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Toward the fair sharing of profit in a supply network formation

Jean-Claude Henneta,∗

, Sonia Mahjouba,b

a LSIS, CNRS-UMR 6168, Université Paul Cézanne, Faculté Saint Jérôme, Avenue

Escadrille Normandie Niémen, 13397 Marseille Cedex 20, France

b FIESTA , ISG Tunis, 41 rue de la liberté, 2000 Le Bardo, Tunisia

Abstract

The design of a supply chain network can be interpreted as a coalition formation problem in cooperative

game theory and formulated as a linear production game (LPG). The companies which are members of the

optimal coalition share their manufacturing assets and resources to produce a set of end-products and globally

maximize their profits by selling them in a market. This paper investigates the possibility of combining the

requirement of coalition stability with a fair allocation of profits to the participants. It is shown that, in general,

the purely competitive allocation mechanism does not exhibit the property of fairness. A technique is proposed

to construct a stable and fair allocation system when the core of the game does not exclusively contain a set of

competitive allocations.

Keywords: supply chain, cooperative game theory, coalitions, linear programming, duality

*Corresponding author. Tel. : +33 491 05 6016 ; fax: +33 491 05 6033,

E-mail address: jean-claude.hennet@lsis.org

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1. Introduction

In the context of modern information and communication networks, many firms are open to

finding new partnerships and penetrating new markets. In the manufacturing sector, several classical

paradigms and tools must be revisited in the light of more adaptability, flexibility and network

reconfigurability. This paper describes a supply network formation process as a network of

enterprises attempting to collaborate and capture a market share for a particular set of products.

The result of such a collaboration may be called an emergent supply network. The basic and

necessary ingredients to enable collaboration to prevail are the existence of a targeted market, and

resources for manufacturing, communication and logistics. This study focuses on the latter success

factor: existing resources could bring more profit if they were used jointly rather than separately. As

a consequence, many enterprises are ready to use their own resources and those of other firms to

produce services and products in the most advantageous quantity and at the lowest cost.

Supply networks are thus characterized by a basic common goal and a highly integrated

operational system, combined with a complex decentralized decisional organization. Such

characteristics are of major concern in game theory, under the classical decomposition into

cooperative (or coalitional) games and non-cooperative (or strategic) games. Models referred to as

‘non-cooperative’ are composed of players with different preference relations or utility functions.

Their actions obey a strategy that takes into account information (generally imperfect) on the other

players’ actions and preferences. As noted in Cachon and Netessine (2004), most supply chain

models based on game theory use a non-cooperative approach. However, cooperative game theory

seems more appropriate to analyze a supply network in its design stage, as it is characterized by

many possibilities for enterprise coalitions and allocation patterns for tasks and rewards. In such a

strategic stage, all the actors are ready to cooperate to create competitive advantages in the market.

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Partner enterprises share a common goal that is clearly identified: penetrate a market segment to

obtain the maximal global expected profit.

A general characteristic of cooperative games is the understanding of the players that they can

obtain a larger global benefit from pooling their resources than by acting separately. This

characteristic is particularly important when resources are scarce, as in the case of spare parts for

aircrafts studied by Kilpi et al. (2009), and when important economies of scale in the availability

service can be achieved by cooperation. Several authors have used cooperative game theory to

represent alliances between retailers in their relation to the market and/or to suppliers. Nagarajan

and Sošić (2008) have studied the benefits expected by retailers from price setting, pooling their

market share and sharing risks. Profit can also be increased by centralizing inventories. The works of

Granot and Sošić (2003), Cachon and Netessine (2004) and Reinhardt and Dada (2005) provide

convincing interpretations of supply chain design problems as cooperative games by focusing on

manufacturing capacity and inventory pooling. In the light of cooperative game theory, a supply

network can be modelled as a coalition of partners pooling their resources and sharing the same

utility function (profit). The partnership building problem can then be modelled as a cooperative

game with transferable utilities (TU-game). Such a game is firstly characterized by global optimization

of the supply chain value (total expected profit). A TU-game can thus be seen as a target model on

which the partners can agree to estimate the maximal value of the chain and the shares of the global

profit acceptable to all of them. Another practical advantage of this model is that it evaluates and

compares different possible coalitions. The results can thus be used as arguments in the supply chain

design stage, to convince the partners to be part of the best possible coalition and set up the joint

venture.

Once the comparative advantage of cooperation has been demonstrated, the key remaining

issue is the choice of the scheme for sharing benefits between the members. Expected profit is

interpreted as the transferable utility that should be allocated to players in a manner that is

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acceptable to all of them and guarantees coalitional stability. In cooperative game theory, several

concepts have been introduced for approaching the stability issue. A necessary condition for the

stability of a coalition is that no set of players is able to increase its members’ profits by forming a

different coalition. The set of payoff profiles that verifies this property is known as the core of the

TU-game (Gillies, 1959). It is the set of non-dominated feasible payoff profiles (also called

imputations) covering all the possible coalitions. Non-emptiness of the core has been shown if the

problem is convex (Shapley and Shubik, 1969). By introducing some restrictions on the possible

deviations from a coalition, several sets have been defined to characterize stable and/or balanced

imputations among the partners of the optimal coalition: stable set, bargaining set, nucleolus, kernel

and the ‘Shapley value’ (see e.g. Osborne and Rubinstein, 1994). In recent years, several imputation

mechanisms have been proposed in the literature on supply chain analysis by cooperative game

theory.

Several authors have proposed the use of the ‘Shapley value’ approach in various contexts

related to supply chain design (Nagarajan and Sošić, 2008). In particular, Reinhardt and Dada (2005)

have studied the case of n firms cooperating by pooling their critical resources. Due to the difficulty

of computing the Shapley value for a large number of players, these authors have proposed a

pseudo-polynomial algorithm to compute the Shapley value allocation of benefits for particular

games, called ‘coalition symmetric’. In addition to its computational complexity, the Shapley value

also has the drawback of not necessarily belonging to the core, as pointed out by Granot and Sošić

(2003). Bartholdi and Kemahlıoglu-Ziya (2005) have also used the Shapley value to allocate profit in a

network composed of two retailers and one supplier. They have shown, in particular, that such an

allocation coordinates the supply chain but may appear unfair to the supplier.

Another example of an alliance between retailers can be found in Guardiola et al. (2007). The

authors analyze the case of a supply chain with one supplier and n retailers selling the same product

in separate markets. Through cooperation, the retailers increase their profit by obtaining a lower

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wholesale price from the supplier. In counterpart, the supplier receives equal side payments from all

the retailers. The maximal value of these payments is computed to guarantee that the allocation

mechanism belongs to the core of the retailers–supplier game. In an almost symmetrical manner, Jin

and Wu (2006) have shown that in online reverse auctions there is one unique strongly stable

suppliers’ coalition that allows for maximizing profit. Several studies have concentrated on the

costing aspects rather than on profit sharing. In particular, Charles and Hansen (2008) have applied

cooperative game theory to global cost minimization and cost allocation in an enterprise network.

They have shown that under classical concave cost functions for all members, the cost allocation

obtained by the activity based costing (ABC) technique is rational and belongs to the core of the

game.

In many practical situations, a supply network can be viewed as a multistage production

system in which the different production stages are performed by different enterprises. Material

requirement planning (MRP) theory (Grübbström, 1999; Grubbström and Huynh, 2006) can then be

used to define and distribute responsibilities and manufacturing orders among the partners. In this

view, the product structure supports the enterprise network organization, especially under an

extended view of the bill of materials (BOM) such as the generic BOM (Lamothe et al., 2005).

Additionally, multistage production by several producers highly differs from multistage production by

a single producer because of the need for coordinating requirements and contracts negotiation. It

also carries new possibilities for increasing the effectiveness of ordering and inventory policies (Arda

and Hennet, 2006). In the design stage, the largest possible set of firms should be investigated for

selecting partners and sharing resource costs and/or rewards in the most efficient manner.

This paper analyzes the problem of optimally choosing the partners to constitute a multistage

production system. By assumption, the firms who own manufacturing resources are ready to make

them available to the supply chain if they obtain sufficient rewards. In such a scheme, some

enterprises may be complementary (if they own complementary resources) and some may compete

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(if they own resources of the same type). In addition, they all compete with each other to obtain the

largest possible share of the global profit. Using MRP theory, this paper shows that the supply

network formation problem can be formulated as a linear production game (LPG). Since the pioneer

works of Shapley and Shubik (1972) and Owen (1975), many results have been obtained for this

particular class of cooperative game. LPGs are characterized by a linear value function to be

optimized by linear programming (LP) under linear constraints relating production to resources under

capacity constraints.

Since the work of Owen (1975), it has been well-known that the core of an LPG is generally not

empty and that a purely competitive allocation scheme, called the Owen set, is contained in the core.

This set is constructed from the optimal solution of the dual linear program, which defines the

shadow prices of the resources. In the Owen set, each player receives the reward corresponding to

the shadow prices of the resources that he owns. Then, if the dual problem has a unique solution,

the Owen set consists of a single point, called the purely competitive allocation (Van Gellekom et al.,

2000). These results have been extended to semi-infinite LP situations, under the assumption of a

finite bound on the maximal profit, in two basic cases. Fragnelli et al. (1999) have considered the

case of a potentially infinite number of end products, and Tijs et al. (2001) have studied the

possibility of an infinite number of transformation techniques.

Based on the previous works of Owen (1975) and Van Gellekom et al. (2000), this paper

explicitly constructs the Owen set of the multistage LPG under study. Then, the quality of the

solution is discussed in terms of stability and fairness. In particular, the existence of partners

obtaining a null profit is considered as an unfair property that decreases the robustness of the

coalition’s stability. The main issue of this paper is to study the existence of fair allocations in the

core of LPGs. Theorem 1 in section 4 gives a sufficient condition for existence and a technique to

construct such allocations. To solve this problem, some side payments are introduced from the

partners with a positive payoff to their associates with a null payoff in the Owen allocation. The

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maximal value of these side payments is computed under the condition of maintaining the allocation

in the core of the game. If this value is strictly positive, as may occur if the competition is not too

severe, then a new set of fair imputations lying in the core is constructed.

Section 2 presents some preliminaries on cooperative game theory. Then, section 3 represents

the multistage supply chain design problem as a cooperative linear production game. The problem is

solved in terms of maximizing profit and forming a coalition with the smallest number of partners.

The Owen set of this game is constructed and analyzed. It is shown via an example that, under the

Owen allocation, some participants of the optimal coalition may receive no payoff. To correct this

drawback as much as possible, the side-payments mechanism is proposed and studied in section 4.

The main result of the paper is stated in Theorem 1. It provides a test of existence and a technique to

construct an efficient, rational and fair allocation. Some conclusions on the applicability of the

method are finally exposed.

2. Some preliminaries on TU-games

Classically, a game involves a finite set of N players, denoted { }N,...,1=N , with N = Card( N ).

A coalition S is a subset of N : NS ⊆ . The set )(NP is the set of all the subsets of N .

In a TU cooperative (or coalitional) game in the sense of Von Neumann and Morgenstern (1944),

each coalition )(NP∈S is characterized by a value function 0)( ≥Sv . The value )(Sv is the

maximal utility (or payoff) that can be obtained by coalition S. Each player N∈i seeks to maximize

his utility function, which is the payoff that he can obtain from belonging to a coalition. By

convention, all the utilities are nonnegative. The minimal utility that a player may obtain is zero. In

this model, the players’ actions within a coalition are implicit. In order to maximize his utility

function, the only possible decision a player can take is to belong to a coalition or not. If, as the result

of the game, coalition S , N⊆S prevails, then each player Si∈ obtains a share 0)( ≥Svi such

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that: )()( SvSvSi

i =∑∈

, and each player j of N not belonging to S ( Sj N \∈ ) has a null payoff:

0)( =Sv j . Notation S\N represents the set of players that belong to N but not to S . All the

utilities are transferable (TU-game) in the sense that they are all shares of the global payoff.

A TU-game is thus simply noted v)(N, . It raises two basic problems:

• global utility maximization: determination of the maximal value function and the coalitions S

for which this value is obtained,

• allocation problem: determination of the endowments of the agents by distributing the global

payoff among them.

As in Osborne and Rubinstein (1994), an S-feasible payoff profile is defined as a vector Siiu ∈)(

such that )(SvuSi

i =∑∈

, and a feasible payoff profile as a vector N∈iiu )( such that )(NN

vui

i =∑∈

.

Let *v be the maximal global payoff of the TU-game v)(N, :

)(*)(

SvvS NP∈

= max . (1)

Consider a feasible payoff profile N∈iiu )( . With every coalition S we associate a payoff )(Su

defined by expression (2):

∑∈

=Si

iuSu )( . (2)

Several properties will be now defined. These properties appear necessary for a payoff profile

to solve a coalitional game.

Property 1: Efficiency (Pareto optimality)

The feasible payoff profile N∈iiu )( is said to be efficient (or Pareto optimal) if and only if

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*)(1

vuuN

ii == ∑

=N . (3)

Another condition for a feasible payoff profile N∈iiu )( to be accepted by all the players is

(coalitional) rationality, as defined below.

Property 2: Rationality

A feasible payoff profile N∈iiu )( is said to be rational if the payoff of every coalition S is larger

than its value )(Sv :

)()( SvSu ≥ )(NPS; ∈∀ S . (4)

The core of the game v)(N, is a game-theoretic concept introduced in Gillies (1959). It is

defined as follows.

Definition 1: Core

The core of a TU-game is the set of feasible payoff profiles Niiu ∈)( that satisfy conditions (3)

and (4). Namely, it is the set of feasible payoff profiles that are both efficient (Pareto optimal) and

rational.

A major issue in the analysis of a TU cooperative game consists in characterizing its core. It can

be noted that equations (2) and (3) and linear inequalities (4) define a closed and convex set.

However, due to the combinatorial growth of the number of coalitions (cardinal of )(NP ) with the

number of players, N, the core is generally difficult to construct and fully explore. The approach that

will be followed in this study for the LPG will rather be to construct a core solution by a known

algorithm and then to locally modify this solution without escaping the core.

An interesting index to characterize the value of a subset N⊆⊆ SS ' is its marginal

contribution in S , denoted )'(SS∆ and defined as follows.

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Definition 2: Relative marginal contribution

The marginal contribution of a subset 'S of a coalition S is:

)'\()()'( SSvSvSS −=∆ (5)

where notation '\ SS represents the set of players that belong to S but not to 'S .

In a classical manner, marginal contributions relative to the so-called ‘grand coalition’ N are

simply called ‘marginal contributions’.

Property 3: Marginal contribution principle

An allocation that lies in the core satisfies the marginal contribution principle:

)()( SSu N∆≤ )(NPS; ∈∀ S . (6)

The proof of this property is straightforward: a feasible payoff profile Niiu ∈)( that lies in

the core satisfies )\()\( SvSu NN ≥ for every coalition S. Then, from

)\()()\()()( SuvSuuSu NNNN −=−= , one derives )\()()( SvvSu NN −≤ .

Hence, a core allocation is such that any coalition cannot obtain a payoff greater than its

marginal contribution. With the objective of studying the core of a TU-game, inequalities (6) are

particularly useful since they provide upper bounds to coalition payoffs, while rationality conditions

(4) provide lower bounds.

The maximal global payoff of the TU-game v)(N, , *v , has been defined by equation (1). As

this value can be obtained for several coalitions, it is possible to introduce a secondary criterion to

define the optimal coalitions.

Definition 3: Optimal coalition

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The optimal cardinality the TU-game v)(N, is: { }*)()(min* vSvScards == . An optimal

coalition of the TU-game v)(N, is a coalition N⊆*S that satisfies *)( * vSv = and *)( sScard = .

The most critical situation for a player in a TU-game is when his utility remains at its minimal

value, zero. In other words, his share of the global payoff, *v , is null. For a given feasible payoff

profile N∈iiu )( , a player who does not receive any payoff is called a ‘null payoff player’ (NPP),

according to the following definition.

Definition 4: Null payoff player (NPP)

An NPP with respect to a feasible payoff profile N∈iiu )( is a player N∈i such that 0=iu .

For every optimal coalition *S , two types of NPPs may be distinguished: players outside and

players inside the optimal coalition *S .

The following properties derive from the definitions above.

Property 4

Under a core allocation profile, any player who does not belong to every optimal coalition is

an NPP.

Proof. *Sj N \∈ Suppose with *S as an optimal coalition. Then, by the efficiency property

of the payoff profile, **)(*)(*)()( vSuSuSuu ==+= NN \ , therefore 0*\=∑

∈ Siiu

N

and, by the

nonnegativity of utilities,

*0 Sju j N \ ∈∀= .

Property 5

Under a core allocation profile, any NPP which belongs to an optimal coalition, *Si∈ satisfies

the two following conditions: {} 0)( =iv and {} 0)(*S >∆ i .

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Proof. {} Niivui ∈∀≥ )( Condition is a well-known property, called ‘individual rationality’. For

a core allocation, this property is verified as a particular instance of rationality property (4), for sets

NS ⊂ with cardinality 1. For an NPP, condition 0=iu implies {} 0)( =iv . To show {} 0)(*S >∆ i , it

suffices to apply definition 5 to note that the reverse condition, {} 0* =∆ )(S i , would contradict the

minimality of coalition *S .

With this background, it is now possible to propose a definition of fairness for payoff profiles.

Definition 5: Fairness

A feasible payoff profile Niiu ∈)( is fair if all the members of an optimal coalition *S obtain a

strictly positive payoff : *0 Siui ∈∀> .

This definition raises the problem of the existence of fair solutions belonging to the core of a

TU-game.

To show that some TU-games cannot admit any fair solution, it suffices to construct games

that possess several different optimal coalitions, optimality meaning that the coalition creates the

maximal utility value, *v , and has minimal cardinality in the set of coalitions S verifying *)( vSv = . It

will now be shown that games with this property can be constructed, as inspired by Aumann (1964),

by replicating one or several players of an optimal coalition.

Consider a production game v)(N, in which the players own resources that they put into

common use to manufacture products that create value. Consider a player i who belongs to an

optimal coalition *S and suppose that he owns resources that are useful but not critical, in the sense

that he possesses all of them in excess of what is required for the optimal production mix. Other

resources that he does not own are critical. Let us now define a new game, { } v),i'(N ∪ , by

introducing a new player, denoted i’, exactly identical to player i.

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Because of the non-criticality of the resources owned by players i and i’, the act of increasing

their quantity does not change the optimal production mix. Hence, the optimal value of the new

game is the same as the optimal value, v*, of the original game. Coalitions *S and {}( ) { }'\* iiS ∪ are

both optimal for the new game. Therefore, Property 4 implies 0' == ii uu for any core allocation u of

the game { } v),i'(N ∪ . And yet player i belongs to the optimal coalition *S , and player i’ to the

optimal coalition {}( ) { }'\* iiS ∪ . Therefore, player i is an NPP in *S and player i’ an NPP in

{}( ) { }'\* iiS ∪ .

As a consequence, a necessary condition for the existence of a fair solution in the core of a TU-

game is the uniqueness of the optimal coalition *S . Note that this is an intrinsic property of the

game, independent from the core allocation considered.

3. Coalition formation for profit maximization

3.1. A multistage model

Consider a network of N firms represented by numbers in the set { }N,...,1=N . These firms

are willing to cooperate to produce commodities and sell them in a market. These commodities

belong to a set of g manufactured products (or families of products) i=1,…,g. Typically, manufacturing

recipes are fixed and well defined. The gozinto graph describes the product structure and has no

cycle. It can then be decomposed into levels: level 0 products are the g final products. Then,

intermediate and primary products are numbered in the increasing order of their level. The level of

product i, for i=g+1,...,n is the maximal number of stages to transform product i into a final product.

Each production stage is supposed to have several input products but only one output product. The

BOM technical matrix, G, is defined as follows: according to the given manufacturing recipe,

production of one unit of product i requires the combination of components j=1,...,n in

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quantities jiG . It can be noted that under a level-consistent ordering of products, matrix G then has

a lower triangular structure (Hennet, 2003).

In the context of MRP theory (Grübbström, 1999; Grubbström and Huynh, 2006), matrix G is

the input matrix. When combined with the lead-time (output) matrix, it can be used to define the

generalized Leontieff inverse of the system. However, the scope here will be restricted to stationary

balance equations under a simple production structure. In this case, the output matrix reduces to the

identity matrix.

Let )(( jixx = be the matrix of the quantities of product i produced (or purchased) at firm j and

Tnyyy )( 1= be the output vector of products during a reference period. The components of this

matrix and vector are the variables of the design problem. For simplicity, quantities per period (or

throughputs) are supposed to be continuous: nNn yx +×+ ℜ∈ℜ∈ , . Due to the sharing of resources, the

problem can be formulated in terms of the global throughput vector, denoted ω and related to

matrix x through the elementary summation relation (7):

1x=ω with 1 being the unit vector of dimension N. (7)

The output vector can be computed from the global throughput vector by the following

relation:

ω)( GIy −= (8)

with I the identity matrix of dimension nn× .

Matrix G being componentwise nonnegative and lower triangular (with 0s on the diagonal),

matrix )( GI − is a regular –M-matrix and 1)( −−GI is lower-triangular (with 1s on the diagonal)

and nonnegative (Berman and Plemmons, 1979). Then the BOM can be expressed as follows:

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yGI 1)( −−=ω . (9)

There are N enterprises which are candidates to be part of the supply chain to be created.

Each candidate enterprise is characterized by its production resources: manufacturing plants,

machines, work teams, robots, pallets, storage areas, etc.

As in Van Gellekom et al. (2000), a coalition S is defined as a subset of the set N of N

enterprises with characteristic vector { }NSe 1,0∈ such that:

if 0)( if 1)(

SjeSje

jS

jS

∉=∈=

. (10)

For the R types of resources considered (r=1,…,R), let rjc be the amount of resource r available

at enterprise j, NRrjcC ×ℜ∈= ))(( , and rim the amount of resource r necessary to produce one unit

of product i, nRrimM ×ℜ∈= ))(( .

Capacity constraints for coalition S are written:

∑∑∈=

≤Sj

rj

n

iiri cm

1ω or equivalently ∑∑

∈=≤

NS(e

jjrj

n

iiri cm )

1ω . (11)

It can be noted that the left-hand side quantities in inequalities (11) fully characterize coalition S.

One originality of the proposed model is that it combines the coalition decision problem

through the choice of vector Se , with the multistage manufacturing model represented by

constraints (9) and (11). The second originality is related to the choice of the objective function that

aggregates the benefits expected from manufacturing activities. The value chain concept introduces

transfer prices for intermediate goods so that each manufacturing stage should be profitable.

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3.2. The Value chain

Let TR ),,( 1 χχχ = be the vector of unit costs for resources. The unit cost for product i, iγ , is

assumed to be the same for all the firms possessing the required resources. It is determined by the

cost resource requirement for producing one unit of product i:

∑=

=R

rriri m

1χγ . (12)

Let Tn ),,( 1 γγγ = be the vector of unit production costs for the products. The set of relations

(12) for all the products can be written in vector form:

χγ TM= . (13)

Prices may be partly or totally exogenous. Let Tnppp )( 1= be the vector of market prices for

the final products.

A necessary and sufficient condition for the global profitability of the supply chain is:

0≥− ωγ TT yp , (14)

or, using relation (9),

0])([ 1 ≥−− − yGIp TT γ . (15)

The prices of final products can be supposed fixed and exogenous. The global profitability

condition (15) can be considered sufficient for the supply chain to be viable.

More restrictive conditions will now be established to allow for a total decomposability of the

multistage manufacturing process. It is now assumed that the prices of intermediate products are

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negotiated in the network so that each manufacturing stage is profitable. Under this more restrictive

requirement, the following inequality should be verified for any product i:

∑=

+≥n

jjijii pGp

1γ

. (16)

Let Tn ),,( 1 πππ = be the vector of unit profits for products. The unit profit iπ associated with

product i is:

∑=

−−=n

jjijiii pGp

1γπ

. (17)

Condition (16) corresponds to the profitability conditions { }nii ,,10 ∈∀≥π that characterize

the value chain. These conditions can be gathered into the following condition in vector form:

0)( ≥−−= γπ pGI . (18)

Condition (18) may appear unnecessarily restrictive. However, it can be used to fix the transfer

prices between products in the network so that each manufacturing stage is intrinsically profitable.

3.3. Global profit maximization

The maximal total payoff, *v , is obtained from the solution of the mixed variables’ Linear

Programming problem (P):

{ } 1,0 ,

)( subject to

,)( Maximize

1

N

1i

Nn

iiT

ey

QeyGIM

yyMpGIv

∈ℜ∈

≤−

>=−−=<

+

−=∑πχ

(P)

with the cardinality of the coalition as the complementary objective to be minimized:

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∑=

=N

jjes

1)( . (19)

The secondary objective can be solved together with the global profit maximization problem by

adding a ‘small’ term ∑=

−N

jjSe

1))((ε with 0 <ε <<1 to v in the objective function of (P). The modified

problem, (P'), is formulated as follows:

{ } 1,0 ,

)( subject to

)( Maximize

1

1 1

Nn

S

n

i

N

jjii

ey

QeyGIM

ey

∈ℜ∈

≤−

−=

+

−

= =∑ ∑επϕ

.

(P')

The term ∑=

−N

jje

1))((ε is said to be ‘small enough’ if the optimal solutions of (P) and (P’) are

identical with respect to the optimal vector *y . This property is achieved for any value of ε smaller

than a threshold value .00 >ε The optimal coalition of lowest cardinality S* is then obtained by

resolution of problem (P'):

{ }{ }.1*,,,1;* =∈= jeNjjS

This coalition is supposed not empty. Accordingly, the maximal value function of the TU-game is

supposed strictly positive. It is exactly computed from the solution of (P’) by: ∑=

+=N

jjev

1*)(** εϕ .

In problem (P), vector e is a vector of binary variables. By definition, problem ( SP ) relates to a

particular coalition NS ⊆ . It is defined by the same constraints as (P), but with the vector of

variables e replaced by the known vector Se given by (10): if 0)( if 1)(

SjeSje

jS

jS

∉=∈=

.

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.

)( 1

nS

iiS

y

QeyGIM

yv

+

−=

ℜ∈

≤−

= ∑

to subject

MaximizeN

1iπ

(PS)

The TU-game v)(N, defined in this section is an LPG as defined in Owen (1975). It is an N-persons

game in which the value )(Sv of a coalition S is obtained as the solution of an LP problem, (PS),

defined by the nonnegative production matrix 1)( −−GIM , the nonnegative resource matrix Q and

the nonnegative unit profit vector π .

Several properties can be derived from this definition. In particular, it is super-additive:

)()()( TSvTvSv ∪≥+ for all disjoint coalitions S and T , and it has a non-empty core (Owen,

1975). Superadditivity implies, in particular, that if *S is an optimal coalition, then *)( vSv = for any

coalition S such that SS ⊆* . In particular, the maximal total payoff, *v can simply be obtained for

the grand coalition, N , by solving problem NP .

3.4. The core of the linear production game

The core of a TU-game is often difficult to determine. Some subsets are of particular importance.

It is a classical result that in exchange economies (markets with transferable payoffs), every

competitive allocation is in the core (Shubik, 1959). For LPGs, the set of competitive allocations has

been characterized by Owen (1975) and called the Owen set by Van Gellekom et al. (2000).

Conversely, for coalitions with a limited number of players, as in the supply chains example, the

core may be substantially larger. On the other hand, the set of competitive allocations often reduces

to a single point. It is thus interesting, in particular regarding improving negotiation convergence and

robustness, to consider payoff allocations that lie in the core but possibly outside the set of

competitive allocations.

20

3.5. The Owen set

The numerical resolution of problem (P’) solves the global utility maximization problem presented

in section 3.3.: it defines the maximal total payoff, v*, and an optimal output vector y*. Problem (P)

can then be replaced by problem ( *SP ), in which all the variables are real numbers.

At this stage, however, the allocation problem, which determines the share of the total payoff to

be distributed to each coalition partner, still remains open.

Consider the dual of ( SP ), denoted ( SD ):

to subject

Minimize

RTR

TT

R

rrrS

),z,(zz

zMGI

zSqw

+

−=

ℜ∈=

≥−

= ∑

1

1

)(

)(

π

.

( SD )

The coefficient of variable rz in the objective function is the quantity of resource r available for

production if coalition S is selected:

∑∑∈=

==Sj

rj

N

jrjjSr cceSq

1)()( . (20)

It can be noted that the set of constraints of ( SD ) is the same for any coalition S. Further, since

the optimal dual variables )(* Szr can be interpreted as shadow prices for resources, they determine

a vector of payoffs, the so-called ‘Owen set’ for this TU-game, which is optimal in the context of a

purely competitive economy (Van Gellekom et al., 2000). Owen (1975) has shown that the Owen set

of an LPG is contained in the core of the LPG.

21

Consider the optimal solution (S*, y*) of problem (P). The purely competitive payoff profile

*)(* Suu = is also called the Owen set or the Owen point, since it is generally a single point. It is

obtained from the solution (wS*, z*(S*)) of ( * SD ) through the following relation:

*)(*1

* r

R

rrjjSj zceu ∑

=

= or equivalently,

∉=

∈=∑=

* if 0*

* if **1

Sju

Sjzcu

j

r

R

rrjj . (21)

Expression (21) shows that the payoff of each player equals the value of his resource bundle

under the marginal price. Moreover, this vector of payoffs forms a subset of the core in this

production game. However, it will be shown that the Owen set allocation has some drawbacks, in

particular a lack of fairness that may induce a corresponding lack of stability robustness against the

objections to coalition S*.

3.6. Example

Consider the BOM of the example in Hennet (2003) with two final products (1 and 2), three

intermediate products (3, 4 and 5), the unit profit vector T)0 0 0 25 22(=π and the technical

matrix

=

0022100300000220000000000

G . Four resources are necessary for the five products at the different

manufacturing stages, with the following requirement matrix

=

11122211000110000011

M .

Ten enterprises are candidates for partnership in the supply chain. The amounts of the four

resources owned by the ten firms are represented in the following matrix:

22

=

20202705503301990137101200200010955000300100000

C .

The optimal total payoff and optimal coalition are obtained from the solution of the LP (P)

(with the additional term to obtain a coalition of lowest cardinality). The maximal total payoff

is 5.87* =v , obtained for T] 0 0 0 3.5 0[* =y . It is obtained by the minimal coalition

{ }9876543* =S and also, indeed, by any coalition containing S*.

The associated purely competitive payoff profile (Owen set) is obtained by formula (21) for

]025.100[* =z :

0] 23.75 11.25 0.00 16.25 8.75 12.50 15.00 0 0[* =u .

In this solution, the endowment of partner 7 is null. Yet, partner 7 is important to the coalition

since for coalition { }7*−S , the total payoff drops down to 46.875! Its individual marginal

contribution in S* is:

{ } { } 40.625)7\*(*)()7(* ≅−=∆ SS vvS .

The reason for a null endowment is that, with partner 7 in the coalition, resource 4 is in excess

and its shadow price falls to 0. The individual marginal contribution of partner 7 in the grand coalition

N is also strictly positive:

{ } { } 28.12559.375\ =−=−=∆ 5.87)7()()7( NNN vv .

From the marginal contribution principle (6), non nullity of { })7(N∆ is a necessary condition

for the existence of a core payoff profile N∈iiu )( with 07 >u .

23

4. Improving payoff functions with side payments

4.1. Some criticisms of the Owen set allocation

The property of not allocating any payoff to the players with resources in excess is inherent in

the Owen set since this solution rule derives from the duality principle in linear programming. In

duality theory, a shadow price indicates the value of one additional unit of the resource associated

with the corresponding primal constraint. Thus, if a dual variable is equal to zero (zr* = 0), this means

that the addition of one unit of resource r has no effect on the optimal objective function. Hence the

allocations based on the values of the optimal dual variables exhibit this property: the players with

scarce resources share the total worth of the coalition between them, and the players with excess

resources get a null payoff.

In light of these shortcomings, the Owen set solution may become critically stable since the

players with excess resources are indifferent as to whether they participate. Such a lack of incentive

toward NPPs is not desirable in supply chain design because it does not provide stability robustness

against objections.

Therefore, even if the allocation mechanism may better reward the players with scarce

resources, allocating a null imputation to the players with all resources in excess may not be the best

solution. The next section explores the possibility of constructing a core solution with a strictly

positive payoff for each player of the optimal coalition.

4.2. Trying to combine fairness with rationality

Due to the criticisms directed at the Owen set as a competitive solution, other practical

allocation schemes will be explored to ensure that the maximal payoff is fairly allocated to the

players of the LPG game. So, the objective is to find an allocation that is fair and belongs to the core

of the LPG. These requirements can be formulated by the following set of constraints:

24

*1

vxn

ii =∑

=

; (22)

N⊂∀≥∑∈

SSvxSj

j )( ; and (23)

*0 S⊂∀> jx j . (24)

As shown in the previous example, the optimal payoff may strongly decrease if a player who only

owns resources that are globally in excess decides not to participate in the cooperative game. This is

why condition (24), denoted ‘fairness property’, has been added to core requirements (22) and (23);

it provides a positive gain to every partner of the optimal coalition. However, the existence of

solutions to the set of conditions (22), (23) and (24) is not guaranteed in general. The purpose of this

section is precisely to characterize the cases when such solutions exist.

By construction, the core of an LPG is not empty if problem ( NP ) has a finite optimal solution

that is strictly positive, 0)( >Nv . This is clear from the fact that an Owen solution exists whenever

the dual problem ( ND ) , or equivalently ( *SD ), has a solution. Then, in order to explore whether the

core of the game contains solutions that also satisfy (24), the following set of imputations can be

constructed.

The purely competitive payoff profile *)(* Suu = has been defined by (21). The set N can be

decomposed into three disjoint subsets, as follows:

*** 210 SSSN ∪∪= (25)

with { }0*;* =∈= *uSii i and 0S , { }0*;* >∈= *uSii i and 1S , *SNS2 −=* .

25

The cardinals of these sets are respectively denoted **,*, 210 sss with, by assumption,

1*,1* ≥≥ 10 ss . This corresponds to the case when the total payoff is strictly positive and some of

the players in the optimal coalition are NPP: their Owen allocation is null.

A new set of imputations, denoted w , is defined as follows:

∈∀==∈∀−=∈∀=+=

*0***

**

2

1

0

S

S

S

iuwiuw

iuw

ii

ii

ii

βαα

, (26)

with

0≥α , 0≥β , ** 10 ss βα = . (27)

−

≤

∑∈

⊂ 1

)(*

mins

Svui

i

S1

S

Nβ with *11 SSS ∩= and )( 1Scard=1s (28)

*min*

ii

u1

S∈<β (29)

Theorem 1

A set of imputations w that satisfies conditions (26)-(29) belongs to the core of the LPG.

Furthermore, there exist strictly positive values of β that satisfy (28)-(29) and define fair imputations

if and only if 0*)(***

*min

1

2~ >

∪−

−= ∑

∈⊂ Sjj

SSSvu

1001

0

S sssss

β with *~ S⊂S and

0** 01 >− 10 ss ss .

Proof

26

• Efficiency : From condition (26), ***1

10N

ss βα −+= ∑∑∈= i

in

ii uw . Then, Property (22) for

imputation w is obtained from relation (27) and the efficiency property of the Owen

imputation ( **1

vun

ii =∑

=): *

1vw

n

ii =∑

=.

• Fairness:

By construction of *1S , 0min*

>∈

ii

u1

S . Then, If 0

)(*

min1

>

−

=

∑∈

⊂ s

Svui

i

S1

S

Nβ , it is possible to

select 0>β such that (22) and (23) are satisfied.

• Rationality:

The Owen imputation belongs to the core and thus satisfies the rationality property:

N⊂∀≥∑∈

SSvuSj

j )(* .

(30)

Consider now the new imputation w and a coalition N⊂S . Then, define *00 SSS ∩= , with

*11 SSS ∩= , *22 SSS ∩= and )( 0Scard=0s , )( 1Scard=1s .

10 ss βα −+= ∑∑∈∈ Sj

jSj

j uw *

If 0≥− 10 ss βα , then condition (25) implies )(SvwSj

j ≥∑∈

for any value of β satisfying

(27), (28) and (29).

27

Consider now the case 0<− 10 ss βα . Using relation (18), this condition can be equivalently

replaced (with 0* ≠0s ) by 0** 01 >− 10 ss ss . Then, condition )(SvwSj

j ≥∑∈

is satisfied for any

value of β that verifies

.)(***

*

1

−

−≤ ∑

∈Sjj Svu

1001

0ssss

sβ (31)

Then, by super-additivity of v , *)()( 210210 SSSvSSSv ∪∪≤∪∪ and the two sets have

the same parameters 1s,0s . Therefore it suffices to test condition (31) for sets such that

0** 01 >− 10 ss ss and *22 ss = . Conversely, if condition (31) for some set S implies 0=β , then

imputation *u is the only one satisfying conditions (26)-(29). �

4.3. Example

In the example of section 2.5, the optimal coalition is { }9876543* =S and the

grand coalition N can be partitioned as follows: *** 210 SSSN ∪∪= with { }7=0S ,

{ }9,8,6,5,4,3=1S and { }10,2,1=2S . The total number of coalitions in N is 10231210 =− .

However, using theorem 1, only 63126 =− coalitions have to be tested to determine the value of

β . In this example, 6*,1* == 10 ss and condition 0** 01 >− 10 ss ss requires 00 =s . Then,

the possible values of 1s ( 1s =1,…,6) generate the 126 − sets for which condition (27) has to be

tested. The minimal bound, 25.1=β is obtained for the set { }10,8,6,2,1* =S and since

iSi

u1

min∈

>β , the value β defines the following imputation which belongs to the core and satisfies

the condition of fairness:

28

0.00] 22.50 10.00 7.50 15.00 7.50 11.25 13.75 0.00 00.0[=w .

The proposed technique has actually constructed the set of core allocations )(βw defined by

parameter β in the interval [ ]25.10 and such that:

0] -23.75 -11.25 6 -16.25 -8.75 -12.50 -15.00 0 ββββββββ 0[)( =w .

For 0>β , the solution )(βw belongs to the core and has the property of fairness. The feasible

interval for β can be used as a negotiation space by the players of the optimal coalition.

4.4. Discussion

The situation described in the numerical example has only 10 players with resource capacities

that are not oversized. It has been shown that, in this particular case, the core is not reduced to a

single point and it is possible to construct imputations that are both fair and rational.

In contrast to this situation, other numerical experiments have been performed with more

players and several owners for each resource. These results show that, in this case, the core of an

LPG game is generally reduced to a single point, which is precisely the Owen competitive allocation.

This result is not surprising if one realizes that in an open market in which many players have similar

abilities and equipment, the core of an LPG tends toward a single point that characterizes the

perfectly competitive situation. Convergence of the core to the Owen set of an LPG has been shown

and characterized by Owen (1975) and Semet and Zamel (1984) for games in which players are

replicated when the number of replications tends to infinity. In a broader context, a well-known

result is that, for large numbers of players, the core of the game tends to be the set of competitive

allocations (Aumann, 1964).

Using the LPG framework, this study has provided a rational explanation for the difficulty in

conciliating economic efficiency with a fair repartition of profit. A related finding is that, in Linear

29

Production Games, cooperation and competition are complementarily entangled rather than

opposed. Many practical examples of cooperative competition, also called “co-opetition”

(Brandenburger and Nalebuff, 1997), can be found in today’s industrial world. A well-known example

is the short-term cooperation that took place between the competing companies IBM and Oracle to

develop ERPs for SMEs. On the one hand, coalition formation between retailers, manufacturers, and

suppliers can provide a competitive advantage on the market because of the increased

manufacturing possibilities generated by the sharing of competence and resources. On the other

hand, competition between owners of similar resources or between manufacturers of similar

products tends to decrease the value of these resources or products. As a consequence, it may

jeopardize the coalitional stability by decreasing the partners’ motivation to cooperate. If a resource

becomes potentially available beyond its necessary level, its marginal value decreases to zero, and so

does the profit allotted to its owners for possessing it. In this respect, a good business strategy for a

company could be to maintain the uniqueness of its products through technical progress and

innovation.

The search for cases when fair repartitions of profits exist also indicates that in order to

maintain good profit margins, an enterprise should become a partner of its complementors and

avoid associations with its competitors. The difficulty is that the same company may be both the

enterprise’s complementor and its competitor. In such a case, a possible path to reach a win–win

situation is not to enter the game as it is, but rather to change its rules, typically through negotiation

with the partners. The current industrial and commercial practices show that many agreements and

contracts are currently established to reach mutually profitable situations, with a particular attention

paid to the business legal regulations to be respected. In terms of practical significance, this study

has thus provided some clues for achieving economic profitability, by identifying some key properties

that generate positive profits: ownership or production of assets that are not easily available

elsewhere and cooperation with firms producing complements of the enterprise’s own products.

30

5. Conclusions

In the light of cooperative game theory, supply networks can be modelled as profit-maximizing

systems creating value in their socio-economic environment, considered as a market. Under the

assumption of a perfect sharing of resources and production capacities in their manufacturing

environment, supply network design problems can be represented as problems of optimal coalition

formation. Then, coalition stability requires efficiency and rationality in the distribution of profit

among the enterprises of the supply network. However, if the manufacturing environment is highly

competitive, only the owners of resources that are marginally scarce receive a strictly positive share

of the profit. A firm does not receive a share of the profit if all its resources are also possessed by

other firms in the manufacturing environment and are globally in excess. Its revenue simply covers its

costs. This result is consistent with the general equilibrium analysis in competitive economies.

However, such profit allocation rules are not motivating and may appear unfair to the firms that

belong to the optimal coalition without owning any scarce resource. This paper has shown that in

moderately competitive manufacturing environments, where resources are not very abundant, the

core of the game, which is the set of efficient and rational profit allocations, is not always restricted

to the purely competitive profit allocation rule. Under these conditions, a set of feasible allocation

rules has been constructed to guarantee a positive profit to all the enterprises in the supply network.

A possible extension of this work could be to use this feasible set as a negotiation set for the firms of

the supply network.

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