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
1
CNRS, LAAS, 7 avenue du colonel Roche, F-31400 Toulouse, France
2
Universit´e de Toulouse, INSA, LAAS, F-F-31400 Toulouse, France
3
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 nding 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,
27
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
Copyright
c
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
1
,... , A
u
,... , A
m
}.
Arcs are distributed among agents. An agent A
u
owns a set of m
u
arcs, denoted E
u
. Each arc (i, j)
belongs to exactly one agent (i.e., E
u
E
v
=
/
0 for
each agent’s pair (A
u
,A
v
) A
2
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
]. q
i, j
(resp.
q
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.
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28
For any circulating flow f
i, j
, it classically holds
f
i, j
q
i, j
with q
i, j
[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
u
, such that (i, j)
E
u
, for increasing q
i, j
by one unit.
The vector C
u
denotes the cost vector incurred by
augmenting the capacity of arcs by the agent A
u
.
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
u
} defines the sharing policy of rewards
among the agents. The A
u
reward for a gain of
one unit of maximum flow equals w
u
π.
We denote by Q
u
, u = 1,.. . ,m the vector of capac-
ities chosen by the agent A
u
for the arcs belonging
to him. Q
Q
u
Q represents the individual
strategy of the agent A
u
. We further refer to
S = (Q
1
,...,Q
m
) as the vector of individual strategies
of all agents. A strategy S
u
denote the strategies
of the (m 1) players, but agent A
u
, such that
S
u
= (Q
1
,Q
2
,..,Q
u1
,Q
u+1
,..,Q
m
).
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
f
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
=
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
u
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
u
for a strategy Q
u
equals:
C
u
(Q
u
Q
u
) =
(i, j)E
u
c
i, j
(q
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
u
(S) of agent A
u
under strategy S is equal
to:
Z
u
(S) = w
u
π (F(S) F
)
(i, j)E
u
c
i, j
(q
i, j
q
i, j
)
(1)
We denote by Z(S) = (Z
1
(S),.. . ,Z
m
(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
1
and A
2
such that E
1
= {b, c, d}
and E
2
= {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
,
q
i, j
],c
i, j
). Costs and capacities are such that
C
AB
= C
BD
= 50, C
AC
= C
CD
= 30, C
BC
= 10 and
q
AB
, q
BD
, q
AC
, q
CD
, q
BC
[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
1
= w
2
=
1
2
.
B
C
Agent A1
Agent A2
,i j
c
A
D
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
1
(S),Z
2
(S),.. . ,Z
m
(S))
s.c.
(i) f
i, j
q
i, j
, (i, j) E
(ii)
(i, j)E
f
i, j
( j,i)E
f
j,i
=
0 i 6= s,t
F , i = s
F , i = t
(iii) q
i, j
q
i, j
q
i, j
, (i, j) E
f
i, j
0, (i, j) E
Where Z
u
(S), u = 1,. . .,m is the profit of agent A
u
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
29
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
u
has to
decide the arc capacity q
i, j
, (i, j) E
u
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
u
(S
) Z
u
(S) for all A
u
, with at least
one inequality being strict.
The set of Pareto optimal strategies is denoted by S
P
.
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
u
, a strategy S = (Q
1
,... , Q
m
) is a Nash
Equilibrium if for any agent A
u
with strategy Q
u
, the
following equation holds:
Z
u
(Q
u
,S
u
) Z
u
(Q
u
,S
u
), Q
u
6= Q
u
(2)
We refer to S
N
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
1
,... , Q
m
) with flow F(S) is a poor strategy if and
only if it exists an agent A
u
and an alternative strategy
Q
u
such that Z
u
(S) < Z
u
(S
) and F(S
) = F(S), where
S
= (S
u
,Q
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
N
S
S
P
.
The set of non-poor strategies will be denoted by
ˆ
S.
Ideally, agents should choose a strategy which sat-
isfy both Pareto optimality and Nash stability (i.e.,
S S
N
T
S
P
). Nevertheless, since S
N
T
S
P
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
u
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
G
f
(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
f
(S) = (V,E
r
) 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
=
q
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
f
(S) from s to t, where e
1
= s and
e
k
= t.
We refer to P as the set of augmenting paths.
The greatest flow augmentation that can be achieved
using P is r
p
= min{r
ij
: (i, j) P}.
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30
An augmenting path in G
f
(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
+
c
i, j
(i, j)P
c
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
f
(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
f
(S) is made of for-
ward arcs (having the opposite direction than the one
in G
f
(S)) and backward arcs (having the same direc-
tion in G
f
(S)). The set of forward and backward arcs
are denoted
P
+
and P
, respectively.
profit(
P) =
(i, j)
P
+
c
i, j
(i, j)
P
c
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
+
,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
u
,
cost
u
(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
u
(P) =
(i, j)P
+
E
u
c
i, j
(i, j)P
E
u
c
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
u
, cost
u
(P) < w
u
× 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
+
,P
} is composed of forward and backward
arcs. If q
i, j
is decreased by one unit, (i, j)
P
+
, and
increased by one unit, (i, j)
P
, the overall flow is
decreased by one unit.
The profit profit
u
(
P) generated by decreasing capac-
ity through a decreasing path, for an agent A
u
, is de-
fined as follows:
profit
u
(
P) =
(i, j)
P
+
E
u
c
i, j
(i, j)
P
E
u
c
i, j
(6)
Definition 8. Profitable decreasing path. A decreas-
ing path
P P is profitable if there is one agent A
u
such that profit
u
(P) > w
u
× 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
u
A , P P such that(i, j) E
u
cost
u
(P) > w
u
× π (7)
A
u
A ,
P P
profit
u
(P) < w
u
× π (8)
Proof. Consider a strategy S and an agent A
u
. If S
is non poor, A
u
can improve its situation only by in-
creasing or decreasing the flow. In the former case, for
an additional unit of flow, A
u
receives w
u
× π. Since,
such increase is profitable to A
u
if and only if there is
an augmenting path P such that cost
u
(P) < w
u
× π. In
the latter case, viceversa, it is profitable for an agent
A
u
to decrease the flow by one unit if and only if there
is a decreasing path P such that profit
u
(P) > w
u
× π.
Therefore, if and only if for no agent any of those con-
ditions holds, no agent A
u
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
31
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
1
= (0,1,0,0, 1) (see Figure 2(a)) with
F(S
1
) = 1 and Z
1
(S
1
) = Z
2
(S
1
) = 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
2
= (1, 1,0,1, 1) (see Figure 2(b))
with F(S
2
) = 2 and Z
1
(S
2
) = Z
2
(S
2
) = 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
1
. In fact,
A
1
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
S
3
= (1, 0,1,0,1) (see Figure 2(c)) with F(S
3
) = 1
and profits Z
1
(S
3
) = 50 and Z
2
(S
3
) = 20, which is
obviously bad for A
2
. Therefore, although the strat-
egy S
2
corresponds to a Pareto Optimum, which leads
to a maximization of agent’s profits, it is not a stable
strategy. Strategy S
1
is a Nash Equilibrium but not
Pareto Optimum. Therefore, in our example there is
no a strategy which is both in S
N
and S
P
. 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.
P
MAMCF
Max F
s.c.
(i) f
i, j
q
i, j
, (i, j) E
(ii)
jP
+
(i)
f
i, j
jP
(i)
f
j,i
=
0 i 6= s,t
F , i = s
F , i = t
(iii) q
i, j
q
i, j
q
i, j
, (i, j) E
(iv) profit
u
(
P) < w
u
× π, P G
f
(S)
f
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
u
(P) greater or
equal to w
u
× 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
f
(S) depends on
the strategy chosen by each agent.
Constraints (iv) impose that no decreasing path
P ex-
ists in solution S with profit
u
(
P) greater or equal to
w
u
× 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
ICORES2014-InternationalConferenceonOperationsResearchandEnterpriseSystems
32
0 1 2 3 4 5 6 7 8 9
A1
A2
A3
0
( ,7)e
1
( ,8)e
2
( , 7)e
3
( ,7)e
4
( , 7)e
5
( ,8)e
6
( ,9)e
8
( ,9)e
9
( ,7)e
18
( , 7)e
10
( ,8)e
19
( ,8)e
17
( ,9)e
26
( , 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
0
,...,a
K1
} of K = 3k positive
integers, such that
K1
l=0
a
l
= k × B and
a
l
]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
0
= s and the last one is the
sink node V
K
= t. An agent A
u
A = {A
1
,... , A
k
}
owns K arcs.
The tail of an arc e
i
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
i
belongs to A
idivK
.
The cost of arc e
i
is c
e
i
= a
imodK
. In other words,
to every positive integer a
l
ζ is associated k paral-
lel arcs with, same head and tail, maximum capac-
ity
q
e
i
= 1 and cost a
l
. The total reward is set to
π = (B+ε)k, ε being an arbitrary small positive value.
The sharing policy is defined by w
u
= 1/k. Therefore,
agent’s unit reward is w
u
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
a
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
u
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
33
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.
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