q ¼ i2N N i . For each S # N q define the demand vector xðSÞ ¼ ðjS \ N i jÞi2N and consider the cooperative game ðN q ; v q Þ defined by v q ðSÞ ¼ CðxðSÞÞ. Denote the Shapley value of this game by /Sh ðN q ; v q Þ. The Aumann–Shapley method assigns to each ðN; q; CÞ 2 P the vector /ASh ðN; q; CÞ defined for all i 2 N by
/ASh i ðN; q; CÞ ¼
X Sh q /j ðN ; v q Þ: j2N i
Notice that by the anonymity of the Shapley value the real numq q bers /Sh j ðN ; v Þ are independent of the choice of the sets N i , and consequently the Aumann–Shapley method is well defined. 3. An axiomatization of the Aumann–Shapley method The first axiomatization presented in this paper is based on two axioms. Before it is introduced some notation is required. Let ðN; q; CÞ 2 P; i 2 N, and x 2 ½0; q such that xi < qi . Then we write @ i CðxÞ ¼ Cðx þ 1i Þ CðxÞ. So @ i CðxÞ denotes the impact on the cost function when agent i consumes one more unit at consumption vector x. That is, @ i CðxÞ is the marginal cost associated with agent i at consumption vector x. Monotonicity: Let ðN; q; CÞ; ðN; q; C 0 Þ 2 P, and i 2 N. If for all x 2 ½0; q such that xi < qi it holds @ i CðxÞ 6 @ i C 0 ðxÞ, then /i ðN; q; CÞ 6 /i ðN; q; C 0 Þ. 1 The domain of C differs from the model of Sprumont (2005), since he considers that the domain of the cost function C is all NN , instead of ½0; q. We make this distinction to provide a more concise and elegant presentation of our paper (see Remark 3 on page 11).
Please cite this article in press as: Albizuri, M. J., et al. Monotonicity and the Aumann–Shapley cost-sharing method in the discrete case. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.03.044
3
M.J. Albizuri et al. / European Journal of Operational Research xxx (2014) xxx–xxx
Cost functions C and C 0 reveal the costs associated with different consumption levels corresponding to agents i in N or equivalently production levels of goods i. Fixing i 2 N, if @ i CðxÞ 6 @ i C 0 ðxÞ at all feasible consumption levels, then it can be said that with the movement from C 0 to C an improvement in production efficiency has taken place with respect to good i. Monotonicity requires that i cannot be penalized in this case, that is, the cost share associated with i cannot increase. So if efficiency with respect to a particular agent increases, then the cost share associated with that agent cannot increase. The continuous version of Monotonicity (together with other axioms) is employed by Young (1985b) and by Monderer and Neyman (1988) to characterize the Aumann–Shapley price mechanism. In the continuous version the first partial derivative of the cost function is employed; the one sided first partial derivative on the boundary of the domain. Notice that with both cost shares and prices Monotonicity is stated in the same way since
/i ðN; q; CÞ 6 /i ðN; q; C 0 Þ if and only if
/i ðN; q; CÞ /i ðN; q; C 0 Þ 6 ; qi qi
that is, for agent i, cost shares and prices satisfy the same inequality. The second axiom that we employ is the one introduced by Sprumont (2005) to characterize the Aumann–Shapley cost sharing method in the discrete case. No Merging or Splitting: Let ðN; q; CÞ 2 P; i 2 N, and I 2 N be such that N \ I ¼ fig. Consider a consumption profile ðN 0 ; q0 Þ, where N 0 ¼ ðN n iÞ [ I; q0 is such that q0j ¼ qj for all j 2 N n i and P P 0 0 0 0 0 0 i0 2I qi0 ¼ qi . Define C 2 CðN ; q Þ by C ðxÞ ¼ C xNni ; i0 2I xi . Then
Theorem 1. The Aumann–Shapley method is the only cost-sharing method that satisfies Monotonicity and No Merging or Splitting. To prove this theorem some previous lemmas and further notation are required. Lemma 1. Let / be a cost-sharing method. If ðN; q; CÞ 2 P is such that CðxÞ ¼ 0 for every x 2 ½0; q, then /i ðN; q; CÞ ¼ 0 for every i 2 N. Proof. This holds due to the Budget Balance condition and the non-negativity of /. h If ðN; q; CÞ 2 P we write C Nni ðxÞ ¼ Cðx; 0i Þ for all x 2 ½0; qNni . The following lemma proves that No Merging or Splitting implies the following property. Zero Independence: Let ðN; q; CÞ 2 P, and i 2 N. If qi ¼ 0 then /i ðN; q; CÞ ¼ 0 and /Nni ðN; q; CÞ ¼ /ðN n i; qNni ; C Nni Þ. That is, if an agent does not demand anything he/she does not have to pay anything and the others pay as if the former agent were not present. This is a very natural property that has already been used by Moulin and Shenker (1994) and by Sprumont (1998). Lemma 2. If / satisfies No Merging or Splitting, then / satisfies Zero Independence. Proof. Let ðN; q; CÞ 2 P and i 2 N such that qi ¼ 0. Assume first that jNj ¼ n P 3. Let j 2 N n i, and write I ¼ fi; jg. Applying No Merging or Splitting to ðN n i; qNni ; C Nni Þ and agent j, it holds that
/j ðN n i; qNni ; C Nni Þ ¼ /i ðððN n iÞ n jÞ [ I; q; CÞ þ /j ðððN n iÞ n jÞ
X /i0 ðN0 ; q0 ; C 0 Þ ¼ /i ðN; q; CÞ: 0
i 2I
This axiom avoids splitting manipulations. It relates two problems with different sets of agents. An agent ‘‘splits’’ into different agents whose total consumption equals the consumption of the original one. So agent i splits into agents in I and consumption of i equals the consumption of agents in I. The other consumptions do not change. Accordingly the cost function does not change since the variable associated with agent i has split into the variables associated with the split agents. The sum of those variables plays the role of the variable associated with the original agent i. No Merging or Splitting requires the cost share of the original agent to be equal to the sum of the cost shares of split agents. Observe that this axiom does not impose any requirement on the cost shares of the remaining agents. Notice also that when i splits into several agents then agent i is also present. So the consumption of i is divided into several quantities and it is assumed that these quantities are associated with i and some new agents. As Sprumont (2005) writes, the counterpart of No Merging or Splitting for price mechanisms with perfectly divisible goods appears in Tauman (1988) as Axiom 500 . There are two characterizations of the Aumann–Shapley price mechanism with perfectly divisible goods by means of stronger axioms than Axiom 500 . The characterization given by Young (1985b) and the one given by Monderer and Neyman (1988). Both characterizations are given for cost functions which are not necessarily increasing. Writing the latter one using Axiom 500 , they prove that Axiom 500 , Symmetry (prices do not depend on the names of the agents), Rescaling (if the demands are rescaled so are the prices) and Monotonicity characterize the Aumann–Shapley price mechanism. So we prove in the present paper that in the discrete case (and considering non-decreasing cost functions) Symmetry and Rescaling can be dropped. We have to say that in the discrete case demands could not be changed in scale in any way, but only with non-negative integers.
[ I; q; CÞ: Since ððN n iÞ n jÞ [ I ¼ N, it can be concluded that
/j ðN n i; qNni ; C Nni Þ ¼ /i ðN; q; CÞ þ /j ðN; q; CÞ for all j 2 N n i:
ð1Þ
That implies
X
/j ðN n i; qNni ; C Nni Þ ¼ ðn 1Þ/i ðN; q; CÞ þ
j2Nni
X
/j ðN; q; CÞ:
j2Nni
Then by the definition of C Nni and Budget Balance condition it holds that
CðqÞ ¼ C Nni ðqNni Þ ¼ ðn 1Þ/i ðN; q; CÞ þ CðqÞ /i ðN; q; CÞ: Hence ðn 2Þ/i ðN; q; CÞ ¼ 0, and consequently /i ðN; q; CÞ ¼ 0. Furthermore from expression (1) it follows /j ðN; q; CÞ ¼ /j ðN n i; qNni ; C Nni Þ for all j 2 N n i as was to be proved. Observe that the case jNj ¼ 1 is trivial, so assume that N ¼ fi; jg; i – j. Let ‘ R N, and write I ¼ fi; ‘g; N 0 ¼ ðN n iÞ [ I, and define q0 ¼ ðq; 0‘ Þ and C 0 2 CðN 0 ; q0 Þ by C 0 ðxÞ ¼ CðxN Þ for all x 2 ½0; q0 . Then jN 0 j ¼ 3 and the result of the first part of the proof can be applied. Since q0i ¼ qi ¼ 0, it holds that /i ðN 0 ; q0 ; C 0 Þ ¼ 0. On the other hand q0‘ ¼ 0, hence /i ðN 0 ; q0 ; C 0 Þ ¼ /i ðN; q; CÞ. The two equalities together give /i ðN; q; CÞ ¼ 0. In addition from the Budget Balance condition it holds that /j ðN; q; CÞ ¼ /j ðN n i; qNni ; C Nni Þ and the proof is complete. h Sprumont (2005) shows that No Merging or Splitting implies the following property. Weak Symmetry: Let ðN; q; CÞ 2 P; i 2 N, and p : N ! N be a bijection. If i 2 N is such that pðjÞ ¼ j for all j 2 N n i, then /pðiÞ pðN; q; CÞ ¼ /i ðN; q; CÞ. According to this axiom, if an agent is renamed his/her cost share does not change. Notice that weak symmetry does not say anything about the agents whose names are not modified. This statement is also valid when the domain of the cost function is ½0; q instead of NN . The proof would be as in Sprumont (2005), and is therefore omitted here.
Please cite this article in press as: Albizuri, M. J., et al. Monotonicity and the Aumann–Shapley cost-sharing method in the discrete case. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.03.044
4
M.J. Albizuri et al. / European Journal of Operational Research xxx (2014) xxx–xxx
Lemma 3 Sprumont, 2005. If / satisfies No Merging or Splitting then / satisfies Weak Symmetry. The next lemma uses the following basis. Let M 2 N such that jMj ¼ 2s 1 for a positive integer s. Define MðsÞ ¼ fR # M : jRj ¼ sg and Mðs 1Þ ¼ fR0 # M : jR0 j ¼ s 1g. Notice that R0 2 Mðs 1Þ if and only if M n R0 2 MðsÞ, and hence jMðs 1Þj ¼ jMðsÞj. For each R 2 MðsÞ define the vector uR 2 RMðs1Þ by
uR ðR0 Þ ¼
0
1; if R R;
ð2Þ
0; otherwise:
Lemma 4. Let ðT [ S; q; CÞ 2 P such that T \ S ¼ £ and the agents in S are symmetric. If / satisfies No Merging or Splitting, and for every bijection p : N ! N such that pðkÞ ¼ k for every k 2 T it holds P P then /i ðT [ S; q; CÞ ¼ k2T /k pðT [ S; q; CÞ ¼ k2T /k ðT [ S; q; CÞ, /j ðT [ S; q; CÞ for every i; j 2 S. Proof. Let M 2 N such that S # M; M \ T ¼ £ and jMj ¼ 2s 1. Define MðsÞ as above and write lðsÞ ¼ jMðsÞj. For each R 2 MðsÞ let pR be a bijection of N such that pR S ¼ R, and pR ðkÞ ¼ k for every k 2 T. Consider the problem pR ðT [ S; q; CÞ (notice that this problem is independent of the particular choice of pR since the agents in S are symmetric). By the Budget Balance condition it follows that i2R
ð3Þ
k2T
for all R 2 MðsÞ. Define Mðs 1Þ also as above. Now for any R0 2 Mðs 1Þ, by Lemma 3, there exists a number cðR0 Þ such that
0 0 /i T [ R0 [ i; pR [i q; pR [i C ¼ cðR0 Þ
for all i 2 M n R0 . Hence expression (3) can be rewritten as
X
cðR n iÞ ¼ CðqÞ
i2R
X /k pR ðT [ S; q; CÞ k2T
for all R 2 MðsÞ. This is a system of lðsÞ linear equations in the lðs 1Þ ¼ lðsÞ variables cðR0 Þ; R0 2 Mðs 1Þ that can be rewritten in the form
uR c ¼ CðqÞ
X
/k pR ðT [ S; q; CÞ for all R 2 MðsÞ;
k2T
where c ¼ ðcðR0 ÞÞ 2 RMðs1Þ and uR 2 RMðs1Þ is defined in expression P P (2). Since k2T /k pðT [ S; q; CÞ ¼ k2T /k ðT [ S; q; CÞ, clearly
!
cðR0 Þ ¼
X 1 CðqÞ /k ðT [ S; q; CÞ s k2T
for every R0 is a solution. Moreover, since uR form a basis for RMðs1Þ the system has a unique solution. Then for all i 2 S, choosing R0 ¼ S n i yields !
/i ðT [ S; q; CÞ ¼
X 1 CðqÞ /k ðT [ S; q; CÞ ; s k2T
and the conclusion follows. h The next two proofs use the fact that each cost function C 2 CðN; qÞ can be expressed as a linear combination of primitive cost functions C r 2 CðN; qÞ (r 2 ½0; q; r – 0). That is, P C ¼ 0–r6q ar C r , where ar 2 R and
C r ðxÞ ¼
1 if x P r; 0
e C¼C
X
e ar Cr :
ð4Þ
0–r6q
An example is given. C ¼ C ð1;0Þ þ C ð0;2Þ C ð1;2Þ . Then,
Let
N ¼ f1; 2g; q ¼ ð1; 2Þ
and
fr 2 ½0; q : 0 – r 6 qg ¼ fr1 ; r 2 ; r3 ; r 4 ; r 5 g; where r1 ¼ ð0; 1Þ; r 2 ¼ ð1; 0Þ; r3 ¼ ð1; 1Þ; r4 ¼ ð0; 2Þ and r 5 ¼ ð1; 2Þ. Hence,
The vectors uR form a basis for RMðs1Þ . The following lemma deals with symmetric agents.
X X /i pR ðT [ S; q; CÞ ¼ CðqÞ /k pR ðT [ S; q; CÞ
function (obviously it is non-decreasing) for which all the agents with the same consumption demand are symmetric, and in particular those whose demand is 1. It is therefore possible to write
otherwise:
P Given this expression C ¼ 0–r6q ar C r , for each q 2 N; 1 6 P q 6 qðNÞ (remind qðNÞ ¼ i2N qi ), define aq ¼ max0–r 6 q : ar , rðNÞ ¼ q e ¼P a r ¼ arðNÞ ar P 0. Also let C a C , that is a cost and e rðNÞ r 0–r6q
ar1 ðNÞ ¼ ar2 ðNÞ ¼ a1 ¼ maxf1; 0g ¼ 1; ar3 ðNÞ ¼ ar4 ðNÞ ¼ a2 ¼ maxf1; 0g ¼ 1; ar5 ðNÞ ¼ a3 ¼ maxf1g ¼ 1; and therefore,
e ¼ C ð0;1Þ þ C ð1;0Þ þ C ð1;1Þ þ C ð0;2Þ C ð1;2Þ : C a r1 ¼ 1; e a r2 ¼ 0, e a r3 ¼ 1; e a r4 ¼ 0 and e a r5 ¼ 0. And thus, Moreover, e (4) holds. Now follow the general case, and consider all the expressions e is symmetric for agents with demand 1, and (4) for C in which C e a r P 0 for all r. Define the index i of C to be the minimum number of terms which appear in the sum of such an expression. Lemma 5. If / is a cost-sharing method that satisfies Monotonicity and No Merging or Splitting, and ðN; q; CÞ 2 P is such that q 6 1N , then /ðN; q; CÞ ¼ /ASh ðN; q; CÞ. Proof. Let ðN; q; CÞ 2 P be such that q 6 1N . By Lemma 2 and since /ASh satisfies Zero Independence, it can be assumed that q ¼ 1N . The lemma can be proved by induction on the index i. If i ¼ 0 then all the agents in N are symmetric, so from Lemma 4, for the case T ¼ ; and S ¼ N, it follows /ðN; q; CÞ ¼ /ASh ðN; q; CÞ. Assume now that / coincides with /ASh whenever the index of C is at most i, and let C have index i þ 1 with expression e Piþ1 e C¼C k¼1 a r k C r k . Let S ¼ fi 2 N : ðr k Þi ¼ 1 for every k ¼ 1; . . . ; i þ 1g and assume e P e that i R S. Define D ¼ C k:ðr k Þi ¼1 a r k C r k . Observe that D is a cost P a rk C rk , where e a rk P 0. The index function because D ¼ C þ k:ðrk Þi ¼0 e P e of D is at most i and @ i DðxÞ ¼ @ i CðxÞ þ @ i k:ðr k Þi ¼0 a r k C r k ðxÞ ¼ @ i CðxÞ for all x 2 ½0; q s. t. xi < qi , so by induction and Monotonicity it follows that ASh /i ðN; q; CÞ ¼ /i ðN; q; DÞ ¼ /ASh i ðN; q; DÞ ¼ /i ðN; q; CÞ:
It remains to show that /i ðN; q; CÞ ¼ /ASh ðN; q; CÞ when i 2 S. But i notice that the agents in S are symmetric in ðN; q; CÞ. Moreover since it has been proved that /i ðN; q; CÞ ¼ /ASh i ðN; q; CÞ when i 2 N n S, Lemma 4 can be applied to obtain /i ðN; q; CÞ ¼ /j ðN; q; CÞ for every i; j 2 S. By the Budget Balance requirement the proof is complete. h Remark 1. Lemma 5 does not follow from Young’s characterization (1985a) of the Shapley value for transferable utility games. Observe that there is a bijection between fðN; 1N ; CÞ : C 2 CðN; 1N Þg and the set of transferable utility games with N as set of players. Indeed, ðN; 1N ; CÞ has associated the transferable utility game defined by v ðSÞ ¼ cð1S Þ for every S # N. Moreover, /ASh ðN; 1N ; CÞ coincides with the Shapley value of v. Young (1985a) characterizes the Shapley value with Monotonicity and Symmetry. However in Lemma 5 we cannot apply that the cost-sharing method satisfies Symmetry. The cost-sharing method satisfies No Merging or Splitting, and therefore Weak Symmetry (Lemma 3). As its name reveals, Weak
Please cite this article in press as: Albizuri, M. J., et al. Monotonicity and the Aumann–Shapley cost-sharing method in the discrete case. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.03.044
M.J. Albizuri et al. / European Journal of Operational Research xxx (2014) xxx–xxx
Symmetry does not imply Symmetry. Indeed, if we look at the formulation of Weak Symmetry we have that pðiÞ ¼ i or pðiÞ R N. However, in the formulation of Symmetry pðiÞ may be any player of N, that is, pðiÞ 2 N. On the other hand, by No Merging or Splitting, if all the agents in N are symmetric then the agents in N have the same allocations. This can be proved applying Lemma 4, taking T ¼ ; and S ¼ N. And obviously this property is weaker than Symmetry. And this is the proof of Theorem 1. Proof. It is clear that /ASh satisfies No Merging or Splitting and Monotonicity. Conversely, let / be a cost-sharing method that satisfies these two properties. For any k ¼ 0; 1; . . ., let DðkÞ ¼ fðN; q; CÞ 2 P : jfi 2 N : qi > 1gj 6 kg, i.e. the set of problems where no more than k agents demand several units. To prove this theorem it must be shown that /ðN; q; CÞ ¼ /ASh ðN; q; CÞ for every ðN; q; CÞ 2 DðkÞ and for every k ¼ 0; 1; 2; . . . Proceed by induction on k. From Lemma 5 it follows that /ðN; q; CÞ ¼ /ASh ðN; q; CÞ for every ðN; q; CÞ 2 Dð0Þ. So let k P 0 be fixed and assume that /ðN; q; CÞ ¼ /ASh ðN; q; CÞ for every ðN; q; CÞ 2 DðkÞ. Now let ðN; q; CÞ 2 Dðk þ 1Þ and prove that /ðN; q; CÞ ¼ /ASh ðN; q; CÞ. By the induction hypothesis it can be assumed that ðN; q; CÞ 2 Dðk þ 1Þ n DðkÞ, i.e. exactly k þ 1 agents demand several units. So consider a problem ðN; q; CÞ 2 Dðk þ 1Þ. By Lemma 2, it can also be assumed that q P 1N . STEP 1: /i ðN; q; CÞ ¼ /ASh ðN; q; CÞ for every i 2 N such that qi P 2. i Indeed, let I 2 N such that N \ I ¼ fig and jIj ¼ qi , and consider a consumption profile ðN 0 ; q0 Þ, where N 0 ¼ ðN n iÞ [ I, q0 is such that q0j ¼ qj for all j 2 N n i and q0j ¼ 1 for all j 2 I. Define C 0 2 CðN 0 ; q0 Þ P by C 0 ðxÞ ¼ C xNni ; i0 2I xi0 . Then by No Merging or Splitting, the induction hypothesis and the definition of /ASh we get
/i ðN; q; CÞ ¼
X X ASh 0 /i0 ðN0 ; q0 ; C 0 Þ ¼ /i0 ðN ; q0 ; C 0 Þ i0 2I
5
Remark 3. If the domain of cost functions C 2 CðN; qÞ is NN instead P of ½0; q, then we cannot write C ¼ 0–r6q ar C r , which is employed to prove Lemma 5 and Theorem 1. However, Monotonicity and No Merging or Splitting can be written as they are in the present work. And the latter axiom implies that /ðN; q; CÞ ¼ /ðN; q; C 0 Þ when P C ¼ C 0 on ½0; q. Hence, /ðN; q; CÞ ¼ / N; q; 0–r6q ar C r . So the proofs of Lemma 5 and Theorem 1 are also valid with the same axioms. Remark 4. As mentioned in the Introduction, in this paper we show that in Sprumont (2005) Additivity and Dummy can be replaced by Monotonicity when No Merging or Splitting is present. Notice that No Merging or Splitting gives rise to a variety of results and plays a key role in the proof of Theorem 1. This axiom implies Zero Independence (Lemma 2), and moreover, Lemma 4 for symmetric agents. Zero Independence, Lemma 4 and Monotonicity imply Lemma 5, which is the start point for the induction hypothesis in the proof of Theorem 1. Finally, No Merging or Splitting allows to prove step 1 in that proof, since the cost shares of an agent can be calculated by means of cost shares of agents with lower demands. All these results and Monotonicity determine the Aumann–Shapley method. Remark 5. Sprumont (2005) provides two variations of the axiomatization for the Aumann–Shapley method. No Merging or Splitting relates cost-sharing problems with different sets of agents. Sprumont (2005) changes this axiom in order to relate cost-sharing problems with equal sets of agents. In the second variation he let agents consume bundles of goods. We point that Additivity and Dummy can also be replaced by Monotonicity in Sprumont’s variations of his main axiomatization.
4. Conclusion
i0 2I
¼ /ASh i ðN; q; CÞ: STEP 2: /i ðN; q; CÞ ¼ /ASh ðN; q; CÞ for every i 2 N such that qi ¼ 1. i Denote N 1 ¼ fi 2 N : qi ¼ 1g. It will be proved that /i ðN; q; CÞ ¼ /ASh i ðN; q; CÞ by induction on the index i of C. If i ¼ 0 then all the agents in N 1 are symmetric. Then by step 1 Lemma 4 can be applied and the conclusion follows. Assume now that / coincides with /ASh whenever the index of C is at most i, and let C have index i þ 1 with expression e Piþ1 e C¼C k¼1 a r k C r k . Let S ¼ fi 2 N 1 : ðrk Þi ¼ 1 for every k ¼ 1; . . . ; i þ 1g and assume e P e that i R S. Define the cost function D ¼ C k:ðr k Þi ¼1 a r k C r k . The index of D is at most i and @ i DðxÞ ¼ @ i CðxÞ for all x 2 ½0; q s. t. xi < qi , so by induction and Monotonicity it follows that ASh /i ðN; q; CÞ ¼ /i ðN; q; DÞ ¼ /ASh i ðN; q; DÞ ¼ /i ðN; q; CÞ:
To finish the proof it remains to show that /i ðN; q; CÞ ¼ /ASh i ðN; q; CÞ when i 2 S. But the agents in S are symmetric in ðN; q; CÞ, and taking into account step 1, and that it has been proved that /i ðN; q; CÞ ¼ /ASh ðN; q; CÞ when i 2 N 1 n S, the proof is complete by i Lemma 4. h Remark 2. If cost functions which are not necessarily nondecreasing (in this case cost shares might be negative) are considered, another axiom must be added to obtain the axiomatization. It suffices to require cost shares to be non-negative if the cost function is non-decreasing. Notice that Lemma 1 holds and the rest too. e is not needed for the proofs, just the expression Furthermore, C P C ¼ 0–r6q ar C r .
In this paper we offer an axiomatization of the Aumann– Shapley method in the discrete case by employing Monotonicity. In doing so, we fill the gap with regard to cost-sharing problems in which goods are perfectly divisible. At the same time we find that weaker axioms are needed in the discrete case. This is revealed by looking at the characterization of the Aumann–Shapley price mechanism introduced by Monderer and Neyman (1988). It is an open question whether axioms could be weakened in the latter case, especially, if Symmetry is not needed as an explicit requirement. On the other hand, there are axiomatizations of other cost-sharing methods in the discrete case in which Additivity and Dummy are employed. For example in Sprumont (2000) and Moulin (1995). It is also an open question whether Monotonicity, possibly with other axioms, could replace the two axioms. Another setting connected with cost-sharing problems is that of bankruptcy problems. Bergantiños and Vidal-Puga (2006) characterize a family of additive rules for discrete bankruptcy problems. It deserves further research to examine the role of the Monotonicity property for these discrete problems. Acknowledgements We thank two anonymous referees for their helpful comments. Financial support from the University of the Basque Country (GIU13/31 and UFI11/51) and the Ministry of Economic Affairs and Competitiveness of Spain (ECO2012-33618) is gratefully acknowledged.
Please cite this article in press as: Albizuri, M. J., et al. Monotonicity and the Aumann–Shapley cost-sharing method in the discrete case. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.03.044
6
M.J. Albizuri et al. / European Journal of Operational Research xxx (2014) xxx–xxx
References Aumann, S., & Shapley, L. (1974). Values of nonatomic games. Princeton, NJ: Princeton Univ. Press. Bahel, E., & Trudeau, C. (2013). A discrete cost sharing model with technological cooperation. International Journal of Game Theory, 42, 439–460. Bergantiños, G., & Vidal-Puga, J. (2006). Additive rules in discrete allocation problems. European Journal of Operational Research, 172, 971–978. Billera, L., & Heath, D. (1982). Allocation of shared costs: A set of axioms yielding a unique procedure. Mathematics of Operations Research, 7, 32–39. Billera, L., Heath, D., & Raanan, J. (1978). Internal telephone billing rates: A novel application of non-atomic game theory. Operations Research, 26, 956–965. Calvo, E., & Santos, J. C. (2000). A value for multichoice games. Mathematical Social Sciences, 40, 341–354. Castaño-Pardo, A., & García-Díaz, A. (1995). Highway cost allocation: An application of the theory of non-atomic games. Transportation Research, 29, 187–203. Chen, C., Iyengar, G., & Moallemi, C. C. (2013). An axiomatic approach to systemic risk. Management Science, 59, 1373–1388. Dubey, P., Neyman, A., & Weber, R. J. (1981). Value theory without efficiency. Mathematics of Operations Research, 6, 122–128. Faria, E., Barroso, L. A., Kelman, R., Granville, S., & Pereira, M. V. (2009). Allocation of firm-energy rights among hydro plants: An Aumann–Shapley approach. IEEE Transactions on Power Systems, 24, 541–551. Haviv, M. (2001). The Aumann–Shapley price mechanism for allocating congestion costs. Operations Research Letters, 29, 211–215. Hougaard, J. L., & Tind, J. (2009). Cost allocation and convex data envelopment. European Journal of Operational Research, 194, 939–947. Krus´, L., & Bronisz, P. (2000). Cooperative game solution concepts to a cost allocation problem. European Journal of Operational Research, 122, 258–271. McLean, R. P., Pazgal, A., & Sharkey, W. W. (2004). Potential, consistency, and cost allocation prices. Mathematics of Operations Research, 29, 602–623. Mirman, L., & Tauman, Y. (1982). Demand compatible equitable cost sharing prices. Mathematics of Operations Research, 7, 40–56.
Monderer, D., & Neyman, A. (1988). Values of nonatomic games: The method of multilinear approximation. In The shapley value: Essays in honor of Lloyd Shapley (pp. 217–234). Cambridge: Cambridge Univ. Press. Moulin, H. (1995). On additive methods to share joint costs. Japanese Economic Review, 46, 303–332. Moulin, H., & Shenker, S. (1994). Average cost pricing versus serial cost sharing: An axiomatic comparison. Journal of Economic Theory, 64, 178–201. Myerson, R. (1980). Conference structures and fair allocation rules. International Journal of Game Theory, 9, 169–182. Pierru, A. (2010). Allocating the CO2 emissions of an oil refinery with Aumann– Shapley prices: A reply. Energy Economics, 32, 746–748. Samet, D., & Tauman, Y. (1982). The determination of marginal cost prices under a set of axioms. Econometrica, 50, 895–909. Samet, D., Tauman, Y., & Zang, I. (1984). An application of the Aumann–Shapley prices for cost allocation in transportation problems. Mathematics of Operations Research, 9, 25–42. Shapley, L. (1953). A value for n-person games. In H. W. Kuhn & W. Tucker (Eds.), Contributions to the theory of games II (pp. 307–317). Princeton, NJ: Princeton Univ. Press. Shubik, M. (1962). Incentives, decentralized control, the assignment of joint costs and internal pricing. Management Science, 8, 325–343. Sprumont, Y. (1998). Ordinal cost sharing. Journal of Economic Theory, 81, 126–162. Sprumont, Y. (2000). Coherent cost-sharing rules. Games and Economic Behavior, 33, 126–144. Sprumont, Y. (2005). On the discrete version of the Aumann–Shapley cost-sharing method. Econometrica, 73(5), 1693–1712. Wang, Y. (1999). The additivity and dummy axioms in the discrete cost-sharing model. Economics Letters, 64, 187–192. Young, H. P. (1985a). Monotonic solutions of cooperative games. International Journal of Game Theory, 14, 65–72. Young, H. P. (1985b). Producer incentives in cost allocation. Econometrica, 53, 757–765.
Please cite this article in press as: Albizuri, M. J., et al. Monotonicity and the Aumann–Shapley cost-sharing method in the discrete case. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.03.044
Contents lists available at ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Decision Support
Monotonicity and the Aumann–Shapley cost-sharing method in the discrete case M.J. Albizuri ⇑, H. Díez, A. Sarachu University of the Basque Country, Faculty of Economics and Business, Department of Applied Economics IV, Bilbao, Spain
a r t i c l e
i n f o
Article history: Received 4 August 2013 Accepted 27 March 2014 Available online xxxx Keywords: Cost sharing Aumann–Shapley method Monotonicity
a b s t r a c t We give an axiomatization of the Aumann–Shapley cost-sharing method in the discrete case by means of monotonicity and no merging or splitting (Sprumont, 2005). Monotonicity has not yet been employed to characterize this method in such a case, by contrast with the case in which goods are perfectly divisible, for which Monderer and Neyman (1988) and Young (1985b) characterize the Aumann–Shapley price mechanism. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction In this paper we provide an axiomatization for the Aumann– Shapley cost-sharing method in the discrete case by employing a monotonicity property. Aumann–Shapley pricing is a method for solving cost-sharing problems in which agents consume a quantity of possibly different goods. Assuming that a finite set N ¼ f1; . . . ; ng denotes the set of agents i; qi is the quantity of good i demanded by agent i and q is the corresponding consumption profile, we want to split CðqÞ among the agents. Goods may be priced instead of sharing the cost CðqÞ. The two approaches are equivalent, and we have chosen the latter, following Sprumont (2005). In the discrete case agents can consume several units of the good. The consumption qi of agent i is a non-negative integer which indicates how many units of good i are consumed by agent i. A simple cost-sharing method for solving such problems is the Shapley– Shubik method introduced by Shubik (1962). Given a coalition S # N of agents, Shubik considers the cost of the consumptions of those agents: v ðSÞ ¼ CðqS ; 0NnS Þ; and calculates the Shapley value (1953) of the transferable utility game v with N as the set of players. This Shapley value gives precisely the assignment by the Shapley– Shubik method. The Aumann–Shapley method is also defined by means of a transferable utility game, but now more information about cost is taken into account. Indeed, costs of intermediate consumptions of agents are considered and not just those associated with consumptions qi . Given an agent i, each unit consumed by i is viewed as a different player, so there is a set N i of jN i j ¼ qi players ⇑ Corresponding author. Tel.: +34 94 601 3805. E-mail addresses: [email protected] (M.J. (H. Díez), [email protected] (A. Sarachu).
Albizuri),
[email protected]
S and the grand set N q ¼ i2N N i . Therefore there are coalitions of players, which are subsets of N q . Given S # N q , dðSÞ ¼ ðjS \ N i jÞi2N represents the units demanded by agents in S and the cost associated with those demands is v q ðSÞ ¼ CðdðSÞÞ. The Shapley value of v q gives allocations for players in Nq . Since each agent i is represented by the elements in N i , the sum of the allocations of these elements is by definition the allocation for agent i provided by the Aumann–Shapley method. The Aumann–Shapley pricing originates in the work of Billera, Heath, and Raanan (1978), when these authors used Aumann and Shapley’s (1974) theory of values for nonatomic games to set the telephone billing rates in order to share the cost of service among the users. Subsequently, Billera and Heath (1982) and Mirman and Tauman (1982) transfer the axiomatic approach introduced by Aumann and Shapley (1974) to characterize this pricing mechanism for cost-sharing problems in which goods are perfectly divisible. Samet and Tauman (1982) offer an axiomatic characterization for the Aumann–Shapley price mechanism based on the one given by Dubey, Neyman, and Weber (1981) for the semivalues; and latterly, McLean, Pazgal, and Sharkey (2004) propose another one using the potential and consistency. We also find many applications of Aumann–Shapley pricing, notably from the pioneering works as the above by Billera et al. (1978) and Samet, Tauman, and Zang (1984) to the contributions made by Castaño-Pardo and García-Díaz (1995), Krus´ and Bronisz (2000) and Haviv (2001), or the more recent works by Faria, Barroso, Kelman, Granville, and Pereira (2009), Pierru (2010) and Chen, Iyengar, and Moallemi (2013). These applications range from the pricing of transportation, water, phone, electricity or highways, to the allocation of firmenergy rights, the allocation of the CO2 emissions, the analysis of
http://dx.doi.org/10.1016/j.ejor.2014.03.044 0377-2217/Ó 2014 Elsevier B.V. All rights reserved.
Please cite this article in press as: Albizuri, M. J., et al. Monotonicity and the Aumann–Shapley cost-sharing method in the discrete case. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.03.044
2
M.J. Albizuri et al. / European Journal of Operational Research xxx (2014) xxx–xxx
systemic risk or the study of multiple commodity situations. Another related work is Hougaard and Tind (2009), where parametric programming is employed to study the Aumann–Shapley pricing. Only more recently the discrete case has been given attention. The Aumann–Shapley method is characterized in two papers. In the first, by Calvo and Santos (2000), the method is characterized by means of Balanced Contributions, as Myerson (1980) does for the Shapley value, and requiring agents who consume nothing to pay zero. In the second Sprumont (2005) employs Additivity, Dummy and No Merging or Splitting. Dummy is the natural extension for cost-sharing problems of the Dummy axiom for transferable utility games. It requires that dummies get zero. An agent i is a dummy if there are no consumption profile whose cost changes if i consumes one more unit. No Merging or Splitting, which is described in detail in Section 3, prevents splitting manipulations. Accordingly, if an agent i splits into several agents (and therefore splits his/her consumption into several consumptions), the split agents have to pay in sum the same quantity as agent i had to. The characterization by Sprumont (2005) can be seen as formed by basic axioms, like the one given by Mirman and Tauman (1982) for the Aumann–Shapley price mechanism with perfectly divisible goods and Shapley’s (1953) characterization of his value. A look at other characterizations for the Aumann–Shapley price mechanism with perfectly divisible goods reveals that Monotonicity has also been employed, as Young (1985a) does when he characterizes the Shapley value. Monotonicity for cost-sharing problems states that if the marginal costs associated with an agent at all consumption levels decrease, then the cost share corresponding to that agent cannot increase. Young (1985b) and Monderer and Neyman (1988) obtain two characterizations for the Aumann–Shapley price mechanism using this axiom. In this paper we provide precisely an axiomatization for the Aumann–Shapley method with Monotonicity. We prove that this method is the only one that satisfies Monotonicity and No Merging or Splitting. Thus, we prove that Additivity and Dummy can be replaced by Monotonicity in the characterization of Sprumont (2005). Observe that the same happens when Young (1985a) characterizes the Shapley value (1953) for transferable utility games with Symmetry and Monotonicity. Young (1985a) proves that Additivity and Dummy can be replaced by Monotonicity in Shapley’s (1953) characterization. Moreover, an examination of the characterizations of the Aumann–Shapley price mechanism given by Young (1985b) and by Monderer and Neyman (1988) reveals that they employ stronger axioms than No Merging or Splitting. Wang (1999) characterizes the methods satisfying Additivity and Dummy in the discrete case. Clearly, all such methods satisfy Monotonicity. In this work we prove that in the presence of No Merging or Splitting, Monotonicity implies Additivity and Dummy. The presence of No Merging or Splitting is crucial for the result. For example, Bahel and Trudeau (2013) study a more general discrete cost sharing model where any two properties among Additivity, Dummy and Monotonicity fail to imply the third. It must be said that, like Sprumont (2005), we deal with nondecreasing cost functions, unlike Young (1985b) and Monderer and Neyman (1988), who use cost functions which are not necessarily non-decreasing. In addition, as Sprumont (2005) does, we assume that cost shares are non-negative. If we considered any cost function we would characterize the Aumann–Shapley method just requiring cost shares to be non-negative if the cost function is nondecreasing. The rest of the paper is structured as follows. Section 2 gives preliminaries, Section 3 presents the axiomatization and the paper ends with conclusions in Section 4 and references.
2. Preliminaries First some notation must be given. Denote by N the set of nonnegative integers, and by N the set of non-empty finite subsets of N. If N 2 N , denote by jNj the cardinality of N, and let NN be the Cartesian product of N with itself jNj times. Coordinates of elements in NN are indexed by the elements of N. If x ¼ ðxi Þi2N 2 NN P M and M # N we write xðMÞ for denotes the i2M xi , and xM 2 N restriction of x to M. If x; y 2 NN we write x 6 y to denote xi 6 yi for all i 2 N. Some distinguished vectors in NN are 0 ¼ ð0; . . . ; 0Þ and for every M # N its indicator 1M , defined by 1M i ¼ 1 if i 2 M i and 1M i ¼ 0 otherwise. We write 1 instead of the more cumbersome 1fig and N n i instead of N n fig. A pair ðN; qÞ defines a consumption profile, where N 2 N is the set of agents and q 2 NN is the list of their consumptions. If q; q0 2 NN , write ½q; q0 for the set fx 2 NN : q 6 x 6 q0 g. For each consumption profile ðN; qÞ, denote by CðN; qÞ the set of cost functions for ðN; qÞ, that is non-decreasing functions C : ½0; q NN ! Rþ such that Cð0Þ ¼ 0.1 A cost-sharing problem is a triple ðN; q; CÞ, where ðN; qÞ is a consumption profile and C 2 CðN; qÞ. The set of all problems is denoted by P. Then a cost-sharing method is a mapping / that assigns to each cost-sharing problem ðN; q; CÞ 2 P a vector /ðN; q; CÞ 2 RNþ P that satisfies the Budget Balance condition: i2N /i ðN; q; CÞ ¼ CðqÞ, that is, that exactly shares all the costs. We give some more notation. Let p : N ! N be a bijection. If pN N 2 N , write pN ¼ fpðiÞ : i 2 Ng. If x 2 RNþ , define px 2 Rþ by ðpxÞpðiÞ ¼ xi for all i 2 N. Moreover if ðN; qÞ is a consumption profile and C 2 CðN; qÞ, define pC 2 CðpN; pqÞ by pCðpxÞ ¼ CðxÞ. Write pðN; q; CÞ ¼ ðpN; pq; pCÞ. Given ðN; q; CÞ 2 P two agents i; j 2 N are said to be symmetric if for any bijection p : N ! N such that pðiÞ ¼ j; pðjÞ ¼ i, and pðkÞ ¼ k for every k 2 N n fi; jg it holds that pðN; q; CÞ ¼ ðN; q; CÞ. Notice that the consumption levels corresponding to i and j must be equal, and the cost function C must be symmetric in their demands. The next step is to formalize the Aumann–Shapley method. Given ðN; q; CÞ 2 P, let fN i gi2N be a family of pairwise disjoint sets S such that jN i j ¼ qi , and let N q ¼ i2N N i . For each S # N q define the demand vector xðSÞ ¼ ðjS \ N i jÞi2N and consider the cooperative game ðN q ; v q Þ defined by v q ðSÞ ¼ CðxðSÞÞ. Denote the Shapley value of this game by /Sh ðN q ; v q Þ. The Aumann–Shapley method assigns to each ðN; q; CÞ 2 P the vector /ASh ðN; q; CÞ defined for all i 2 N by
/ASh i ðN; q; CÞ ¼
X Sh q /j ðN ; v q Þ: j2N i
Notice that by the anonymity of the Shapley value the real numq q bers /Sh j ðN ; v Þ are independent of the choice of the sets N i , and consequently the Aumann–Shapley method is well defined. 3. An axiomatization of the Aumann–Shapley method The first axiomatization presented in this paper is based on two axioms. Before it is introduced some notation is required. Let ðN; q; CÞ 2 P; i 2 N, and x 2 ½0; q such that xi < qi . Then we write @ i CðxÞ ¼ Cðx þ 1i Þ CðxÞ. So @ i CðxÞ denotes the impact on the cost function when agent i consumes one more unit at consumption vector x. That is, @ i CðxÞ is the marginal cost associated with agent i at consumption vector x. Monotonicity: Let ðN; q; CÞ; ðN; q; C 0 Þ 2 P, and i 2 N. If for all x 2 ½0; q such that xi < qi it holds @ i CðxÞ 6 @ i C 0 ðxÞ, then /i ðN; q; CÞ 6 /i ðN; q; C 0 Þ. 1 The domain of C differs from the model of Sprumont (2005), since he considers that the domain of the cost function C is all NN , instead of ½0; q. We make this distinction to provide a more concise and elegant presentation of our paper (see Remark 3 on page 11).
Please cite this article in press as: Albizuri, M. J., et al. Monotonicity and the Aumann–Shapley cost-sharing method in the discrete case. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.03.044
3
M.J. Albizuri et al. / European Journal of Operational Research xxx (2014) xxx–xxx
Cost functions C and C 0 reveal the costs associated with different consumption levels corresponding to agents i in N or equivalently production levels of goods i. Fixing i 2 N, if @ i CðxÞ 6 @ i C 0 ðxÞ at all feasible consumption levels, then it can be said that with the movement from C 0 to C an improvement in production efficiency has taken place with respect to good i. Monotonicity requires that i cannot be penalized in this case, that is, the cost share associated with i cannot increase. So if efficiency with respect to a particular agent increases, then the cost share associated with that agent cannot increase. The continuous version of Monotonicity (together with other axioms) is employed by Young (1985b) and by Monderer and Neyman (1988) to characterize the Aumann–Shapley price mechanism. In the continuous version the first partial derivative of the cost function is employed; the one sided first partial derivative on the boundary of the domain. Notice that with both cost shares and prices Monotonicity is stated in the same way since
/i ðN; q; CÞ 6 /i ðN; q; C 0 Þ if and only if
/i ðN; q; CÞ /i ðN; q; C 0 Þ 6 ; qi qi
that is, for agent i, cost shares and prices satisfy the same inequality. The second axiom that we employ is the one introduced by Sprumont (2005) to characterize the Aumann–Shapley cost sharing method in the discrete case. No Merging or Splitting: Let ðN; q; CÞ 2 P; i 2 N, and I 2 N be such that N \ I ¼ fig. Consider a consumption profile ðN 0 ; q0 Þ, where N 0 ¼ ðN n iÞ [ I; q0 is such that q0j ¼ qj for all j 2 N n i and P P 0 0 0 0 0 0 i0 2I qi0 ¼ qi . Define C 2 CðN ; q Þ by C ðxÞ ¼ C xNni ; i0 2I xi . Then
Theorem 1. The Aumann–Shapley method is the only cost-sharing method that satisfies Monotonicity and No Merging or Splitting. To prove this theorem some previous lemmas and further notation are required. Lemma 1. Let / be a cost-sharing method. If ðN; q; CÞ 2 P is such that CðxÞ ¼ 0 for every x 2 ½0; q, then /i ðN; q; CÞ ¼ 0 for every i 2 N. Proof. This holds due to the Budget Balance condition and the non-negativity of /. h If ðN; q; CÞ 2 P we write C Nni ðxÞ ¼ Cðx; 0i Þ for all x 2 ½0; qNni . The following lemma proves that No Merging or Splitting implies the following property. Zero Independence: Let ðN; q; CÞ 2 P, and i 2 N. If qi ¼ 0 then /i ðN; q; CÞ ¼ 0 and /Nni ðN; q; CÞ ¼ /ðN n i; qNni ; C Nni Þ. That is, if an agent does not demand anything he/she does not have to pay anything and the others pay as if the former agent were not present. This is a very natural property that has already been used by Moulin and Shenker (1994) and by Sprumont (1998). Lemma 2. If / satisfies No Merging or Splitting, then / satisfies Zero Independence. Proof. Let ðN; q; CÞ 2 P and i 2 N such that qi ¼ 0. Assume first that jNj ¼ n P 3. Let j 2 N n i, and write I ¼ fi; jg. Applying No Merging or Splitting to ðN n i; qNni ; C Nni Þ and agent j, it holds that
/j ðN n i; qNni ; C Nni Þ ¼ /i ðððN n iÞ n jÞ [ I; q; CÞ þ /j ðððN n iÞ n jÞ
X /i0 ðN0 ; q0 ; C 0 Þ ¼ /i ðN; q; CÞ: 0
i 2I
This axiom avoids splitting manipulations. It relates two problems with different sets of agents. An agent ‘‘splits’’ into different agents whose total consumption equals the consumption of the original one. So agent i splits into agents in I and consumption of i equals the consumption of agents in I. The other consumptions do not change. Accordingly the cost function does not change since the variable associated with agent i has split into the variables associated with the split agents. The sum of those variables plays the role of the variable associated with the original agent i. No Merging or Splitting requires the cost share of the original agent to be equal to the sum of the cost shares of split agents. Observe that this axiom does not impose any requirement on the cost shares of the remaining agents. Notice also that when i splits into several agents then agent i is also present. So the consumption of i is divided into several quantities and it is assumed that these quantities are associated with i and some new agents. As Sprumont (2005) writes, the counterpart of No Merging or Splitting for price mechanisms with perfectly divisible goods appears in Tauman (1988) as Axiom 500 . There are two characterizations of the Aumann–Shapley price mechanism with perfectly divisible goods by means of stronger axioms than Axiom 500 . The characterization given by Young (1985b) and the one given by Monderer and Neyman (1988). Both characterizations are given for cost functions which are not necessarily increasing. Writing the latter one using Axiom 500 , they prove that Axiom 500 , Symmetry (prices do not depend on the names of the agents), Rescaling (if the demands are rescaled so are the prices) and Monotonicity characterize the Aumann–Shapley price mechanism. So we prove in the present paper that in the discrete case (and considering non-decreasing cost functions) Symmetry and Rescaling can be dropped. We have to say that in the discrete case demands could not be changed in scale in any way, but only with non-negative integers.
[ I; q; CÞ: Since ððN n iÞ n jÞ [ I ¼ N, it can be concluded that
/j ðN n i; qNni ; C Nni Þ ¼ /i ðN; q; CÞ þ /j ðN; q; CÞ for all j 2 N n i:
ð1Þ
That implies
X
/j ðN n i; qNni ; C Nni Þ ¼ ðn 1Þ/i ðN; q; CÞ þ
j2Nni
X
/j ðN; q; CÞ:
j2Nni
Then by the definition of C Nni and Budget Balance condition it holds that
CðqÞ ¼ C Nni ðqNni Þ ¼ ðn 1Þ/i ðN; q; CÞ þ CðqÞ /i ðN; q; CÞ: Hence ðn 2Þ/i ðN; q; CÞ ¼ 0, and consequently /i ðN; q; CÞ ¼ 0. Furthermore from expression (1) it follows /j ðN; q; CÞ ¼ /j ðN n i; qNni ; C Nni Þ for all j 2 N n i as was to be proved. Observe that the case jNj ¼ 1 is trivial, so assume that N ¼ fi; jg; i – j. Let ‘ R N, and write I ¼ fi; ‘g; N 0 ¼ ðN n iÞ [ I, and define q0 ¼ ðq; 0‘ Þ and C 0 2 CðN 0 ; q0 Þ by C 0 ðxÞ ¼ CðxN Þ for all x 2 ½0; q0 . Then jN 0 j ¼ 3 and the result of the first part of the proof can be applied. Since q0i ¼ qi ¼ 0, it holds that /i ðN 0 ; q0 ; C 0 Þ ¼ 0. On the other hand q0‘ ¼ 0, hence /i ðN 0 ; q0 ; C 0 Þ ¼ /i ðN; q; CÞ. The two equalities together give /i ðN; q; CÞ ¼ 0. In addition from the Budget Balance condition it holds that /j ðN; q; CÞ ¼ /j ðN n i; qNni ; C Nni Þ and the proof is complete. h Sprumont (2005) shows that No Merging or Splitting implies the following property. Weak Symmetry: Let ðN; q; CÞ 2 P; i 2 N, and p : N ! N be a bijection. If i 2 N is such that pðjÞ ¼ j for all j 2 N n i, then /pðiÞ pðN; q; CÞ ¼ /i ðN; q; CÞ. According to this axiom, if an agent is renamed his/her cost share does not change. Notice that weak symmetry does not say anything about the agents whose names are not modified. This statement is also valid when the domain of the cost function is ½0; q instead of NN . The proof would be as in Sprumont (2005), and is therefore omitted here.
Please cite this article in press as: Albizuri, M. J., et al. Monotonicity and the Aumann–Shapley cost-sharing method in the discrete case. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.03.044
4
M.J. Albizuri et al. / European Journal of Operational Research xxx (2014) xxx–xxx
Lemma 3 Sprumont, 2005. If / satisfies No Merging or Splitting then / satisfies Weak Symmetry. The next lemma uses the following basis. Let M 2 N such that jMj ¼ 2s 1 for a positive integer s. Define MðsÞ ¼ fR # M : jRj ¼ sg and Mðs 1Þ ¼ fR0 # M : jR0 j ¼ s 1g. Notice that R0 2 Mðs 1Þ if and only if M n R0 2 MðsÞ, and hence jMðs 1Þj ¼ jMðsÞj. For each R 2 MðsÞ define the vector uR 2 RMðs1Þ by
uR ðR0 Þ ¼
0
1; if R R;
ð2Þ
0; otherwise:
Lemma 4. Let ðT [ S; q; CÞ 2 P such that T \ S ¼ £ and the agents in S are symmetric. If / satisfies No Merging or Splitting, and for every bijection p : N ! N such that pðkÞ ¼ k for every k 2 T it holds P P then /i ðT [ S; q; CÞ ¼ k2T /k pðT [ S; q; CÞ ¼ k2T /k ðT [ S; q; CÞ, /j ðT [ S; q; CÞ for every i; j 2 S. Proof. Let M 2 N such that S # M; M \ T ¼ £ and jMj ¼ 2s 1. Define MðsÞ as above and write lðsÞ ¼ jMðsÞj. For each R 2 MðsÞ let pR be a bijection of N such that pR S ¼ R, and pR ðkÞ ¼ k for every k 2 T. Consider the problem pR ðT [ S; q; CÞ (notice that this problem is independent of the particular choice of pR since the agents in S are symmetric). By the Budget Balance condition it follows that i2R
ð3Þ
k2T
for all R 2 MðsÞ. Define Mðs 1Þ also as above. Now for any R0 2 Mðs 1Þ, by Lemma 3, there exists a number cðR0 Þ such that
0 0 /i T [ R0 [ i; pR [i q; pR [i C ¼ cðR0 Þ
for all i 2 M n R0 . Hence expression (3) can be rewritten as
X
cðR n iÞ ¼ CðqÞ
i2R
X /k pR ðT [ S; q; CÞ k2T
for all R 2 MðsÞ. This is a system of lðsÞ linear equations in the lðs 1Þ ¼ lðsÞ variables cðR0 Þ; R0 2 Mðs 1Þ that can be rewritten in the form
uR c ¼ CðqÞ
X
/k pR ðT [ S; q; CÞ for all R 2 MðsÞ;
k2T
where c ¼ ðcðR0 ÞÞ 2 RMðs1Þ and uR 2 RMðs1Þ is defined in expression P P (2). Since k2T /k pðT [ S; q; CÞ ¼ k2T /k ðT [ S; q; CÞ, clearly
!
cðR0 Þ ¼
X 1 CðqÞ /k ðT [ S; q; CÞ s k2T
for every R0 is a solution. Moreover, since uR form a basis for RMðs1Þ the system has a unique solution. Then for all i 2 S, choosing R0 ¼ S n i yields !
/i ðT [ S; q; CÞ ¼
X 1 CðqÞ /k ðT [ S; q; CÞ ; s k2T
and the conclusion follows. h The next two proofs use the fact that each cost function C 2 CðN; qÞ can be expressed as a linear combination of primitive cost functions C r 2 CðN; qÞ (r 2 ½0; q; r – 0). That is, P C ¼ 0–r6q ar C r , where ar 2 R and
C r ðxÞ ¼
1 if x P r; 0
e C¼C
X
e ar Cr :
ð4Þ
0–r6q
An example is given. C ¼ C ð1;0Þ þ C ð0;2Þ C ð1;2Þ . Then,
Let
N ¼ f1; 2g; q ¼ ð1; 2Þ
and
fr 2 ½0; q : 0 – r 6 qg ¼ fr1 ; r 2 ; r3 ; r 4 ; r 5 g; where r1 ¼ ð0; 1Þ; r 2 ¼ ð1; 0Þ; r3 ¼ ð1; 1Þ; r4 ¼ ð0; 2Þ and r 5 ¼ ð1; 2Þ. Hence,
The vectors uR form a basis for RMðs1Þ . The following lemma deals with symmetric agents.
X X /i pR ðT [ S; q; CÞ ¼ CðqÞ /k pR ðT [ S; q; CÞ
function (obviously it is non-decreasing) for which all the agents with the same consumption demand are symmetric, and in particular those whose demand is 1. It is therefore possible to write
otherwise:
P Given this expression C ¼ 0–r6q ar C r , for each q 2 N; 1 6 P q 6 qðNÞ (remind qðNÞ ¼ i2N qi ), define aq ¼ max0–r 6 q : ar , rðNÞ ¼ q e ¼P a r ¼ arðNÞ ar P 0. Also let C a C , that is a cost and e rðNÞ r 0–r6q
ar1 ðNÞ ¼ ar2 ðNÞ ¼ a1 ¼ maxf1; 0g ¼ 1; ar3 ðNÞ ¼ ar4 ðNÞ ¼ a2 ¼ maxf1; 0g ¼ 1; ar5 ðNÞ ¼ a3 ¼ maxf1g ¼ 1; and therefore,
e ¼ C ð0;1Þ þ C ð1;0Þ þ C ð1;1Þ þ C ð0;2Þ C ð1;2Þ : C a r1 ¼ 1; e a r2 ¼ 0, e a r3 ¼ 1; e a r4 ¼ 0 and e a r5 ¼ 0. And thus, Moreover, e (4) holds. Now follow the general case, and consider all the expressions e is symmetric for agents with demand 1, and (4) for C in which C e a r P 0 for all r. Define the index i of C to be the minimum number of terms which appear in the sum of such an expression. Lemma 5. If / is a cost-sharing method that satisfies Monotonicity and No Merging or Splitting, and ðN; q; CÞ 2 P is such that q 6 1N , then /ðN; q; CÞ ¼ /ASh ðN; q; CÞ. Proof. Let ðN; q; CÞ 2 P be such that q 6 1N . By Lemma 2 and since /ASh satisfies Zero Independence, it can be assumed that q ¼ 1N . The lemma can be proved by induction on the index i. If i ¼ 0 then all the agents in N are symmetric, so from Lemma 4, for the case T ¼ ; and S ¼ N, it follows /ðN; q; CÞ ¼ /ASh ðN; q; CÞ. Assume now that / coincides with /ASh whenever the index of C is at most i, and let C have index i þ 1 with expression e Piþ1 e C¼C k¼1 a r k C r k . Let S ¼ fi 2 N : ðr k Þi ¼ 1 for every k ¼ 1; . . . ; i þ 1g and assume e P e that i R S. Define D ¼ C k:ðr k Þi ¼1 a r k C r k . Observe that D is a cost P a rk C rk , where e a rk P 0. The index function because D ¼ C þ k:ðrk Þi ¼0 e P e of D is at most i and @ i DðxÞ ¼ @ i CðxÞ þ @ i k:ðr k Þi ¼0 a r k C r k ðxÞ ¼ @ i CðxÞ for all x 2 ½0; q s. t. xi < qi , so by induction and Monotonicity it follows that ASh /i ðN; q; CÞ ¼ /i ðN; q; DÞ ¼ /ASh i ðN; q; DÞ ¼ /i ðN; q; CÞ:
It remains to show that /i ðN; q; CÞ ¼ /ASh ðN; q; CÞ when i 2 S. But i notice that the agents in S are symmetric in ðN; q; CÞ. Moreover since it has been proved that /i ðN; q; CÞ ¼ /ASh i ðN; q; CÞ when i 2 N n S, Lemma 4 can be applied to obtain /i ðN; q; CÞ ¼ /j ðN; q; CÞ for every i; j 2 S. By the Budget Balance requirement the proof is complete. h Remark 1. Lemma 5 does not follow from Young’s characterization (1985a) of the Shapley value for transferable utility games. Observe that there is a bijection between fðN; 1N ; CÞ : C 2 CðN; 1N Þg and the set of transferable utility games with N as set of players. Indeed, ðN; 1N ; CÞ has associated the transferable utility game defined by v ðSÞ ¼ cð1S Þ for every S # N. Moreover, /ASh ðN; 1N ; CÞ coincides with the Shapley value of v. Young (1985a) characterizes the Shapley value with Monotonicity and Symmetry. However in Lemma 5 we cannot apply that the cost-sharing method satisfies Symmetry. The cost-sharing method satisfies No Merging or Splitting, and therefore Weak Symmetry (Lemma 3). As its name reveals, Weak
Please cite this article in press as: Albizuri, M. J., et al. Monotonicity and the Aumann–Shapley cost-sharing method in the discrete case. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.03.044
M.J. Albizuri et al. / European Journal of Operational Research xxx (2014) xxx–xxx
Symmetry does not imply Symmetry. Indeed, if we look at the formulation of Weak Symmetry we have that pðiÞ ¼ i or pðiÞ R N. However, in the formulation of Symmetry pðiÞ may be any player of N, that is, pðiÞ 2 N. On the other hand, by No Merging or Splitting, if all the agents in N are symmetric then the agents in N have the same allocations. This can be proved applying Lemma 4, taking T ¼ ; and S ¼ N. And obviously this property is weaker than Symmetry. And this is the proof of Theorem 1. Proof. It is clear that /ASh satisfies No Merging or Splitting and Monotonicity. Conversely, let / be a cost-sharing method that satisfies these two properties. For any k ¼ 0; 1; . . ., let DðkÞ ¼ fðN; q; CÞ 2 P : jfi 2 N : qi > 1gj 6 kg, i.e. the set of problems where no more than k agents demand several units. To prove this theorem it must be shown that /ðN; q; CÞ ¼ /ASh ðN; q; CÞ for every ðN; q; CÞ 2 DðkÞ and for every k ¼ 0; 1; 2; . . . Proceed by induction on k. From Lemma 5 it follows that /ðN; q; CÞ ¼ /ASh ðN; q; CÞ for every ðN; q; CÞ 2 Dð0Þ. So let k P 0 be fixed and assume that /ðN; q; CÞ ¼ /ASh ðN; q; CÞ for every ðN; q; CÞ 2 DðkÞ. Now let ðN; q; CÞ 2 Dðk þ 1Þ and prove that /ðN; q; CÞ ¼ /ASh ðN; q; CÞ. By the induction hypothesis it can be assumed that ðN; q; CÞ 2 Dðk þ 1Þ n DðkÞ, i.e. exactly k þ 1 agents demand several units. So consider a problem ðN; q; CÞ 2 Dðk þ 1Þ. By Lemma 2, it can also be assumed that q P 1N . STEP 1: /i ðN; q; CÞ ¼ /ASh ðN; q; CÞ for every i 2 N such that qi P 2. i Indeed, let I 2 N such that N \ I ¼ fig and jIj ¼ qi , and consider a consumption profile ðN 0 ; q0 Þ, where N 0 ¼ ðN n iÞ [ I, q0 is such that q0j ¼ qj for all j 2 N n i and q0j ¼ 1 for all j 2 I. Define C 0 2 CðN 0 ; q0 Þ P by C 0 ðxÞ ¼ C xNni ; i0 2I xi0 . Then by No Merging or Splitting, the induction hypothesis and the definition of /ASh we get
/i ðN; q; CÞ ¼
X X ASh 0 /i0 ðN0 ; q0 ; C 0 Þ ¼ /i0 ðN ; q0 ; C 0 Þ i0 2I
5
Remark 3. If the domain of cost functions C 2 CðN; qÞ is NN instead P of ½0; q, then we cannot write C ¼ 0–r6q ar C r , which is employed to prove Lemma 5 and Theorem 1. However, Monotonicity and No Merging or Splitting can be written as they are in the present work. And the latter axiom implies that /ðN; q; CÞ ¼ /ðN; q; C 0 Þ when P C ¼ C 0 on ½0; q. Hence, /ðN; q; CÞ ¼ / N; q; 0–r6q ar C r . So the proofs of Lemma 5 and Theorem 1 are also valid with the same axioms. Remark 4. As mentioned in the Introduction, in this paper we show that in Sprumont (2005) Additivity and Dummy can be replaced by Monotonicity when No Merging or Splitting is present. Notice that No Merging or Splitting gives rise to a variety of results and plays a key role in the proof of Theorem 1. This axiom implies Zero Independence (Lemma 2), and moreover, Lemma 4 for symmetric agents. Zero Independence, Lemma 4 and Monotonicity imply Lemma 5, which is the start point for the induction hypothesis in the proof of Theorem 1. Finally, No Merging or Splitting allows to prove step 1 in that proof, since the cost shares of an agent can be calculated by means of cost shares of agents with lower demands. All these results and Monotonicity determine the Aumann–Shapley method. Remark 5. Sprumont (2005) provides two variations of the axiomatization for the Aumann–Shapley method. No Merging or Splitting relates cost-sharing problems with different sets of agents. Sprumont (2005) changes this axiom in order to relate cost-sharing problems with equal sets of agents. In the second variation he let agents consume bundles of goods. We point that Additivity and Dummy can also be replaced by Monotonicity in Sprumont’s variations of his main axiomatization.
4. Conclusion
i0 2I
¼ /ASh i ðN; q; CÞ: STEP 2: /i ðN; q; CÞ ¼ /ASh ðN; q; CÞ for every i 2 N such that qi ¼ 1. i Denote N 1 ¼ fi 2 N : qi ¼ 1g. It will be proved that /i ðN; q; CÞ ¼ /ASh i ðN; q; CÞ by induction on the index i of C. If i ¼ 0 then all the agents in N 1 are symmetric. Then by step 1 Lemma 4 can be applied and the conclusion follows. Assume now that / coincides with /ASh whenever the index of C is at most i, and let C have index i þ 1 with expression e Piþ1 e C¼C k¼1 a r k C r k . Let S ¼ fi 2 N 1 : ðrk Þi ¼ 1 for every k ¼ 1; . . . ; i þ 1g and assume e P e that i R S. Define the cost function D ¼ C k:ðr k Þi ¼1 a r k C r k . The index of D is at most i and @ i DðxÞ ¼ @ i CðxÞ for all x 2 ½0; q s. t. xi < qi , so by induction and Monotonicity it follows that ASh /i ðN; q; CÞ ¼ /i ðN; q; DÞ ¼ /ASh i ðN; q; DÞ ¼ /i ðN; q; CÞ:
To finish the proof it remains to show that /i ðN; q; CÞ ¼ /ASh i ðN; q; CÞ when i 2 S. But the agents in S are symmetric in ðN; q; CÞ, and taking into account step 1, and that it has been proved that /i ðN; q; CÞ ¼ /ASh ðN; q; CÞ when i 2 N 1 n S, the proof is complete by i Lemma 4. h Remark 2. If cost functions which are not necessarily nondecreasing (in this case cost shares might be negative) are considered, another axiom must be added to obtain the axiomatization. It suffices to require cost shares to be non-negative if the cost function is non-decreasing. Notice that Lemma 1 holds and the rest too. e is not needed for the proofs, just the expression Furthermore, C P C ¼ 0–r6q ar C r .
In this paper we offer an axiomatization of the Aumann– Shapley method in the discrete case by employing Monotonicity. In doing so, we fill the gap with regard to cost-sharing problems in which goods are perfectly divisible. At the same time we find that weaker axioms are needed in the discrete case. This is revealed by looking at the characterization of the Aumann–Shapley price mechanism introduced by Monderer and Neyman (1988). It is an open question whether axioms could be weakened in the latter case, especially, if Symmetry is not needed as an explicit requirement. On the other hand, there are axiomatizations of other cost-sharing methods in the discrete case in which Additivity and Dummy are employed. For example in Sprumont (2000) and Moulin (1995). It is also an open question whether Monotonicity, possibly with other axioms, could replace the two axioms. Another setting connected with cost-sharing problems is that of bankruptcy problems. Bergantiños and Vidal-Puga (2006) characterize a family of additive rules for discrete bankruptcy problems. It deserves further research to examine the role of the Monotonicity property for these discrete problems. Acknowledgements We thank two anonymous referees for their helpful comments. Financial support from the University of the Basque Country (GIU13/31 and UFI11/51) and the Ministry of Economic Affairs and Competitiveness of Spain (ECO2012-33618) is gratefully acknowledged.
Please cite this article in press as: Albizuri, M. J., et al. Monotonicity and the Aumann–Shapley cost-sharing method in the discrete case. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.03.044
6
M.J. Albizuri et al. / European Journal of Operational Research xxx (2014) xxx–xxx
References Aumann, S., & Shapley, L. (1974). Values of nonatomic games. Princeton, NJ: Princeton Univ. Press. Bahel, E., & Trudeau, C. (2013). A discrete cost sharing model with technological cooperation. International Journal of Game Theory, 42, 439–460. Bergantiños, G., & Vidal-Puga, J. (2006). Additive rules in discrete allocation problems. European Journal of Operational Research, 172, 971–978. Billera, L., & Heath, D. (1982). Allocation of shared costs: A set of axioms yielding a unique procedure. Mathematics of Operations Research, 7, 32–39. Billera, L., Heath, D., & Raanan, J. (1978). Internal telephone billing rates: A novel application of non-atomic game theory. Operations Research, 26, 956–965. Calvo, E., & Santos, J. C. (2000). A value for multichoice games. Mathematical Social Sciences, 40, 341–354. Castaño-Pardo, A., & García-Díaz, A. (1995). Highway cost allocation: An application of the theory of non-atomic games. Transportation Research, 29, 187–203. Chen, C., Iyengar, G., & Moallemi, C. C. (2013). An axiomatic approach to systemic risk. Management Science, 59, 1373–1388. Dubey, P., Neyman, A., & Weber, R. J. (1981). Value theory without efficiency. Mathematics of Operations Research, 6, 122–128. Faria, E., Barroso, L. A., Kelman, R., Granville, S., & Pereira, M. V. (2009). Allocation of firm-energy rights among hydro plants: An Aumann–Shapley approach. IEEE Transactions on Power Systems, 24, 541–551. Haviv, M. (2001). The Aumann–Shapley price mechanism for allocating congestion costs. Operations Research Letters, 29, 211–215. Hougaard, J. L., & Tind, J. (2009). Cost allocation and convex data envelopment. European Journal of Operational Research, 194, 939–947. Krus´, L., & Bronisz, P. (2000). Cooperative game solution concepts to a cost allocation problem. European Journal of Operational Research, 122, 258–271. McLean, R. P., Pazgal, A., & Sharkey, W. W. (2004). Potential, consistency, and cost allocation prices. Mathematics of Operations Research, 29, 602–623. Mirman, L., & Tauman, Y. (1982). Demand compatible equitable cost sharing prices. Mathematics of Operations Research, 7, 40–56.
Monderer, D., & Neyman, A. (1988). Values of nonatomic games: The method of multilinear approximation. In The shapley value: Essays in honor of Lloyd Shapley (pp. 217–234). Cambridge: Cambridge Univ. Press. Moulin, H. (1995). On additive methods to share joint costs. Japanese Economic Review, 46, 303–332. Moulin, H., & Shenker, S. (1994). Average cost pricing versus serial cost sharing: An axiomatic comparison. Journal of Economic Theory, 64, 178–201. Myerson, R. (1980). Conference structures and fair allocation rules. International Journal of Game Theory, 9, 169–182. Pierru, A. (2010). Allocating the CO2 emissions of an oil refinery with Aumann– Shapley prices: A reply. Energy Economics, 32, 746–748. Samet, D., & Tauman, Y. (1982). The determination of marginal cost prices under a set of axioms. Econometrica, 50, 895–909. Samet, D., Tauman, Y., & Zang, I. (1984). An application of the Aumann–Shapley prices for cost allocation in transportation problems. Mathematics of Operations Research, 9, 25–42. Shapley, L. (1953). A value for n-person games. In H. W. Kuhn & W. Tucker (Eds.), Contributions to the theory of games II (pp. 307–317). Princeton, NJ: Princeton Univ. Press. Shubik, M. (1962). Incentives, decentralized control, the assignment of joint costs and internal pricing. Management Science, 8, 325–343. Sprumont, Y. (1998). Ordinal cost sharing. Journal of Economic Theory, 81, 126–162. Sprumont, Y. (2000). Coherent cost-sharing rules. Games and Economic Behavior, 33, 126–144. Sprumont, Y. (2005). On the discrete version of the Aumann–Shapley cost-sharing method. Econometrica, 73(5), 1693–1712. Wang, Y. (1999). The additivity and dummy axioms in the discrete cost-sharing model. Economics Letters, 64, 187–192. Young, H. P. (1985a). Monotonic solutions of cooperative games. International Journal of Game Theory, 14, 65–72. Young, H. P. (1985b). Producer incentives in cost allocation. Econometrica, 53, 757–765.
Please cite this article in press as: Albizuri, M. J., et al. Monotonicity and the Aumann–Shapley cost-sharing method in the discrete case. European Journal of Operational Research (2014), http://dx.doi.org/10.1016/j.ejor.2014.03.044