16603606

Experiments on robustness and

deception in a coalition

formation model

M. V. Belmonte, R. Conejo,

J. L. Perez-de-la-Cruz

and F. Triguero

Dpt. Lenguajes y Ciencias de la Computacion, ETSI Informatica, Universidad de Malaga, Malaga 29071, Spain

SUMMARY

In the last few years coalition formation algorithms have been proposed as a possible way of modeling autonomous agent cooperation in multi-agent systems. This work is based on a previously proposed coalition formation model founded on game theory for a class of task-oriented problems that guarantees an optimum task allocation and a stable profit division. In this paper we study two properties of the model that are very important for application in real-life scenarios: robustness and tolerance to an agent’s misbehavior. First, we study the robustness of this model as regards the effect the agent’s failure has on the resultant profits of the coalition formation. Secondly, we also study the coalition formation model in the presence of misbehaving agents. Agents have some kind of execution autonomy, and they can deceive or mislead each other when they reveal their information, if they believe this will give them more profits. Copyright c2005 John Wiley & Sons, Ltd.

KEY WORDS: multi-agent systems; coalitions; game theory; deception-free; robustness; task-oriented domains

INTRODUCTION

In recent years it has become clear that computer systems do not work in isolation. In order to carry out their tasks, these computer systems have to cooperate and coordinate their activities with other systems and people. Coalition formation is an important mechanism for cooperation in multi-agent

systems (MASs) [1–6]. Autonomous agents forming coalitions may improve their profits and abilities

to satisfy their goals, sharing their resources and distributing their tasks. The desired goals of a coalition formation process include the following:

∗_{Correspondence to: J. L. Perez-de-la-Cruz, ETSI Informatica, Universidad de Malaga, Malaga 29071, Spain.}

†_{E-mail: [email protected]}

Contract/grant sponsor: Spanish Government, CICYT; contract/grant number: TIC 2000-1370-C04-02

(i) maximize the coalition profit or utility;

(ii) divide the total utility among agents in a stable way, such that the agents in the coalition are not motivated to abandon it—stable payment configuration; and

(iii) do it within a reasonable amount of time and with a reasonable amount of computational effort.

A coalition formation model—based on game theory—was presented in [7], allowing cooperation

among autonomous, rational and self-interested agents in a class of superadditive task-oriented

domains [8]. In these domains agents are in charge of performing certain tasks, but they do it at different

costs, depending on the agent and the type of task. However, it is possible to form a coalition among

agents with a new re-distribution of the tasks that may allow them to obtain benefits. That model [7]

(i) maximizes the benefits and abilities of agents to satisfy their goals;

(ii) guarantees optimum task allocation and, according to the core, stable profit distribution among the coalition members; and

(iii) has a polynomial complexity in terms of number of agents.

In this paper we present two empirical studies of this coalition formation model. The first study analyses the robustness of the model in view of the failure of one or more system agents. This is an important issue, due to the fact that a service provided by a MAS must be maintained, sometime badly, when faced with the failure of processors, communication networks or agents. The second experiment addresses another interesting issue: since agents have some kind of execution autonomy and are self-interested, they can deceive or mislead each other when they reveal their information if they believe that they will obtain more profits by doing so. So it is necessary to study the effects of deception and manipulation on the model.

The rest of the paper is organized as follows: in the next section we briefly present the proposed coalition formation model. Then we study the model behavior in the face of failure of one or more system agents. In the following section we describe one of the possible manipulations the system agents can carry out when they reveal their private information and test its effects on the model’s stability. We finish with some conclusions and a proposal for future works.

COALITION FORMATION MODEL

In this section we summarize the main features of the model presented in [7].

We start with a motivating example. Let us assume that there are a number of geographically dispersed hospitals in a certain region. Each hospital is responsible for the treatment of its district’s patients, and it has the necessary resources to treat the various patients’ pathologies. However, each pathology treatment has a different cost in each hospital (due to different staff costs, staying costs, infrastructure costs, etc.). Thus, it is possible to form a hospital coalition, in such a way that patients are transferred among coalition members with the goal of minimizing costs. Obviously, the costs of transferring patients must also be taken into account. These costs include physical transfer of patients, staying in the hospital, future patient post-treatment controls, etc. Each pathology treatment is a different type of task. The estimated number of patients that have to be treated in each hospital is the initial amount of the task (obviously, it is an integer number; however, if the number is large, we can approximate the problem by means of real magnitudes). The cost of a pathology treatment

is the unitary execution task. Finally, it can be assumed that each hospital has a maximum capacity for each pathology treatment, and the initial amount of the task is always less than this maximum capacity. Note that costs of ‘physical’ processes (i.e. transfer of patients) will be much larger than communication costs.

Let us abstract now from this concrete domain and let us assume that there are several agents in charge of performing certain tasks, and that each agent has an Internet connection. The agents communicate their intention of participating in the coalition formation process to a middle agent (matchmaker). This agent, which we will call the coalition directory, provides a yellow pages service to the agents interested in these services. The interested agents send their identification to the coalition directory and the type of task or service that they can perform. Then, the coalition directory automatically adds a generic software component, called a mediator, to the agents, which have previously registered in the coalition directory. The mediator provides the agents with the necessary services or interfaces to participate in a coalition formation process.

From this moment on, any registered agent may initiate a coalition formation process. With this purpose the initiator agent sends a message to another system component called a coalition manager. This agent controls the actual number of active coalition formation processes (only an active coalition formation process for each type of task is possible at the same time).

The agents who accept the invitation must send additional information to the initiator agent, since it is in charge of calculating the task allocation and the payment division among the participant agents. This additional information includes maximum capacity, unitary execution cost, initial amount of tasks that they have to perform and their physical location in order to calculate the transfer cost among

each pair of agents. In [9] a software architecture is described that implements this coalition formation

model on the FIPA-OS agent platform.

More formally, let us consider a set ofnagentsN = {1,2, . . . , n}that can communicate, such that

each agenta_imust perform a certain initial amountt_i0of task units (a task is considered as an infinitely

divisible job). Agenta_i can perform a maximum of k_i task units (t_ia ≤ k_i) at a unitary cost ofc_i.

We will define the surplus capacityh_i ofa_ibyh_i =k_i−t_i0(0≤h_i).

Agents can change the amounts of tasks initially assigned to each one; lett_ij ∈Rbe the number of

task units transferred froma_itoa_j. However, to allow this new deal, they must incur a transfer cost; we

will assume that the cost of transferring a task unit froma_i toa_jisc_ij. In this way, the unitary profit,

bij, attached to the transfer of a task unit betweena_i anda_jisb_ij =c_i −c_j −c_ij andB_ij =b_ij×t_ij

is the transfer profit. On the other hand, we will assume that coalition formation costs (communication and coordination) are negligible compared with actual task transfer costs (this assumption leads to a superadditive problem, i.e. a problem in which the global coalition is always the most profitable).

Coalition formation is often studied using the concept of characteristic function games [10].

The characteristic function or coalitional value,v(S), of a game with a set of playersN is the total

utility or profit that the members ofScan attain by coordinating and acting together. In our domain the

coalitional value of any coalition will be stated as

v(S)=

i,j∈N,i=j

Bij=

i,j∈N,i=j

bij×tij∗ (1)

wheret_ij∗ are the task transferences obtained from an optimum task allocationT∗. In this way, the

A set of transfers is feasible if and only if the following conservation and capacity constraints hold

for every agenta_i:

• the task assigned after the transfers is not negative, i.e.

0≤t_i0−

j≤n,i<j

tij+

j≤n,j<i

tji (2)

• the task assigned after the transfers is not greater than the total capacityk_i =t_i0+h_i, i.e.

t0

i −

j≤n,i<j

tij+

j≤n,j<i

tji≤ki (3)

The optimal value can be calculated as

max

i,j∈N,i<j

bij∗tij (4)

subject to constraints (2) and (3), i.e. as the solution of a linear program on(n(n−1))/2 non-negative

variablest_ij(i, j ≤n, i < j)and 2×nconstraints (this is a slightly modified version of the program

presented in [7]). Note that there are algorithms (see, for example, [11]) that solve this problem in time

polynomial with respect to the number of constraints/variables, i.e. the number of agents.

Payoff division is the process of dividing the utility or benefits of a coalition among its members. The problem is to distribute the utility of the coalition in a stable way, so the agents in the coalition are not motivated to abandon it. Game theory provides different concepts (core, kernel, Shapley value, etc.)

for the stability of coalitions [10]; however, these concepts identify if a concrete payment division,x, is

stable, but they do not explain how to reach the stablex; and it is computationally complex to directly

check their definitions for a givenx. Informally speaking, a payment configuration lies inside the core

when no subgroup of agents is motivated to change their allegiances, because the value of the new coalition would not be greater than the sum of payments that they are receiving in the present payment configuration.

The proposed payoff division scheme is calculated by means of the concept of marginal profit. Thus, the payment to each agent will be given by the marginal profit with respect to the resource supplied by the agent, and multiplied by the quantity of this resource. Since the concept of marginal profit is really

that of partial derivative, the payment vectorx, according to formula (1), will be computed as follows:

xi =ti∂V_∂t i +

∂V

∂hihi (i=1, . . . , n) (5)

In [7] we prove that this payment vector lies inside the core and that it can be computed in just one

step.

EXPERIMENTS ON ROBUSTNESS

Fault tolerance in MAS

A MAS, like any other distributed system, is susceptible to the same faults that any distributed system is susceptible to: processor failures, communication failures, deterioration of the efficiency

and software bugs. However, in spite of the modular nature of this kind of system—failures may be isolated contributing to the avoidance of their propagation—the non-determinist nature of the agents, the dynamic environment, the interconnectedness of the agents and the lack of any central control point make it very difficult to prevent possible fault states, and thus make fault handling behavior

unforeseeable [12]. In addition, since MASs rely on collaboration among agents, the failure of one of

the agents involved can bring the whole computation to a dead end.

Surprisingly, a large part of actual distributed multi-agent frameworks and their applications do not

consider this possibility of failures in a systematic way [13]. The main reason is that much of the

experimentation with these systems is done using reliable and closed agent environments, which do not need to handle failures while the system is running. However, when MASs are developed in an open environment, where agents from various organizations interact in the same MAS and the system is distributed over many hosts, more attention must be paid to fault tolerance. In an open system the

agents can appear and disappear, the tasks may constantly change [14], the agents may be malevolent

or poorly designed and the processors may be overloaded and fail. Thus, in order to avoid this type of

failure, a system must be capable of coping with them [12,13]. In general, there are several degrees of

adaptation to failure. For example, Burns and Wellings [15] establish the following levels or degrees

of fault tolerance:

(i) full fault tolerance, where the system continues to operate without significant loss of functionality, even in the presence of faults,

(ii) graceful degradation, where the system maintains operation with some loss of performance, (iii) fail-safe, where vital functions are preserved while others may fail.

Some multi-agent platforms propose solutions linked to failures, but most of them are problem specific. For instance, several works address the complex problem of maintaining agent

cooperation [13,16]. In general, it has been shown that the replication of data, agents and/or

computation is the only efficient way to achieve full fault tolerance in distributed systems [17].

However, in practice replication is not always feasible because it is costly and the multiplication of the agents involved in the MAS can lead to excessive overhead. Only specific agents which are identified as crucial to the application should be rendered fault tolerant, and the scheme used for this purpose

should be carefully selected [12,18].

We will focus on graceful degradation, which, according to Foner [19], is one of the main attributes

that must be considered when examining agent behavior. The agents (systems) work best when they exhibit graceful degradation in cases of communication mismatch, domain mismatch or other kinds of faults. If most of a task can still be accomplished, the result is generally better than failing to accomplish the full task, and gives the user more trust in the agent’s performance.

In our case, we will assume that an agent’s failure means that after forming the coalition and before

starting its duties, agenta_i stops working and sends no other message (silent failure). This fact will

yield two consequences.

(i) Sincet_i0disappears from the system, the total amount of tasks performed will decrease. However,

this fact is independent of the coalition formation mechanism and hence is not addressed in this paper.

(ii) Since the configuration of agent resources (initial tasks and capacity excesses) has changed, a new computation of task transfers and payoff division is needed. This problem must be addressed by the coalition formation mechanism.

An obvious alternative could be to calculate the optimal transferences among the surviving agents again, and to distribute the new profit according to the algorithm mentioned in the previous section. However, in general this will be impossible: if task transfers are executed in an asynchronous way (which is usually the case), some of them will have been transferred and, moreover, already performed according to the prior arrangement.

For that reason, we propose an incremental alternative payoff division model and experimentally check if the degradation of the generated profit is significant with respect to the optimal solution. Note that this way the robustness we are studying is the one related to the utility generated by the coalition formation mechanism with respect to the optimal achievable utility.

Alternative algorithm and experimental design description

In order to study the behavior of our coalition formation model in the presence of the failure of one or more participant agents, we have developed a simulation tool to experiment with our model. This tool calculates optimum task allocation and a stable payoff division according to our coalition formation model.

Experiments were carried out for a number of agents varying from 10 to 50, with a failure probability ranging from 0.1 to 0.5. The input data for the problem were generated in a random way in a specific interval and following uniform distribution. The data consisted of the following.

1. k_i: a random value is generated from a uniform distribution in the interval

[MIN-CAP..MAXCAP].

2. c_i: a random value is generated from a uniform distribution in the interval

[MIN-COST..MAXCOST].

3. t_i: this value is stated asβ×k_i, whereβis a value generated randomly from a uniform distribution

in the interval [0..1].

4. c_ij: in order to establish it, all the participant agents are placed in a rectangle with vertices(0,0)

and(Xmax, Ymax). The transferring costs are generated proportionally to the Euclidean distance

between each pair of agents. The triangular inequality constraint (see the following section) is thus satisfied. In addition, these costs must also be proportional to the unitary costs in order to avoid model degradation.

These parameters are easily configurable. For the simulation, the selected values were

[MINCAP..MAXCAP]=[0..100], [MINCOST..MAXCOST]=[5..25] andXmax=20,Ymax=20.

The experimental design compares the optimum model results, which produce a stable payoff division in the sense of the core, with the results of an alternative payoff division scheme. This schema makes a division of the utility loss—due to the failure of the agents—that is proportional to the initial utility obtained by them in the optimum allocation scheme prior to the failure.

Alternative payoff division model

The goal of this model is to compute a new payment vectorxfor thenparticipant agents after the

failure of one or more of these agents, but without applying the optimum model of task allocation and payment division again. Note that this alternative schema is essentially heuristic and does not guarantee an optimum assignment of tasks nor a stable payment vector in the sense of the core.

1.Goal: computation of a new payment vector,x=(x₁, x₂, . . . , x_n)after the failure of an agent called falling 2.Input: optimum payment vector, before the agent failure,x=(x1, x2, . . . , xn), optimum task transfers for all system agents,thj, h, j∈N, and the attached profit to each task transfer for all system agents,Bhj, h, j∈Nand the falling agent falling

3.Output: alternative payment vectorx=(x₁, x₂, . . . , x_n)

1. Determine the coalition S in which falling participates 2. S = {t(h,j) such that h = falling or j = falling} 3. if S is empty

then x’ <- x

else A <- the addition of coalition profits in which falling participates B <- the addition of transfer profits

in which falling participates if A = B

then for all i in S, x’(i) <- x else C <- B - x(falling)

x’(falling) <- 0 if C <> 0

then for every i<>falling

d(i) <- a part of C proportional to x(i) x’(i) <- x(i) - d(i)

Figure 1. Algorithm for the alternative payoff division model.

In order to establish the new assignment of tasks, the original one is perturbed in a conservative way, i.e. scheduled transfer is maintained unless the silent agent is involved in it. If transfers are executed in an asynchronous way (which is usually the case) this is an unavoidable requirement. On the other hand, the utility loss is not assigned to the agents directly connected to the silent agent, but shared among all

those that remain active by discounting from each one a percentageαof its initial utilityxi (its utility

prior to the failure) proportional toxi. The algorithm in Figure1is applied to each of the falling agents,

following its falling order.

Report of results

We have performed 100 runs of the simulation for each combination number of agents/failure

probability. The results reported are averages of the runs. The following magnitudes are represented in

the graphs (ordinate axis).

• Optimal utility (global)Uopt(Figures2(a)and(b)). This is the utility generated by the algorithm

presented in [7].

• Utility degradation (global) (Figure 3(a)). Let us call Usub the utility generated by the

(a)

(b) 0

500 1000 1500 2000 2500 3000 3500

10 20 30 40 50 number of agents

optimal utility (global)

0,1 0,2 0,3 0,4 0,5

0 500 1000 1500 2000 2500 3000 3500

0,1 0,2 0,3 0,4 0,5 failure probability

optimal utility (global)

10 20 30 40 50

Figure 2. Optimal utility.

• Utility degradation (maximum per agent)(Figure3(b)). For each agenta_i, letx_ibe its payoff

computed by the algorithm presented in [7] and letx_ibe its payoff computed by the alternative

mechanism. We compute the maximum ofx_i−x_i,∗x=max(x_i−x_i).

• % Utility degradation (global)(Figure4). It isu=100×U/Uopt.

Figures 2(a) and (b) show two graphs plotting the behavior of the optimal utility (global) Uopt.

The abscise axis represents the number n of agents (Figure 2(a)) and the failure probability f

(Figure2(b)). It is clear thatUoptincreases withnand decreases withf.

In Figures 3(a) and (b), the abscise axis representsf. The ordinate axis represents the utility

degradation (global) U (Figure 3(a)) and the utility degradation (maximum per agent) ∗x

(Figure3(b)). A line is drawn for every value ofn. For everyn, these magnitudes (U and∗x)

do not increase indefinitely withf; they reach a maximum for a certain value off (depending onn)

and then decrease. In this way, we could assume that the degradation of utility due to the alternative mechanism is bounded. However, this conclusion must be clarified by taking into account the results

(a) (b) 0 100 200 300 400 500 600 700 800

0,1 0,2 0,3 0,4 0,5 failure probability

utility degrad. (global)

10 20 30 40 50 0 50 100 150 200 250 300

0,1 0,2 0,3 0,4 0,5 failure probability utility(max.deg r.per agent) 10 20 30 40 50

Figure 3. Utility degradation.

0 5 10 15 20 25 30 35 40 45 50

0,1 0,2 0,3 0,4 0,5

failure probability % u tility d e g ra d .(g lo b a l) 10 20 30 40 50

This clarification is done in Figure 4. Here the abscise axis represents f and the ordinate axis

represents the percentage of utility degradation (global)u. A line is drawn for every value ofn.

The graph shows that, for everyn,uincreases (almost lineally) withf, so no bound can be fixed for

the relative degradation of the utility.

These results show that a serious problem can arise when a large fraction of the total population of agents fail, and the others have started the scheduled jobs in such a way that it is impossible to start

a new round of negotiations. In fact, the relative degradation is above 45% forf =0.5 andn =50.

Even with a low failure probability (f = 0.1), the relative degradation can reach 12% forn = 50.

Therefore, from the point of view of the generation of global utility, it seems sensible to enforce the fulfillment of the tasks by means of social engineering methods (penalties, metagames, etc.).

A STUDY OF INSINCERE AGENTS

Insincerity in the agent domain

An important goal in the development of negotiation mechanisms consists of designing protocols in which the effects of deception and manipulation can be constrained. Because of agents’ execution autonomy and self-interest, they can deceive or mislead each other if they believe that they will thus obtain more profits.

Rosenschein and Zoltkin were pioneers in the study of the effects of insincerity on the development

of negotiation protocols for task-oriented domains (TODs) [8]. They considered that the assumption of

full information, i.e. that the information about tasks, capabilities and costs of each agent is accessible to the others, is a very strong requirement that plays a critical role in the analysis of negotiation protocols for this domain. The reason is that this assumption allows agents to make manipulations or deceptions when they declare their information to the others. In this sense, there should exist a clear distinction between the private and the public behavior of an agent. An agent’s public behavior corresponds to the protocol being used in the negotiation, while its private decisions correspond to the strategy that it is using in the negotiation. Since the protocol is publically known, it should be possible to verify that the agent’s public behavior complies with the protocol’s rules. On the other hand, it is more difficult to monitor the agent’s private decision-making or the strategies that the agent uses in choosing its actions.

Concretely, Rosenschein and Zoltkin [8] study how the agents might choose to act in TODs when

they reveal their private information about their tasks. They explore the issue of non-manipulability negotiation mechanisms examining three dimensions.

(i) Three types of lies that agents can tell in TODs: hiding tasks—an agent may hide tasks; phantom tasks—an agent may declare tasks that do not exits and that cannot be created under the request of other agents; and decoy tasks—an agent may declare tasks that do not exist but that can be generated under petition.

(ii) Three types of deals over which the agents can negotiate: pure deals, mixed deals and all-or-nothing deals.

They found that in most general TODs (general, subadditive) and by choosing the appropriate deal type, many kinds of lies may be profitable for agents.

In the same direction Teschet al. [20] described the applicability of arbitration protocols in bilateral

bargaining situations, and analyse of their robustness against manipulations. Other approaches based

upon enforcing mechanisms have been proposed by Shehory and Kraus [2] and Sandholmet al. [3].

In our case, we will suppose that agents make manipulations or lie when they reveal their private information. Due to the fact that agents are self-interested, lies arise when agents can obtain a greater share by lying. We have implemented a simulation tool—similar to the one described in the preceding section—to compute the payments obtained by an agent when it lies and when it does not lie. With this tool we have performed a set of experiments to study if lying is profitable.

Experimental design description

Let us assume that transfer costs c_ij satisfy the triangular inequality, i.e. that for all i, j ∈ N,

cij+cjk ≥cik(note that every graph can be transformed into a graph satisfying the triangular inequality

in polynomial time by computing optimal paths and adding new arcs). It can be proved [9] that in such

a case there is no agenta_i and optimal transferstij andtik such thattij ×tik < 0. As a result, the

set of agents is partitioned into five subsets:full producers,fringe producers,full consumers,fringe

consumersandnon-participants. Non-participants neither transfer nor receive tasks. Producer agents

are those that only transfer tasks and consumer agents are those that only receive tasks. A full producer will transfer its entire task and a full consumer will receive a task that fills its entire capacity surplus. On the other hand, a fringe producer will not transfer its entire task and a fringe consumer will not

consume its entire surplus. In the following, full producers will simply be calledproducersand full

consumers will simply be calledconsumers.

In this section we describe one of the possible manipulations that system agents can carry out when they reveal their private information. Concretely, we will suppose that any system agent lies to other

agents about its real value ofc_i. The agent may think that a cost increment or decrement could increase

its profits.

With the goal of testing the effects of this manipulation on the ‘stability’ of the coalition formation model, we have designed another simulation tool. The simulation tool calculates optimum task allocation and a stable payoff division among the system agents. The simulation was tested for a number of agents varying from 10 to 50, with a lying probability ranging from 0.1 to 0.5. The input data for the problem are similar to those studied in the preceding section.

In relation to the manipulation of c_i, the tool allows us to increase or decrease the cost of any

agent. The manipulated cost will be generated in a random way as a percentage (10%, 20%,. . .) over

(larger or smaller) the original cost. This quantity will be limited by the parameters MAXCOST and MINCOST. In addition, the simulation tool computes the lying bonus obtained by the insincere agent

when it deceived during the revelation ofci. The lying bonus,PM, is stated in the following way: let

cv be the original cost of the insincere agent, and lettvandxv be respectively the quantity of task

units and the utility assigned to the agent after the execution of the task allocation and payoff division

algorithm with the original cost,c_v. In addition, letc_mbe the manipulated or lying cost, andt_mand

xm be respectively the quantity of task units and the utility assigned to the agent after the execution

obtained by the agent revealing its original cost, true balance,B_v, and the total profit obtained by the

agent revealing the manipulated cost, lying balance,B_m.

1. If the insincere agent is a consumer or fringe consumer agent,B_v andB_mare calculated in the

following way:

Bv= −tvcv+tvcv+xv=xv (6)

Bm= −tmcv+tmcm+xm=tm(cm−cv)+xm (7) In both cases, the benefit obtained by the agent is stated as the addition of its benefit (money)

less the work that it has to carry out.B_vis calculated as the addition of the utility obtained by the

agent,x_v, plus the quantity of money paid by the producer agent to the lying agent for its work,

tvcv(given that the lying agent is a consumer, it carries out the task transferred by a producer

agent), less the work that the lying agent has to carry out,t_vc_v. ForB_mthe agent’s profit will be

the addition of its utility,x_m, plus the quantity of money paid by the producer agent,t_mc_m; and

its work will bet_mc_v. So,

PM =Bm−Bv=tm(cm−cv)+xm−xv (8)

2. If the lying agent is a producer or fringe producer agent,B_vandB_mare stated as follows:

Bv= −tvcv+tvcv+xv=xv (9)

Bm=tmcv−tmcm+xm=tm(cv−cm)+xm (10) In both cases, the total benefit obtained by the agent is calculated as the addition of its profit (money) less the work that it has to carry out, or, in this case, the money that it must send to the consumer agent to execute its task, given that the lying agent is a producer agent. The reasoning

is analogous to the previous case, andPM will be

PM =Bm−Bv=tm(cv−cm)+xm−xv (11)

Report of results

We have performed several hundred simulations, and the following tables show the results obtained.

TableIsummarizes the results obtained when the lying agent manipulates, increasing itsc_i. The table

shows the effect of this kind of manipulation on the different types of system agents (consumer, fringe consumer, producer, fringe producer). The possible results for this manipulation in any type of agent may be a positive, a negative or a null lying bonus. If in the table the corresponding square is empty, it means that for that agent type and with the considered manipulation—increase in the cost—a lying bonus of that type will never be obtained. Otherwise, if the square is not empty, it means that it is possible to obtain a bonus of that type, and, in addition, inside the square the new possible status of

the lying agent after the manipulation is shown. TableIIis the analogue of table for a lying agent that

decreasesc_i.

From the results mentioned above we can conclude that a positive lying bonus may be obtained in two situations.

1. When the lying agent is a consumer or fringe consumer agent and the manipulation is an increase

Table I. The lying agent increases its cost.

Lying agent PositiveP_M NegativeP_M NullP_M

Consumer Fringe consumer Producer Consumer Fringe producer

Non-participant Fringe consumer

Fringe consumer Fringe consumer Producer Consumer Fringe producer

Non-participant Consumer

Producer Producer

Fringe producer Producer

Fringe producer

Table II. The lying agent decreases its cost.

Lying agent PositiveP_M NegativeP_M NullP_M

Consumer Consumer

Fringe consumer Consumer

Fringe consumer

Producer Fringe producer Consumer Producer Fringe consumer

Non-participant Fringe producer

Fringe producer Fringe producer Consumer Producer Fringe consumer

Non-participant Producer

2. When the lying agent is a producer or fringe producer agent and the manipulation is a decrease

inc_i.

Even in these situations a positive lying bonus is not obtained in all cases. The consumer/producer

agent or the fringe consumer/fringe producer agent may increase/decrease itsci and obtain a null or

negative lying bonus. The lying agent will only obtain a positive bonus when the consumer/producer agent or the fringe consumer/fringe producer agent is transformed after the manipulation into a fringe consumer/fringe producer.

Table III. Bonus for lying agents.

Lying agent PositiveP_M (%) NegativeP_M (%) NullP_M(%)

(a)

Consumer 5.81 62.7 31.39

Producer 2.43 78 17.07

(b)

Fringe consumer 18.18 81.81 0

Fringe producer 35 60 5

(c)

Total consumers 7.2 64.9 27.8

Total producers 13.3 73.3 13.3

TableIII(a)shows the rates obtained from the simulations. In the rows we give the situation of the

lying agent before the manipulation (producer or consumer), and the columns show the possible results

of the manipulation or lying bonus (positive, negative or null), after an increase inc_i when the lying

agent is a consumer or a decrease when it is a producer. The ratio obtained for each kind of agent and for each possible result of the lying bonus is shown in each table cell. The results show that for producer and consumer agents, the rate of positive lying bonuses is lower than the negative or null rates (for consumer agents it is 5.81% compared with 94.09%, and for producers 2.43% compared with 95.07%).

The rates obtained when the manipulation consists of an increase inc_i when the lying agent is a

fringe consumer or a decrease when it is a fringe producer are shown in TableIII(b). In this case again,

the percentage of positive lying bonuses is lower than the negative or null rates. The results are the following: for fringe consumer agents, 18.81% compared with 81.81%, and for fringe producers 35%

compared with 65%. However, if we compare the results of TableIII(a)with those of TableIII(b), in

(b) the ratio of positive lying bonuses is larger than in (a). The reason is that in case (a) the lying agents,

in order to obtain a positiveP_M, have to modify their situations after an increase/decrease in theirc_i

(changing their situations to fringe agents, producers or consumers). But, in case (b) the lying agents

do not have to modify their situations after an increase/decrease in theirci, with the goal of obtaining

a positivePM.

In TableIII(c)we show a summary of the results obtained in the two previous cases. In the row

called ‘total consumers’ we group the results of fringe consumers and consumers, and in the row ‘total producers’ we give the results of fringe producers and producers. These results show that the probability of a lying agent obtaining a greater profit or a positive lying bonus after a manipulation of

ci is very small in comparison with the probability of obtaining a null profit or even of losing profit

with respect to the situation before the manipulation (for the consumer agents 7.2% compared with 92.8% and for the producers 13.3% compared with 86.7%).

Let us assume that an agenta_i isignorant, i.e.a_iis not aware of the initial taskst_j0, costsc_j, cjiand

capacitiesk_j (j =i) (theglobal situation). We have shown that the only possible situations in which

ai could obtain a positiveP_M by lying are produced:

(i) whena_i (if sincere) is a consumer or fringe consumer,a_i reveals an increasedc_i, anda_i is a

fringe consumer in the manipulated situation;

(ii) whena_i (if sincere) is a producer or fringe producer,a_i reveals a decreasedc_i, anda_i is a fringe

producer in the manipulated situation.

However, even in these situations the empirical study shows that the probability of obtaining an additional profit is much lower than the probability of losing profit. So, with this type of manipulation,

the rational behavior for an ignoranta_i is to be honest with respect to itsc_i, and to comply with the

proposed interaction mechanism.

In the protocol presented in [9] and described in the second section of this paper, every agent is

ignorant with the exception of the initiator, i.e. the agenta_kthat performs the computations. Therefore,

that protocol is stable for non-initiator agents. However, the initiator is not ignorant; it could compute its additional profit and eventually lie. So, the basic protocol must be modified as follows in order to ensure stability even for the initiator agent:

1. the initiatora_kis chosen at random;

2. akreveals its resources and costs to the rest of the agents;

3. then all agents send their data toak, which perform the computations.

In this way, both the initiator and the rest of the agents must be honest, since they reveal their data when they do not know the global situation.

However, it could be argued that strategic behavior can arise when the agents have uncertain or conjectural knowledge about the global situation. Deeper research should be carried out in order to study the stability under these circumstances.

CONCLUSIONS

In this paper we have presented two empirical studies on the robustness of a coalition formation model in the face of the failure of one or more system agents, and on the effects of deception and manipulation on the model.

We have studied the performance of the model when agents fail at random. Experimental results show that utility degradation is large when silent agents are ignored and a simple heuristic procedure is applied to reallocate the remaining utility. For this reason, future works should include an extension of the study, for example computing the expected utility of a silent agent under a penalty schema, or in an iterated process of negotiation.

We have also studied the stability of the coalition formation model in the face of an agent’s deception in the revelation of its unitary execution cost. We have shown that the rational behavior for an ignorant agent is to be honest with respect to its cost and to comply with the proposed interaction mechanism. Future works could include an extension of the study, for example considering other types of manipulations: about the number of tasks to carry out (hidden tasks or phantom tasks), or about the maximum quantity of tasks that the agent can perform, or even simultaneous manipulations of several agents.

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