An Extended Multi-agent Coalitions Mechanism with Constraints

Souhila Arib

and Samir Aknine

Laboratoire Quartz, EISTI, France

Laboratoire LIRIS, Universit

e Claude Bernard, Lyon 1, France

Keywords:

Multi-agent Systems, Coalition Formation, Coordination, Negotiation.

Abstract:

In multiple realistic scenarios, limited agent capabilities may negatively affect task performance. To overcome

this, multi-agent cooperation may be required. Many studies have focused on cooperative task performance.

To facilitate such cooperation, we develop and evaluate a coalition mechanism that enables agents to partici-

pate in concurrent tasks achievement in competitive situations, in which agents have several constraints. We

consider a set of self-interested bounded-rational agents, each of which has a set of tasks, that leads an agent

to achieving its goal. The agents have not a priori knowledge about the preferences of their opponents. All

the agents have their speciﬁc constraints and this information is private. In this paper, we do not deal with the

negotiation protocol but just introduce a new coalition formation mechanism (C F M ) that imposes minimal

sharing of private information to ease negotiations. Speciﬁcally, we only require that agents share preferences

over their constraints. The agents negotiate for coalition formation over these constraints, that may be re-

laxed during negotiations. They start by exchanging their constraints and making proposals, which represent

their acceptable solutions, until either an agreement is reached, or the negotiation terminates. We explore two

techniques that ease the search of suitable coalitions: we use a constraint-based model and a heuristic search

method. We describe a procedure that transforms these constraints into a structured graph on which the agents

rely during their negotiations to generate a graph of feasible coalitions. This graph is therefore explored by a

Nested Monte-Carlo search algorithm to generate the best coalitions and to minimize the negotiation time.

1 INTRODUCTION

Forming coalitions of agents which are able to effec-

tively perform tasks is a key issue for many practi-

cal applications. This paper mainly focuses on self-

interested agents which aim to form coalitions with

other agents as they cannot reach their objectives in-

dividually. Several methods have been developed to

control the behaviors of the agents involved in such

process (Shehory and Kraus, 1998). However few

mechanisms cope with the dynamic of the constraints

of the agents in such contexts. Indeed, these con-

straints can gradually be revealed, and relaxed by the

agents at different moments of the negotiation in or-

der to meet the requirements of their opponents and

thus to ease the convergence. Some coalition meth-

ods have been developed to determine the optimal

coalitions and take into account the constraints of the

agents involved in the coalition process. These meth-

ods have addressed important issues such as com-

putational complexity and heuristics approaches for

the optimal coalition structure generation, (Rahwan

et al., 2011), (Voice et al., 2012) , (Ramchurn et al.,

2010). In this paper, we present an extended version

of the mechanism presented in (Arib et al., 2015).

We focus on contexts where agents neither have the

same utility functions, nor they reveal these functions.

Thus, it is infeasible to precisely estimate a priori

the corresponding utility of each agent for each fea-

sible proposal of solution with current optimal coali-

tions search algorithms. The issues with processing

the constraints of the agents in the negotiation phase

for the coalition formation deserve a particular at-

tention and a deep study. Yet, only few works pro-

pose a mechanism to deal with the dynamic of such

constraints while agents negotiate them. Note that,

since we consider that the agents are self-interested

and do not share their information and computations,

our aim is not to identify the optimal solution of the

coalitions, but to ease the convergence to an agreed

common solution for these agents. Our main contri-

bution is not to deal with the negotiation protocol it-

self but propose a new mechanism that enables agents

to negotiate and form coalitions. This mechanism is

Arib, S. and Aknine, S.

An Extended Multi-agent Coalitions Mechanism with Constraints.

DOI: 10.5220/0008969001990207

In Proceedings of the 12th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2020) - Volume 1, pages 199-207

ISBN: 978-989-758-395-7; ISSN: 2184-433X

199

based on three main abstractions: a constraint graph,

a coalition graph and a Nested Monte-Carlo search

method. First, we develop a constraints based graph

which handles the revealed constraints of the agents.

This graph of constraints can be used to specify differ-

ent types of constraints relations, such as a constraints

ordering over potential decision outcomes. Building

upon this, we transform this representation into a ﬂat

representation of coalitions in the graph of coalitions.

Each level of this graph allows generating a set of

possible coalitions and in this set the agent selects

the best coalitions that can be accepted. This graph-

ical representation of constraints and coalitions spec-

iﬁes constraints relations in a relatively compact, in-

tuitive, and structured manner. In the next sections,

we ﬁrst deﬁne the problem and link it to other ex-

isting problems, so that approximate solution tech-

niques and anytime heuristics that provide increas-

ingly better solutions if given more time can be re-

used. We advise new solutions that allow agents use

a nested Monte-Carlo search algorithm (Cazenave,

2009) which ﬁnds the best coalitions that maximize

the utility of each agent. Nested Monte-Carlo search

methods address the problem of guiding the search to-

wards better states when there is no available heuris-

tic. These methods use nested levels of random games

in order to guide the search of coalitions. These al-

gorithms have been studied theoretically on simple

abstract problems and applied successfully to several

games (Gelly and Silver, 2007). Speciﬁcally, this pa-

per advances the state of the art in the following ways.

We advise new anytime heuristics to ﬁnd approximate

solutions fast, we empirically evaluate our algorithm

and show that it computes (in less than 600 millisec-

onds) 689 proposals of solutions for non-trivial prob-

lems involving up to 30 agents and 50 tasks. Thus,

our work encompasses essential aspects of the coali-

tion formation, from the coalition model, negotiation,

and an anytime heuristic. The reminder of the paper is

organized as follows. Section 2 brieﬂy describes the

related works. Section 3 introduces some preliminar-

ies and the case study. Section 4 presents the coalition

formation mechanism, and a ﬁnal section will con-

clude the work with a summary of the contributions.

2 RELATED WORK

In game-theoretic perspective, coalitional games with

constraints have been addressed by a number of

works. However, none of these mechanisms is able to

model agents’ negotiations for reaching joint agree-

ments. The authors in (Yang et al., 2016) addressed

the problem of coalition formation in small cell net-

works. Small cells can mitigate the co-tier interfer-

ence within a coalition and thus increase the system

capacity. (Bistaffa et al., 2017) adopted a coopera-

tive game theoretic approach to deal with the prob-

lem of social ridesharing. Based on a social network

representation of the set of commuters, they proposed

an algorithm to form coalitions and arrange one-time

rides at short notice which is based on two princi-

pales. First, the optimization problem of forming the

coalitions that minimise the cost of the overall sys-

tem, for which they restrict the feasible coalitions

by means of a graph-constrained coalition formation

model, allowing to specify both spatial and tempo-

ral preferences. Second, they address the payment al-

location aspect of ridesharing. In (Demange, 2009),

the authors proposed a game-theoretical study and fo-

cus on strategic, core-related issues rather than com-

putational analysis of the coalition formation. This

work is more close to (Rahwan et al., 2011) where

authors proposed a constrained coalition formation

model and an algorithm for optimal coalition struc-

ture generation. They developed a procedure that

transforms the speciﬁed set of constraints, making it

possible to identify all the feasible coalitions. Build-

ing upon this, they provide an algorithm for optimal

coalition structure generation. (Skibski et al., 2016)

have introduced the k-coalitional games. The authors

have proposed an extension of the Shapley value for

these games, and studied its axiomatic and compu-

tational properties. The authors in (Michalak et al.,

2010) have considered only the issue of representing

coalitional games in multi-agent systems with exter-

nalities in coalition formation. They have proposed

a new representation which is based on Boolean ex-

pressions. Their aim was to construct much richer

expressions that allow for capturing externalities in-

duced upon coalitions. (Voice et al., 2012) addressed

the problem of coalition formation with sparse syner-

gies where the set of feasible coalitions is constrained

by the edges of a graph. Their aim is to check whether

the knowledge of the topology of the underlying so-

cial or organizational context graph could be used

to speed up coalition enumeration and structure gen-

eration. (Ramchurn et al., 2010) deﬁned the prob-

lem of allocating coalitions of agents to spatially dis-

tributed tasks with workloads and deadlines so as to

maximize the total number of tasks completed over

time. Nevertheless, these works have not deeply ad-

dressed the constraints of the agents in the proposed

models or specify how agents negotiate over them to

reach agreements. Constraints on coalition sizes have

been considered for coalition structure value calcula-

tion (Sandholm et al., 1999), (Rahwan et al., 2007),

(Rahwan et al., 2009). However, the semantics of

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

200

these constraints has not been used on the same level

as it is done in this paper. Work in (Wang et al.,

2016) introduced a novel mathematical framework

from cooperative games to model and solve cooper-

ative scenarios where network devices have to be-

come more autonomous and cooperate with one an-

other, in two main classes of these games, namely,

interference management and cooperative spectrum

sensing. Such cooperative mechanisms involve the

simultaneous sharing and distribution of resources

among a number of overlapping cooperative groups

or coalitions. (Ieong and Shoham, 2005), (Conitzer

and Sandholm, 2006) developed succinct and expres-

sive representations for coalitional games. In all the

algorithms discussed so far, the main focus was on

maximizing the social welfare, where agents consider

every possible subset of agents as a potential coali-

tion. Such formalism could be used to encode the

constraints, but this is not the main concern of the

constrained C F M mechanism considered in this pa-

per. Negotiation is, mostly, about needed resources to

perform tasks. In (Cao et al., 2013), the agents are as-

sumed to have the same set of tasks and a unique com-

mon goal. The authors in (Arib and Aknine, 2011)

addressed situations where agents plan their activities

dynamically and use these plans to coordinate their

actions and search for the coalitions to be formed.

These studies have not considered the coalition for-

mation with constraints changing requirement, which

makes those approaches not suitable for our problem.

In other hand, multi-agent negotiation has received

much attention both in the context of coalition forma-

tion and in other contexts (An et al., 2011), (Rochlin

and Sarne, 2014), (Sofer et al., 2016), (Kang, 2005).

Indeed, negotiation is an important interaction mech-

anism that may allow a group of agents to reach a

certain goal, particularly in situations where agents

have several possibilities to explore. The authors in

(An et al., 2011) considered interrelated negotiations,

where selﬁsh agents have to efﬁciently coordinate

their negotiation with multiple resource providers to

acquire multiple resources in order to accomplish a

high level task. (Sofer et al., 2016) used negotia-

tion as an exploration process, over a set of several

possible alternatives, in situations where the negotia-

tors have no information on the value of each alterna-

tive. Therefore, negotiation is assumed to be limited

in time and a discount factor is used to make nego-

tiation converge toward a solution. Another work in

which agents have to explore different alternatives to

ﬁnd a solution is that of (Rochlin and Sarne, 2014).

The agents explore several opportunities available to

them, following a costly exploration process, with-

out revealing the beneﬁt with which they are asso-

ciated. Thus, agents’ goal is not to maximize the

overall beneﬁt, but rather to maximize each agent’s

own beneﬁt. In (Jonge and Sierra, 2015), the authors

have introduced a new heuristic-based negotiation al-

gorithm where many self-interested agents have non-

linear utility functions. In this work negotiation is,

mostly, about needed resources to perform tasks. In

our approach, negotiation is about groups of tasks and

agents to perform them.

3 PRELIMINARIES AND CASE

STUDY

To illustrate the coalition formation mechanism we

propose, let us consider a carpooling example, where

some travellers want to move from a city to another,

and they want to share their means of transportation.

Each traveller formulates to his agent the goals to be

achieved. For example ”I want to go from NY to

Boston”, his constraints as departure time, duration

of the travel, and unit price of seat. To solve this

problem, the agents have to deal with all the con-

straints and preferences over those of their associated

travellers in order to enable them to share transporta-

tion. Agents negotiate for the coalitions to form to

decrease the unit price of seat, increase the number of

passengers, etc. They can step aside in favor of other

agents, if an agreement can be found. More formally,

consider a set of agents N = {a

,...,a

}, a set of

actions A = {b

,...,b

} and a set of constraints

= {c

,...,c

}. The agents of N need to exe-

cute the actions of A by satisfying the constraints in

The constraints are deﬁned as intervals, for instance

as an example: departure time: D ∈ [10a.m.,12a.m.],

travel duration in hours: T ∈ [1H,2H] and price:

P ∈ [20,25]. The agents’ preferences are represented

using a preference relation  for those they want to

share a car with, for instance a



(for agent a

is preferred to a

). We consider a coalition c as

a nonempty subset of N (c ⊆ N ). We deﬁne C as

the set of all possible coalitions. For a coalition c

to be formed, each agent a

in c should get a cer-

tain satisfaction. This satisfaction is deﬁned by a util-

ity function u

: C 7→ R . Note that a coalition is ac-

ceptable for agent a

if it is preferred over, or equiv-

alent to a reference coalition, u

(re f ), which corre-

sponds to the minimal guaranteed gain of the agent

during the negotiation. A solution of the negotia-

tion for each agent a

introduces a coalition struc-

ture denoted CS

which is deﬁned on N with its as-

sociated utility u

(CS

). CS

contains a set of coali-

tions {c

,.., c

} to be formed for the set of actions

An Extended Multi-agent Coalitions Mechanism with Constraints

201

⊆ A where a

is involved. Furthermore, for every

∈ [1,q], c

⊆ N and c

performs a set of actions

⊆ A and ∀(x,y) ∈ [1,q]

,x 6= y,A

∩ A

= ∅,

=1,..,q

⊆ A and

=1,..,q

⊆ N . The set of

all coalition structures is denoted S.

4 COALITION FORMATION

MECHANISM (C F M )

In order to satisfy the goals they have to achieve,

the agents perform negotiations on the coalitions they

want to form. So, the CF M requires an analysis step

of constraints that agents exchange in order to guide

the choice of the coalitions and a step of generating

coalition structures from these constraints. Constraint

analysis relies on constructing a graph of constraints

and coalition generation is based on the mapping of

the constraints to possible coalitions in a coalition

graph. Exploring the search graph of coalitions to-

ward better states is based on a Nested Monte-Carlo

algorithm.

4.1 Constraint Graph

An effective technique for solving a coalition forma-

tion problem is a heuristic search through abstract

problem spaces. The ﬁrst problem space can be rep-

resented by a directed connected graph, where nodes

correspond to constraint sets and edges correspond to

actions (cf. Figure 1). The constraint graph may in-

clude many paths from the start to any node. Since the

agents are self-interested, to search among the con-

straints to deal with in the coalitions, every agent con-

structs its own graph of constraints based on its own

constraints and those revealed by other agents during

this negotiation. Given a set of constraints that must

be satisﬁed by an agent to execute a set of actions and

starting from the source node labeled with {b: ∅, c

∅ }, initially there are not constraints and actions as-

sociated with this source node, let us deﬁne a graph

denoted G(c

,b) as follows.

Deﬁnition 1. Given a node labeled {b: ∅, c

: ∅},

the constraint graph G(c

,b) is a directed connected

graph, containing all possible nodes of constraints

represented by intervals, labeled {X

,..., X

} for each

action b

,1 ≤ i ≤ m, that has to be executed by the

agent. Each node has a utility labeled u, and directed

edges from this node are labeled {b

,..., b

} where

1 ≤ j .. k ≤ m.

A constraint graph gathers, the most preferred

constraints’ intervals in its nodes. At the root node,

no action and constraint are added. Each node gen-

erates a ﬁnite set of child nodes which correspond to

the accepted sets of constraints, where the ﬁrst node

of the graph is an outgoing node and the last nodes are

incoming nodes. This constraint graph is built follow-

ing a preference rate on the intervals of constraints.

Let us consider two agents, a

which has its own

interval X and receives from a

an interval Y . The

agent a

wants to create a new interval Z that meets its

constraints and those of a

, {Z |= a

}, by merging its

interval and the one received from a

We will adopt the convention of the left and right

endpoints of an interval X by X and X, respectively

(Moore et al., 2009).

First, a

tests if X ⊆ Y . Thus, if Y 6 X and X 6 Y ,

it will get X ⊆ Y and Z = X; else the agent tests if

X ∩Y and calculates the new interval Z. If X ∩Y = ∅

there are no points in common with a

. Otherwise,

Z = {max{X ,Y }, min{X ,Y }} and the agents tests if

Z complies with its actions. If a

does not choose

this interval, it calculates Z = X ∪Y which means the

union between X and Y and tests if it complies with

its actions. For more details about the operations

over the intervals see (Moore et al., 2009). Based on

this graph, constraint analysis consists for an agent

of comparing and grouping its constraints and those

received from others. A natural constraint graph anal-

ysis involves constructing and linking optimal nodes.

Constraints are gathered based on their relations into

sets represented in the nodes of this graph. Each level

of the graph of constraints refers to an action to be

performed by a coalition. The advantage of the sug-

gested method consists in directing the search of the

solutions of coalitions towards primary constraints,

i.e., important constraints to satisfy, thus, reducing

search complexity. To move from one node of this

graph to another, an action is added to the graph. The

utility of a move, which labels the corresponding edge

in the search space, is the utility of the action when

it is added and performed by the coalition. A solu-

tion path represents a particular succession order of

the added actions, and the width of that order is the

sum of the edge utilities on the solution path.

To construct the constraint graph and to search

for feasible proposals agents use Algorithm 1. Let

us consider constructing the constraint graph by the

agent a

on our previous example using these algo-

rithms. First, assume that agent a

started a nego-

tiation with agents a

to a

and in which each of

these ﬁve agents revealed certain of its constraints.

The actions that have to be executed are: b

which correspond respectively to: go from NY to

Amherst, ﬁnd a hotel room in Amherst, and go

from Amherst to Boston. The constraints identiﬁed

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

202

Figure 1: An example of a graph of constraints against dif-

ferent actions of the agent a

. The child nodes are created

based on Algorithm 1 and each node is labeled with its as-

sociated utility.

by a

for the action b

are: D ∈ [10a.m.,01p.m.],

T ∈ [1H,2H] and P ∈ [20, 25] and for b

are: D ∈

[01p.m., 02p.m.], T ∈ [1H, 2H] and P ∈ [20,25]. The

nodes in the ﬁrst level of the graph assemble pos-

sible sets of constraints concerning the action b

compares its own constraints and those received

from these agents and creates new intervals of con-

straints X

(cf. Algorithm 1). Let us consider again

the agent a

who received these intervals of con-

straints from the agent a

concerning the action b

D ∈ [10a.m.,11a.m.], T ∈ [1H, 3H] and P ∈ [20, 35].

Applying the algorithm 1, D ∈ [10a.m.,11a.m.] ⊆

[10a.m.,01p.m.] so Z = [10a.m.,11a.m.] |= (a

Thus, a

selects the interval Z. For T ∈ [1H, 2H] ⊆

[1H, 3H], Z = [1H,2H] |= (a

), so a

chooses

T ∈ [1H,2H]. For P ∈ [20, 25] ⊆ [20,35], Z =

[20,25] |= (a

). These results are resumed in the

Figure 1. On the left of this ﬁgure, agent a

repre-

sents the ﬁrst node created for the action b

by the

ordered set of intervals D ∈ [09a.m.,11a.m.], T ∈

[1H, 3H] and P ∈ [20,23]. We notice in this node

that a

chooses the interval [09a.m., 11a.m.] even if

its departure time is not completely included in D ∈

[10a.m.,01p.m.] because it has a good proposal of the

seat price. D ∈ [10a.m., 11a.m.], T ∈ [1H,2H] and

P ∈ [20, 25] are associated with the second one and

D ∈ [11a.m., 01p.m.], T ∈ [4H,5H] and P ∈ [20,19]

with the last child. So, for each action, a

generates

the different possible intervals of constraints that sat-

isfy its action b

. We observe that the constraints in

each child node are created taking into account the

end of execution of the antecedent action. So, to

generate intervals of constraints that satisfy the ac-

tion b

i+1

, the agent takes into account the end time

of b

. This allows the agent to manage the rela-

tions between the actions that have to be executed.

In this example, in the second level of the graph the

agent a

identiﬁed for the action b

these intervals:

D ∈ [01p.m.,02p.m.], T ∈ [2H,5H] and P ∈ [20, 25].

The beginning of b

is in [01p.m.,02p.m.] because b

begin

Require: Node(b

);

Loop;

if X

|= b

∧ X

2 b

i+1

then

Generate (X

,...,X

) |= b

i+1

;

Split(b

) into (b

i+1

/y : 1..k

);

for b

i+1

will be executed after b

if (X

,...,X

) ∩ (X

) = φ then

Create child nodes (b

i+1

: (X

,...,X

));

end

else

break

end

else

break

end

End Loop;

end

Algorithm 1: Constraints linking algorithm.

ends at the latest at 01p.m. The nodes are generated

following the Algorithm 1. The dashed arcs show

that nodes can share the same child nodes and the

red and bold ones show the most preferred path from

the root node to the last one, they result from the

Monte-Carlo exploration (detailed below). From the

Algorithm 1, every agent a

which has to negotiate

to execute an action b

while satisfying its constraints

, chooses intervals of constraints: X

) |= a

. It

then creates child nodes for the feasible intervals that

satisfy b

. Each node created can be split under ap-

propriate restrictions to other child nodes. The agent

starts with a node, labeled X

, for the action b

For each action b

i+1

that must be executed after b

and needs negotiation, a

creates the new intervals for

i+1

: (X

,..., X

), and splits the node b

to the

child nodes b

i+1

: (X

,..., X

). The agent a

uses

this procedure until no action needs negotiation.

A notable detail of the constraints search space

construction is that a solution is measured by its max-

imum path utility. We use an additive utility function,

where a path is evaluated by summing its edge util-

ities. For each iteration, feasible solutions are only

explored if their utility is not under a certain refer-

ence situation u

(re f ). If an iteration is completed

without ﬁnding a new possible solution, then all so-

lutions provide less utility than that of the reference

situation, u

(re f ), thus, u

(re f ) may be decreased

and the search is repeated. To optimize the search

time for the new coalitions to propose or to accept,

agents use the Nested Monte-Carlo (NMC) algorithm

(Cazenave, 2009).

In the ﬁrst step of the mechanism, the NMC ex-

plores each level of the graph of constraints and stores

the best path of constraints that satisﬁes the agent a

An Extended Multi-agent Coalitions Mechanism with Constraints

203

The idea is to explore a graph randomly from a given

position in the graph, agents use a function which

plays the move in the position and returns the result-

ing position.

4.2 Coalitions Processing

Based on its graph of constraints and the resulting

path solution, each agent a

constructs a search graph

of coalitions for all the actions that have to be exe-

cuted. Each node of this graph gathers a set of coali-

tions that comply with the corresponding node in the

graph of constraints. Each level of this graph con-

cerns an action. To decide on the coalition to attach

to each node, a

uses the information gathered from

other agents, i.e. on the coalitions they propose during

their negotiations. a

chooses from it the coalitions to

propose and negotiate with others.

As during their negotiations the agents exchange

their proposals of coalitions, each agent updates its

sets of coalitions in its own graph. Agents perform

a graph search to achieve these goals and the out-

put of this search is a list of proposals of coalitions,

one for each action. At any step in the search, a path

corresponds to a set of actions that have to be exe-

cuted. Once the preferred path is identiﬁed in the

constraint graph, the agent a

generates partially the

graph of coalitions that satisﬁes at best the path of

constraints and completes this graph during the ne-

gotiation phase. Hence, to complete the graph, the

agent a

adds the coalition proposal received in the

corresponding level or tries to generate a new feasible

proposal of coalition that takes into account revealed

preferences of other agents. Therefore, a

generates

a new arc in the graph and checks possible connec-

tions with the other nodes of the graph. In case that

the agents changed their constraints associated to an

action, then a

tries to generate other intervals in the

graph of constraints. Then, a

re-explores this graph

with the NMC algorithm. a

chooses a new path and

completes the graph of coalitions with new coalitions

if the old ones do not meet the constraints of the new

path. Let us consider again the agent a

that con-

structs its coalition graph to satisfy the path which is

identiﬁed from its constraint graph. Figure 2 shows

this graph.

5 SOME PROPERTIES OF THE

MECHANISM

Let U be the set of actions that are already used on a

path p:

Figure 2: An example of a search graph against different

actions of the agent a

. Each node gathers possible coali-

tions that satisfy its path of constraints, where each coali-

tion c may contain a set of variables of agents x

,{i > 0 and

∈ N }. These variables are instantiated as the negotiation

progresses.

U =

[

m/b

∈p

(1)

A path ends when no action can be added. This occurs

when for every proposal, some of its actions have al-

ready been used on the path (∀m

∈ U). A utility

that an agent a

gets from one path p is:

(p) =

∑

∈p

) (2)

Proposition 2. Every feasible coalition structure

CS ∈ S is represented in the search graph by exactly

one path from the root to a node. If the children

of a node are the proposals such that: (1) they in-

clude the actions that have not been used on the path

yet, (2) they do not include actions that have already

been used on the path; formally, for any node, Θ, of

the search graph, children(Θ) = {b ∈ {b

,..., b

},b∩

U =

0}.

Proof. We ﬁrst prove that each relevant structure

CS ∈ S is represented by at most one path from the

root to a node.

The ﬁrst condition of the proposition leads to the

fact that a coalition structure can only be generated in

one order of actions on the path. Thus, there can not

exist more than one path for a given coalition struc-

ture.

What still has to be shown is that each relevant

coalition structure is represented by some path from

the root to a node in the graph. Assume for contra-

diction that some relevant coalition structure CS ∈ S

is not. Then, at some point, there has to be a coalition

in that structure such that this coalition has an action

which is not on the path, but that coalition does not

belong to the path.

To summarize, in the search graph, a path from

the root to a leaf corresponds to a relevant structure of

coalitions. Each relevant structure CS is represented

by exactly one path in the graph of coalitions.

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Lemma 3. Every feasible coalition structure CS con-

tains exactly one coalition from level 1 and at most

one coalition from every level of the graph.

Proof. From the proposition 2, in the search graph, a

path from the root to a leaf corresponds to a relevant

structure of coalitions. Each relevant structure CS is

represented by exactly one such path in the coalition

graph, and each path contains exactly one coalition

from each level.

Proposition 4. The number of leaves in the graph

search is no greater than L

with L = 2

n−1

. Fur-

thermore, the number of levels in the graph search

(excluding the root) is not greater than m. The num-

ber of nodes in the graph search is not greater than

m+1

− 1.

Proof. The depth of the graph is at most m since every

node on a path uses up at least one action. Let C (b

)

be the set of coalitions that is created to execute the

action b

. Let L = |C (b

)| the number of possible

coalitions created for the action b

in one level.

There are at most 2

n−1

combinations of agents i.e.

coalitions, so L is not greater than 2

n−1

. Therefore,

the number of leaves in the graph is not greater than

m times the number of coalitions in each level (L ∗L ∗

L ∗ ... ∗ L, m times), so not greater than L

Next we prove that the number of nodes is not

greater than L

m+1

−1. The number of children of each

node is not greater than L because we have not greater

than 2

n−1

possibilities of coalitions. Therefore, the

number of nodes is:

∑

i=1

, the maximization of this

sum gives: (2

n−1

)

+...+(2

n−1

)

= (2

n−1

)

m+1

−1 =

m+1

− 1.

6 EXPERIMENTAL EVALUATION

We have carried out an experimental evaluation of

the CFM in a testbed implemented in JAVA. In these

experiments, we have tested different sets of agents.

Each agent is randomly attributed between 30 and 50

actions (i.e., 30 ≤ |b

| ≤ 50) and the constraints for

each action are chosen randomly between 10 to 100

(i.e., 10 ≤ |C

| ≤ 100). At each step of the nego-

tiation, agents propose, accept, refuse proposals until

agreements are reached or the deadline for the process

is met. To evaluate the performance of our mecha-

nism, we recorded the number of solutions reached,

the time taken to reach such solutions, and the num-

ber of proposals sent to reach these solutions. Our

experiments are repeated 100 times. We do not com-

pare our method to some methods seeking optimal

Figure 3: Taken time to reach a solution with and without

NMC.

Figure 4: Number of solutions.

coalitions since we do not make a priori computation

of coalitions before the negotiation but coalitions are

computed gradually during the negotiation. We also

consider that the agents do not have the same util-

ity functions which are not known by others. Thus,

agents do not necessarily look for an optimal solu-

tion. They try to converge to an acceptable solution

in their negotiations since convergence is difﬁcult to

address with self-interested agents. Figure 3 shows

that when NMC is not used, the time taken to reach

a solution is greater, compared to that based on NMC

using 100 constraints and 50 actions. This is because

our algorithm allows agents to speed up their search

and thus, to explore relevant solutions.

In what follows, we evaluate the inﬂuence of the

NMC on the number of the solutions during the ne-

gotiation process while we use 100 constraints and 50

actions. We notice from Figure 4 that the number of

agreements reached after negotiation is more signiﬁ-

cant in the cases where the NMC is used. These re-

sults clearly show that using the NMC enables agents

to get more possible coalitions, and thus, more feasi-

ble alternatives in a reasonable time limit.

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205

7 CONCLUSION

In this paper, we addressed the problem of coali-

tion formation with constraints, where self-interested

agents have individual alternative sets to reach dif-

ferent goals.We introduced a new coalition forma-

tion mechanism enriched with several principles to

deal with the constraints of the agents and a Nested

Monte-Carlo based search algorithm. Thus, each

agent may have several possible solutions represented

in the form of sequential interdependent coalitions.

Our mechanism aims to allow each agent to take into

account the dependencies among its tasks, which lead

to inter-dependencies among possible coalitions, and

to keep an overall view of all of its possible solu-

tions throughout the coalition formation process. We

have detailed how the constraints are modeled as a

graph and how this graph is explored using the Nested

Monte-Carlo search. From the graph of constraints,

each agent gets its most preferred path of constraints

and constructs a coalition graph that is used to gener-

ate the coalitions to negotiate. We have detailed some

proprieties that the graphs satisfy. Then we have pre-

sented an empirical evaluation of the proposed mech-

anism.

REFERENCES

An, B., Lesser, V., and Sim, K. (2011). Strategic agents for

multi-resource negotiation. Autonomous Agents and

Multi-Agent Systems, 23:114–153.

Arib, S. and Aknine, S. (2011). A plan based coalition

formation model for multi-agent systems. In Pro-

ceedings of the 2011 IEEE/WIC/ACM International

Conferences on Web Intelligence and Intelligent Agent

Technology - Volume 02, WI-IAT ’11, pages 365–368.

Arib, S., Aknine, S., and Cazenave, T. (2015). Nested

monte-carlo search of multi-agent coalitions mecha-

nism with constraints. In MIWAI.

Bistaffa, F., Farinelli, A., Chalkiadakis, G., and Ramchurn,

S. (2017). A cooperative game-theoretic approach to

the social ridesharing problem. Artiﬁcial Intelligence,

246:86–117.

Cao, J., Wang, H., and Wang, X. (2013). A distributed al-

gorithm for agent coalition formation with complex

tasks. pages 127–132.

Cazenave, T. (2009). Nested monte-carlo search. In IJCAI,

pages 456–461.

Conitzer, V. and Sandholm, T. (2006). Complexity of

constructing solutions in the core based on synergies

among coalitions. Artif. Intell., 170.

Demange, G. (2009). The strategy structure of some coali-

tion formation games. Games and Economic Behav-

ior, pages 83–104.

Gelly, S. and Silver, D. (2007). Combining online and of-

ﬂine knowledge in UCT. In ICML ’07, pages 273–

280.

Ieong, S. and Shoham, Y. (2005). Marginal contribution

nets: A compact representation scheme for coalitional

games. In Proceedings of the 6th ACM Conference on

Electronic Commerce, pages 193–202. ACM.

Jonge, D. and Sierra, C. (2015). Nb3: a multilateral negoti-

ation algorithm for large, non-linear agreement spaces

with limited time. Autonomous Agents and Multi-

Agent Systems, 29.

Kang, B.-K. (2005). Optimal stopping problem with dou-

ble reservation value property. European Journal of

Operational Research, page 765–785.

Michalak, T., Marciniak, D., Szamotulski, M., Rahwan, T.,

Wooldridge, M., McBurney, P., and Jennings, N. R.

(2010). A logic-based representation for coalitional

games with externalities. In Proceedings of the 9th

International Conference on Autonomous Agents and

Multiagent Systems: Volume 1 - Volume 1, AAMAS

’10, pages 125–132.

Moore, R. E., Kearfott, R. B., and Cloud, M. J. (2009). In-

troduction to Interval Analysis.

Rahwan, T., Michalak, T., Elkind, E., Faliszewski, P.,

J.Sroka, Wooldridge, M., and Jennings, N. R. (2011).

Constrained coalition formation. In AAAI.

Rahwan, T., Ramchurn, S. D., Dang, V. D., and Jennings,

N. R. (2007). Near-optimal anytime coalition struc-

ture generation. In IJCAI, pages 2365 – 2371.

Rahwan, T., Ramchurn, S. D., Dang, V. D., Jennings, N. R.,

and Giovannucci, A. (2009). An anytime algorithm

for optimal coalition structure generation. J. Artif. Int.

Res., 34(1):521–567.

Ramchurn, S. D., Polukarov, M., Farinelli, A., Truong, C.,

and Jennings, N. R. (2010). Coalition formation with

spatial and temporal constraints. In AAMAS, pages

1181–1188.

Rochlin, I. and Sarne, D. (2014). Utilizing costly coordi-

nation in multiagent. Multiagent and Grid Systems,,

10:23–49.

Sandholm, T., Larson, K., Andersson, M., Shehory, O.,

and Tohme, F. (1999). Coalition structure genera-

tion with worst case guarantees. Artiﬁcial Intelligence,

111:209–238.

Shehory, O. and Kraus, S. (1998). Methods for task alloca-

tion via agent coalition formation. Artif. Intell., pages

165–200.

Skibski, O., Matejczyk, S., Michalak, T. P., Wooldridge,

M., and Yokoo, M. (2016). k-coalitional cooperative

games. In Proceedings of the 2016 International Con-

ference on Autonomous Agents, AAMAS ’16, pages

177–185. International Foundation for Autonomous

Agents and Multiagent Systems.

Sofer, I., Sarne, D., and Hassidim, A. (2016). Negotiation in

exploration-based environment. Autonomous Agents

and Multi-Agent Systems, 30:724–764.

Voice, T., Ramchurn, S., and Jennings, N. (2012). On

coalition formation with sparse synergies. In AAMAS,

pages 223–230.

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

206

Wang, T., Song, L., Han, Z., and Saad, W. (2016). Overlap-

ping coalition formation games for emerging commu-

nication networks. IEEE Network, 30:46–53.

Yang, G., Esmailpour, A., Cao, Y., and Nasser, N. (2016).

A novel coalitional structure generation algorithm for

interference mitigation in small cell networks. pages

1–4.

An Extended Multi-agent Coalitions Mechanism with Constraints

207