A Multi-Agent Min-Cost Flow problem with Controllable Capacities

Complexity of Finding a Maximum-ﬂow Nash Equilibrium

Nadia Chaabane Fakhfakh

1,2

, Cyril Briand

1,3

and Marie-Jos´e Huguet

1,2

CNRS, LAAS, 7 avenue du colonel Roche, F-31400 Toulouse, France

Universit´e de Toulouse, INSA, LAAS, F-F-31400 Toulouse, France

Universit´e de Toulouse, UPS, LAAS, F-31400 Toulouse, France

Keywords:

Multi-Agent, Network Flow, Nash Equilibrium, Complexity, Min-Cost Flow.

Abstract:

A Multi-Agent Minimum-Cost Flow problem is addressed in this paper. It can be seen as a basic multi-agent

transportation problem where every agent can control the capacities of a set of elementary routes (modeled

as arcs inside a network), each agent incurring a cost proportional to the chosen capacity. We assume that a

customer is interesting in transshipping a product ﬂow from a source to a sink node through the transportation

network. It offers a reward that is proportional to the ﬂow that the agents manage to provide. The reward is

shared among the agents according to a pre-established policy. This problem can be seen as a non-cooperative

game where every agent aims at maximizing its individual proﬁt. We take interest in ﬁnding stable strategies

(i.e., Nash Equilibrium) such that no agent has any incentive to modify its behavior. We show how such

equilibrium can be characterized by means of augmenting or decreasing path in a reduced network. We also

focus on the problem of ﬁnding a Nash equilibrium that maximizes the ﬂow value and prove its NP-hardness.

1 INTRODUCTION

Multi-agent network games have become a promising

interdisciplinary research area with important links

to many application ﬁelds such as transportation,

project scheduling, computer network, etc. In these

applicative areas, decision processes often involve

several actors, each one having its own autonomy, its

own objectives and its own constraints. These actors,

often referred to as agents, aim at maximizing their

own proﬁts, provided a global objective should be

fulﬁlled. This kind of problem, called multi-agent

optimization, can be met in many real-life problems

such as transportation networks, supply chain man-

agement, web services, production, etc. The nature

and complexity of network optimization problems

change signiﬁcantly when the multi-agent context is

considered. Besides optimizing a global objective, a

solution should also satisfy additional criteria related

to multi-agent games. In fact, on the one hand, a

solution should optimize the agent’s objective and, on

the other hand, should also be stable in the sense of

Nash (i.e.; no agent is able to improve its proﬁt, to the

detriment of the others). Those additional features

are connected with multi-objective optimization and

Game Theory, respectively. This work sits at the

crossroads between two disciplines, namely multi-

agent systems and social networks. The former ties in

with distributed resolution of multi-agent problems,

while the latter is connected to game theory, which

formalizes the multi-agent optimization problem with

strategic game between different agents.

Recently, some researchers have paid attention to a

particular multi-agent network problem: the Multi-

Agent Project Scheduling (MAPS) problem. In the

seminal work of (Evaristo and Fenema, 1999), the

authors proposed a special framework for distributed

projects, with costs and rewards shared among agents.

In an earliest work (De et al., 1997), the authors

consider a MAPS problem where each agent can

control the duration of its activities at a given cost.

The project activities and precedence constraints are

classically modeled by an activity-on-arc graph. A

reward is offered to agents when they manage to

ﬁnish the project earlier than expected, as proposed in

(Fernandez, 2012). It was demonstrated in (Agnetis

et al., 2013) and (Briand et al., 2012a) that ﬁnding a

Nash equilibrium minimizing the project makespan

is NP-hard in the strong sense . Moreover, based

on the concepts of increasing and decreasing cuts,

as deﬁned in (Kamburowski, 1994), and, on duality

between maximum ﬂow and minimum cut problems,

Chaabane Fakhfakh N., Briand C. and Huguet M..

A Multi-Agent Min-Cost Flow problem with Controllable Capacities - Complexity of Finding a Maximum-ﬂow Nash Equilibrium.

DOI: 10.5220/0004765500270034

In Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems (ICORES-2014), pages 27-34

ISBN: 978-989-758-017-8

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

Briand et al. (2012b) proposed an efﬁcient integer

linear program formulation for this problem (Briand

et al., 2012b).

Another important application to the network opti-

mization is the well-known Network Flow theory

(Ford and Fulkerson, 1958). Several algorithms have

been developed in order to ﬁnd a maximum ﬂow in

a network. Ford and Fulkerson (1956) were the ﬁrst

to develop a clever algorithm based on the duality

between minimum cut and maximum ﬂow (Ford and

Fulkerson, 1956). Later on, efﬁciency improvements

were proposed see eg; (Edmunds and Karp, 1972),

(Goldberg and Tarjan, 1986), etc. The minimum cost

maximum ﬂow problem, which is equivalent to the

minimum cost circulation problem, is solvable in

polynomial time (Tardos, 1985).

As regards to social networks, the prediction of

agents’ behavior is of interest. Several papers focus

on games associated with various forms of networks,

see (Tardos and Wexler, 2007) for an overview. In

a recent work, Apt and Markakis (2011) studied the

complexity of ﬁnding a Nash Equilibrium for the

multi-agent social networks with multiple products,

in which the agents, inﬂuenced by their neighbors,

can choose one out of several alternatives (Apt and

Markakis, 2011).

Speciﬁcally, this work considers a transportation

network that involve a set of agents, each one being

in charge of a part of the network. It is assumed

that each agent is able to control the transportation

capacities of its arcs. A lot of features of this work

are inspired by the multi-agent project scheduling

problems, as presented in (Briand et al., 2012a),

especially concerning the reward sharing policy.

In fact, the outcome of an agent depends on its

own strategy and on the satisfaction of a customer,

which depends on the network ﬂow. As proposed

by Fernandez (2012) (Fernandez, 2012), we assume

that the customer gives a reward proportional to the

maximum-ﬂow that can circulate inside the network.

This reward is shared among agents according to

some ratios predeﬁned in the network design phase

(Cachon and Lariviere, 2005).

To the best of our knowledge, the research presented

here is an original way of presenting a transportation

problem using multi-agent network ﬂow with con-

trollable arcs capacities. One important application to

the problem proposed in this paper is the distributed

control of transportation networks, like trafﬁc, water,

where the road of the network are distributed among

several agents which can control the amount of

product or water to circulate on the network.

This paper mainly discusses the complexity of ﬁnding

a Nash Equilibrium that maximizes the ﬂow in the

network.

The paper is organized as follows: Section 2 deﬁnes

formally the Multi-Agent Minimum-Cost Flow prob-

lem and introduces some important notations. There-

after, Section 3 introduces the duality between efﬁ-

ciency and stability of a strategy and presents some

important deﬁnitions and properties. In Section 4 and

5, we illustrate some basic notions for the single agent

and the multi-agent cases, respectively. In Section 6,

an example is provided to illustrate the notions intro-

duced in previous sections. Section 7 deals with the

complexity of the problem of ﬁnding a Nash equilib-

rium with bounded ﬂow. Finally, conclusions and fu-

ture directions are drawn in Section 8.

2 PROBLEM STATEMENT AND

NOTATIONS

We focus on a Minimum-Cost Flow problem under

a Multi-Agent context. This problem will be further

referred to as MA-MCF. Considering a transportation

network with limited arc capacities, this problem con-

sists in sending a maximum amount of products from

a source node to a sink node, at minimum cost. In

this work, a major assumption is that arc capacities

are controlled by agents, each arc being assigned to a

speciﬁc agent.

2.1 Problem Deﬁnition

The MA-MCF problem can be described as a tuple

< G,A,Q,Q,C, π,W >, where:

• G = (V,E) is a ﬂow network. V is the set of nodes,

s,t ∈ V being the source and the sink nodes of the

ﬂow network G, respectively. E is the set of arcs,

each one having its capacity and receiving a ﬂow.

An arc e from node i to node j is denoted by e =

(i, j).

• A is a set of m agents: A = {A

,... , A

Arcs are distributed among agents. An agent A

owns a set of m

arcs, denoted E

. Each arc (i, j)

belongs to exactly one agent (i.e., E

∩ E

0 for

each agent’s pair (A

) ∈ A

such that u 6= v).

• q

i, j

is the capacity of arc (i, j) which takes value in

an interval [q

i, j

]. q

i, j

(resp.

i, j

) is the normal

(resp. maximum) arc capacity. Q = (q

i, j

)

(i, j)∈E

and

Q = (q

i, j

)

(i, j)∈E

referred to as the vectors of

normal and maximum arc capacities, respectively.

ICORES2014-InternationalConferenceonOperationsResearchandEnterpriseSystems

For any circulating ﬂow f

i, j

, it classically holds

i, j

≤ q

i, j

with q

i, j

∈ [q

i, j

• C = {c

i, j

} is the vector of costs where c

i, j

is the

unitary cost incurred by agent A

, such that (i, j) ∈

, for increasing q

i, j

by one unit.

The vector C

denotes the cost vector incurred by

augmenting the capacity of arcs by the agent A

• π referred to as a reward given by a ﬁnal client.

This reward is proportional to the maximum ﬂow

that can circulate from s to t.

• W = {w

} deﬁnes the sharing policy of rewards

among the agents. The A

reward for a gain of

one unit of maximum ﬂow equals w

π.

We denote by Q

, u = 1,.. . ,m the vector of capac-

ities chosen by the agent A

for the arcs belonging

to him. Q

≤ Q

≤ Q represents the individual

strategy of the agent A

. We further refer to

S = (Q

,...,Q

) as the vector of individual strategies

of all agents. A strategy S

−u

denote the strategies

of the (m − 1) players, but agent A

, such that

−u

= (Q

,..,Q

u−1

u+1

,..,Q

Given a strategy S, F(S) denotes the maximum ﬂow

that can circulate on the network ﬂow given the

current values of capacities. It is equal to the sum of

ﬂow circulating in the forward arcs of source node

(i.e., F =

∑

(s, j)∈E

s, j

). F

corresponds to the maxi-

mum ﬂow when capacities q

i, j

are set to q

i, j

, in other

words, the largest possible ﬂow at zero cost.

F is the

maximum ﬂow that can circulate when capacities are

set to their maximum values (q

i, j

). Therefore,

for any strategy S, it holds that F

≤ F(S) ≤ F.

It is assumed, in this paper, that the share of reward

among agents w

did not depend on the arcs used

by agents. It is nevertheless possible to extend this

work to the case in which the reward depends on how

much the resource owned by each agent is used at

optimum.

The cost incurred by agent A

for a strategy Q

equals:

− Q

) =

∑

(i, j)∈E

i, j

− q

i, j

)

With respect to the above payment scheme, the total

reward given for a circulating ﬂow F(S) under strat-

egy S is π (F(S) − F

The proﬁt Z

(S) of agent A

under strategy S is equal

to:

(S) = w

π (F(S) − F

) −

∑

(i, j)∈E

i, j

− q

i, j

)

(1)

We denote by Z(S) = (Z

(S),.. . ,Z

(S)) the overall

proﬁt vector.

Example of a MA-MCF Network

The network ﬂow G(V,E) displayed in Fig. 1 is com-

posed of ﬁve arcs E = {a, b,c,d,e} distributed be-

tween two agents A

and A

such that E

= {b, c, d}

and E

= {a,e} (their assigned arcs are represented

with plain and dotted arcs, respectively). The set of

vertex is V = {A,B,C,D} where the source node is

A and sink node is D. Each arc in the graph 1 is

denoted by the interval of normal and maximum ca-

pacities, and by the cost of increasing arc capacities

([q

i, j

],c

i, j

). Costs and capacities are such that

= C

= 50, C

= C

= 30, C

= 10 and

, q

∈ [0,1]. When increasing arc

capacities which leads to one additional unit of ﬂow

circulating, a ﬁnal client gives reward π = 120 which

will be shared between agents following the sharing

policy w

= w

Agent A1

Agent A2

,i j

Figure 1: Problem description of example 1.

2.2 Mathematical Formulation

Each agent should choose the capacities of its arcs,

having the objective of maximizing its own proﬁt.

The problem can be formalized as the following

multi-objective mathematical program:

Max (Z

(S),Z

(S),.. . ,Z

(S))

s.c.

(i) f

i, j

≤ q

i, j

, ∀(i, j) ∈ E

(ii)

∑

(i, j)∈E

i, j

−

∑

( j,i)∈E

j,i







0 ∀i 6= s,t

F , i = s

−F , i = t

(iii) q

i, j

≤ q

i, j

≤

i, j

, ∀(i, j) ∈ E

i, j

≥ 0, ∀(i, j) ∈ E

Where Z

(S), u = 1,. . .,m is the proﬁt of agent A

given by the equation (1) for each strategy S.

Constraints (i) represent the capacity constraints.

AMulti-AgentMin-CostFlowproblemwithControllableCapacities-ComplexityofFindingaMaximum-flowNash

Equilibrium

Constraints (ii) impose the conservation of the ﬂow.

The aim of this problem is to ﬁnd an overall strategy

S that maximizes agents’ proﬁt. Each agent A

has to

decide the arc capacity q

i, j

, ∀(i, j) ∈ E

in order to

maximize its proﬁt.

3 EFFICIENCY VS. STABILITY

A strategy is said efﬁcient if it corresponds to a

Pareto-optimal solution with respect to the above

multi-objective program. The notion of Pareto opti-

mality is concerned with social efﬁciency (Ehrgott,

2005). A Pareto strategy is preferred to any other

strategy dominated by it.

Deﬁnition 1. Pareto Optimality: A strategy S is

Pareto-optimal if it is not dominated by any other

strategy S

′

. In other words, it does not exist any strat-

egy S

′

such that Z

′

) ≥ Z

(S) for all A

, with at least

one inequality being strict.

The set of Pareto optimal strategies is denoted by S

On the other hand, a strategy is stable if there is no in-

centive for any agent to modify its decision in order to

improve its proﬁt. The stability of a strategy ensures

that agents can trust each other. It is connected to the

notion of Nash equilibrium in non-cooperative game.

Deﬁnition 2. Nash Equilibrium: given a sharing re-

ward policy w

, a strategy S = (Q

,... , Q

) is a Nash

Equilibrium if for any agent A

with strategy Q

′

, the

following equation holds:

−u

) ≥ Z

′

−u

), ∀Q

′

6= Q

(2)

We refer to S

as the set of Nash equilibria.

Let us also deﬁne the concept of a poor strategy.

This concept will be useful for characterizing prop-

erly Nash equilibria.

Deﬁnition 3. Poor Strategy: A strategy S =

,... , Q

) with ﬂow F(S) is a poor strategy if and

only if it exists an agent A

and an alternative strategy

′

such that Z

(S) < Z

′

) and F(S

′

) = F(S), where

′

= (S

−u

′

In other words, S is a poor strategy if and only if

one agent is able to increase its proﬁt by changing

unilaterally its strategy (modifying the capacity of

some of its arcs), without modifying the overall ﬂow

in the network, nor the proﬁts of other agents. It is

obvious that for any poor strategy S, S 6∈ S

The set of non-poor strategies will be denoted by

Ideally, agents should choose a strategy which sat-

isfy both Pareto optimality and Nash stability (i.e.,

S ∈ S

). Nevertheless, since S

can be

empty, such a strategy is not always attainable. In

this case, we are looking for a Nash equilibrium that

is as efﬁcient as possible with respect to the customer

viewpoint. A Nash equilibrium that maximizes the

ﬂow circulating is indeed suitable both for maximiz-

ing the total reward and the customer satisfaction.

The aim of this study is to ﬁnd an optimal strategy

proﬁle S

∗

such that the solution is a Nash Equilibrium

that maximizes the ﬂow circulating, the share of re-

ward among agents w

being ﬁxed.

Assumptions: For sake of simplicity, it is assumed

throughout this paper, that q

i, j

= 0. Therefore, the

initial minimum circulating ﬂow at zero cost is equal

to F

= 0. This assumption does not modify the fun-

damental results of this work.

4 THE SINGLE-AGENT CASE

This section presents some basic properties related to

classical network ﬂow theory. In the single agent case

(all the arcs belongto a single agent), a non-poor strat-

egy S for a given ﬂow F(s) is a strategy that mini-

mizes the overall cost. Such minimization problem is

well-identiﬁed in the literature as the minimum-cost

ﬂow problem (Busacker and Gowen, 1961).

Let us recall in the following section how the total

ﬂow can be either increased or decreased, at minimum

cost, using increasing or decreasing paths. These no-

tions will be used in section 5.

4.1 Increasing the Max-Flow

Given a ﬂow F(S) for strategy S, we are interested in

increasing the ﬂow value at minimum cost. For this

purpose, we recall the well-known notion of an aug-

menting path based on the concept of residual graph

(S), which is deﬁned below.

Deﬁnition 4. Residual Graph: Given a network G =

(V,E) and a ﬂow F(S), the corresponding residual

graph G

(S) = (V,E

) is deﬁned as follows: each arc

(i, j) ∈ E, having a maximum capacity

q(i, j) and a

ﬂow f

i, j

in G, is replaced by two arcs (i, j) and ( j, i).

The arc (i, j) has cost c

i, j

and residual capacity r

i, j

− f

i, j

and the arc ( j,i) has cost c

j,i

= −c

i, j

and

residual capacity r

j,i

= f

i, j

Deﬁnition 5. Augmenting Path: An augmenting path

is a path P in G

(S) from s to t, where e

= s and

= t.

We refer to P as the set of augmenting paths.

The greatest ﬂow augmentation that can be achieved

using P is r

= min{r

: (i, j) ∈ P}.

ICORES2014-InternationalConferenceonOperationsResearchandEnterpriseSystems

An augmenting path in G

(S) is made of forward arcs

(having the same direction in G) and backward arcs

(having the opposite direction than the ones in G).

The set of forward and backward arcs are denoted P

and P

−

, respectively.

The cost of augmenting the ﬂow by one unit using

the augmenting path P ∈ P is denoted cost(P). It is

expressed as follows:

cost(P) =

∑

(i, j)∈P

i, j

−

∑

(i, j)∈P

−

i, j

(3)

4.2 Decreasing the Max-Flow

When considering the problem of decreasing the ﬂow

at minimum cost in the network, we introduce the new

concept of decreasing path.

Deﬁnition 6. Decreasing Path: a decreasing path P

is a path in G

(S) from nodet to node s through which

the ﬂow can be decreased.

We refer to

P as the set of decreasing paths.

Similarly, a decreasing path in G

(S) is made of for-

ward arcs (having the opposite direction than the one

in G

(S)) and backward arcs (having the same direc-

tion in G

(S)). The set of forward and backward arcs

are denoted

and P

−

, respectively.

profit(

P) =

∑

(i, j)∈

i, j

−

∑

(i, j)∈

−

i, j

(4)

5 THE MULTI-AGENT CASE

In the multi-agent context, one agent can decrease (or

increase) unilaterally its arc capacities to improve its

proﬁt. In this context, we introduce the concept of

proﬁtability of an augmenting or a decreasing path

and provide a characterization of Nash equilibrium.

5.1 Increasing the Max-Flow

Let us introduce the notion of proﬁtable augmenting

path. In a similar way, in the multi-agent context, an

augmenting path is composed by a set of forward and

backward arcs P = {P

−

} such that if q

i, j

is in-

creased by one unit ∀(i, j) ∈ P

and decreased by one

unit ∀(i, j) ∈ P

−

, it is possible to increase the overall

ﬂow by one unit.

The cost of an augmenting path for agent A

cost

(P), is deﬁned as the net change in cost ﬂow for

one unit of ﬂow augmentation throughout this path. It

is expressed as follows:

cost

(P) =

∑

(i, j)∈P

∩E

i, j

−

∑

(i, j)∈P

−

∩E

i, j

(5)

Deﬁnition 7. Proﬁtable augmenting path. An aug-

menting path P ∈ P is said proﬁtable path for all

agents if, for every agent A

, cost

(P) < w

× π.

This means that through a proﬁtable augmenting

path, increasing the ﬂow by one unit, is proﬁtable for

all the agents owning the arcs of the path.

5.2 Decreasing the Max-Flow

Now, the notion of proﬁtable decreasing path is intro-

duced. In the multi-agent context, a decreasing path

P = {P

−

} is composed of forward and backward

arcs. If q

i, j

is decreased by one unit, ∀(i, j) ∈

, and

increased by one unit, ∀(i, j) ∈

−

, the overall ﬂow is

decreased by one unit.

The proﬁt profit

(

P) generated by decreasing capac-

ity through a decreasing path, for an agent A

, is de-

ﬁned as follows:

profit

(

P) =

∑

(i, j)∈

∩E

i, j

−

∑

(i, j)∈

−

∩E

i, j

(6)

Deﬁnition 8. Proﬁtable decreasing path. A decreas-

ing path

P ∈ P is proﬁtable if there is one agent A

such that profit

(P) > w

× π.

In other words, through a proﬁtable decreasing

path, decreasing the ﬂow by one unit is proﬁtable for

one agent, to the detriments of the others.

In the multi-agent context, it is important to char-

acterize strategies in which an agent can decrease or

increase the overall ﬂow. Therefore, it is important

to ﬁnd proﬁtable augmenting paths in order to in-

crease ﬂow without generating decreasing paths that

are proﬁtable for some agent, hence preserving stabil-

ity.

Proposition 5.1. Nash Equilibrium.

For a given non-poor strategy proﬁle S, S is a Nash

Equilibrium if and only if:

• ∀A

∈ A , ∀P ∈ P such that(i, j) ∈ E

cost

(P) > w

× π (7)

• ∀A

∈ A , ∀

P ∈ P

profit

(P) < w

× π (8)

Proof. Consider a strategy S and an agent A

. If S

is non poor, A

can improve its situation only by in-

creasing or decreasing the ﬂow. In the former case, for

an additional unit of ﬂow, A

receives w

× π. Since,

such increase is proﬁtable to A

if and only if there is

an augmenting path P such that cost

(P) < w

× π. In

the latter case, viceversa, it is proﬁtable for an agent

to decrease the ﬂow by one unit if and only if there

is a decreasing path P such that profit

(P) > w

× π.

Therefore, if and only if for no agent any of those con-

ditions holds, no agent A

can individually improveits

proﬁt, and S is a Nash equilibrium.

AMulti-AgentMin-CostFlowproblemwithControllableCapacities-ComplexityofFindingaMaximum-flowNash

Equilibrium

Figure 2: A multi-agent network ﬂow with two agents and ﬁve arcs.

6 ILLUSTRATIVE EXAMPLE

Let us come back to the previous example (section

2.1) to illustrate the optimality-stability duality of a

strategy.

The initial ﬂow on the network is equal to its min-

imum value F

= 0. It is possible to increase it along

the proﬁtable augmenting path (A-C-D), which leads

to the strategy S

= (0,1,0,0, 1) (see Figure 2(a)) with

F(S

) = 1 and Z

) = Z

) = 30. From this strat-

egy, it is still possible to increase the ﬂow along the

proﬁtable augmenting path (A-B-D), which leads to

the strategy proﬁle S

= (1, 1,0,1, 1) (see Figure 2(b))

with F(S

) = 2 and Z

) = Z

) = 40. From this

strategy, we observe that there exists a proﬁtable de-

creasing path (D− B − C − A) from sink node D to

source node A which is proﬁtable for agent A

. In fact,

can improve its own proﬁt, by decreasing back

the ﬂow on b and d by one unit and increasing the

ﬂow on arc c by one unit. This leads to the strategy

= (1, 0,1,0,1) (see Figure 2(c)) with F(S

) = 1

and proﬁts Z

) = 50 and Z

) = − 20, which is

obviously bad for A

. Therefore, although the strat-

egy S

corresponds to a Pareto Optimum, which leads

to a maximization of agent’s proﬁts, it is not a stable

strategy. Strategy S

is a Nash Equilibrium but not

Pareto Optimum. Therefore, in our example there is

no a strategy which is both in S

and S

. The mo-

tivation of this paper is to search for a Nash-stable

solution which is as efﬁcient as possible, i.e., which

maximizes F(S).

7 PROBLEM COMPLEXITY

In this section, we discuss the complexity of ﬁnding a

Nash equilibrium that maximizes the ﬂow in the net-

work. This problem can be described by the following

mathematical model.

MA−MCF

Max F

s.c.

(i) f

i, j

≤ q

i, j

, ∀(i, j) ∈ E

(ii)

∑

j∈P

(i)

i, j

−

∑

j∈P

−

(i)

j,i







0 ∀i 6= s,t

F , i = s

−F , i = t

(iii) q

i, j

≤ q

i, j

≤

i, j

, ∀(i, j) ∈ E

(iv) profit

(

P) < w

× π, ∀P ∈ G

(S)

i, j

≥ 0, ∀(i, j) ∈ E

Constraints (i), (ii) and (iii) are the same as the

one of the multi-objective mathematical formulation

presented above 2. They represent the constraints

of arcs capacities and ﬂow conservation, respectively.

Constraints (iv) impose that no decreasing path

P ex-

ists in solution S with proﬁt profit

(P) greater or

equal to w

× π. In other words, it represent the con-

straints for a solution to being Nash stable. Even if

the constraint (iv) is linear, we notice that the num-

ber of possible paths in the residual graph can grow

exponentially. Moreover, a non-linearity can be rec-

ognized since the residual graph G

(S) depends on

the strategy chosen by each agent.

Constraints (iv) impose that no decreasing path

P ex-

ists in solution S with profit

(

P) greater or equal to

× π (See Proposition 5.1). Increasing paths do not

need to be bounded since since the network ﬂow is

maximized.

7.1 Finding a Nash Equilibrium with

Bounded Flow

We consider the decision problem to ﬁnd a strategy

which is a Nash equilibrium, with a ﬂow greater than

a given value. This problem can be deﬁned as follows.

Nash-Equilibrium Bounded Flow:

Instance: a tuple < G,A ,Q

,Q,C,π,W > as deﬁned in

section 2 and an integer Φ

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0 1 2 3 4 5 6 7 8 9

( ,7)e

( ,8)e

( , 7)e

( ,7)e

( , 7)e

( ,8)e

( ,9)e

( ,10)e

( ,9)e

( ,7)e

( , 7)e

( ,8)e

( ,9)e

( , 9)e

…

Figure 3: Reduction from 3-PARTITION problem with k = 3.

Problem: Is there a Nash Equilibrium stra-

tegy proﬁle S such that F(S) > Φ?

Proposition 7.1. Problem NE-Bounded Flow is

strongly NP-complete.

Proof. The NP-completeness of this problem can be

proved using a reduction from the well-known 3-

partition problem, which is known to be NP-complete

in the strong sense (Garey and Johnson, 1979).

3-Partition:

Instance: a set ζ = { a

,...,a

K−1

} of K = 3k positive

integers, such that

∑

K−1

l=0

= k × B and

∈]B/4,B/2]

Problem: Deciding whether ζ can be partitioned into

k subsets so that the sum of integers in each

subset is equal to B?

An instance of the MA-MCF problem with control-

lable capacities can be generated from an arbitrary in-

stance of the 3-partition problem as follows.

From the 3-partition problem instance, we build up a

network G with k× K arcs and K + 1 nodes where the

ﬁrst one is source node V

= s and the last one is the

sink node V

= t. An agent A

∈ A = {A

,... , A

}

owns K arcs.

The tail of an arc e

isV

idivK

, its head is V

(idivK)+1

Between nodes V

idivK

and V

idivK+1

, there are k paral-

lel arcs, indexed from i to i+ K step k, each of them

belonging to a speciﬁc agent: arc e

belongs to A

idivK

The cost of arc e

is c

= a

imodK

. In other words,

to every positive integer a

∈ ζ is associated k paral-

lel arcs with, same head and tail, maximum capac-

ity

= 1 and cost a

. The total reward is set to

π = (B+ε)k, ε being an arbitrary small positive value.

The sharing policy is deﬁned by w

= 1/k. Therefore,

agent’s unit reward is w

π = B + ε, identical for all

agents. The objective is to determine whether it exists

a Nash strategy such that F(S) > Φ?

For instance, Figure 3 illustrates the resulting ﬂow

network obtained from the 3-partition instance de-

ﬁned by k = 3, ζ = {7,8,7, 7,7,8, 9,10, 9} and B =

24. We have k = 3 agents and K ∗ k = 27 arcs. Be-

tween nodes i and i+ 1, we ﬁnd k = 3 arcs with cost

i+1

. The problem is to ﬁnd, whether it exists, a Nash

strategy such that the ﬂow is strictly greater than 0. In

that case, using the path with bold arcs allows to ob-

tain a one-unit total ﬂow, which is a Nash equilibrium

since every agent does not pay more than its part of

reward (w

π = B+ ε = 24+ ε). Any equivalent stable

path is also a solution to the original 3-Partition prob-

lem.

Let us prove this last property in a general way.

Consider the strategy S

where all arcs have normal ca-

pacity, q

i, j

= 0. The resulting ﬂow obviously equals

to F(S) = 0. With respect to S, we observe that an

agent can increase the ﬂow by the amount δ ∈]0,1],

increasing the capacities of all its arcs by the same

amount δ. However, doing so, the agent pays kBδ and

only gains (B + ε)δ. Hence, the new strategy is not

proﬁtable and cannot be a Nash equilibrium. In order

to obtain a Nash equilibrium, the total cost incurred

by each agent for increasing its arc capacities must

not exceed B, otherwise at least one agent will be in-

terested in decreasing back its capacities.

Due to the topology of the network, in order to in-

crease the ﬂow, exactly K = 3k arcs must be involved

in an augmenting path. In any Nash equilibrium strat-

egy with ﬂow strictly greater than 0, the augmenting

path having to be proﬁtable for every agent, it must be

made of exactly three arcs per agent. The total com-

pression cost for every agent equals exactly B.

8 CONCLUSIONS

This paper presents a new game theory framework for

a multi-agent ﬂow network problem with controllable

capacities. We consider that a ﬁnal customer gives a

reward, shared among agent, for any additional unit

of ﬂow circulating in the network. Each agent has

the possibility to modify the capacities of its arcs at

a given cost. We particularly point out the notions of

efﬁciency and stability of a strategy. We also prove

that ﬁnding a Nash Equilibrium with maximum ﬂow

AMulti-AgentMin-CostFlowproblemwithControllableCapacities-ComplexityofFindingaMaximum-flowNash

Equilibrium

is NP-hard in the strong sense. Further works are on-

going to linearize the mathematical model of ﬁnding a

Nash Equilibrium as a Mixed Integer linear program-

ming. Distributed heuristics able to ﬁnd a Nash equi-

librium are also under study.

ACKNOWLEDGEMENTS

This work was supported by the ANR project no.

ANR-13-BS02-0006-01 named Athena.

REFERENCES

Agnetis, A., Briand, C., Billaut, J.-C., and Sucha, P. (2013).

Nash equilibria for the multi-agent project scheduling

problem with controllable processing times. Technical

report, LAAS-CNRS.

Apt, R. K. and Markakis, E. (2011). Diffusion in social

networks with competing products. In Proceedings of

the 4th international conference on Algorithmic game

theory, SAGT’11, pages 212–223. Springer-Verlag.

Briand, C., Agnetis, A., and Billaut, J. (2012a). The multia-

gent project scheduling problem : complexity of ﬁnd-

ing an optimal nash equilibrium. In Proceedings of

the 13th International Conference on Project Manage-

ment and Scheduling, Leuven, Belgium, pages 106–

109.

Briand, C., Sucha, P., and Ngueveu, S. (2012b). Min-

imisation de la dur´ee totale d’un projet multi-agent :

recherche du meilleur ´equilibre de nash. In Proceed-

ings of the 13th edition of ROADEF, pages 111–112.

Busacker, R. and Gowen, P. (1961). A Procedure for Deter-

mining a Family of Minimal-Cost Network Flow Pat-

terns. Defense Technical Information Center, Johns

Hopkins University, 15 edition.

Cachon, G. and Lariviere, M. (2005). Supply chain coordi-

nation with revenue-sharing contracts: Strengths and

limitations. International journal of Management Sci-

ence, 51(1):30 – 44.

De, P., Dunne, E., Ghosh, J., and Wells, C. (1997). Com-

plexity of the discrete time-cost tradeoff problem for

project networks. European journal of Operations Re-

search, 45(2):302–306.

Edmunds, J. and Karp, R. (1972). Theoretical improve-

ments in algorithmic efﬁciency for netowrk ﬂow prob-

lems. Journal of the Association for Computing Ma-

chinery, 19:248–264.

Ehrgott, M. (2005). Multicriteria Optimization. Springer,

Berlin, second edition edition.

Evaristo, R. and Fenema, P. V. (1999). A typology of

project management: emergence and evolution of new

forms. International Journal of Project Management,

17(5):275–281.

Fernandez, M. E. (2012). A game theoretical approach to

sharing penalties and rewards in projects. European

Journal of Operational Research, 216(3):647–657.

Ford, L. and Fulkerson, D. (1956). Maximum ﬂow through

a network. Canadian Journal of Mathematics, 8:399–

404.

Ford, L. and Fulkerson, D. (1958). Flows in networks.

Princeton University Press, New Jersey.

Garey, M. and Johnson, D. (1979). Computers and

Intractability: A Guide to the Theory of NP-

Completeness. W.H. Freeman & Co, New York, USA.

Goldberg, A. and Tarjan, R. (1986). A new approach to the

maximum ﬂow problem. In Proceedings of the eigh-

teenth annual ACM symposium on Theory of Comput-

ing (STOC’86), pages 136–146. New york.

Kamburowski, J. (1994). On the minimum cost project

schedule. International Journal of Management Sci-

ence (Omega), 22(4):401– 407.

Tardos, E. (1985). A strongly polynomial minimum cost

circulation algorithm. International journal Combi-

natorica, 5(3):247–255.

Tardos, E. and Wexler, T. (2007). Network formation games

and the potential function method. Algorithmic Game

Theory, Cambridge University Press.

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