RESOURCE ALLOCATION PROBLEMS ON NETWORKS
Maximizing Social Welfare using an Agent-based Approach
Antoine Nongaillard and Philippe Mathieu
LIFL, Universit
´
e Lille 1, Villeneuve d’Ascq c
´
edex, France
Keywords:
Multi-agent system, Resource allocation, Emergence, Negotiation, Network.
Abstract:
Numerous applications can be formulated as an instance of resource allocation problems. Different kinds of
solving techniques have been investigated, but the theoretical results cannot always be applied in practice due
to inappropriate assumptions. Indeed, in these studies, agents are most of the time omniscient and/or have
complete communication abilities. These hypotheses are not satisfied real life applications. practice. We
propose in this paper a distributed mechanism leading to optimal solutions with respect to a more realistic
environment. Agents only have limited perceptions and knowledge. Using local negotiations, they elaborate
themselves optimal allocations, which can be viewed as emergent phenomena. We show that negotiations
between individually rational agents lead to sub-optimal states in the society, and we propose a more suitable
decision-making criterion, the sociability, leading to socially optimal solutions. Our method provides a se-
quence of transactions leading to optimal allocations, according to any communication networks, when four
different welfare objectives are considered.
1 INTRODUCTION
Resource allocation problems arouse a great inter-
est in the computer science community since such
problems can be encountered through countless ap-
plications in real life. Centralized approaches as well
as distributed approaches have been investigated to
solve efficiently resource allocations problems. Ac-
cording to centralized solving techniques, all infor-
mation are gathered in a single place in order to deter-
mine the best allocation. These kind of techniques
suit well to the solving of applications like combi-
natorial auction whereas distributed solving methods,
on which we focus in this paper, suit better to appli-
cations where privacy or dynamism is required for
instance. Many studies on distributed frameworks
have been performed (e.g. (Sandholm, 1998; Dunne
et al., 2005; Chevaleyre et al., 2010)) in which au-
thors aim to characterize the solutions that can be
achieved. Indeed, these studies focus on the exis-
tence of transaction paths, on their length or on the
properties satisfied by a solution depending on the set-
tings of the solving method. However, these studies
do not consider restrictions on agent communications
but it is not satisfied in many applications, like the
ones based on peer-to-peer networks or on social net-
works.Former studies focus on the characterization of
solutions but did not focus on the mechanism required
to achieve such solutions. Restricted contact networks
have been considered in different contexts (de Weerdt
et al., 2007)but our objective also differs. Indeed, we
seek to design the negotiation settings that should be
used in order to achieve a socially optimal state within
agent societies.
We propose in this paper a solving method based
on more realistic assumptions. Agents initially have
very limited knowledge: their preferences, their re-
source bundle and a list of neighbors. Starting from
this state, agents negotiate and try to trade their re-
sources thanks to local transactions satisfying their
acceptability criterion. We design the settings leading
negotiation processes to optimal solutions using local
transactions between agents, according to any kind
of contact networks. Four different welfare functions
have been considered. In each case, we first evalu-
ate the quality of our solutions compared to solutions
provided by centralized techniques. Then, we esti-
mate the impact of networks topology in order to de-
termine the characteristics penalizing or favoring the
efficiency of agent negotiations.
This paper is organized as follows. Section 2 de-
scribes the basic parameters on which are based our
model. Section 3 presents our solving approach while
Section 4 describes the experimental protocol. Fi-
206
Nongaillard A. and Mathieu P..
RESOURCE ALLOCATION PROBLEMS ON NETWORKS - Maximizing Social Welfare using an Agent-based Approach.
DOI: 10.5220/0003159702060211
In Proceedings of the 3rd International Conference on Agents and Artificial Intelligence (ICAART-2011), pages 206-211
ISBN: 978-989-8425-41-6
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

nally, Section 5 successively details the results we ob-
tain for utilitarian and egalitarian societies.
2 ISSUES ON AGENT
NEGOTIATIONS
2.1 Definitions and Assumptions
We focus on distributed mechanisms to solve reallo-
cation problems. A distributed solving process starts
from an initial allocation, which evolves, step by step,
thanks to local negotiation between agents, until the
achievement of optimal allocations
Related to the resource nature, we choose to con-
sider unique and atomic resources which are not
shareable. Agents cannot alter the resources they
own, they are only able to trade them. Let A be the
set of all possible allocations.
We propose to consider several parameters, on
which is based the definition of agent. An agent
is defined with a bundle describing the owned re-
sources, the preferences used to evaluate the agent
satisfaction, a behavior specifying how agents inter-
act, an acceptability criterion on which the agent de-
termines if a deal is profitable, and a neighborhood
representing the communication abilities.
Agents express preferences over the resource set,
which are used to determine their individual welfare
(Doyle, 2004). We choose to usean additive util-
ity function. The satisfaction of an agent a
i
to own
a set ρ of resources can be computed as: u
a
i
(ρ) =
∑
r
a
∈ρ
u
a
i
(r
a
).
Essential notions have been defined in this section.
Based on them, we can design the behaviors leading
agents to trade efficiently their resources. A question
can be raised: how can we evaluate a negotiation pro-
cess?
2.2 Evaluation of a Negotiation Process
Since an objective is to identify the negotiation set-
tings leading agent negotiations to optimal solutions,
the absolute efficiency must be evaluated.The quality
of two allocations can be compared thanks to the no-
tions of the social choice theory (Arrow et al., 2002).
Maximizing the utilitarian welfare is equivalent to
maximize the average individual welfare in a popula-
tion. The utilitarian welfare associated with an alloca-
tion A ∈ A can be defined as sw
u
(A) =
∑
a
i
∈P
u
a
i
(R
a
i
).
The maximization of the egalitarian welfare tends to
reduce inequalities in the population. It can be defined
as sw
e
(A) = min
a
i
∈P
u
a
i
(R
a
i
). The Nash product con-
siders the welfare of the whole population and reduces
the inequalities among agents at the same time. It can
be viewed as a compromise between the utilitarian
and the egalitarian welfare: sw
n
(A) =
∏
a
i
∈P
u
a
i
(R
a
i
).
Finally, the elitist welfare, which only considers the
welfare of the richest agent in the population, is de-
fined as follows: sw
e`
(A) = max
a
i
∈P
u
a
i
(R
a
i
).
For each social welfare notions, the optimal value
can be determined or estimated by means of central-
ized algorithms, as suggested in (Nongaillard et al.,
2008) but they are not detailed here. The optimal val-
ues, provided by these algorithms, are used as refer-
ences to determine the absolute efficiency of a nego-
tiation process.
Other facets of negotiations must also be consid-
ered. The impact of the social graph topology can be
evaluated, in order to determine the cost of consider-
ing restrictions on agent communication abilities.
The topological sensitivity can be evaluated
thanks to the standard deviation among the social val-
ues achieved at the end of negotiation processes. A
large deviation means that the negotiation process is
very sensitive to the graph topology, and thus the
quality of provided solutions significantly varies ac-
cording to the initial conditions.
2.3 The Social Graph: An Important
Issue?
Since agents communication abilities are usually not
restricted in allocation problems, it is legitimate to
investigate the importance of such a parameter. Ne-
gotiation processes, which lead to optimal solutions
according to complete communication abilities (i.e.
based on complete contact networks), may only lead
to solutions far from the optimum, when communica-
tions are restricted.
Proposition 1 (Social graph impact). Independently
of the objective function considered, the achievement
of optimal resource allocations cannot be guaranteed
if a restricted social graph is considered.
Proof. Let us prove this proposition using a counter-
example, based on a population of 3 agents P =
{a
0
,a
1
,a
2
} and a set of 3 resources R = {r
1
,r
2
,r
3
},
where the aim is to maximize the utilitarian welfare.
The agents preferences and the social graph with the
initial resource allocation are described in Figure 1.
For instance, agent a
1
associates the utility values 1
with resources r
1
and r
3
. Let us consider a topology
where agent a
1
can communicate with agents a
0
and
a
2
while they can only negotiate with him. The initial
allocation is A = [{r
1
}{r
2
}{r
3
}].
RESOURCE ALLOCATION PROBLEMS ON NETWORKS - Maximizing Social Welfare using an Agent-based
Approach
207

Table 1: Example of agents preferences.
Population P
Resource Set R
r
1
r
2
r
3
a
0
3 1 9
a
1
1 4 1
a
2
10 2 3
Agents only perform transactions increasing their
own utility. According to such conditions and to the
social graph, no transaction can be performed. Only
two exchanges are possible, but both lead to a de-
crease of the individual welfare of at least one par-
ticipant. The exchange of r
1
and r
2
, or the exchange
of r
2
and r
3
penalizes both participants. However, the
current allocation is suboptimal. The exchange of r
1
and r
3
, which leads to an increase of both participants’
utility, is not possible since agent a
0
and agent a
2
can-
not communicate. Hence, restrictions on agents com-
munication abilities may prevent the achievement of
optimal solutions.
Proposition 2 (Negotiation order). Independently to
the objective function which is considered, the order
in which agents negotiate with each other can prevent
the achievement of optimal resource allocations.
Proof. Let us prove this proposition using a counter-
example, based on a population of 3 agents P =
{a
0
,a
1
,a
2
} and a set of 3 resources R = {r
1
,r
2
,r
3
},
where the aim is to maximize the utilitarian welfare.
The agents preferences are described in Table 2.Let us
consider a topology where agent a
1
can communicate
with agents a
0
and a
2
while they can only negotiate
with him. The initial allocation is A = [{r
1
}{r
2
}{r
3
}].
Table 2: Example of agents preferences.
Population P
Resource Set R
r
1
r
2
r
3
a
0
2 10 4
a
1
5 3 9
a
2
2 7 1
Let us assume that agent a
1
initiates a negotia-
tion. Depending on which neighbor the initiator se-
lects to negotiate first, the negotiation process can end
with sub-optimal allocations instead of optimal ones.
If agent a
1
first chooses agent a
0
as partner, the ex-
change leads to a sub-optimal allocation from which
the negotiation process cannot leave. However, if
agent a
2
is selected first, the negotiation process ends
on a socially optimal allocation. Hence, the optimum
can only be achieved using a specific order of negoti-
ation.
Thus, the social graph represents an important is-
sue since its topology may prevent the achievement
of optimal allocations in practice. The influence of
the contact network on the efficiency of negotiation
processesmust not be omitted as it has been done in
former studies.
3 SOLVING APPROACH
CHARACTERISTICS
3.1 Transaction
During negotiation processes, the resource allocation
evolves step by step by means of local transactions
among agents. Only bilateral transactionsare consid-
ered in this paper. A transaction can be viewed as the
association of the two participants’ offer. It is a pair
δ
a
j
a
i
hu,vi = (ρ
δ
a
i
,ρ
δ
a
j
), where the initiator a
i
offers a set
ρ
δ
a
i
of u resources and its partner a
j
offers a set ρ
δ
a
j
of
v resources.
This representation can model transactions from
any class, like the ones described in (Sandholm,
1998), using restrictions on the number of offered re-
sources. During a gift, the initiator offers one resource
and its partner provides nothing: it is then equivalent
to a h1,0i-deal.
3.2 Acceptability Criterion
The acceptability criterion is used to determine
whether a deal is profitable or not. The individual ra-
tionality is the most widely used criterion in the liter-
ature. It specifies that agents can only accept transac-
tions (transforming their bundle of resources R
a
i
into
R
a
j
) increasing their individual welfare:u
a
i
(R
a
j
) >
u
a
i
(R
a
i
).
With respect to the social criterion, agents accept
transactions (changing the initial allocation A into A
0
)
that will not harm the society sw(A
0
) > sw(A)
The sociability is a criterion centered on the so-
cial welfare value, which is a global notion. This
value can only be determined thanks to the welfare of
all agents. Such conditions cannot be satisfied since
agents have only local information. However, to know
the evolution of the welfare value is sufficient to deter-
mine if a transaction penalize the society. These com-
putations can be restricted to the local environment of
agents, by considering the remaining population as a
constant.
3.3 Agent Behavior
Behaviors define agents from an external point of
view. They describe how agents interact with each
ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence
208

other, i.e. how they negotiate. During a negotiation,
each agent makes and receives offers, checks their ac-
ceptability according to its own criterion. If a trans-
action is acceptable for every participant, it is per-
formed. Otherwise, agents have to decide who has to
modify its offer according to their behavior, and thus
the negotiation continues.
Let us assume that agent a
i
∈ P initiates a negoti-
ation and proposes an offer to of its partner a
j
∈ N
a
i
previously selected. Both offers correspond to a bi-
lateral transaction δ
a
j
a
i
. If both agents consider this
transaction acceptable, it is performed. However, if
one participant rejects the offer, three alternatives can
then be considered:
• agent a
i
gives up and ends the negotiation;
• agent a
i
changes the selected partner;
• agent a
i
changes its offer or asks to change its
partner’s offer.
Determining the order of these actions is an im-
portant issue. Many behaviors have been imple-
mented and tested, but only the most efficient one is
presented here. Agents always sorts the list of pos-
sible subsets they can offer according to their prefer-
ences. The initiator can then offer the least penalizing
subset first. The initiator a
i
∈ P can change partners
as well as offers during a negotiation process. Such
an agent behavior is called frivolous flexible.
According to this behavior, if an acceptable trans-
action exists somewhere in the neighborhood, it will
necessarily be identified. The neighborhood should
be shuffled between two negotiations in order to mod-
ify the order in which neighbors are considered, and
thus avoid a bias.
4 SIMULATIONS AND
PROTOCOL
Simulations are characterized by the number of agents
and by the mean number of resources per agent.
During the experiments presented in this paper, 50
agents are negotiating 250 resources according to dif-
ferent settings. Agents can be either rational or so-
cial. Agents negotiate according to a negotiation pol-
icy, which is characterized by the size of agents’ of-
fers: h1, 1i means that agents can only perform swaps
whereas “up to h2, 2i” means that agents can propose
up to two resources. It can also be explicitly writ-
ten as: T = {h1, 0i, h0, 1i, h1, 1i, h2, 1i,h1,2i,h2,2i}.
Simulations are performed on social graphs that be-
long to different classes: complete, grids, Erd
˝
os-
R
´
enyi and small worlds.In this study, the link prob-
ability p varies from 0.05 up to 1.0. Each simulation
is iterated 100 times from different initial resource al-
locations randomly generated, in order to evaluate the
topological sensitivity. Utility functions and initial re-
source allocations are randomly generated according
to a uniform probability distribution.
5 BILATERAL NEGOTIATIONS
This section is dedicated to the evaluation of negotia-
tion processes according to two welfare notions. First,
for each welfare notions, the efficiency is evaluated,
by a comparison with the optimal social value, as well
as the topological sensitivity. Tables 3 and 4 presents
the efficiency of negotiation processes based on dif-
ferent negotiation policy and on different classes of
social graphs. These tables contain the proportion of
the optimal welfare value that can be achieved (left-
side of the cells). The greater is the proportion, the
closer to optima are the resulting allocations. The
deviation (right-side of the cells) shows the propor-
tion according to which may vary the solution qual-
ity. For instance, in Table 3, negotiation processes
based on a grid where rational agents negotiate using
δh1,1i transactions only end on social values repre-
senting 79.0% of the optimumDepending on the ini-
tial resource allocation, the welfare value achieved
may vary of 1.6%.
Then, the impact of the graph connectivity is eval-
uated. The topology of a contact graph greatly affects
the resource traffic and the negotiation efficiency. The
larger are agent neighborhoods, the denser are social
graphs, and the easier is the resource traffic. The
probability p for a link to exist between nodes from
any pair can be modified. Figures 1 to 2 show the
evolution of the welfare value in time.
5.1 Utilitarian Case
Independently of the contact network’s topology, ra-
tional negotiation processes always lead to weaker al-
locations than social negotiation processes. The re-
strictive character of the acceptability criterion affects
the quality of the provided solution. When consider-
ing complete social graphs, different negotiation pol-
icy always lead to optimal resource allocations. How-
ever, the use of large offers leads to important addi-
tional costs.
Negotiation processes lead to allocations associ-
ated with up to 98.9% of the optimal welfare value
when Erd
˝
os-R
´
enyi graphs are considered. Only
91.4% of the optimum is achieved when small-worlds
are considered. In an Erd
˝
os-R
´
enyi graph, the proba-
bility for a link to exist between any pair of nodes
RESOURCE ALLOCATION PROBLEMS ON NETWORKS - Maximizing Social Welfare using an Agent-based
Approach
209

Table 3: Utilitarian efficiency (%) and its deviation (%) according to the class of social graphs.
Social graph Rational policy Social policy
class h1,1i up to h2,2i h1,0i h1,1i up to h1,1i up to h2,2i
Full 96.6 0.3 97.0 0.2 100 0 98.3 0.2 100 0 100 0
Grid 79.0 1.6 81.3 1.3 86.2 0.9 85.3 1.1 86.1 0.9 86.1 0.9
Erd
˝
os-R
´
enyi 94.8 0.5 95.0 0.4 98.9 0.1 97.1 0.2 98.9 0.1 98.9 0.1
Small World 80.8 2.0 84.8 1.3 91.4 0.8 90.0 1.0 90.2 0.8 90.3 0.8
is constant, while in small-worlds, the larger is the
number of neighbors, the higher is the probability to
link this agent. Many agents have only one neigh-
bors, and the resource traffic is unequally distributed.
Then, bottlenecks, i.e., agents that block the resource
circulation, may appear. When grids are considered,
social negotiation processes achieve up to 86.2% of
the optimum. A weak mean connectivity handicaps
the resource traffic and hence the achievement of so-
cially efficient allocations.
The more restricted are social graphs, the weaker
is the negotiation efficiency, and the higher is the de-
viation. In all cases, the standard deviation observed
among the social values achieved remains small. It
means that when the utilitarian welfare is considered,
the topology has not a significant impact for a given
class. The deviation is higher with rational negoti-
ations since they restrict more the resource traffic,
which then influences on the solution quality. The
more restricted is the resource traffic, the higher is the
standard deviation, and thus more important become
the initial resource allocation.
Figure 1 shows the impact of the social graph con-
nectivity on the efficiency within a population negoti-
ating using social gifts. It represents the evolution of
the utilitarian welfare value in time.
30000
35000
40000
45000
50000
55000
60000
65000
0 500
Utilitarian welfare value
Computation Time (ms)
p=0.05
p=0.1
p=0.3
p=0.5
p=1.0
Figure 1: Utilitarian value vs. computation time according
to the mean connectivity.
This figure shows that a weak linking probabil-
ity, which corresponds to small neighborhoods, leads
to a welfare value far from the optimum. The grad-
ual increase of the probability p leads to larger wel-
fare values and to more time-consuming negotiations.
Larger neighborhoods facilitate the resource circula-
tion by offering a larger number of possible transac-
tions to all agents. The impact becomes really signifi-
cant when p < 0.3. Above this threshold, the resource
circulation is sufficient to achieve socially interesting
allocations.
5.2 Egalitarian Case
Table 4 shows that, generally, negotiations among ra-
tional agents achieve unfair allocations. Indeed, ra-
tional negotiations end quite far (only 20%) from the
optimal welfare value. The standard deviation is also
very important (up to 73% of deviation on small-
worlds). Thus, the rationality criterion is definitively
not well-adapted to solve egalitarian problems effi-
ciently. It restricts the set of possible transactions too
much and throws negotiation processes into local op-
tima. Generosity is hence an essential feature in order
to achieve fair allocations.
Even using on complete graphs, no social negoti-
ation policy can guarantee the achievement of egali-
tarian optima. Whereas social gifts are well adapted
to the solution of utilitarian problems, they do not suit
to egalitarian negotiations. Only 78.5% of the opti-
mum can be achieved in the best cases. Indeed, after
a finite number of transactions, agents can not give
any additional resource without becoming poorer than
their partners. Negotiations based on social swaps
lead to severely sub-optimal resource allocations with
an efficiency of at most 24.1% on complete social
graphs. The inherent constraints of swap transactions
prevent the modification of the resource distribution,
which penalizes a lot egalitarian negotiations. When
both gifts and swaps are allowed, the negotiation effi-
ciency is really close to the optimum. Larger bilateral
transactions improve only a little the fairness among
agents, but are much more expensive to determine.
Social graphs of weaker mean connectivity like
grids lead negotiation processes to socially weaker al-
locations. When small-worlds are considered, the re-
source traffic is restricted by the large number of agent
leaves, leading to a larger deviation. Indeed, some re-
sources may be trapped in the bundle of agent leaves.
Similarly to utilitarian negotiations, Figure 2
ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence
210

Table 4: Egalitarian efficiency (%) and its deviation (%) according to the class of social graphs.
Social graph Rational policy Social policy
class h1,1i up to h2,2i h1,0i h1,1i up to h1,1i up to h2,2i
Full 19.3 62.9 20.8 73.9 78.5 1.8 24.1 28.7 99.9 0.3 99.9 0.3
Grid 13.9 71.3 14.6 80.2 66.2 4.1 23.6 29.6 80.2 1.8 80.6 1.7
Erd
˝
os-R
´
enyi 17.4 71.9 20.2 76.8 77.3 2.2 23.8 27.3 96.1 6.8 96.6 6.5
Small World 13.1 73.0 13.9 77.5 63.8 10.4 23.4 27.8 78.1 9.4 78.2 10.5
0
250
500
750
1000
1250
0 500 1000 1500
Egalitarian welfare value
Computation Time (ms)
p=0.05
p=0.1
p=0.3
p=0.5
p=1.0
Figure 2: Egalitarian value vs. computation time according
to the mean connectivity.
shows that a large linking probabilityleads to larger
welfare value. Indeed, larger neighborhoods facilitate
the resource circulation by offering larger numbers of
possible transactions to all agents. The impact of the
connectivitybecomes really significant below p ≤ 0.5.
Thus, the impact of the mean connectivity in a contact
graph is more important in a egalitarian society than
in a utilitarian one.
6 CONCLUSIONS
Many applications can be represented by an alloca-
tion problem but only a limited number of studies fo-
cus on the mechanism require to achieve the great-
est results in a realistic context. Centralized solving
techniques may be really inefficient, due to scalabil-
ity issues or unadapted as soon as restrictions on agent
communications are considered. Studies investigat-
ing distributed methods remain often theoretical and
do not consider plausible assumptions. Indeed, agents
are omniscient most of the time and are able to nego-
tiate with all agents within the population. Neverthe-
less, such an ideal context is not satisfied in many real
life applications.
In this paper, we propose a distributed mechanism
based on local interactions, leading a negotiation pro-
cess to socially optimal allocations, according to a
more realistic context. Agents have only limited per-
ceptions and knowledge. They have to reorganize
themselves the resource allocation using local trans-
actions, achieving socially optimal allocations (or so-
cially close allocations if the need arises) by emer-
gence. All kinds of contact network can be used to
restrict agents’ communication abilities. We identify
the characteristics favoring and penalizing the negoti-
ation efficiency according to different negotiation set-
tings and different welfare notions. We show that ne-
gotiations between rational agents (widely used in lit-
erature) are not efficient, and we provide the negotia-
tion settings to use in order to achieve allocation that
are optimal for the society when four different welfare
notions are considered.
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