Optimization of Multiple Related Negotiation through Multi-Negotiation Network

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1 Optimization of Multiple Related Negotiation through Multi-Negotiation Network Fenghui Ren 1,, Minjie Zhang 1, Chunyan Miao 2, and Zhiqi Shen 3 1 School of Computer Science and Software Engineering University of Wollongong, Australia {fr510,minjie}@uow.edu.au 2 School of Computer Engineering 3 School of Electrical and Electronic Engineering Nanyang Technological University, Singapore {ASCYMiao,zqshen}@ntu.edu.sg Abstract. In this paper, a Multi-Negotiation Network (MNN) and a Multi- Negotiation Influence Diagram (MNID) are proposed to optimally handle Multiple Related Negotiations (MRN) in a multi-agent system. Most popular, state-of-theart approaches perform MRN sequentially. However, a sequential procedure may not optimally execute MRN in terms of maximizing the global outcome, and may even lead to unnecessary losses in some situations. The motivation of this research is to use a MNN to handle MRN concurrently so as to maximize the expected utility of MRN. Firstly, both the joint success rate and the joint utility by considering all related negotiations are dynamically calculated based on a MNN. Secondly, by employing a MNID, an agent s possible decision on each related negotiation is reflected by the value of expected utility. Lastly, through comparing expected utilities between all possible policies to conduct MRN, an optimal policy is generated to optimize the global outcome of MRN. The experimental results indicate that the proposed approach can improve the global outcome of MRN in a successful end scenario, and avoid unnecessary losses in an unsuccessful end scenario. 1 Introduction Negotiation is a significant methodology for autonomous agents to reach mutual beneficial agreements in multi-agent systems [1]. People can study and classify negotiations through different aspects. For instance, by considering the number of negotiated issues, negotiations can be classified as single issue negotiations and multiple issue negotiations [1]. In general, single issue negotiations focus on bargains involving only one attribute, while multiple issue negotiations containmorethanonenegotiatedissues. By considering the number of negotiators, negotiations can also be classified as bilateral negotiations and multilateral negotiations [2]. A bilateral negotiation is performed between only two negotiators, while a multilateral negotiation considers opportunity and competition from other negotiators. By considering negotiation environments, negotiations can be classified as static negotiations and dynamic negotiations [3]. In a static negotiation, the negotiation environment is relatively fixed, and can be fully observed and The primary author is a Ph.D candidate. Y. Bi and M.-A. Williams (Eds.): KSEM 2010, LNAI 6291, pp , c Springer-Verlag Berlin Heidelberg 2010

2 Optimization of Multiple Related Negotiation through MNN 175 expected by negotiators. While in a dynamic negotiation, the negotiation environment is changed out of a negotiator s control and can only be partially observed and expected by negotiators. Also, some hybrid partitions combine the above criteria together. For example, Sycara et al. [2] introduced a three-level nest view on negotiations. Through our studies, it is found that all of these classifications focus only on a negotiation with a sole goal, but do not consider Multiple Related Negotiations (MRN) with different goals. However, in complex negotiation environments, one agent may perform more than one negotiation with different opponents for different goals at same time. Sometimes, these goals are not independent, and these MRN are somehow related. For instance, in a scheduling problem, the negotiation result on the deadline of an early occurring event will definitely impact the negotiation on the starting time of a later occurring event. The negotiation result between a mortgagor and a banker on a mortgage will determine the mortgagor s reservation in the negotiation with a real estate agent on a property s price. In order to cover related negotiations, we introduce a three-level hierarchic model in Figure 1 to represent different types of negotiations. Fig. 1. Three levels hierarchic view of agent negotiation The first level is named as the Bilateral Level, which covers static, bilateral, and multiple issues negotiations. In this level, agents focus on sophisticated negotiation with only one opponent. In order to achieve an optimal outcome, agents may adopt different procedures [1], strategies [4], equilibriums [5] and preferences [6] in negotiation based on individual interest. The second level is named as the Multilateral Level, which covers dynamic, multilateral, and multiple issue negotiations. In this level, agents negotiate with more than one opponent synchronously. Researches on this level may pay attention to negotiation pattern selections [7], multilateral negotiation protocols [8] and negotiations in open and dynamic environments [3]. Both the first and second levels focus on one negotiation with a solo goal. The third level is named as the Multi-Negotiation Level, which normally pays attention to MRN. Negotiations in this level have different goals, and the goals are somehow related. Each single negotiation in the third level can be represented by a Bilateral Level negotiation or a Multilateral Level negotiation. Significant achievements have been reached in agent negotiation in the first two levels with state-of-the-art techniques and approaches. However, very few of them

3 176 F. Ren et al. consider the third level and can handle MRN properly [9,10]. Most existing approaches just separate these MRN and treat each of them individually. The major disadvantage of dealing with those negotiations separately is that if related negotiations are not considered together, the negotiation outcomes may not be optimized or even be damaged for some cases. In order to solve the problem mentioned in MRN, we introduce a Multi- Negotiation Network (MNN) and a Multi-Negotiation Influence Diagram (MNID) in this paper. Firstly, MRN are represented by MNN. Secondly, MNN is extended to MNID. The joint success rate and the joint utility by considering all related negotiations in the MNID is calculated. Thirdly, an optimal policy to conduct MRN is calculated for the MNID, by considering both the joint success rate and the joint utility, to optimize the negotiation outcome of MNID. 2 A Multi-Negotiation Network 2.1 Construction of a MNN In this subsection, we introduce notations and the procedure to construct a MNN based on Bayesian Networks [11]. Let a four-tuple < G, R, P, Φ > indicate a MNN, where G =(V, E) is a directed acyclic graph, set R indicates the restriction function between two related negotiations, P is a set of success rates, and Φ is a set of the utility functions. V is a finite, nonempty set of vertices and each vertex indicates a negotiation in a MNN, and E is a set of ordered pairs of distinct elements of V. Each element of E is called a restriction edge with a direction to represent a dependency relationship of two related negotiations, (i.e. a link with arrow between two vertices in a MNN). For example, if a pair (V i,v j ) E, we say that there is an edge from V i to V j.inother words, V j depends on V i,andv i is one of V j s parent. We use function r ij : Φ j Φ j (r ij R, Φ j Φ) to indicate the restriction between V i and V j. If there is no restriction between V i and V j, r ij is nil. (That means two negotiations V i and V j are independent and there is no impact on each other.) p i = P (V i pa(v i )) (p i P, p i [0, 1]) indicates the success rate of V i,wherepa(v i ) are the parents of V i. Φ i (Φ i Φ) indicates the utility function of Negotiation V i. Figure 2 is an example of a MNN. In this example, the set of vertices in the MNN is V = {X, Y, Z, W}, the set of edges is E = {(X, Y ), (X, Z), (Y,Z), (Y,W), (Z, W)}, the set of restriction functions is R = {r xy,r xz,r yz,r yw,r zw }, the set of probabilities is P = {P (X),P(Y X),P(Z X, Y ), P (W Y,Z)}, and the set of utility functions is Φ = {Φ x,φ y,φ z,φ w }. From a MNN, an agent can view its related negotiations based on dependency relationships of these negotiations. The basic procedure to construct a MNN includes the following three steps. Step 1: to represent each negotiation by a unique vertex V i (V i V) and to assign a utility function Φ i (Φ i Φ)forV i. Of course, the agent can modify the utility function anytime; Step 2: to generate all restriction edges, which belong to E, between each two related negotiations; and to define restriction functions R in the form of r ij : Φ j Φ j (r ij R).

4 Optimization of Multiple Related Negotiation through MNN 177 The restriction function indicates how an ongoing or accomplished negotiation impacts another ongoing negotiation. An accomplished negotiation will not be impacted by other negotiations anymore. For instance, if a buyer synchronously performs two negotiations between a banker and a real estate agent under the condition that the buyer s reservation on a property s price depends on the mortgage, then there is a restriction edge from the mortgage negotiation to the property negotiation. If the negotiation between the buyer and the banker is completed first, its impact on the negotiation between the buyer and the real estate agent will be fixed. However, if the negotiation between the buyer and the real estate agent is completed firstly, the mortgage negotiation will not have any further impact on the property negotiation. In a MNN, if Negotiation V i is an independent negotiation, other negotiations will have no impact on its utility function Φ i. Otherwise, if Negotiation V i has K dependent negotiations, then its utility function will be modified by the consideration of all impacts from these K dependent negotiations as follows: r 1i... r Ki : Φ i Φ i (1) Let us take the MNN in Figure 2 as an example. Negotiation X has no dependent negotiations, its utility function does not need a modification; Negotiation Y is dependent on Negotiation X, its utility function is modified as Φ y = r xy (Φ y ); Negotiation Z is dependent on Negotiations X and Y, its utility function is modified as Φ z = r xz (r yz (Φ z )); and Negotiation W is dependent on Negotiation Y and Z, it utility function is modified as Φ w = r yw (r zw (Φ w )). Fig. 2. A Multi-Negotiation Network Step 3: to define the success rate p i (p i P) for Negotiation V i. The success rate p i indicates how likely an agent s latest offer will be accepted by its opponents in the remaining negotiation rounds. Suppose that in negotiation round t, an agent s latest offer is represented as a utility vector (Φ i (t),φ i (t) o ), and one of its opponents offers is a utility vector (u t o,u t a). The agent s latest offer generates a payoff of Φ i (t) for itself and Φ i (t) o for the opponents; and the opponent s offer generates a payoff of u t o for itself and u t a for the agent. Let u w denote the worst possible utility, (a conflict utility) for the agent. If the subjective probability of the agent obtaining u w is p w, then the agent will insist on its offer when its expected utility is greater than the opponent s offer, ie., [(1 p w )Φ i (t)+p w u w ] u t a (2)

5 178 F. Ren et al. According to the above inequality, the highest conflict probability that the agent may encounter with the opponent in the next negotiation round is the maximum value of p w as follows: p w = Φ i(t) u t a (3) Φ i (t) u w Let τ be the negotiation deadline and t be the current round, then the conflict probability that the agent may encounter with the opponent by considering all remaining rounds can be estimated as follows: ( Φi (t) u t ) τ t a p w = (4) Φ i (t) u w Consequently, the aggregated conflict probability that the agent may encounter before the deadline by considering all opponents in Negotiation V i is: p a = ( Si s=1 (Φ i(t) u t s ) (Φ i (t) u w ) Si ) τ t (5) where S i is the number of opponents in Negotiation V i. Therefore, for Negotiation V i, the worst success rate p i that the agent s offer (Φ i (t)) will be accepted by at least one opponent before the deadline is: p i =1 p a =1 ( Si s=1 (Φ i(t) u t s) (Φ i (t) u w ) Si ) τ t (6) In this subsection, we introduced the concept and notations for a MNN and steps to construct a MNN. It must be pointed out that a MNN can be dynamically modified according to changes of the negotiation environment. In the following subsections, we explain how to dynamically update a MNN. 2.2 Updating of a MNN Since negotiation environments can be highly complex and dynamic in real-world situations, agents may need some modifications on their MRN in order to respond to changes in negotiation environments. Such modifications may include the following cases: starting a new negotiation, terminating an ongoing negotiation, adjusting utility functions, adjusting restriction functions, changing negotiation opponents etc. When Fig. 3. Multi-Negotiation Network update

6 Optimization of Multiple Related Negotiation through MNN 179 these changes happen, agents should immediately update their MNNs. In this subsection, we introduce two major operations and suggest other operations incorporating several major changes on MNN updating. Starting a Negotiation. Assume that there are i related negotiations. If a new negotiation is commenced by an agent, a vertex V i+1 should be inserted into the MNN to indicate the new negotiation. Also, the agent should define a utility function Φ i+1 for Negotiation V i+1, and specify restriction edges between all existing negotiations and Negotiation V i+1. If there is a restriction edge from an existing Negotiation V i to the new Negotiation V i+1, restriction functions r i(i+1) should be specified, and the utility function Φ i+1 should be modified according to this restriction. If there is a restriction edge from the new Negotiation V i+1 to an existing Negotiation V i, then Negotiation V i s success rate p i and utility function Φ i should also be updated. In Figure 3(a), an example of adding new Negotiation Z in a MNN is demonstrated. Terminating a Negotiation. If an ongoing Negotiation V i is terminated by an agent, no matter whether Negotiation V i is successful or failed to reach an agreement, we use lower case letters on Negotiation V i s caption to indicate that the negotiation is in a final state. Meanwhile, the success rate for Negotiation V i is set to 1 for a successful negotiation or to 0 for a failed negotiation. For any Negotiation V j which Negotiation V i depends on, the restriction function r ji is set to nil and Negotiation V i s utility function Φ i is replaced by a constant to indicate the payoff of Negotiation V i. For any Negotiation V k which depends on Negotiation V i, the restriction function r ik is eventually fixed and its impact on Negotiation V k s utility function is also fixed. In Figure 3(b), an example of terminating an ongoing Negotiation Z in a MNN is demonstrated. Other Operations. Besides the previous two situations, agents may modify some ongoing negotiations without adding or deleting any negotiation. For example, an agent may modify its negotiation strategy for a negotiation when the number of opponents in the negotiation is changed. An agent can modify its utility function according to its new expectation on negotiation outcome. An agent may delete an existing restriction between two related negotiations or generate a new restriction between two independent negotiations. In Figure 3(c), an example of these operations in a MNN is demonstrated. 3 Decision Making in a MNN Because a MNN may contain more than one negotiations, and these negotiations are processed concurrently, whether to accept or reject an offer or even quit from an ongoing negotiation involves a decision making process during negotiations. An agent s decision on a single negotiation may impact its other negotiations or even the whole MNN. This section introduces an efficient procedure which can help agents to make advisable decisions for each negotiation in a MNN in order to optimize the outcome of the MNN by considering both joint utility and success rate.

7 180 F. Ren et al. 3.1 Multi-Negotiation Influence Diagram Suppose there are I negotiations in a MNN =< G, R, P, Φ >. The decision problem in the MNN is how to make an advisable decision policy for all related negotiation in order to optimize the outcome of the MNN. A decision policy is a set of decisions that the agent makes for all negotiations in the MNN. In general, agents could have three typical decisions on an ongoing negotiation, which are (1) to accept the best offer from opponents, (2) to reject alloffersandsend a counter-offerand (3) to quit the negotiation. If a MNN contains I negotiations, the number of total decision policies for the MNN is I 3, and each policy will generate different outcomes for the MNN. In order to model the relationship between decision policies and corresponding global outcomes, we propose Multi-Negotiation Influence Diagram. Fig. 4. A Multi-Negotiation Influence Diagram A MNID can be defined by a six-tuple < G, R, P, Φ, D,U >,whereg, R, P, Φ are same as in a MNN, and set D indicates a negotiation policy and U indicates the joint utility of the MNID by considering all related negotiations. D i = {a, r, q} (D i D) indicates three possible decisions for each Negotiation V i in the MNID, where a indicates accept, r indicates reject, andq indicates quit. A MNN can be extended to a MNID by adding a rectangular node D i for each Negotiation V i and one diamond node U for the whole MNN. Also, the edge from each Decision D i to the corresponding Negotiation V i and edges from all Decision D i and Negotiation V i to node U should be added. In Figure 4, a MNID is illustrated. Let u(d) be the joint utility of the MNID based on decisions D, andp(d) indicates the joint success rate, and EU(D) indicates the expected utility, then I u(d) = u i (D i ) w i (7) p(d) = i=1 I P (V i pa(v i ),D i ) (8) i=1 EU(D) =p(d) u(d) (9) where w i ( I i=1 w i =1) is the preference on Negotiation V i, u i (D i ) is the utility of Negotiation V i by performing Decision D i,andp(v i pa(v i ),D i ) is the success rate of Negotiation V i by considering all dependent negotiations and Decision D i. Finally, the optimal policy for a MNID is calculated as follows: π =argmax(eu(d)) (10) D

8 Optimization of Multiple Related Negotiation through MNN 181 Fig. 5. The MNN and MNID for the experiment 4 Experiment 4.1 Experiment Setup Suppose that Agent b s global goal is to get a mortgage and to purchase a property with the mortgage, so Agent b needs to perform two negotiations. The first negotiation, (mortgage negotiation), is processed between Agent b and two bankers (Opponents o m1, o m2 ) on the issues of mortgage amount and interest rate. The second negotiation, (property negotiation), is processed between Agent b and two real estate agents (Opponents o p1, o p2 or Opponents o p3, o p4 ) on the issue of the property price. It is assumed that Agent b believes that the property negotiation depends on the result of the mortgage negotiation. In Figure 5, the MNN and MNID for Agent b are displayed. Circle nodes M and P indicate the mortgage negotiation and the property negotiation, respectively. Rectangular nodes D M and D P are decisions on two negotiations, respectively. Diamond node U is the joint utility of the MNID. We adopt equal weighting between these two negotiations, so w m = w p =0.5. Because Agent b cannot afford a property price which is higher than the mortgage amount, the restriction from mortgage negotiation to property negotiation is r mp, which indicates the reserved property price is the mortgage amount. Negotiation parameters for the two negotiations are listed in Table 1 and Table 2, respectively. Because mortgage negotiation contains two issues, (i.e. mortgage amount and interest rate), we adopt package deal procedure [1] for this multi-issue negotiation and equally weight two issues. We demonstrate experimental results in two scenarios, ie. a successful scenario, (Scenario A) and an unsuccessful scenario, (Scenario B). In Scenario A, Agent b negotiates with Opponents o m1, o m2, o p1 and o p2, while in Scenario B, Agent b negotiates with Opponents o m1, o m2, o p3 and o p4. Table 1. Parameters for mortgage negotiation Agent Initial Offer Reserved Offer Deadline Agent b (500k, 5%) (300k, 7%) 10 Opponent o m1 (310k, 6.9%) (450k, 5.2%) 15 Opponent o m2 (330k, 6.5%) (500k, 5.5%) Scenario A (a Successful Scenario) In Scenario A, Agent b negotiates with Opponents o m1 and o m2 for mortgage negotiation, (the first negotiation) and with Opponents o p1 and o p2 for property negotiation,

9 182 F. Ren et al. Table 2. Parameters for property negotiation Agent Initial Offer Reserved Offer Deadline Agent b 200k depends 12 Opponent o p1 550k 330k 15 Opponent o p2 500k 350k 9 Opponent o p3 650k 450k 10 Opponent o p4 630k 470k 11 (a) mortgage (b) property Fig. 6. Negotiations using NDF approach for scenario A (the second negotiation). Firstly, we adopt the NDF approach [4] to sequentially process mortgage negotiation and property negotiation by using the linear concession strategy, and the outcomes of the two negotiations are illustrated in Figure 6. Let letter a indicate accept, letter r indicate reject and letter u indicate utility. For instance, the legend u1a (or u1r) indicates the utility of the first negotiation by accepting (or rejecting) opponents offers. In Scenario A, both negotiations successfully reached an agreement by adopting the NDF negotiation model. The utility of mortgage negotiation is 0.5, and the utility for property negotiation is Because these two negotiations are equally weighted, the overall utility is Fig. 7. Negotiations using MNN approach for scenario A The negotiation outcomes by using the proposed MNN approach are illustrated in Figure 7. Both the mortgage negotiation and property negotiation are synchronously processed. Agent b s reserved price in property negotiation is dynamically updated in each negotiation round according to the latest offer from mortgage negotiation. Let letter e indicate expected utility. For instance, Legend e1a2r indicates the expected utility of the MNID by accepting the best offer from opponents in mortgage negotiation

10 Optimization of Multiple Related Negotiation through MNN 183 and rejecting all opponents offers in property negotiation. The expected utility for the MNID is illustrated in Figure 7. It can be seen that before round-6,thecurvee1r2r leads to the highest expected utility; from round-6 to round-8, thecurvee1a2r leads to the highest expected utility; after round-8, thecurvee1a2a leads to the highest expected utility. Therefore, in order to maximize the outcome of the MNID, Agent b should reject all opponents offers in both negotiations in the first five rounds. At round-6, Agent b should accept the best offer from opponents in mortgage negotiation but keep on bargaining in property negotiation until round-8. At round-9, Agent b should accept the best offer in property negotiation. By adopting such a decision policy, the utility of mortgage negotiation is increased to 0.58 and the utility of property negotiation is increased to 0.26, so the global utility is increased to 0.42, which is 20% more than the result from the NDF approach. The result of Scenario A indicates that if the global goal of related negotiations can be achieved, the proposed approach can improve the negotiation outcome through considering both joint success rate and joint utility. With comparison to sequential negotiation processes, the proposed approach can synchronously process all related negotiations and dynamically optimize the global outcome. 4.3 Scenario B (an Unsuccessful Scenario) In Scenario B, Agent b negotiates with Opponents o m1 and o m2 in mortgage negotiation and with Opponents o p3 and o p4 in property negotiation. Also, we adopt the NDF approach (linear concession strategy) to sequentially process mortgage negotiation and property negotiation. The outcomes of the two negotiations are illustrated in Figure 8. In contrast to Scenario A, Agent b successfully completes mortgage negotiation, but fails property negotiation. In this case, the result of mortgage negotiation is meaningless or even has a negative impact by considering the global goal of related negotiations. That is because without purchasing a property, the approval of a mortgage proposal can only lead to an unnecessary cost on mortgage interest and a penalty from the bank. Therefore, if Agent b is not absolutely sure that the global goal of its related negotiations can be finally achieved, it is not efficient to process these negotiations sequentially. (a) mortgage (b) property Fig. 8. Negotiations with NDF approach for scenario B However, if we employ the proposed approach for Scenario B, the outcome will be different. In Figure 9, we illustrate the experimental results by adopting the proposed approach. In order to avoid partially reaching the global goal, Agent b can only select

11 184 F. Ren et al. policies between curves e1a2a and e1r2r (see Figure 9), which means to accept or reject both negotiations together. It can be seen that before round-8, curvee1r2r exceeds the curve e1a2a. At round-8,curvee1a2a can bring more utility to Agent b than curve e1r2r. It seems that Agent b can accept opponents offers in both negotiations at round-8. However, because Agent b cannot purchase a property whose price is higher than the mortgage amount, so the utility of property negotiation must be greater than 0. At round-8, by accepting the best offer from opponents, Agent b will lose utility by 0.17, so Agent b cannot reach agreement in both negotiations at round-8. However,if Agent b stays on curve e1r2r at round-8, the expected utility will be a negative number as well in round-9. Therefore, in order to avoid any loss, Agent b can not choose neither to accept nor to reject both negotiations at round-8,but to quit from negotiations without achieving any agreement with any opponent. So Agent b does not need to worry about the unnecessary interest and the penalty from the bank anymore. The results of Scenario B indicate that if the global goal of related negotiations can not be achieved, then the proposed approach can help agents to avoid unnecessary losses caused by the sequential procedure. Fig. 9. Negotiations with MNN approach for scenario B 5 Related Work X. Zhang and V. Lesser [10] proposed a meta-level coordination approach to solve negotiation chain problems in semi-cooperative multi-agent systems. In a complex negotiation chain scenario, agents need to negotiate concurrently in order to complete their goals on time. The order and structure that negotiations occur may impact the expected utility for both individual agents and the whole system. A pre-negotiation approach is introduced to transfer meta-level information, such as starting time, deadlines and durations of negotiation. By using this information, agents can estimate successful probability of negotiations, and model the flexibility of negotiations. However, the success probability in their work is calculated based only on a predefined time schedule, while the success rate in our work is dynamically calculated by considering an agent s possible payoffs in all related negotiations. Proper and Tadepalli [9] proposed an assignment-based decomposition approach by employing the Markov Decision Process (MDP) to solve an optimal decision making problem in assignment decomposition between multiple collaborative agents. A centralized controller which has relevant information about the states of all agents is assumed in their approach. The approach contains two levels, where the upper level focuses on

12 Optimization of Multiple Related Negotiation through MNN 185 task assignment, and the lower level focuses on task execution. The centralized controller solves the assignment problem through a searching algorithm and solves the task execution problem through coordinated reinforcement learning. The difference between our approach and their approach is that (1) we do not assume a centralized controller, and (2) we optimize the outcome of a global object through balancing its payoff and opportunity. 6 Conclusion In the real world, an agent may need to process several related negotiations in order to reach a global goal. Most of state-of-the-art approaches perform these related negotiations sequentially. However, because the result of the latter negotiation is not predictable by using a sequential procedure, agents cannot optimally execute all negotiations in correct sequential order. The motivation of our approach is to solve such a problem and handle MRN concurrently. Firstly, MNN is proposed to represent MRN by considering several key features. Secondly, MNID is proposed to handle an agent s possible decisions by employing the expected utility. Lastly, through comparing expected utilities between all possible policies, an optimal policy is generated to optimize the outcome of MRN. The experimental results indicate that the proposed approach improved an agent s global utility of MRN in a successful end scenario, and avoid unnecessary losses for the agent in an unsuccessful end scenario. References 1. Fatima, S., Wooldridge, M., Jennings, N.: An Agenda-Based Framework for Multi-Issue Negotiation. Artificial Intelligence 152(1), 1 45 (2004) 2. Li, C., Giampapa, J., Sycara, K.: Bilateral Negotiation Decisions with Uncertain Dynamic Outside Options. IEEE Trans. on Systems, Man, and Cybernetics, Part C 36(1), (2006) 3. Ren, F., Zhang, M., Sim, K.: Adaptive Conceding Strategies for Automated Trading Agents in Dynamic, Open Markets. Decision Support Systems 46(3), (2009) 4. Faratin, P., Sierra, C., Jennings, N.: Negotiation Decision Functions for Autonomous Agents. J. of Robotics and Autonomous Systems 24(3-4), (1998) 5. Rubinstein, A.: Perfect Equilibrium in a Bargaining Model. Econometrica 50(1), (1982) 6. Fatima, S., Wooldridge, M., Jennings, N.: An Analysis of Feasible Solutions for Multi-Issue Negotiation Involving Nonlinear Utility Functions. In: Proc. of 8th Int. Conf. on AAMAS 2009, pp (2009) 7. Brzostowski, J., Kowalczyk, R.: On Possibilistic Case-Based Reasoning for Selecting Partners for Multi-Attribute Agent Negotiation. In: Proc. of 4th Int. Conf. on AAMAS 2005, pp (2005) 8. Hemaissia, M., Seghrouchni, A., Labreuche, C., Mattioli, J.: A Multilateral Multi-Issue Negotiation Protocol. In: Proc. of 6th Int. Conf. on AAMAS 2007, pp (2007) 9. Proper, S., Tadepalli, P.: Solving Multiagent Assignment Markov Decision Processes. In: Proc. of 8th Int. Conf. on AAMAS 2009, pp (2009) 10. Zhang, X., Lesser, V.: Meta-Level Coordination for Solving Negotiation Chains in Semi- Cooperative Multi-Agent Systems. In: Proc. of 6th Int. Conf. on AAMAS 2007, pp (2007) 11. Jensen, F., Nielsen, T.: Bayesian Networks and Decision Graphs, 2nd edn. Springer, Heidelberg (2001)