Mathematical properties of the discrete

pseudo-average cost sharing method

Jean Derksa and Juan Carlos Santosba

Department of Knowledge Engineering, Maastricht University,

P.O. Box 616 6200 MD, Maastricht, The Netherlands. e-mail:jean.derks@maastrichtuniversity.nlb

Departamento de Economıa Aplicada IV, University of the Basque Country UPV/EHU,

Lehendakari Agirre, 83. 48015 Bilbao. Spain. e-mail: carlos.santos@ehu.es

February 28, 2013

Abstract

We study the pseudo-average cost method in discrete cost allocationproblems. This solution was introduced by Moulin (1995). We first show itto be a fixed flow method. Next we characterize it by two standard axioms- additivity and zero charge for zero demand - together with the irrelevanceof dummy changes property and a version of strong monotonicity. We endby offering a formula for computing the method.

JEL classification: C70, C71, D70Keywords: cost sharing, pseudo-average method, Shapley-Shubik method,

flow system.

1 Introduction

This paper is related to the application of cooperative game theory to costsharing problems. In particular the discrete model of cost sharing is consid-ered, i.e. goods produced in indivisible units. Surveys of this problem andits different solutions can be found in Moulin (1995, 2002), Moulin and Spru-mont (2007) and Koster (2009). The Shapley-Shubik method (Shubik, 1962),the discrete Aumann-Shapley method (Moulin, 1995, Nouweland et al., 1995),the serial cost sharing method (Moulin and Shenker, 1992, Moulin, 1995) andthe pseudo-average method (Moulin, 1995) are the cost sharing methods mostwidely studied in this literature.1

All these methods are extensions of the Shapley value (Shapley 1953) forcooperative TU games (each cooperative TU game can be considered as a costsharing problem). Other solutions recently studied for this model are fixed-path

1Sprumont (1998) proposes a different extension of the serial method for these problems:he calls the new proposal the cross-subsidizing serial method and the proposal by Moulin(1995) the subsidy-free serial method.

1

methods (Moulin and Vohra, 2003), quasi-proportional methods and fixed-flowmethods (Moulin and Sprumont, 2005).

Several characterizations of these methods can be found in the surveys ofMoulin (2002) and Moulin and Sprumont (2007). The latter is mentioned fre-quently in this paper, where it is denoted by M-S (2007).

The only method that does not have a proper axiomatic characterizationis the pseudo-average method (Moulin, 1995, does not offer a characterizationof this method). Albizuri et al. (2002) prove that the pseudo-average methodcoincides with a solution for multichoice games: the extended Shapley valueproposed by Derks and Peters (1993) and studied by Klijn et al. (1999). Larreaand Santos (2007) propose a characterization based on Myerson’s (1980) idea ofbalanced contributions. Using an adaptation of this axiom (which they call themean balanced contributions axiom), the standard restriction of nonnegativityand cost coverage suffices to characterize the pseudo-average method.

In this paper we characterize the pseudo-average cost sharing method byusing elementary principles and compare it with two well known solutions inthis context: the discrete Aumann-Shapley method and the Shapley-Shubikmethod. Two standard axioms - additivity and zero charge for zero demand -and two further axioms - irrelevance of dummy changes and strong proportionalmonotonicity - are used. The irrelevance of dummy changes property is usedby Sprumont (2008) to characterize fixed flow methods and nearly serial meth-ods. Strong proportional monotonicity is an adaptation to this context of thestrong N -monotonicity axiom used by Arin and Feltkamp (2012) to study theSR solution for TU games (a serial rule for balanced games). This characteriza-tion precisely show the common properties of the pseudo-average method andthe Shapley-Shubik method and reveal small differences between them. TheAumann-Shapley method differs from the pseudo-average method in the irrele-vance of dummy changes property.

Finally we calculate a formula for the pseudo-average method. By adaptingthe proposed formula in Derks and Peters (1993) to this context, we obtain itin terms of the marginal contributions. We also prove that the pseudo-averagemethod is a fixed flow method.

The paper is organized as follows: the preliminaries are in Section 2. InSection 3 we show that the pseudo-average method is a fixed flow method. Thecharacterization theorem is given in Section 4. The formula is in Section 5.

2 Preliminaries

The discrete cost sharing model offered in M-S (2007) is considered.2 LetU be the universe of goods and N = {1, 2, . . . , n} a finite set, ∅ 6= N ⊂U . The set of nonnegative integers is denoted by Z+={0, 1, 2, ...}, and ZN+ ={x ∈ ZN : xi ≥ 0

}. For all q ∈ ZN+ and S ⊆ N , let qS ∈ ZN+ be defined by

2Equivalent to the model proposed by Moulin (1995), Nouweland et al. (1995) and Spru-mont (2005).

2

qSi = qi if i ∈ S and qSi = 0 if i ∈ N\S; in the particular case S = {j} , wewrite qj instead of q{j}; the vector qS ∈ ZS+ is defined by (qS)i = qi for eachi ∈ S; moreover we denote q (N) =

∑j∈N qj . The unit vector in ZN+ is denoted

by eN = (1, . . . , 1) and the zero vector by θN = (0, . . . , 0), or by θ when noconfusion is possible. Given two vectors x, y ∈ ZN+ , x ≤ y means that xj ≤ yj ,for all j ∈ N ; and x < y if x ≤ y and x 6= y.

A cost sharing problem is a pair (q, C) where q ∈ ZN+ is the consumptionprofile and the cost function C is a nondecreasing mapping C : ZN+ → <+ suchthat C(θ) = 0.

A cost sharing method is a mapping η that assigns to each problem (q, C)a vector of nonnegative cost shares η(q, C) ∈ <N+ satisfying the budget bal-ance condition

∑i∈N ηi(q, C) = C(q). Note that by definition

∑i∈N ηi(θ, C) =

C (θ) = 0, and ηi(q, C) > 0, for all q and i. Then ηi(θ, C) = 0.

The cost sharing methods most frequently studied in the relevant litera-ture are the Shapley-Shubik method, the discrete Aumann-Shapley method,the (subsidy-free) serial method and the pseudo-average method. These fourmethods are linear solutions. For each cost sharing problem (q, C) they aredefined as follows:

Shapley-Shubik method, ϕShSh

ϕShSh (q, C) = Sh (N,Cq) ,

where Sh denotes the Shapley value, and the cooperative game (N,Cq) is definedby Cq (S) = C

(qS), for each coalition S ⊆ N. That is, the Shapley-Shubik

method is the Shapley value of the cooperative game where every good is viewedas a single player.

Aumann-Shapley method, ϕASh

ϕAShi (q, C) =∑j∈Ni

Shj (M,Γq) ,

where M =⋃i∈N Ni, with |Ni| = qi, and the cooperative game (M,Γq) is

defined, for each S ⊆ M, by Γq (S) = C (|S ∩N1| , . . . , |S ∩Nn|) . That is, theAumann-Shapley method is the Shapley value of the cooperative game whereeach unit of each good by is regarded as a single player.

For the sake of simplification we define the serial and pseudo-average meth-ods by using a family of cost sharing problems that resemble unanimity gamesfor TU games. Fixed N , the family {(q, Cs)} s∈ZN

+

θ<s≤qis defined as follows:

Cs(t) =

{1, if s ≤ t0, otherwise.

Note 1 It is straightforward to show that each cost sharing problem can beexpressed uniquely as a linear combination of elements of this family.3

3See for example Moulin (1995), page 312.

3

Serial method, ϕS . Let M (s) = {i ∈ N : si ≥ sj for all j ∈ N}. This linearsolution is defined on this family by:

ϕSi (q, Cs) =

{ 1|M(s)| , if i ∈M(s),

0, if i /∈M(s).

That is, for these simple games, the serial method divides the cost equallyamong the goods with the highest demand.

Pseudo-average method, ζ, is the linear solution defined on the family by:

ζi (q, Cs) =si∑

j∈Nsj, for each i ∈ N.

That is, the pseudo-average method divides the cost proportionally to thedemand of each good.

Now we present a simple example, which is used again below, to show someof the differences between these methods.

(Game A) Two families want to get together for a week’s vacation in aprivately rented house on the beach. The price of this house is 1000 dollars aweek. The price for two weeks is 1800 dollars. One family is contemplating thepossibility of staying another week. If so, how much should each family pay?

One family suggests applying the proportional rule,(2 · 1800

3,

1800

3

)= (1200, 600)

In the other family there is a person who knows about game theory and costsharing. The cost sharing problem ((2, 1) , C) is built up as follows

C(1, 0) = C (0, 1) = C (1, 1) = 1000 and C(2, 0) = C (2, 1) = 1800

Applying these methods for the allocation of the total cost between thefamilies face the following cost sharings:

Shapley-Shubik method

ϕShSh ((2, 1), C) = (1300, 500) .

Aumann-Shapley method

ϕAShi ((2, 1), C) =

(4400

3,

1000

3

)

Serial methodϕSi ((2, 1), C) = (1300, 500)

4

Pseudo-average method

ζi ((2, 1), C) = (1300, 500) .

They agree on (1300, 500) and the first family will stay another week.(Game B) The next year the two families want to return to the same house.

Unfortunately they can not coincide on the same dates. One family sees thatthe house has new prices: 1000 dollars a week and a special offer of three weeksfor only 2100 dollars. This family suggests the following plan (which is feasible):book the house for three weeks; for the first two weeks it will be occupied byone family, and for the third by the other. In this case, how much should eachfamily pay?

They repeat the calculations. The proportional rule(2 · 2100

3,

2100

3

)= (1400, 700)

Now the cost sharing problem ((2, 1) , C) is

C(1, 0) = C (0, 1) = 1000, C(1, 1) = C (2, 0) = 2000 and C (2, 1) = 2100

We obtain the following cost sharingsShapley-Shubik method

ϕShSh ((2, 1), C) = (1550, 550) .

Aumann-Shapley method

ϕAShi ((2, 1), C) = (1400, 700)

Serial methodϕSi ((2, 1), C) = (1100, 1000)

Pseudo-average method

ζi ((2, 1), C) = (1400, 700) .

They think that the proportional rule is the most appropriate in this caseand they agree on (1400, 700).

Note that only the pseudo-average rule gives the allocation chosen in bothcases.

5

3 ζ is a fixed-flow method

In this section, we prove that the pseudo-average method is a fixed-flow method.We first introduce three basic axioms. The first two axioms are well known, thethird was proposed by Sprumont (2008).

Let η be a cost sharing method.

Axiom 1 Additivity

For all (q, C) and (q, C ′)

η(q, C + C ′) = η(q, C) + η(q, C ′).

This axiom also implies that the cost sharing method only depends upon therestriction of C to the rectangle [θ, q] (M-S 2007).

Axiom 2 Zero charge for zero demand

For all (q, C) , if qi = 0, then

ηi(q, C) = 0.

This very natural condition is well known. A strong version of this axiom isindependence of zero demands, which appears in some characterizations of theserial method and the discrete Aumann-Shapley method (Moulin and Shenker,1994, Sprumont, 1998, 2005).

The third axiom, is inspired by the dummy axiom. Sprumont (2008) usesthis axiom to characterize fixed flow methods and nearly serial methods.

Axiom 3 Irrelevance of dummy changes

For all (q, C) and all θ ≤ x ≤ q, if C(qi, xN\{i}

)= C

(qi +m,xN\{i}

)for

all m ∈ Z+, then η(q +m · ei, C) = η(q, C).This means that costless units are added without affecting the final cost

allocation.Flow methods are the family of cost sharing methods that satisfy additivity

and separability (Wang, 1999, M-S, 2007).4 It can be immediately verified thatthe pseudo-average method satisfies additivity and separability. Hence, thepseudo-average method is a flow method, and then there exists a unique flowsystem f (., q) associated with this method, that is

ζi (q, C) =∑θ≤t<qti<qi

fi (t, q)[C(t+ ei)− C(t)

],

where f (., q) : [θ, q[ → <N+ is a mapping that satisfies:

4Separability means that for all (q, C) , if C(x) =∑

i∈N ci (xi), θ ≤ x ≤ q, then ηi (q, C) =ci (qi), for all i ∈ N.

6

fi (t, q) = 0 (if ti = qi) ,∑i∈N

fi (θ, q) = 1, and∑i∈N

fi (t, q) =∑i∈Nti>0

fi(t− ei, q

).

Fixed-flow methods are characterized by two more requirements: monotonicityand unambiguous externalities (M-S, 2007). Sprumont (2008) proposes a simpleaxiomatization of fixed-flow methods: a cost sharing method η satisfies additiv-ity, zero charge for zero demand and irrelevance of dummy changes if and onlyif η is a fixed-flow method.

Theorem 4 The pseudo-average method is a fixed-flow method.

Proof. From Sprumont (2008) it is sufficient to prove that the pseudo-averagemethod satisfies additivity, zero charge for zero demand and irrelevance ofdummy changes. By definition, the pseudo-average method satisfies additiv-ity and zero charge for zero demand. To check irrelevance of dummy changes,let (q, C) be such that C

(qi, xN\{i}

)= C

(qi +m,xN\{i}

)for all θ ≤ x ≤ q and

m ∈ Z+. By note 1, for some (as)θ<s≤q+m·ei , as ∈ <:

C(x) =∑

θ<s≤q+m·eiasCs(x), and C(x) = C

(min {xi, qi} , xN\{i}

).

Note that Cs(min {xi, qi} , xN\{i}

)= 0 for s � q, s ≤ q+m·ei, and Cs

(min {xi, qi} , xN\{i}

)=

Cs(xi, xN\{i}

)for s ≤ q.

Hence

C(x) =∑

θ<s≤q+m·eiasCs

(min {xi, qi} , xN\{i}

)=∑θ<s≤q

asCs(xi, xN\{i}

).

Then, from additivity, and taking into account that

ζ (q, Cs) = ζ(q +m · ei, Cs

)if θ < s ≤ q, we have

ζ(q +m · ei, C) = ζ

q +m · ei,∑θ<s≤q

asCs

= ζ

q, ∑θ<s≤q

asCs

= ζ(q, C).

4 A characterization of ζ

In this section we introduce a new axiom - strong proportional monotonicity -and characterize the pseudo-average method with the help of this new axiom.By using a version of this axiom - strong equal monotonicity - we present a newcharacterization of the Shapley-Shubik method.

7

Axiom 5 Strong proportional monotonicity

For all (q, C) and (q, C ′), if C (x) = C ′ (x) for all θ ≤ x < q, then for alli, j ∈ N, qi 6= 0 6= qj ,

ηi (q, C ′)− ηi (q, C)

qi=ηj (q, C ′)− ηj (q, C)

qj.

Axiom 6 Strong equal monotonicity

For all (q, C) and (q, C ′), if C (x) = C ′ (x) for all θ ≤ x < q, then for alli, j ∈ N, qi 6= 0 6= qj ,

ηi (q, C ′)− ηi (q, C) = ηj (q, C ′)− ηj (q, C) .

The interpretation is the following: negotiations related to the allocation ofthe cost of a project take place before it is even started. But it is most unusualfor the final cost of a project to be known as early as the first discussions; thegeneral rule is rather that it exceeds the forecasts. The strong (proportionaland equal) monotonicity property demands that each player participates in anyrise or lowering in the total cost.

These two properties are the natural translation to cost sharing problems ofthe strong N -monotonicity axiom,5 used by Arin and Feltkamp (2012) to studythe SR solution (a monotonic core solution, on the class of veto balanced TUgames, defined by a serial rule principle). Calleja et al. (2012) use the equalsurplus division property, that is a strong version of the strong N -monotonicityaxiom.

It can also be seen as an adaptation to this context of the coalitional indepen-dence axiom,6 used by Hernandez-Lamoneda et al. (2008) to study solutionswithout dummy axiom for TU games. For tree games, Mishra and Talman(2010) use a similar axiom (strong symmetry).

Theorem 7 A cost sharing method η satisfies additivity, zero charge for zerodemand, irrelevance of dummy changes and strong proportional monotonicityif and only if η is the pseudo-average method.

Proof. We first show that the pseudo-average method ζ satisfies the axioms.From the previous theorem we need only prove that it satisfies the strong pro-portional monotonicity axiom.

5Strong N -monotonicity: if for all TU games v, w, if for all S 6= N , v(S) = w(S) andv(N) < w(N), then for all i, j ∈ N, ϕi(w)− ϕi(v) = ϕj(w)− ϕj(v) ≥ 0.

6Coalitional independence: ϕi(w) − ϕi(v) = ϕj(w) − ϕj(v) for all TU games (N, v) and(N,w) that only differ in S, and i, j ∈ S or i, j ∈ N\S.

8

Let (q, C) and (q, C ′) such that C (x) = C ′ (x) for all θ ≤ x < q. ObviouslyC ′ = C + (C ′ (q)− C (q))Cqq . Additivity of the pseudo average method impliesthat

ζ(q, C ′) = ζ(q, C) + ζ(q, (C ′ (q)− C (q))Cqq

).

Then for all i ∈ N, qi 6= 0,

ζi(q, C′) = ζi(q, C)+ζi

(q, (C ′ (q)− C (q))Cqq

)= ζi(q, C)+(C ′ (q)− C (q))

qiq (N)

That isζi(q, C

′)− ζi(q, C)

qi= (C ′ (q)− C (q))

1

q (N)

Hence the pseudo-average method satisfies the strong proportional mono-tonicity axiom.

Unicity. Using the additivity axiom, it is sufficient to prove that η = ζon the family {(q, Cs)}θ<s≤q .7 Fix s, θ < s ≤ q. Applying the irrelevance ofdummy changes axiom repeatedly to each i ∈ N such that si 6= qi we have that

η (q, Cs) = η (s, Cs) .

Moreover the cost sharing problem (s, Cs) can be written as follows

(s, Cs) = (s,0) + (s, Cs)

where (s,0) is the zero cost problem.8

If si = 0, zero charge for zero demand implies that

ηi (s, Cs) = 0 = ζi (s, Cs) .

If si 6= 0, sj 6= 0 the strong proportional monotonicity axiom implies that

ηi (s, Cs)− ηi (s,0)

si=ηj (s, Cs)− ηj (s,0)

sj

Henceηi (s, Cs)− 0

si=ηj (s, Cs)− 0

sj

Taking into account that∑ni=1 ηi (s, Cs) = Cs (s) = 1, and case si = 0, we

deduce thatηi (s, Cs) =

sis(N)

= ζi (s, Cs) = ζi (q, Cs)

and the proof is complete.If we impose that any change in the final cost should be divided equally

among the goods then the Shapley-Shubik method is the only cost sharingmethod that satisfies all the above axioms. Formally:

7It can be easily shown that additivity plus positivity (non-negative cost shares) implieslinearity. See Aczel (1966).

8It is easy to see that for this problem, each player must pay zero (for any cost sharingmethod).

9

Theorem 8 A cost sharing method η satisfies additivity, zero charge for zerodemand, irrelevance of dummy changes and strong equal monotonicity if andonly if η is the Shapley-Shubik method.

Proof. We first show that the Shapley-Shubik method ϕShSh satisfies theaxioms. It is well known that this is a fixed flow method, hence it satisfiesadditivity, zero charge for zero demand and irrelevance of dummy changes. Sowe only need to prove that it also satisfies the strong equal monotonicity axiom.Following the same lines as in the proof of the previous theorem, additivity ofthe Shapley-Shubik method implies that

ϕShSh(q, C ′) = ϕShSh(q, C) + ϕShSh(q, (C ′ (q)− C (q))Cqq

).

Then for all i ∈ N, qi 6= 0,

ϕShShi (q, C ′) = ϕShShi (q, C) + ϕShShi

(q, (C ′ (q)− C (q))Cqq

)= ϕShShi (q, C) + (C ′ (q)− C (q))

1

|N(q)|where N (q) = {i ∈ N : qi 6= 0}.That is

ϕShShi (q, C ′)− ϕShShi (q, C) = (C ′ (q)− C (q))1

q (N)

Hence the Shapley-Shubik method satisfies the strong equal monotonicityaxiom.

Unicity.Following the same lines as in the proof of the previous theorem, we have:If si = 0, zero charge for zero demand implies that

ηi (s, Cs) = 0 = ϕShShi (s, Cs) .

If si 6= 0, sj 6= 0 the strong equal monotonicity axiom implies that

ηi (s, Cs)− ηi (s,0) = ηj (s, Cs)− ηj (s,0)

Henceηi (s, Cs) = ηj (s, Cs)

And taking into account that∑ni=1 ηi (s, Cs) = Cs (s) = 1, and case si = 0,

ηi (s, Cs) =1

N(s)= ϕShShi (s, Cs) = ϕShShi (q, Cs) .

Remark 9 The axioms are independent.

10

In Theorem 7:Strong proportional monotonicity. The Shapley-Shubik method satisfies all

the axioms except strong proportional monotonicity.Irrelevance of dummy changes. The discrete Aumann-Shapley method sat-

isfies all the axioms except the irrelevance of dummy changes axiom.Zero charge for zero demand. Consider the additive method η defined on

the family {(q, Cs)}θ<s≤q by:

ηi (q, Cs) =si∑j∈N sj

if |{j ∈ N : sj = 0}| 6= 1, and by

ηi (q, Cs) = C(q), if si = 0, and ηi (q, Cs) = 0, if si 6= 0.

if |{j ∈ N : sj = 0}| = 1.This method satisfies all the axioms except zero charge for zero demand.Additivity. A method satisfying all the axioms except additivity is the fol-

lowing:Let Q1 be the set of all cost sharing problems (q, C) such that |N | = 2,

q ≥ (1, 1), C (1, 0) 6= 0 6= C (0, 1) , and satisfying

C(x, y) = C(min {1, x} ,min {1, y}), for each (0, 0) ≤ (x, y) < q.

We define recursively Qk+1, k ≥ 1, as follows: (q, C) ∈ Qk+1 if there existsr = r (q, C) , r ≤ q, such that (r, C) ∈ Qk and C(s) = C(min {s1, r1} ,min {s2, r2}),for each θ ≤ s < q.

The method η is defined by:

η (q, C) =

(C(1, 1)

2+C(q)− C(1, 1)

q1 + q2q1,

C(1, 1)

2+C(q)− C(1, 1)

q1 + q2q2

)

if (q, C) ∈ Q1, by

η (q, C) = η (r (q, C) , C) +

(C(q)− C(r (q, C))

q1 + q2q1,

C(q)− C(r (q, C))

q1 + q2q2

)

if (q, C) ∈ Qk+1,for some k ≥ 1, and

η = ζ

otherwise.In Theorem 8:Strong equal monotonicity. The pseudo-average method satisfies all the ax-

ioms except strong equal monotonicity.Irrelevance of dummy changes. The additive method η defined on the family

{(q, Cs)}θ<s≤q by:

11

ηi (q, Cs) = ζi (q, Cs) =si∑j∈N sj

if s 6= q, and ηi (q, Cq) = ϕShSh (q, Cq) , satisfies all the axioms except theirrelevance of dummy changes axiom.

Zero charge for zero demand. Same idea that used for Theorem 7.Additivity. Same idea that used for Theorem 7, that is, allocate C(q)/2 in

those problems.

5 A formula for ζ

This section seeks to find a formula for computing the pseudo-average method.Moulin (1995) only defines this method for a basis and no general expression forit exists in the relevant literature. The formula that we present here is an av-erage of marginal costs at the demand profiles below the actual demand. Thisshows again that the pseudo-average method is a flow method. We likewisecalculate the flow system associated with the pseudo-average method. Never-theless, proving from the formula that it is a fixed-flow method is no simpletask.

Given a cost sharing problem (q, C) , and θ < t ≤ q, we denote by kq(t)the number of unsaturated goods in N , that is, kq(t) = |Kq(t)| where Kq(t) ={j ∈ N : tj 6= qj} . The pseudo-average method computes the cost shares in theproblem (q, C) according to the formula

ϕi (q, C) =∑t≥θ

t+ei≤q

kq(t)∑r=1

(−1)r−1

(kq(t)− 1r − 1

)ti + 1

t (N) + r

[C(t+ ei)− C(t)

],

for each i ∈ N.To clarify this expression, we consider the case N = {1, 2} and q = (a, b) ∈

Z2+. For each (x, y) ∈ Z2

+, θ < (x, y) ≤ (a, b) , we define P 1 (x, y) = xx+y . We fix

x, x 6= 0. To move from x − 1 to x there are b + 1 links (flows): (x− 1, 0) to(x, 0) , (x− 1, 1) to (x, 1) , . . . , (x− 1, b) to (x, b) . The coefficients of these flowsare:

f1 ((x, y) , q) =

{P 1 (x, y) if y = bP 1 (x, y)−

∑y<z≤b

f1 ((x, y) , q) if y < b.

Obviously∑

0≤y≤bf1 ((x, y) , q) = P 1 (x, 0) = 1.

The idea of this flow system is the following: If x is fixed, what is theprobability of link from (x− 1, y) to (x, y)? The starting point is (x, b) . Fromthis point, decreasing the level, there are two links: (x− 1, b) to (x, b) , and(x, b) to (x, b− 1) . The probability of these links is proportional to the levels,

12

hence f1 ((x, b) , q) = xx+b . In the next step we are at the point (x, b− 1) . Again,

the probability of the link from (x− 1, b− 1) to (x, b− 1) is proportional to thelevels, but now we have to subtract the probability that the link from x− 1 tox has occurred in the previous step. And so on until b = 0, where the processends.

This interpretation of the flow system can be extended naturally to the casen > 2.

To compare this flow system with the flow systems associated with theShapley-Shubik and serial methods, we return to the second case in the ex-ample given in Section 2: game B, 1000 dollars each week and three weeks for2100 dollars. The flows are (for good 1):

Shapley-Shubik method

ϕShSh1 (q, C) =1

2[C(1, 0)− C(0, 0)] +

1

2[C(2, 0)− C(1, 0)] +

1

2[C(1, 1)− C(0, 1)] +

1

2[C(2, 1)− C(1, 1)] =

1

21000 +

1

21000 +

1

21000 +

1

2100 = 1550.

Serial method

ϕS1 (q, C) =1

2[C(1, 0)− C(0, 0)] +

1

2[C(1, 1)− C(0, 1)] +

[C(2, 1)− C(1, 1)] =1

21000 +

1

21000 + 100 = 1100.

Pseudo-average method

ζ1 (q, C) =1

2[C(1, 0)− C(0, 0)] +

1

2[C(1, 1)− C(0, 1)] +

1

3[C(2, 0)− C(1, 0)] +

2

3[C(2, 1)− C(1, 1)] =

1

21000 +

1

21000 +

1

31000 +

2

3100 = 1400.

Theorem 10 For each cost sharing problem (q, C) , the pseudo-average methodζ (q, C) coincides with ϕ (q, C) .

Proof. Since the pseudo-average method ζ (q, C) is a flow method, que havethat ζi (q, C) =

∑t≥θ

t+ei≤q

fi (t, q)[C(t+ ei)− C(t)

].Then, it is sufficient to prove

that

fi (t, q) =

kq(t)∑r=1

(−1)r−1

(kq(t)− 1r − 1

)ti + 1

t (N) + r.

We follow the model of Derks and Peters (1993) on the aplication of theMoebius transformation in cooperative games. Applying the definition of thepseudo-average method to the family {(q, Cs)}θ<s≤q, we obtain, for each i ∈ N ,

sis(N)

= ζi (q, Cs) =∑t≥θ

t+ei≤q

fi (t, q)[Cs(t+ ei)− Cs(t)

]=

∑t≥θ

t+ei≤q

fi (t, q)mist

13

with mist = Cs(t+ ei)−Cs(t) = 1 if ti = si − 1, and s− ei ≤ t, and mi

st = 0otherwise.

Therefore, denoting ζi ={

sis(N)

}ei≤s≤q

, fi = {fi (t, q)} t≥θt+ei≤q

and Zi the

transpose of the matrix(mist

)ei≤s≤q

t≥θ,t+ei≤q, we have ζi = fiZ

i. The matrix Zi

is invertible, and its inverse has the following coefficients: zist = (−1)|S|−1

forS ⊆ N, i ∈ S, s = t+ eS , and zist = 0 otherwise (with eS =

∑j∈S e

j).The coefficients {fi (t, q)}t≥θ,t+ei≤q are now provided by

fi (t, q) =(ζiZ

i)s=t+ei

=∑

S⊆N,i∈St+eS≤q

(−1)|S|−1 ti + 1

t(N) + |S|

and the proof is completed.AcknowledgementsThe authors gratefully acknowledge financial support from the University

of the Basque Country UPV/EHU (project IT-293-07 and UFI11/51) and theSpanish MICINN (project ECO2012-33618).

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