The additivity and dummy axioms in the discrete cost sharing model

Economics Letters 64 (1999) 187–192

The additivity and dummy axioms in the discrete cost sharing model YunTong Wang ´ ´ ´ , 3150 avenue Jean-Brillant, Montreal ´ , H3 T 1 N8, Departement de Sciences Economiques , Universite´ de Montreal Canada Received 22 October 1998; received in revised form 17 March 1999; accepted 12 April 1999

Abstract The paper considers the discrete cost sharing model first studied in Moulin’s paper [Moulin, H., 1995. On additive methods to share joint costs. Japanese Economic Review 46, 303–332]. It shows that the set of additive rules satisfying the dummy axiom is the set of all convex combinations of the path generated rules.  1999 Elsevier Science S.A. All rights reserved. Keywords: Cost sharing; Additivity; Dummy JEL classification: D63; C71

1. Introduction We consider the following cost sharing problem. A production facility is shared by a finite number of agents i 5 1,...,n. Each demands a quantity qi of a personalized good, and the total production cost expressed by C(q1 ,...,qn ) must be equitably divided among the n agents. There are three particular formulations of the above cost sharing problem, depending on the assumptions about the demand profile q 5 (q1 ,...,qn ). The first model (Model 1) assumes that all individual demands are either 0 or 1. This is equivalent to the standard model of cooperative games with transferable utility. The second model (Model 2) assumes that all demands are real numbers. This corresponds to the well-known Aumann-Shapley pricing model (see the survey by Tauman (1988)). The third model (Model 3), first considered by Moulin (1995), generalizes the first model by assuming that the demand profile is a list of non negative integers. These three models have been studied. Stemming from Shapley’s (1953) seminal work on the Shapley-value theory for the

E-mail address: [email protected] (Y. Wang) 0165-1765 / 99 / $ – see front matter PII: S0165-1765( 99 )00080-4

 1999 Elsevier Science S.A. All rights reserved.

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cooperative games, the axiomatic approach has been intensively used to select and characterize various cost sharing rules. This paper reconsiders Model 3. We reexamine the implications of the two basic axioms: Additivity and Dummy, on the cost sharing rules. For Model 1, Weber (1988) showed that the additivity (to be precise, he uses a stronger axiom called linearity) and dummy axioms characterize the class of random order values. For Model 2, Friedman and Moulin (1997) provided a representation formula for all cost sharing rules meeting Additivity and Dummy axioms. Recently, Friedman (1998) and Haimanko (1998) provided characterizations in terms of the path generated rules for Model 2. For Model 3, Moulin (1995) characterized the class of cost sharing rules satisfying Additivity, Dummy and Demand Monotonicity. In this paper, we provide a characterization of the entire class of rules satisfying the Additivity and Dummy axioms for Model 3. We show that these cost sharing rules can be generated by the convex combinations of the path generated rules (Theorem 1).

2. The model and the axioms The model is essentially the same model discussed by Moulin (1995). The only difference here is that we will fix the demand profile in the discussion. The reader will immediately see that the model with variable demand profile allows for exactly the same result. Denote N 5 h0,1,2,...j. Let n [ N, n . 0 and N 5 h1,...,nj be the set of agents. Let q [ N n be a demand profile, which is fixed throughout this paper. Denote [0, q] the interval: 0 # t # q in N n . A cost function is a mapping C:[0, q] → R such that C(0) 5 0 and C(t) # C(t9) if t # t9. The set of all cost functions is denoted by #. A (cost sharing) problem is a pair (q;C) where C [ #. Since we fixed the demand profile we simply call C [ # a problem. A (cost sharing) rule x is a mapping from # into R n1 satisfying the budget balance condition ox i (C) 5 C(q). A few more notations will be used. The interval ]0,q] means the set [0,q]\h0j and ]0,q[ 5 [0,q]\h0,qj. If t [ [0,q] and S # N is a coalition, denote t S the restriction of t to S. We write t 5 (t S ,t 2S ) instead of t 5 (t S ,t N 2S ) and i instead of hij. Denote A(t) 5 hi [ Nut i . 0j the set of active agents at t. Let 1 i stand for the vector in N n whose jth component is 1 if j 5 i and 0 otherwise. Given a problem C and t i . 0, we define ≠ i C(t) 5 C(t) 2 C(t 2 1 i ). The paper only considers the cost sharing rules satisfying the following two axioms: Additivity and Dummy. A rule x is additive if x(C1 1 C2 ) 5 x(C1 ) 1 x(C2 ) for any C1 ,C2 [ #. It satisfies the dummy axiom if x i (C) 5 0 whenever ≠ i C(t) 5 0 for every t [ [0,q] such that t i . 0. The Additivity and Dummy axioms imply the following representation result. It corresponds to Friedman and Moulin’s (1997) representation lemma for Model 2 and to Weber’s (1988) Theorem 2 for Model 1. The proof of this lemma is omitted here and is available from the author upon request. Lemma 1. ((Representation Lemma)) A rule x is additive and satisfies the dummy axiom if and only if, for each i [ A(q),1 there exists a unique mapping mi :[1 i ,q] → R 1 such that x i (C) 5

O

mi (t)≠ i C(t) for each C [ #,2

t[[1 i , q ]

1 2

For i [ ⁄ A(q), x i (C)50. They are implied by the budget balance condition. See footnote 3. In general, each mi (i [ A(q)) depends on q. Since q is fixed in this paper we write mi instead of m qi for each i [ A(q).

(1)

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and

O O

mi (t i ,s 2i ) 5 1 for each t [ ]0,q].3

(2)

i [ A(t )s 2i [[t 2i , q 2i ]

The collection m 5 ( mi ) i [ A( q) is called the weight system associated with the rule x.

3. The main theorem: The characterization by the path generated rules In this section, based on Lemma 1, we further show that any additive rule satisfying the dummy axiom must be a convex combination of the so-called path generated rules defined below. This alternative characterization is much more intuitive than the representation lemma. Definition 1. A path to q is a mapping P:h0,1,...,q(N)j → [0,q] such that • P(0) 5 0, • For each k [ h0,1,...,q(N)j, P(k) is identical to P(k 2 1) in all coordinates but one, say, the ith, for which Pi (k) 5 Pi (k 2 1) 1 1, where q(N) 5 o N qi . Denote 3 the set of all paths to q. Definition 2. A path generated rule, generated by a path P [ 3, is a rule which charges each agent the sum of his marginal costs along the path. More precisely, let P be a path. The P 2 path generated rule is defined as follows: if C is a problem and qi . 0, we compute for every t i [ [1,qi ] the unique integer k 5 :k(t i ) for which Pi (k 2 1) 5 t i 2 1 and Pi (k) 5 t i , and then charge agent i P

x i (C) 5

O

4

≠ i C(P(k(t i ))).

t i [[1, q i ]

Note that the path generated rules satisfy the Additivity and Dummy axioms. And their associated weight systems consist of vectors with components of 0 or 1 only. Denote CP the set of all the path generated rules w.r.t. the path set 3. Denote CSM the set of all the cost sharing rules satisfying the Additivity and Dummy axioms. Then, CP is a subset of CSM. From Lemma 1, we observe that each rule in CSM corresponds uniquely to a weight system, which is a vector in the Euclidean space of dimension o n 1 qi Pj ±i (q j 1 1). Therefore, the sets CP and CSM can be thought of as subsets in this vector space. Then, it can be shown that the set CSM is a convex

3 4

This ensures the budget balance condition. This expression should be understood to be zero if qi 50.

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compact set and the set CP is a subset of the set of extreme points 5 of CSM, ext(CSM), i.e., CP # ext(CSM) (we omit the proof). Therefore, we conclude that conv(CP) # CSM. 6 Our main theorem, given in the following, states that the converse inclusion, CSM # conv(CP), is also true. Theorem 1. The cost sharing rules satisfying the Additivity and Dummy axioms are the convex combinations of the path generated rules x P (P [ 3 ), i.e., CSM 5 conv(CP).

(3)

The proof of the theorem, i.e., the proof of CSM # conv(CP), relies on several lemmas. Lemma 2. Let x be an additive rule satisfying the dummy axiom. Let mi (i [ A(q)) be the weight system associated with x (by Lemma 1). Then, for every t [ ]0,q[,

O

mi (t) 5

i [ A( q)> A(t )

O

mi (t 1 1 i )

(4)

i [ A( q)>hi ut i ,q i j

and

O m (1 ) 5 1, O m (q) 5 1. i

i

(5)

i

(6)

i [ A( q)

i [ A( q)

This lemma is in Sprumont (1998). It follows directly from the representation lemma by applying it to the following two types of cost functions: C *t (s) 5 1 if s $ t and 0 otherwise, C ** t (s) 5 1 if s . t and 0 otherwise. Note that if we interpret the weights mi (t) (i [ A(q)) at any intermediate node t as flows, then Eq. (4) says that the incoming flow at t equals the outgoing flow from t. Eqs. (5) and (6) say respectively that there is one unit of flow coming out from the ‘origin’ 0, and going into the ‘sink’ q. Therefore, the weight system associated with a cost sharing rule can be regarded as a ‘flow’ running through a ‘network’. Formally, a network is a pair G 5 (V,E), in which V 5V1
The definition of extreme point is the following: A point c [ A (A is a convex set) is an extreme point of A if whenever a,b [ A, and c 5 a / 2 1 b / 2, then a 5 b. 6 The notation ‘conv’ refers to the convex hull.

Y. Wang / Economics Letters 64 (1999) 187 – 192

E(t) 5 he i (t)ui [ A(q)j

191

(7)

and define E 5 < t []0, q ] E(t). Let f and c(e) (e [ E) be non-negative real numbers. Given these numbers, a feasible flow on G is a set of numbers m (e) (e [ E) such that f t50 t [ ]0,q[ m (e i (t 1 1 )) 2 m (e i (t)) 5 0 i [ A( q)> A(t ) i [ A( q)>hi ut i ,q i j 2f t5q

O

i

O

5

(8)

with capacity constraints 0 # m (e) # c(e), e [ E.

(9)

Our proof of Theorem 1 also relies on the following two lemmas from the integer programming theory (see Garfinkel and Nemhauser, 1972, pages 66–74). Lemma 3. (Garfinkel and Nemhauser, 1972) The constraint matrix corresponding to the flow constraints Eqs. (8) and (9) is totally unimodular 7 . Lemma 4. (Garfinkel and Nemhauser, 1972) If A is totally unimodular, then the extreme points (if any) of S(b) 5 hxuAx 5 b,x $ 0j are integer vectors for any arbitrary integer vector b $ 0. We are now ready to prove that CSM # conv(CP). Let x [ CSM. We observe that the weight system m associated with x is a feasible flow for the numbers c(e) 5 1 (e [ E) and f 5 1. This directly follows from Lemma 2. Indeed, let m (e i (t)) 5 mi (t) in the flow constraints Eqs. (8) and (9). From Lemma 2, the constraints (8) are satisfied for f 5 1 and c(e) 5 1 (e [ E). The constraints Eq. (9) are satisfied due to Lemma 1’s budget balance condition. In particular, we also observe that the weight systems associated with the path generated rules correspond to the ‘unit flows’, namely, those feasible flows m for which m (e) is zero or one for each e [ E. Conversely, a unit flow m defines a path to q:P 5 ht [ [0,q]ut 5 0 or there exists an i [ A(q) s.t. m (e i (t)) 5 1j (check that P is indeed a path), which generates a rule x P (Definition 2). Now, consider the set of all feasible flows with c(e) 5 1 (e [ E) and f 5 1 defined by Eqs. (8) and (9). It is a compact convex set and can be spanned by its extreme points. However, its extreme points, by Lemmas 3 and 4, are integer vectors, and in our case, vectors with components of zero or one. So these extreme points are the unit flows. This proves that the weight system m associated with x is a convex combination of the unit flows. Accordingly, the cost sharing rule x is a convex combination of the path generated rules. This completes our proof of the theorem.

7

An integer m 3 n matrix A is totally unimodular if every square, nonsingular submatrix B of A has determinant 1 or 2 1.

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Acknowledgements I would like to thank Yves Sprumont, Eric Friedman, Herve´ Moulin and an anonymous referee for their helpful discussions or comments.

References Aumann, R.J., Shapley, L., 1974. Values of Nonatomic Games, Princeton University Press. Friedman, E., 1998. Paths and consistency in additive cost sharing. Mimeo, Rutgers University. Friedman, E., Moulin, H., 1997. Two methods to share joint costs or surplus. Mimeo, Duke University. Garfinkel, R.S., Nemhauser, G.L., 1972. Integer Programming, John Wiley. Haimanko, O., 1998. Partially symmetric values. Mimeo, Hebrew University, Jerusalem. Moulin, H., 1995. On additive methods to share joint costs. Japanese Economic Review 46, 303–332. Shapley, L.S., 1953. A value for n-person games. In: Kuhn, H.W., Tucker, A.W. (Eds.), Contributions to the Theory of Games II. Annals of Mathematics Studies, Vol. 28, pp. 307–317. Sprumont, Y., 1998. Coherent cost sharing. Mimeo. University of Montreal. Tauman, Y., 1988. The Aumann-Shapley prices: a survey. In: Roth, A. (Ed.), The Shapley Value, Cambridge University Press, Cambridge. Weber, R., 1988. Probabilistic values for games. In: Roth, A. (Ed.), The Shapley Value, Cambridge University Press, Cambridge.