COOPERATION MECHANISM FOR A NETWORK GAME

Alon Grubshtein and Amnon Meisels

Dept. of Computer Science, Ben Gurion University of the Negev, Beer-Sheva, Israel

Keywords:

Multi agent systems, Self interest, Cooperation, Cost of cooperation.

Abstract:

Many real world Multi Agent Systems encompass a large population of self interested agents which are con-

nected with one another in an intricate network. If one is willing to accept the common axioms of Game

Theory one can assume that the population will arrange itself into an equilibrium state. The present position

paper proposes to use a mediating cooperative distributed algorithm instead. A setting where agents have to

choose one action out of two - download information or free-ride their neighbors’ effort - has been studied

recently. The present position paper proposes a method for constructing a Distributed Constraint Optimization

Problem (DCOP) for a Network Game. The main result is that one can show that by cooperatively minimiz-

ing the constructed DCOP for a global solution all agents stand to gain at least as much as their equilibrium

gain, and often more. This provides a mechanism for cooperation in a Network Game that is beneﬁcial for all

participating agents.

1 INTRODUCTION

A key attribute of any Multi Agent System (MAS) is

the level of collaboration that agents are expected to

adopt. When agents share a common goal it is natu-

ral for agents to follow a fully cooperative protocol.

A common goal can be the election of a leader, ﬁnd-

ing shortest routing paths, or searching for a globally

optimal solution to a combinatorial problem (Meisels,

2007). When the involved parties (agents) have differ-

ent and conﬂicting goals, competition and self interest

are the natural behaviors one expects to ﬁnd.

In its most basic form, a fully cooperative model

involves a set of agents attempting to satisfy or opti-

mize a common global objective. The most important

aspect of such a model is that the actions taken by the

agents do not bring into consideration the impact of

the globally optimal solution on the state of the in-

dividual agent (Maheswaran et al., 2004; Grubshtein

et al., 2010; Meisels, 2007).

In a fully competitive scenario it is common to

assume that participants are only willing to take ac-

tions which improve (or do not worsen) their gains.

In some situations agents can reach an equilibrium

state from which no agents would care to devi-

ate (Roughgarden, 2005; Meir et al., 2010). The efﬁ-

ciency of equilibria states with respect to the global

objective has been studied intensively in the last

decade (Roughgarden, 2005).

The central question that is at the focus of

the present study is the interplay between the self-

interests of agents and their cooperation towards a

common goal. Speciﬁcally, towards the increase of

some global gain. A rich and applicable frame-

work for this study is the domain of Network Games

(NGs) (Jackson, 2008; Galeotti et al., 2010). File

sharing and ad hoc P2P networks are good exam-

ples of network games. The existence of a Bayesian

Nash Equilibrium (BNE) for the NG introduced in

Section 2 was shown by (Galeotti et al., 2010). The

present paper proposes a method for securing cooper-

ation among agents in a network game, by guarantee-

ing improved gains to each agent. This approach is

related to former work on Cooperation Games (Grub-

shtein and Meisels, 2010). The proposed method

is based on the construction of an Asymmetric Dis-

tributed Constraints Optimization Problem (ADCOP)

(Grubshtein et al., 2010) from the given network

game and cooperatively solving it.

Former attempts to introduce cooperation into

competitive games fall into several categories. Strong

empirical evidence to the beneﬁts and emergence of

cooperation in the iterated prisoners’ dilemma tour-

nament was reported in (Axelrod, 1984). Petcu et.

al present a cooperative distributed search algorithm

which faithfully implements the VCG mechanism

for the problem of efﬁcient social choice by self-

interested agents in DCOP search (Petcu et al., 2008).

An approach for achieving cooperation among com-

petitive agents is described in (Monderer and Tennen-

336

Grubshtein A. and Meisels A..

COOPERATION MECHANISM FOR A NETWORK GAME.

DOI: 10.5220/0003276303360341

In Proceedings of the 3rd International Conference on Agents and Artiﬁcial Intelligence (ICAART-2011), pages 336-341

ISBN: 978-989-8425-41-6

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

holtz, 2009), which introduce mediators to achieve

stronger equilibria among competitive agents.

The present study is unique in proposing a dis-

tributed search method that guarantees the improve-

ment of the personal gain of each agent. It uses the

deﬁnition of the Cost of Cooperation (Grubshtein and

Meisels, 2010), which deﬁnes a measure of individual

loss from cooperation. It is shown that for a known

family of network games this cost is negative(see sec-

tion 3). That is, a complex real world setting is pre-

sented where one can prove that cooperation produces

better individual gains to all participating agents.

2 DOWNLOAD/FREE-RIDE

NETWORK GAME

Consider a set of users wishing to download large

amounts of information from a remote location. Users

may receive the information from other users in their

local neighborhood (if these neighbors have it) or di-

rectly download it from a central hub. The process

of downloading data from the hub requires signiﬁ-

cant bandwidth and degrades performance (resource

consumption, battery, etc). In contrast, sharing infor-

mation with peers does not result in any signiﬁcant

impact on the users. Information exchange between

users is limited to directly connected peers only (no

transfer of information to second degree neighbors).

The entire interaction can be captured by a graph

G = (V, E) where each edge e

speciﬁes a neighbor-

hood relationship and a vertex i ∈ V represents a user.

denotes the action (e.g., assignment) made by user

i, and x

(i) is the joint action taken by all of i’s neigh-

bors.

The degree of i will be denoted by d

While the network itself (the graph G = (V, E)) is

not known to all agents, the total number of partici-

pants n = |V| is known and so is the (ﬁxed) probabil-

ity p for an edge to exist between any two vertices.

Such graphs G = (n, p) are known as Poisson Ran-

dom Graphs or an Erd¨os - R´enyi network (Jackson,

2008). Figure 1 is an example of such a network with

n = 8 and p = 0.45 (8 participants and 13 edges).

For simplicity, the gain from receiving informa-

tion is unity and the cost of downloading it is c.

Users are only aware of their immediate peers and are

known to be self interested. That is, users decide to ei-

ther download the relevant information from the hub

(take action D) or wait for one of their neighbors to

download it (take action F ), and do so in a way which

We speciﬁcally refrain from the common x

−i

notation

to represent a joint action by all players but i to emphasize

that player i is only affected by the set of her neighbors.

Figure 1: An interaction graph with n = 8 and p = 0.45.

maximizes their own utility function:

, x

(i)) =











1− c if x

= F

1 ∃ j ∈ x

(i) s.t x

= D

0 otherwise

That is, if the user exerts effort and downloads the in-

formation her gain is 1− c. If, on the other hand, she

does not download but one of her peers does, her gain

is 1. Finally, if neither her nor any of her peers down-

load the information her gain is 0. We assume that the

interaction between users is a one shot interaction (i.e.

users are assumed to be at some decision crossroad).

2.1 The Game Theoretic Approach

As mentioned in section 1, the common approach

taken when self interested entities are present, is to in-

voke Game Theory (Osborne and Rubinstein, 1994).

By accepting the fundamental axioms of game theory

one is able to predict possible outcomes of an inter-

action. These outcomes are standardly assumed to be

equilibrium points (in terms of agents’ actions) from

which no agent is willing to deviate. The best known

solution concept is the Nash Equilibrium (NE) point,

but there are many other equilibrium points which

ﬁt different aspects of the interaction (cf. (Osborne

and Rubinstein, 1994)). For example, in the above

D/F network game the equilibrium concept used is

the Bayesian Nash Equilibrium (BNE). The BNE cap-

tures the incomplete information of all participants

(each knows only its own neighbors).

This is the approach taken in a recent work by Ga-

leotti et. al (Galeotti et al., 2010) for a similar inter-

action. They describe our example of a network game

and cast it to many other types of interactions: vacci-

nate or not, research new technology or wait for the

competitors to do it, etc. In the domain of computer

science one can think of remote rovers belonging to

different agencies exploring the same area, of ad-hoc

network participation, and of P2P networks.

Let us describe the network game that is at the

focus of the present study in more detail. ¿From the

point of view of each participant in the game (i.e. a

user) there are two possible strategies: D (download),

COOPERATION MECHANISM FOR A NETWORK GAME

337

F (free ride). Representing this game by a traditional

n-dimensional matrix is clearly intractable and hence

a graphical game representation is used. A graphical

game (Kearns et al., 2001) is a succinct representation

in which each vertex u represents a party and the set

of edges emanating from it represents its relations to

a subset of the vertices. These vertices are the only

vertices which affect u. When the underlying graph

of a game is the complete graph this representation is

equivalent to the n-dimensional matrix representation.

Although the payoff structure is known to all par-

ticipants, the degrees of neighbors (e.g., their connect-

edness) are unknown. Based on the global probabil-

ity p for an edge, each participant can calculate the

probability for a randomly selected neighbor to be of

degree k:

Q(k; p) =



n−2

k−1



(k−1)

(1− p)

(n−k−1)

Galeotti et. al use this information to ﬁnd a threshold

Bayesian Nash Equilibrium. They deﬁne a parameter

t which is the smallest integer such that:

1−

∑

k=1

Q(k; p)

≥ 1− c

In the unique BNE that they ﬁnd for the network

game, any participant of degree k ≤ t selects strategy

D and any participant of degree k > t selects F .

D D

0.3

Figure 2: The strategies of all participants in the BNE and

the corresponding gains.

For example, consider the graph G = (8, 0.45) il-

lustrated in Figure 1 and a cost value of c= 0.7. From

the above equation we calculate the threshold t = 2.

As a result, in the BNE, a

and a

will select strategy

D while all others will assign F . The BNE assign-

ments and gains are depicted in Figure 2.

3 COOPERATIVE MECHANISM

FOR SELF INTERESTED

AGENTS

Despite the success game theory has had in many

MAS problems and applications it suffers from two

important drawbacks:

1. It often fails to predict human behavior (e.g., the

iterated prisoners’ dilemma (Axelrod, 1984)).

2. Results are often inefﬁcient (i.e. equilibria points

are not necessarily pareto efﬁcient) (Nisan et al.,

2007).

The former point induces interesting research (of-

ten related to psychology) which may produce new

solution concepts (Halpern and Rong, 2010). The

present paper focuses on the latter point. More specif-

ically, we follow the ideas presented in (Grubshtein

and Meisels, 2010) which deﬁne the Cost of Cooper-

ation and Cooperation Games:

Deﬁnition 1. An agent’s Cost of Cooperation (CoC)

with respect to a global objective function f(x) is de-

ﬁned as the difference between the lowest gain that

the agent can get in any equilibria of the underlying

game (if any, otherwise zero) and the lowest gain an

agent receives from a (cooperative) protocol’s solu-

tion x that maximizes f(x).

Deﬁnition 2. A game is a Cooperation Game (CG) if

there exists a solution for which the CoC (with respect

to some f(x)) of all agents is non positive.

In other words, It is beneﬁcial for all agents to co-

operate in search for an optimal solution, rather then

play their NE strategy (as competitive agents are ex-

pected to act) (Grubshtein and Meisels, 2010).

For a game that satisﬁes the above two deﬁnitions

one can propose a mechanism which will provide the

following guarantee: if all participants agree to fol-

low the mechanism, they are each expected to gain

at least as much as they would have gained had they

played their NE strategy.

The present study focuses on Network Games and

on the D/F game in particular. The main result of

the paper is a proof that the D/F game is a Coop-

eration Game (CG). This enables a clear mechanism

that solves a generated Asymmetric DCOP (ADCOP)

(Grubshtein et al., 2010) and can act as a natural

choice of strategy for all participants of the game. An

ADCOP is an extension to the DCOP paradigm which

addresses individual agent’s gains and captures game-

like interactions between agents. An ADCOP can be

viewed as a form of a cooperative graphical game.

The proposed framework for the mechanism

makes the distinction between users and agents.

ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence

338

When referring to users one refers to self interested

entities seeking to maximize their gain. Each of the

users employs an agent - an application entity which

follows a cooperative protocol on the one hand, but is

expected to faithfully represent its user’s preferences.

In the proposed setting agents represent the users, but

are not under the users’ direct control (i.e. users’ en-

trust the agents with the ability to choose for them, but

are unable to control or affect the agents’ decisions).

An important feature of the proposed approach

is its inherent cooperation. The desired solution it

seeks is not necessarily a stable one in the game the-

oretic sense. Nonetheless, rational, self interested

users would rather let their agents choose an action

for them knowing in advance that agents’ cooperation

will yield results which are at least as good as they

would get if they play selﬁshly.

The distributed cooperativeprotocol for the agents

which will provide the above guarantee to the users

solves the following ADCOP:

• A set of Agents A = {a

, a

, ..., a

} - each holds a

single variable and corresponds to a user.

• A set of Domains D = {D

, D

, ..., D

} for the

variables held by the agents. Each domain con-

sists of only two possible values: {D, F }.

• A set of asymmetric constraints C. Each con-

straint c

is identiﬁed with a speciﬁc agent (e.g.

agent i) and deﬁned as a (d

+ 1)-arity constraint

acting over all of agent’s i neighbors. The set of

agents involved in constraint c

is A

= a

∪x

(i).

That is, the ADCOP has n constraints and each

agent contributes to d

+ 1 different constraints.

The costs associated with each constraint that add

up to the total cost of c

for agent i are summarized

in the table in Figure 3.

Costs of c

if ∀a

∈ A

, x

= F

(all agents of the constraint assign F )

for each a

∈ A

with d

> t and

∀a

∈ x

( j), d

> t that assign x

= D

(n− 1)n

for each a

∈ A

with d

> t and

∃a

∈ x

( j), d

≤ t that assign x

= D

for each a

∈ A

with d

≤ t

which assigns x

= D

Figure 3: The constraint’s costs for each combination of

assignments.

Note: costs of the DCOP’s solutions (full or partial)

are completely unrelated to the agents’ gains.

Let us now proceed to prove that ﬁnding a mini-

mal assignment (an optimal solution) for the ADCOP

results in gains which are at least as high as the actual

gains resulting from the BNE, for all agents .

Let the optimal solution to the ADCOP be denoted

with x. We use the term null neighborhood to deﬁne

a situation in which an agent and all of its directly

connected peers assign F as their action.

Lemma 1 (No null neighborhoods). In the optimal

solution to the ADCOP there exists at least one agent

in the local neighborhood of each agent which assigns

D (speciﬁcally, the agent itself can assign D).

Proof. Assume by negation that there exists an agent

with a null neighborhood in x. This means that the

cost of c

(the constraint originating from a

) is n

. Let

′

be a complete assignment which differs from x in

the assignment of an agent a

∈ c

. That is, x

′

= D.

As a result the previously null neighborhood of a

x is not part of x

′

One can compute an upper bound on the cost of

′

by assigning this change to an agent a

with d

n−1 (where n−1 > t) which has at least one neighbor

∈ x

( j) of degree d

≤ t (i.e. a

appears in all n

constraints of the problem). The resulting cost of x

′

COST(x

′

) = COST(x) − n

+ (n− 1)n

= COST(x) − n

which implies that COST(x

′

) < COST(x) in contra-

diction to the optimality of x.

An important implication of Lemma 1 is that in

the optimal solution to the ADCOP none of the agents

receive a payoff of 0.

Lemma 2. The gain of agents a

with degree d

≤ t in

x is at least as high as its gain in the BNE.

Proof. In the BNE a

’s gain is exactly 1-c (d

≤ t and

hence a

assigns D). The only possible lower gain

is 0. However, this is the gain of an agent with a

null neighborhood and hence (following Lemma 1)

its gain must be at least 1-c when cooperating (either

it will assign D or one of its peers will).

Lemma 3. The gain of agent a

with degree d

> t in

x is at least as high as its gain in the BNE.

Proof. Due to its degree, a

’s BNE gain is never1−c.

When all its neighbors a

∈ x

(i) are of degree d

> t

its BNE gain is 0, and when at least one neighbor is

of degree d

≤ t its BNE gain is 1. Since the ADCOP

gain is always higher than 0 (Lemma 1) we only con-

sider the case when a

’s gain is 1 in the BNE. This is

only attainable when a

has at least one neighbor a

with degree d

≤ t. Thus, to conclude our proof we

have to show that a

’s ADCOP gain is not 1− c.

By the problem description this gain can only oc-

cur when a

assigns D in x. This assignment incurs a

COOPERATION MECHANISM FOR A NETWORK GAME

339

positive contribution on the ADCOP only when it pre-

vents the existence of a null neighborhood for a

, or

when it prevents it for one (or more) of a

’s neighbors.

If a

has a null neighborhood which it would like

to prevent, there must be a neighbor a

∈ x

(i) (with

degree d

≤ t) which assigns F in the ADCOP. In this

case we deﬁne a new solution x

′

, which differs from

x in the assignments of a

and a

(i.e., x

′

= F and

′

= D). x

′

’s cost is:

COST(x

′

) = COST(x) − (d

+ 1)(n − 1)n + d

The highest cost x

′

can take will be when d

= 1 and

= n− 1 in which case:

COST(x

′

) = COST(x) − 2(n − 1)n + (n − 1)

= COST(x) − (n− 1)(2n− 1)

implying that a

does not assign D to prevent a local

null neighborhood in the optimal solution.

When a

assigns D to prevent the existence of a

null neighborhood for one of its neighbors, we clas-

sify the neighbor’s type:

1. The neighbor a

has at least one neighbor a

with

degree d

≤ t. We generate the assignment x

′

which x

= F and x

= D. As before the cost of x

′

is lower than that of x since a

contributes a single

unit per each constraint it is involved in (at most n

units) whereas the assignment change of a

lowers

the cost by at least (n− 1)n units in contradiction

to the optimality of x.

2. The neighbor a

has no neighbor a

with degree

≤ t. Assigning x

= D and x

= F to a

lowers

the resulting cost since the contribution of a

the cost is bounded by n

(it incurs a cost of d

to each of the d

+ 1 constraints it is involved in),

whereas the gain from changing the assignment of

is (d

+ 1)(n − 1)n.

This means that even if all neighbors are of the sec-

ond type (which always incur greater costs), and they

are all connected to all other agents, the cost of the

modiﬁed solution is:

COST(x

′

) =COST(x)−(d

+1)(n−1)n+d

·n(n−1)

Hence we conclude that that the gain of a

will not

be reduced from 1 in the BNE to 1 − c (or 0) in the

ADCOP’s optimal solution.

Theorem 1 (Gain guarantee). The optimal solution

to the ADCOP described above results in gains which

are at least as high as those achieved in the BNE.

Proof. Directly follows from Lemmas 1-3.

Figure 4 demonstrates the results of applying and

solving the ADCOP on our previous example. Agents

, a

and a

assign F and do not incur any cost.

Agent a

and a

incur a cost of 1 to c

and c

(to c

and c

in case of a

) and agents a

and a

incur a

cost of 3 (their degree) to the 4 constraints they are

involved with (c

, c

and c

, c

) resulting

in a cost of 28.

D D

0.3

Figure 4: Strategies of all participants in the ADCOP and

the corresponding gains. Red indicates higher gains than

those received in the BNE.

Note that the ADCOP’s solution dictates an as-

signment in which 4 agents (namely a

, a

and a

)

increase their gain. None of the agents gained less

than their BNE gain depicted in Figure 2.

4 DISCUSSION

In a general combinatorial problem, maximizing the

total sum of gains (also known as a “utilitarian”

scheme, or the social choice) provides no guarantees

to the individual participant. Depending on the nature

of the problem, achieving the optimal social solution

requires that some participants agree to an extremely

low gain (Moulin, 1991). On the other hand, the

equilibrium solution suggested by Game Theory for

problems involving only self interested parties may

be greatly improved when agents cooperate (e.g., the

“prisoners’ dilemma” (Axelrod, 1984; Monderer and

Tennenholtz, 2009)).

The present paper proposes a cooperative, qual-

ity guaranteeing framework to overcome this prob-

lem. The proposed framework and mechanism are

proven to provide strong guarantees for a general net-

work game (the Download/Free-ridegame) which can

be cast into many diverse applications.

In the proposed framework, agents acting on be-

half of users adopt a cooperative behavior (coopera-

tively solve an ADCOP) to achieve a solution which

pareto dominates the Bayesian Nash Equilibrium.

Thus, users are always expected to gain at least as

much as in the BNE. One of the beneﬁts of this ap-

ICAART 2011 - 3rd International Conference on Agents and Artificial Intelligence

340

proach is that it justiﬁes the use of cooperative algo-

rithms as mechanisms for instances where the Nash

Equilibrium is not readily known.

The state of a Network Game in which agents do

not fully know the whole network is natural to a dis-

tributed scenario in which information is often local.

The analogous situation in ADCOPs is one in which

agents only communicate with their neighbors during

search and are not aware of more remote agents. The

corresponding family of distributed search algorithms

(local search algorithms) are not guaranteed to ﬁnd

optimal results but may be scaled to large populations

of agents (Grubshtein et al., 2010; Maheswaran et al.,

2004; Zhang et al., 2005).

It is not clear that our previous guarantee can be

satisﬁed in a setting where agents employ a local

search algorithm: Consider the interaction of Fig-

ure 1 and a set of agents participating in a stochastic

search (Zhang et al., 2005). The initial assignment

of all agents is F . As a result, all agents consider a

change of assignment to D in the next round. Speciﬁ-

cally, Agents A

and A

may change their assignments

while the rest of the agents stochastically avoid any

change. The resulting solution is a local minima from

which agents will not deviate (similar to an equilibria

from which users will not deviate). In this converged

solution our guarantee is violated - agent A

’s gain is

reduced from 1 to 1−c. Nonetheless, a simple manip-

ulation to distributed hill climbing algorithms such as

MGM (Maheswaran et al., 2004) can result in solu-

tions which provide the desired guarantee.

The framework and methods proposed in the

present position paper form a mechanism that enables

the self driven desires and goals of users to be solved

by a cooperative system of computerized agents. We

believe that this is a natural mechanism for many

user applications which interact with their environ-

ment and with other users. In such settings, coop-

eration between agents is the natural action only if it

can provide a strong and realistic guarantee regarding

the expected gain to each user. A guaranteed nega-

tive CoC provides a suitable incentive for cooperation

- securing a gain which is at least as high as the worst

possible gain attained by the user.

ACKNOWLEDGEMENTS

This work was supported by the Lynn and William

Frankel center for computer science and the Paul

Ivanier Center for Robotics.

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