Coalition formation based on marginal contributions and the Markov process

Decision Support Systems 57 (2014) 355–363

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Decision Support Systems journal homepage: www.elsevier.com/locate/dss

Coalition formation based on marginal contributions and the Markov process Stephen Shaoyi Liao a,⁎, Jia-Dong Zhang b, Raymond Lau a, Tianying Wu a a b

Department of Information Systems, City University of Hong Kong, Hong Kong Department of Computer Science, City University of Hong Kong, Hong Kong

a r t i c l e

i n f o

Article history: Received 23 March 2012 Received in revised form 2 July 2013 Accepted 29 September 2013 Available online 18 October 2013 Keywords: Coalition formation Coalition structure generation Payoff division Shapley values Markov process

a b s t r a c t With competition intensifying in the globalized economy, an increasing number of firms are forming coalitions or alliances to improve purchasing efficiency and reduce operating costs in various industries. Forming such coalitions or alliances has become a key research challenge in two important kinds of decision support systems, namely group support systems and negotiation support systems, since the number of possible coalitions is very large in most cases. Most of the existing research on coalition formation focuses on generation of optimal structures alone. Nevertheless, self-interested agents, who are mainly concerned with their own benefits, usually determine whether to join a coalition on the basis of payoffs they can possibly get from the coalition. Accordingly, in this paper, we propose a novel method of coalition formation to enable agents to improve their own benefits based on marginal contributions and the Markov process. Our method considers both coalition structure generation and payoff division which are two primary concerns of group and negotiation support systems. By using a real-world scenario, we give an example of formation of retailer coalitions to illustrate the proposed method. Finally, it is experimentally showed that the method proposed in this paper is effective and efficient, compared with other existing methods. The coalitions generated by our algorithms can significantly increase most agents' payoffs. The managerial implication of our research is that firms can apply the proposed method to identify the most beneficial coalition network with their business partners. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Coalition formation in the cooperative game theory is an important step that can help improve social welfare and individual agents' interests in two important kinds of decision support systems, namely group support systems and negotiation support systems [4]. It has been studied by many researchers from different perspectives with different methods [2,27–29,32]. Coalition formation is always a big challenge given the strategic conflicts among the participating agents in group decision-making or negotiation processes within a wide variety of political, economic, and social settings [12,15]. Coalition formation process involves three main activities, that is, coalition structure generation, payoff calculation and payoff division among members. While payoff division and coalition structure generation are important issues that need to be studied independently [5,27,28], it is desirable to extend those studies by focusing on both simultaneously in the context of self-interested agents [30]. Most of the existing research works have considered only the coalition structure generation when dealing with the coalition formation problem [30,27,29]. A coalition C is a subset of agent set N, N = {1, …,n}. Such a coalition can improve the performance of individual agents and/or the system as ⁎ Corresponding author. E-mail addresses: [email protected] (S.S. Liao), [email protected] (J.-D. Zhang). 0167-9236/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.dss.2013.09.019

a whole, especially when tasks cannot be performed by a single agent or when a group of agents performs the tasks more efficiently [26]. A value v(C), called the payoff in this paper, indicates how beneficial coalition C would be if it was formed and v is called the characteristic function. The coalition structure, usually denoted by CS, is a partition of the agent set. Dealing with the problem of coalition formation, most of the extant research has considered the value of the coalition structure CS as V(CS) = ΣC ∈ CSv(C) and coalition structure CS∗ where V(CS∗) = argmax ΣC ∈ CSv(C) as the optimal one. Obviously, such a viewpoint is reasonable when agents in the game are cooperative. What concerns the agents is maximization of social welfare while payoff division is a non-issue. However, what concerns self-interested agents is how to enhance their own incomes. A coalition may earn a good payoff as a whole but a self-interested agent may not participate in it if the payoff division is not satisfactory to that particular agent. Instead, the agent may join another coalition that might be earning a smaller aggregate payoff but the agent can get a higher payoff. Thus we consider coalition formation from the perspective of individual agents. In this paper, we call coalitions formed by our method individual agents' benefits based coalitions (ICs) and the corresponding coalition structure individual preference based coalition structure (ICS) (refer to Definition 3.4). Against this background, we construct an integrated theory that encompasses coalition structure generation and payoff division where we suppose v is known during the model description process. This

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paper discusses coalition formation from the perspective of individual agents based on marginal contributions and the Markov process. Whether a coalition is formed depends on payoffs member agents can get from the coalition. The Shapley value [31], as a method of payoff division, is always non-empty and unique in all types of cooperative games. On the other hand, the Shapley value reflects the fairness well since individual agents get their payoffs according to their contributions to the corresponding coalition. Therefore, we take the Shapley value as the payoff division method in this paper. Agents choose coalitions from which they can get higher Shapley values. In this paper, against the exponential growth of coalition structure space O (nn) with agents' number n [28], a method of pruning the search space is given at first. The method involves forming superior coalitions and discarding strictly inferior coalitions simultaneously, based on the verified relationship between Shapley values and marginal contributions. Theoretically, this method reduces the search space drastically and the search space is pruned to O (2n − 1) in the worst case. Experimental analyses show that the search space is reduced by more than 99.9% when the number of agents is more than 7. The following endeavor is to find the optimal coalition structure (namely, ICS) from the pruned search space. It is still a complex task to solve the ICS from the perspective of individual agents even though the search space has been narrowed drastically. Fortunately, the process of individual agents' transitions among different coalition structures is a typical Markov process, where a current state transits into future states with certain distribution probabilities [16]. So we model individual agents' transitions among the coalition structures by the Markov process and search the ICS based on probability distribution of Markov states. However, the typical way of figuring out Markov states' probability distribution is random sampling, which is complex and time-consuming [17,19]. Fortunately, the Markov chain modeled in this paper has some excellent properties that allow us to obtain the probability distribution by solving the stationary distribution vector. The number of firms considering coalitions to enhance competitiveness and reduce operating costs is growing [20,13]. The illustrative example given involves a one-supplier several-retailers' two-echelon supply chain where retailers form coalitions (called retailer-coalitions) to buy products from the supplier. Numerical analyses show that coalitions increase most retailers' payoffs. Moreover, analyses of results of simulation experiments in the context of the retailer-coalition show that the search space is cut down to less than 0.1% when there are more than 7 agents in the cooperative game. Comparison of running time of different coalition formation ways shown in Table 3 indicates that the method proposed in this paper can be carried out efficiently. Moreover, from the payoff divisions shown in Fig. 5 we can see that the formed ICs can increase agents' incomes remarkably. Importantly, group and negotiation support systems can leverage our novel computational method to generate effective coalition structures and maximize individual agents' benefits during group decisionmaking or negotiation processes. Moreover, our efficient method can also help develop practical solutions given the complexity of multiagent negotiations [10,14]. For instance, in the context of groupbuying, the complexity of the problem space involved may well exceed a buyers capacity and capability of processing information related to multiple parties in a timely manner. Fortunately, our proposed method enables buyers to efficiently determine the optimal buying groups so that they are able to maximize their total surplus, as illustrated by a real-world example in later section. The main contributions of this paper can be summarized as follows: • First of all, we prune search space of coalition structures by forming superior coalitions and discarding strictly inferior coalitions simultaneously, based on the verified relationship between Shapley values and marginal contributions (Section 3).

• Further, we identify and show some important properties of the Markov process for modeling individual agents' transitions among the coalition structures. These properties enable us to efficiently and correctly solve the probability distribution of Markov states and then find the optimal coalition structure, i.e. the individual preference based coalition structure (ICS) (Section 4). • Finally, the proposed method is illustrated by a real world example and is experimentally evaluated by comparing it with three benchmark methods, which shows the efficiency and effectiveness of our method (Sections 5, 6). The rest of the paper is structured as follows. Section 2 describes some related works. Section 3 introduces the algorithm for pruning search space based on the concept of marginal contributions. Section 4 models individual agents' transition process among coalition structures by the Markov process and finds the ICs by figuring out the probability distribution of Markov states. Section 5 gives a numerical example of a retailer-coalition. Section 6 describes experimental analyses and Section 7 concludes.

2. Related work Coalition formation has received considerable attention in recent years. As mentioned in Section 1, the coalition formation process mainly includes three activities [29]: calculation of coalitions' payoffs, generation of the coalition structure, and division of the payoffs. These three activities interact. For example, the coalition that an agent wants to join depends on the portion of the payoff that the agent would get. However, in the long run it would be desirable to construct an integrated theory that encompasses all three activities [30]. Most of the existing work concentrates on coalition structure generation only when considering the problem of forming coalitions [30,28,27,29,32]. The researchers take coalitions in the coalition structure CS∗, where V(CS∗) = argmax ΣC ∈ CSv(C), as the optimal coalitions. A lot of algorithms have been developed from this viewpoint, including the improved dynamic programming algorithm (IDP) [27], heuristic algorithms [32], anytime optimal algorithms (IP-Uniform and IP-Normal) [30,29] and Markov-based algorithms [17]. The Markov based algorithms have three advantages in comparison to the others. First, the coalition structure in each period of the Markov chain is endogenously determined, which allows us to study how coalitions evolve over time [3]. Second, the conflicts among agents are solved and the agents are grouped into coalitions according to their satisfaction degrees [17]. Third, the Markov chain combines two questions of stability with explicit monitoring of coalition formation [6]. Nevertheless, payoff division is a significant activity since all that the agents want to do is to improve their own incomes, i.e. agents are selfinterested. Fortunately, a variety of methods have been developed to divide payoffs [8,9,14,2]. We compare two most well-known methods: the Shapley value [31,18] and the Banzhaf value [1,22] to emphasize the strengths of the Shapley value. First, the Shapley value is perfectly consistent and theoretically elegant but the Banzhaf value is a mere gimmick with no coherent theoretical underpinnings. Second, the Banzhaf value violates equiprobability in a very queer way and, as a result, a certain coalition is vulnerable to the defection of only one particular member [8]. Finally, the Shapley value possesses many excellent properties such as existence, uniqueness, fairness, monotonicity and practical usefulness. Several applications use Shapley values for payoff division [5,11]. In particular, [11] analyzed payoff division by using Shapley value in the context of “centralizing inventory in supply chains”. In their work, coalition structure generation is not considered with the assumption that the supply chain has been formed. However, just considering the payoff division is not sufficient in most of the scenarios since discussing payoff division before knowing what coalitions are formed is meaningless.

S.S. Liao et al. / Decision Support Systems 57 (2014) 355–363

In this paper, we discuss coalition structure generation and payoff division simultaneously. Aiming at this problem, [7] investigated a non-cooperative coalitional bargaining game. In this game, during the bargaining rounds, agents compete for the right to make an offer by investing resources. In such a context, pure stationary sub-game perfect equilibrium payoffs coincide with the core, if the core exists. The coalition formation investigation given in this paper is applicable in games where there must be no discounting or, alternatively, little discounting and a discrete offer space. Otherwise, no stationary pure strategy equilibrium exists. Research works discussed above mainly concentrate on theoretical research. Coalition formation, which is much considered in intelligentagent systems, has also been widely studied in different application backgrounds, such as in business environments [14,23,24]. In the case of a buyer who assembles a product from outsourced components, n suppliers form coalitions to bargain with the assembler sequentially, based on the Nash bargaining game [24]. The aim of the supplier coalitions is to improve suppliers' profit. Nevertheless, the sequential negotiation process increases the bargaining power of the assembler, which is not beneficial for suppliers. On the other hand, members get equal shares of the coalition's profit, which is not reasonable in many business scenarios. Moreover, agents who sell substitutable products form coalitions and then the formed coalitions compete against one another to determine price and inventory [23]. This is a specific discussion covering a special scenario. The formation of coalitions is discussed based on parameters of price and inventory. Obviously, such a specific discussion is not adaptable any longer as the considerations are not price and inventory. Some general method is needed for scenarios with similar characteristics. Given what has been discussed above in this paper, we focus on formation of coalitions from perspectives of individual agents' benefits by encompassing coalition structure generation and payoff division. In our discussion, the coalition formation process is modeled based on Shapley values and the Markov process. We can extend the usage of our method to different applications by modeling real-world scenarios. On the other hand, as different from [7], we use the Shapley value as the method of payoff division, which is adaptable in various games. This means we do not need to worry about the case where the core is empty and there exists discounting in the game.

3. Pruning search space based on marginal contributions

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Definition 3.1. [32] The marginal contribution of agent i to coalition C, written as Mt (C;v), is given by Mi(C;v) = v(C) − v(C ∖ {i}). Definition 3.2. [32] The Shapley value of agent i, written as φi(v), is given by φi ðvÞ ¼ ∑ ðjCj−1Þ!n!ðn−jCjÞ!M i ðC; vÞ, where C represents the potential C

coalitions containing agent i. From Definitions 3.1 and 3.2, a marginal contribution is the difference between two coalitions' payoffs while agent i's Shapley value depends on marginal contributions of i to all potential subcoalitions which contain i. Characterizations of Shapley values are explained very well by three properties [31]: Symmetry. The Shapley value is independent of the sequence in which agents join the coalition. Validity. The sum of all coalition members' payoff division value is n

equal to the coalition's maximal payoff, that is, ∑φi ðvÞ ¼ vðNÞ , i¼1

where n is the number of agents and φi(v) is agent i's Shapley value. Additivity. If v and v′ are two different games, then φi(v + v′) = φi(v) + φi(v′), which means agents can get the same Shapley values when playing two games simultaneously as when they play the two games sequentially. After the Shapley value that an agent can get form a certain coalition is determined, we can define the superior coalition and the ICS as follows. Definition 3.3. A coalition C is a superior coalition if ∃ i ∈ C such that ∀ C′ ∈ Ω, i ∈ C′, C′ ≠ C, φi(C) ≥ φi(C′), where Ω is the set of all potential coalitions. Otherwise, coalition C is a strictly inferior coalition, since each agent has a better coalition. Definition 3.4. A player (i.e. agent) i is satisfied with a coalition C if player i can get more payoff in C# compared to all other coalitions that include player i. The number of players satisfied with coalition structure CS is denoted by SN(CS). We say a coalition structure CS# is ICS if SN(CS#) = max(SN(CS)). Coalitions in CS# are called ICs. According to Definition 3.3, strictly inferior coalitions cannot be preferred by individual agents since better coalitions that offer higher Shapley values are available to each agent. Therefore, we only need to maintain superior coalitions from which at least one agent can get the highest possible Shapley value. However, how do we find out these superior coalitions efficiently? For this problem, in order to avoid calculating Shapley values directly (since that is time-consuming), we present the relationship between Shapley values and marginal contributions in Theorem 3.1.

As mentioned before, the search space for coalition formation grows exponentially as the number of agents increases. It is intuitive to narrow the search space as much as possible. It means we just focus on subspaces which contain coalitions preferred by individual agents. Whether a coalition is formed depends on member agents' Shapley values obtained from the coalition. So we should prune the search space from the perspective of individual agents' Shapley values. However, computing Shapley values is time consuming. We are motivated to search some way to reduce the time and effort required for calculation of Shapley values. Fortunately, we can verify the relationship between Shapley values and marginal contributions (Section 3.1) and then prune the search space by comparing marginal contributions, avoiding calculation of Shapley values (Sections 3.2 and 3.3).

Theorem 3.1. For two same size coalitions C1 = {s11, …,s1k} and C2 = {s21, …,s2k}, and C1 ∖ {s1k} = C2 ∖ {s2k}, C = C1∪{s2k} = C2∪{s1k}. ∀ i ∈ {s11, …,s1k} (and j ∈ {s21, …,s2k}), if for any coalition C1l ⊂ C1, i ∈ C1l, (and C2l ⊂ C2, j ∈ C2l), we have Mi(C1l∪{s2k}; v) ≥ Mi(C1l;v) (and Mi(C2l∪{s1k}; v) ≥ Mi(C2l;v)). Then φi(v(C)) ≥ φi(v(C1)) (and φi(v(C)) ≥ φi(v(C2))).

3.1. Relationship between Shapley values and marginal contributions

Proof. For any i ∈ C1, according to the definition of Shapley value, we have:

Note that in this paper we have assumed that the characteristic function v of coalitions is known. In practice, the way the payoff is calculated depends on the problem under investigation. Here, we present formal definitions of marginal contributions and Shapley values.

φi ðvðC 1 ÞÞ ¼

1 1!ðk−2Þ! X M ðfig; vÞ þ Mi ðfijg; vÞ þ … k i k! j∈s ; j≠i 1

m!ðk−mÞ! 1 ∑M i ðC 1m ; vÞ þ … þ M i ðC 1 ; vÞ þ k! k

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together with 3

1 1!ðk−1Þ! 4 X M ðfig; vÞ þ M ðfijg; vÞ þ Mi ðfis2k g; vÞ5 kþ1 i ðk þ 1Þ! j∈s ; j≠i i

φi ðvðC ÞÞ ¼

“1” {1, 2}

{1} 2

{1, 3} {2}

“12” {1, 2, 3}

1

þ… þ

 i ðm−1Þ!ðk þ 1−mÞ! h ∑M i ðC 1m ; vÞ þ ∑Mi C 1ðm−1Þ∪s2k ; v ðk þ 1Þ!

{3}

þ… þ

1 M ðC; vÞ: ðk þ 1Þ! i

l=1

“2” {2, 3} l=3

l=2 Fig. 1. A 3-agent cooperative game.

On changing the form of this formula, we have  φi ðvðC ÞÞ ¼

 1 1!ðk−1Þ! M i ðfig; vÞ þ M i ðis2k ; vÞ kþ1 ðk þ 1Þ!   ðm−1Þ!ðk þ 1−mÞ! m!ðk−mÞ! þ… þ  ∑M i ðC 1m ; vÞ þ ∑M i ðC 1m ∪s2k ; vÞ ðk þ 1Þ! ðk þ 1Þ! þ…

ðk−1Þ!1! 1 þ M ðC ; vÞ þ M ðC; vÞ: ðk þ 1Þ! i 1 kþ1 i

Then we can obtain the inequality, based on the condition Mi(C1j∪ l; v) ≥ Mi(C1j;v), as follows: φi ðvðC ÞÞ≥

  1 1!ðk−1Þ! ðm−1Þ!ðk þ 1−mÞ! m!ðk−mÞ! Mi ðfig; vÞ þ … þ þ þ ðk þ 1Þ ðk þ 1Þ! ðk þ 1Þ! ðk þ 1Þ!   ðk−1Þ!1! 1 1 þ M ðC ; vÞ ¼ Mi ðfig; vÞ ∑Mi ðC 1m ; vÞ þ … þ ðk þ 1Þ! kþ1 i 1 k

½

þ… þ



ðm−1Þ!ðk−mÞ! 1 ∑Mi ðC 1m ; vÞ þ … þ Mi ðC 1 ; vÞ ¼ φi ðvðC 1 ÞÞ: k! k

So that φi(v(C)) ≥ φi(v(C1)). We can also prove that φi(v(C)) ≥ φi (v(C2)), in the same way. Based on Theorem 3.1, we can form superior coalitions based on marginal contributions instead of Shapley values. Let's explain this through the following example. Suppose C1 = {1,2}, C2 = {1,3}, v({1}) = v({2}) = v({3}) = 1, v(1,2}) = v({2,3}) = v({1,3}) = 3, and v({1,2,3}) = 6. For C1 and agent 1, we have M1({1}∪{3}, v) =v({1,3}) − v({3}) = 2 N M1({1},v) = 1, and M1({1,2}∪{3}, v) = v({1,2,3}) − v({2,3}) = 3 N M1({1,2},v) = v({1,2}) − v({2}) = 2, so agent 1 meets the condition in Theorem 3.1. Similarly, we can show that all other agents meet the condition. According to Theorem 3.1, we obtain φi(v(C))≥φi(v(C1)) and φi(v(C))≥φi(v(C2)), where i={1,2,3} and C={1,2,3}. Subsequently, inferior coalitions C1 and C2 are discarded and the superior coalition C is formed, since each agent can get more payoffs from C than C1 or C2. Note that here we only compute marginal contributions using Definition 3.1rather than Shapley values using Definition 3.2. 3.2. Forming the superior coalitions According to Theorem 3.1, superior coalitions can be obtained based on marginal contributions. Concretely, the process of formation of superior coalitions is as follows. Agents are numbered 1, 2, …, n. Superior coalitions are formed and strictly inferior coalitions are discarded, level by level. Size-1 coalition set is denoted by Level 1, and Size-2 coalition set is denoted by Level 2, and so on. We start from Level 1, until no larger size superior coalitions exist. Obviously, there are n levels at the most. We denote level by l, that is, 1 ≤ l ≤ n. Coalitions in the same level are of the same size. Coalitions in each level are grouped according to member agents. If we describe coalitions Ci and Cj in Level l as Ci = {ai1 … ail} and Cj = {aj1 … ajl}, respectively, then Ci and Cj will be in the same groups if Ci − ail = Cj − ajl(i ≠ j). There are C(n − 1, l − 1) groups in Level l. Groups in Level l have (n − l + 1) coalitions at most. Here, the group is marked with a unique label “ai1 … ai(l − 1)”.

For example, a 3-agent cooperative game is shown in Fig. 1 in which coalitions are grouped in each level. There are 3 levels in this game. Each level is grouped according to the label marked in upper left corner. Let us take Level 2 as an example, where there are C(2,1) = 2 groups in all. And the group labels in Level 2 are “1” and “2”. To form a superior coalition in Level l, we should consider l coalitions of size-(l − 1) in Level (l − 1). Only groups with more than one coalition have the chance to form a new larger and superior coalition. So we just need to consider groups with more than one coalition in each level. For instance, if the group marked by “a1 … a(l − 1)” has more than one coalition, then we consider coalitions in this group pair-wise, sequentially. Suppose group “a1 … a(l − 1)” has 4 coalitions, denoted by C1, C2, C3 and C4. Then consider different possible combinations of coalitions in the sequence as “C1C2”, “C1C3”, “C1C4”, “C2C3”, “C2C4” and “C3C4”. For each combination pair “CiCj”, other (l − 1) partner coalitions are found. Consider the group marked with “a1′ … a(l − 2)′ail” sequentially. Herein, “a1′ … a(l − 2)′” is the lexicographical order of (l−2) combinations of “ai1 … ai(l − 1)”. In these groups, Coalition P is the partner coalition if P ⊂Ci∪Cj. For example, if Ci = {1234} , and Cj = {1235}, then we consider the lexicographical order of 2-combination of “123” and can get partner coalitions in groups marked by “124”, “134” and “234”. After getting these partner groups' labels, we can determine the indices of partner groups through the following formula directly (indices of the groups begin with 1): ′

l−2

1 þ C a′ −1 þ 1

′

ð j−1Þ −2 l−1 a1 …a X X

j¼2

i¼0

ðl−1Þ− j

C n−a′

ð j−1Þ

−2−i

  if a′1 …a′ð j−1Þ −1N0 :

We just need to consider the conditions of (a1′ − 2) ≥ 0 and (aj − a(j − 1) − 2) ≥ 0. For example, for the group marked “124”, we only have a3 − a2 − 2 = 0, so the index is “1 + C(((5 − 1) − (1 − 1) − 3),(4 − 1 − 3)) = 2”. Considering the index of the group marked “134” in the same way, we have “1 + C(((5 − 1) − (1 − 1) − 2),(4 − 1 − 2)) = 3” and that of “234” can be obtained by “1 + C((5 − 2),(4 − 2)) = 4”. If any considered group is empty, stop the process and consider other coalitions pairwise. Else, find coalition P. If all l coalitions are found and the combining conditions are satisfied, add coalition Ci∪ Cj to group “ai1 … ail” in Level (l + 1) and discard all l coalitions from the superior coalitions set. Procedures for forming superior coalitions above are described in Algorithm 1. The main idea of Algorithm 1 is that an individual agent a2k can join coalition C1 if and only if each individual agent in C1 can get a higher Shapley value in C1∪ a2k, as well as a2k. We can see from Algorithm 1 that to get a new coalition C in Level l, l coalitions in Level (l − 1) will be discarded from C(l − 1), that is, the number of finally generated superior

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coalitions is C(n,⌊n/2⌋) ≤ 2n − 1 at the most. The main operation in Algorithm 1 is comparison and the number of comparisons in Algorithm 1 is n2 ⋅ C(n − 1, ⌊n/2⌋)) at the most. So the computational complexity of Algorithm 1 is O(2n).

359

4. Forming the optimal coalition structure from the perspective of individual agents based on the Markov process Though the search space is pruned drastically by Algorithms 1 and 2, techniques are still needed to find the ICS in the pruned search space. The main idea of our method is forming coalitions from the perspective of individual agents. In this section, we search ICs in the pruned search space by considering individual agents' benefits.

3.3. Generating the superior coalition structures From Algorithm 1 we can obtain 2(n − 1) superior coalitions at the most. In this subsection, based on the superior coalitions obtained from Algorithm 1, we generate the corresponding superior coalition structure in which the ICS is included. First, a theorem is introduced. Theorem 3.2. The coalition structure preferred by individual agents includes at least one superior coalition. Proof. Suppose Theorem 3.2 fails, that is, individual agents' benefits based coalition structure CS# is composed of strictly inferior coalitions only. From the forming process of the superior coalitions we can see that for a discarded strictly inferior coalition C ∈ CS# there exists a superior coalition C′, C ⊂ C′ and ∀ i ∈ C, φi(v(C′)) ≥ φi(v(C)). That is, ∃C     ′ ′ ′ S# , C ′ ∈CS# , SN CS# ≥SN CS# . According to the concept of ICS, ICS = max(SN(CS)), CS# should not be the coalition structure we have solved. That is, the ICS includes at least one superior coalition. In other words, the ICS is included in the superior coalition structures generated from superior coalitions obtained from Algorithm 1. The generation algorithm of the superior coalition structures is shown in Algorithm 2. The main operation in Algorithm 2 is intersection and l ⋅ (n−l+1) ⋅ m intersections are needed at the most, where m is the number of superior coalitions obtained from Algorithm 1. We have shown that the ICS includes at least one superior coalition. In other words, the number of superior coalition structures generated by Algorithm 2 is m at the most. (n − 1) We know that m≤2 , so the computational complexity of Algorithm n 2 is O(2 ).

We know that the Shapley value of a coalition is unique. Agents cannot change their Shapley values unless they join other coalitions. So what agents can do to improve their payoffs is transiting from one coalition to another. Fig. 2 shows an example of individual agents' transition processes among coalition structures {{1, 2, 3}, {4, 5}}, {{1, 3, 4}, {2, 5}} and {{1, 3, 5}, {2, 4}}, which are outputs of Algorithm 2. Direction of the arc represents the transition direction of agents on the arc. For example, the arc from {{1, 3, 5}, {2, 4}} to {{1, 2, 3}, {4, 5}} represents that agents 4 and 5 in {{1, 3, 5}, {2, 4}} have motivation to transit to {{1, 2, 3}, {4, 5}} since they can get higher Shapley values from the latter coalition. Obviously, whether an agent has transition motivation depends on what it obtains from the current coalition. The Markov process is a stochastic process in which a current state transits into future states with varying distribution probabilities. Different agents in a coalition structure have different transfer wishes. For example, agents 1 and 2 are in the same coalition structure CS0. Agent 1 wants to

{{1, 2, 3}, {4, 5}} (5) {{1, 3, 4}, {2, 5}}

(4, 5) (1, 3, 4) (1, 3, 4)

(1) {{1, 3, 5}, {2, 4}}

Fig. 2. Transfer wishes of agents between coalition structures.

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transfer into coalition structure CS1 as it can get more in CS1. But agent 2 wants to transit into coalition structure CS2 since it can get more in CS2. The current coalition structure may transit into other coalition structures with different probabilities. Taking the coalition structures as the Markov states, agents' transition between coalition structures is a typical Markov process. We model agents' transitions between coalition structures by the Markov process and find the ICS by solving the Markov states' probability distribution.

Definition 4.3. A Markov state wi is periodic if the greatest common divisor, d(i), of set {d|d ≥ 1, P(d) ii N 0} ≠ ϕ and d(i) N 1. Otherwise wi is aperiodic. ∞

ðlÞ

Definition 4.4. A state wi is positive recurrent if ηi ¼ ∑lf ii b∞. l¼1

Definition 4.5. A Markov chain is irreducible if ∀ wi, wj ∈ Ω, i ≠ j, we have fij N 0 and fji N 0. 4.2. Forming the ICS

4.1. Modeling agents' transfer by Markov process Obviously, as pointed out earlier, individual agents' transitions among coalition structures is a Markov process. A coalition structure is considered as a Markov state. Each agent has the right to transit from one coalition structure to any other to get a higher Shapley value. The probability of transition from coalition structure wl to wk (coalition structure and state is used interchangeably in the following discussions) can be described as P(Xn = wk|Xn − 1 = wl, Xn − 2 = wn − 2, …, X0 = w0) = P(Xn = wk|Xn − 1 = wl). For simplicity, we denote P(Xn = wk|Xn − 1 = wl) by Plk. We obtain the Plk between wk and wl from individual agents' preferences of benefits given, by Definition 4.1. Definition 4.1. Individual preference βi(wl|wk) is a quantitative description of an individual agent's transition trend. It can be calculated as [17]: βi(wl|wk) = 1 for φi(wk) ≥ φi(wl), and βi(wl|wk) = 0 otherwise. Integrating all agents' βi(wl|wk)(i = 1, …, n), we can obtain the n

transition probability Plk from state wl to state wk: P lk ¼ ∑βi ðwl jwk Þ= i¼1

 n  ∑ ∑βi ðwl jwk Þ , where Ω is the search space composed of coalition

w j ∈Ω

i¼1

structures generated by Algorithm 2. It is obvious that ∑ P lj ¼ 1. Herein, wj

Plk represents the probability of transition from state wl to wk in one step. We denote this step's probability by P(1) lk instead of Plk. According to the stochastic property of the Markov process, wl can also transit to wk in two steps or any m steps (m N 2). We denote the two-step probability (2) as P(2) lk , where Plk = (Xn = wk|Xn − 2 = wl, Xn − 1). Based on probability theories of the Markov process [16,19], the m-step probability P(m) lk can be obtained as: P(m) lk = (Xm = wk|Xk ≠ wl, k = 1, 2, …, m − 1, X0 = wl). (i) Let flk = ∑∞ i = 1 Plk denote the probability of the process starting from state wl ever transiting to state wk. To catch the stochastic property of Markov states quantitatively, we give the concept of absorption coefficient. Definition 4.2. The absorption coefficient of state wk, AC(wk), is used to measure wk quantitatively. It can be calculated [17]: AC ðwk Þ ¼ ∑ f jk . w j ∈Ω

The greater the AC(wk) is, the larger will be the number of agents satisfied with state wk. When the number of transition steps is large enough, we can say that the state with the largest AC is the state with which most agents feel satisfied. And we choose such a state as the ICS. We denote the greatest absorption coefficient by max-AC for simplicity. According to Definition 4.2, to solve the state with the max-AC, we need to figure out the probability distributions of Markov states. It is well known that to solve the probability distributions of Markov states is an NP-hard problem. The existing typical way of solving Markov states' probability distribution is random sampling [17], which is time consuming and is not feasible in situations where timing is necessary. Fortunately, the Markov process modeled in this paper has some important properties that help obtain the states' probability distribution by solving linear equations. Here, some relevant concepts are introduced first.

First of all, a state in the Markov process can be in one of the cases shown in Fig. 3. According to the model described in Section 4.1, we can see that the case shown in Fig. 3(a) is the situation where there is only one coalition structure and we know that this coalition structure is the one in level n comprising only one coalition N. As for the case shown in Fig. 3(b), it is the situation where all agents get the minimal values. Obviously, the coalition structure in such a case cannot be the max-AC coalition structure, so we can delete the case from the search space. As for the case shown in Fig. 3(c), all agents in w3 obtain the highest Shapley values. Consistent with Algorithm 2, w1 is the only coalition structure comprising only one coalition N, which means the inflow is ϕ. In other words, the case shown in Fig. 3(a) and that shown in Fig. 3(c) are the same as in our model. The search space on which we focus comprises the case shown in Fig. 3(d). Summing up the above, the search space obtained from Algorithm 2 may be one of the two cases: (1) Search space composed of only one coalition structure of case w1 (or w3); or (2) Search space composed of coalition structures of case w4. As for Case (1), the only effective coalition structure is the one we solve. For Case (2), we should figure out the probability distribution of the Markov process composed of states shown in Fig. 3(d). It is well known that a typical way of figuring out the probability distribution of Markov states is random sampling, which is time consuming. Fortunately, the states in our search space have some fine properties (Theorem 4.1). Theorem 4.1. The Markov chain generated by the states in the search space is aperiodic, positive recurrent and irreducible. Proof. (1) According to Definition 4.1, all states in the search space are aperiodic since ∀ wi ∈ Ω, P(1) ii N 0, that is, d(i) = 1.(2) The states in our search space are all positive recurrent since for ∀ wi ∈ Ω, ηi = (2) (1) (1) q (1) (1) (q − 1) P(1) + ii + 2(Pii ) + … b qPii (1 − (Pii ) )/(1 − Pii ) b ΓPii (1 + Pii (1) P(1) )b 3qP b ∞, where q is the number of coalition structures in the ii ii search space.(3) We denote the inflow states set of wi by IN(wi) and outflow states set by OUT(wi). The IN(wi) ≠ ϕ and OUT(wi) ≠ ϕ since the search spaces are composed of coalition structures shown in Fig. 3(d). For ∀ wi, wj ∈ Ω, i ≠ j, we test IN(wj). If wi ∈ IN(wj), then we have path “wi → wj”. Else, we test IN(IN(wj)), until wi is included in a certain inflow states set and then we have path “wi → wl → wj”, l ≠ i, l ≠ j. We know that the testing process stops after a specified number of steps since the number of states in the search space is finite. Conversely, we can also find the path from wj to wi by testing the

1

1

P44

P22 w1

w2 outflow

(a)

(b)

w3 inflow

(c)

w4 inflow

outflow

(d)

Fig. 3. Different cases of states in the Markov process modeled in Section 4.1. Inflow is the coalition structure set that transits into the state, and outflow is the coalition structure set that the state transits to.

S.S. Liao et al. / Decision Support Systems 57 (2014) 355–363

OUT(wj) in the same way. In other words, ∀ wi, wj ∈ Ω, i ≠ j, we have fij N 0 and fji N 0. That is, the states in the search space are irreducible. Based on Theorem 4.1, we can get the probability distribution of the Markov states by solving a system of linear equations because a finite, aperiodic and irreducible Markov chain has a unique stationary distribution vector Π, which can be calculated by ΠP = Π, where P is the matrix of probabilities of one-step transitions between states. The forming process of the ICS can be shown by Algorithm 3.

361

Given the production cost of the product, order processing cost and other factors, the supplier cannot afford to reduce price below pmax. For simplicity, we suppose that the time when the supplier needs to ship a certain quantity of the product can be obtained by ddr ¼ QDA , which means the supplier can handle DA units of product in one unit time. Tracing the characteristic function shown in [13], a retailercoalition's payoff can be calculated by the following formula, C

based on the price discount and the deadline: vðC Þ ¼ ∑ i∈C    þλ ; where d wip ðdci −dcÞ  di þ wis 1−dd þd is the price discount ci dd r

i

i

Coalitions in the structure obtained from Algorithm 3 are the ICs we solve in this paper. We can see that the complexity of Algorithm 3 is O(q2), where q is the number of states in search space Ω. Therefore, computational cost of our method is far less than O(Mq2) of the method in [17], where M ≥ (q − 2)2q − 2. On the other hand, the method in [17] uses random sampling to solve the probability distribution of Markov states so as to form the optimal coalition structure. Unfortunately, it does not ensure that it finds the global optimal coalition structure. In a word, the proposed method theoretically outperforms the method presented by [17] in both efficiency and effectiveness. 5. An illustrative example Sellers and buyers in supply chains usually negotiate price, quantity and schedule [21,25]. In the context of a one-supplier several-retailers supply chain, the supplier produces and ships items to retailers who fulfill the demand of their end-customers. In this illustrative example, we assume that retailers are concerned about price, quantity and schedule. A retailer-coalition is, therefore, measured by these three factors. In our example, from the retailers' point of view, we study a single supplier multiple cooperative retailers model, in which retailers form coalitions to buy the product from the supplier. Suppliers often encourage retailers to buy more by offering price discounts. Given the inventory cost, which is a large part of the total cost, retailers tend not to change their order strategies since the present order strategies are best for them. If retailers form coalitions to buy together, they can enjoy price discounts without changing the present ordering strategies. On the other hand, we consider the schedules in the form of deadlines. Suppose there is a retailer set R = {r1,r2, …,rn} and these n retailers order the products from the same supplier. We suppose that the supplier can provide any quantity of the product ordered by the retailers. Each retailer ri is expressed as a five-tuple (qi, di, ddi, wip, wis), where qi is the quantity of product ordered by retailer i, di is the time needed to ship the product from the supplier to retailer i, ddi is the deadline set by retailer i, wip is agent i's preference of price discount and wis is agent i's preference of delivery time and they should satisfy wip + wis = 1. A retailer-coalition is denoted by C. The payoff of C is determined by quantity, price discount and delivery time, which are relevant to the quantity of the product ordered by coalition C. We can obtain the coalition's quantity by Q C ¼ Σri ∈C qi . The price discount principle, given by the supplier, is complex and is relevant to factors such as retailers' current order quantity, fixed order processing cost, production cost, retailers' annual order quantities and so on. For simplicity, we make a quantity-based price discount function as in [17]: dc equates 1 for 0 ≤ QC b Qs, 1−QQ−Q for Qs ≤ QC b Qmax, and Pmax for Qmax ≤ QC. C

s

max

obtained by retailer i when it orders the product alone, and (ddr + di) is the basic time needed to ship the product and λ can be seen as an indicator of the time the coalition needs to distribute the product to retailers in the coalition. Suppose all retailers face order quantity q uniformly distributed over [0, QD] and deadline dd uniformly distributed over [0,dl]. Also, we make such assumptions as Qs =100, Qmax =1000, pmax =0.9, [0,QD]=[0,120], [0,dl] = [0,25], λ = 0.2|C|, DA = 80. Eight retailers are randomly generated (Table 1). By using Algorithms 1 and 2, we obtain three superior coalition structures for these eight retailers: CS1 = {{2,3,4,5},{1},{6},{7},{8}}, CS2 = {{1,5},{2,3,4},{6},{7},{8}} and CS3 = {{2,3,4},{5,8},{1},{6},{7}}. Probabilities of transition between these three coalition structures can be calculated; the transition probability matrix is P =≤, [1/2, 1/4, 1/4; 7/23, 8/23, 7/23; 7/20, 1/4, 2/5]. Calculating the stationary distribution Π through the equation ΠP = Π, we obtain the stationary distribution Π = [Π(CS1) = 0.3965, Π(CS2) = 0.2840, Π(CS3) = 0.3195]. CS1, which has the maximal distribution probability 0.3965, is the solved max-AC coalition structure. That is, the corresponding ICs are {2, 3, 4, 5}, {1}, {6}, {7} and {8}. The retailers' payoffs in each superior coalition structure are shown in Table 2. Remarkably, Retailer 1 gets the highest Shapley value in CS2 when it forms coalition with Retailer 5. Retailers 2, 3, 4 and 5 can get the highest payoffs in CS1 by forming coalition {2, 3, 4, 5}. In this coalition, Retailer 5 obtains 1016.9468% more than when it works alone and 915.1982% more than when it forms coalition {1, 5} with Retailer 1. Retailer 8, by cooperating with Retailer 5, can get the highest payoff. Nevertheless, from the view point of Retailer 5, what it gets from coalition {5, 8} is 848.2091% less than that from {2, 3, 4, 5}. In other words, Retailer 5 participating in coalition {2, 3, 4, 5} can get the highest payoff, and Retailers 2, 3 and 4 also get the highest payoffs. Coalitions {1, 5} and {5, 8} are not in the set of ICs since Retailer 5 is not satisfied with these two coalitions. For Retailers 6 and 7, forming coalitions may damage their own benefits or those of their partners. These causes make them not to form any coalition; they get the same payoff in each coalition structure. To sum up, six retailers are satisfied with CS1. Three retailers are satisfied with CS2 and three retailers are satisfied with CS3. In other words, the formed coalitions are {2, 3, 4, 5}, {1}, {6}, {7} and {8}. 6. Simulation results In this section, we simulate the process of forming of the max-AC coalition structure with different numbers of agents in the context of the retailer-coalition described in Section 5. All simulations described in this section were carried out in the environment of: Intel(R)

Table 1 The eight retailers generated randomly (ID denotes the retailer's identity). ID

qi

di

ddi

wip

wis

ID

qi

di

ddi

wip

wis

1 2 3 4

16 84 84 23

3 14 4 4

17 24 13 24

0.35 0.65 0.91 0.43

0.65 0.35 0.09 0.57

5 6 7 8

94 1 18 19

4 4 1 10

13 8 14 17

0.77 0.98 0.03 0.30

0.23 0.017 0.97 0.70

362

S.S. Liao et al. / Decision Support Systems 57 (2014) 355–363

Table 2 The retailers' payoff situations in each superior coalition structures.

CS1

ID

Shapley values

2 3 4 5 6 7

929.86% 2384.64% 55.80% 1016.95% Alone Alone

CS2

CS3

ID

Shapley values

1 6 7 6 7 8

2.59% Alone Alone Alone Alone 8.67%

Core(TM) i5-2410M CPU 2.30 GHz, 4.00 GB of RAM, Windows 7 operating system, and all algorithms were coded in JAVA 1.6. According to Section 5, each retailer is expressed by a five-tuple (q, d, dd, wp, ws). All five parameters are generated randomly in our simulations. Herein, q is uniformly distributed over [0, 120]. dd is uniformly distributed over [0, 25] and d is a random integer number which satisfies d b dd. Both wp and ws are randomly generated over [0, 1] and wp + ws = 1. On the other hand, we set Qs = 100, Qmax = 1000, Pmax = 0.9, λ = 0.2|C|, DA = 80. In the context of retailer coalitions, we give an exact numerical example in Section 5. In this section, we mainly test the effectiveness of the superior coalition structure generation algorithm (Algorithms 1 and 2) and the efficiency of the whole coalition formation process. Results of simulation of the superior coalition structure generation algorithm are shown in Table 3. Each group is tested 100 times and then the results are averaged. For example, we test 100 times the case where there are 3 retailers. Then average the results and obtain the average number 1.26, of the CS in the pruned search space. We can see that the search space is pruned effectively. When the number of retailers is larger than 7, pruned coalition structures are up to 99.9%, that is, less than 0.1% of coalition structures remain in the search space. Furthermore, we compare performances of different methods, given different numbers of agents (from number 15 to number 25). Notably, the time required in our method (Fig. 4) is the overall time consumed, that is, the total time consumed for applying Algorithms 1, 2 and 3. Besides, the running time of all other methods is the time spent for finding the optimal coalition structure. Among these, IDP is the Improved Dynamic Programming (IDP) algorithm. IP-Uniform and IPNormal are two anytime optimal algorithms. We should clarify that the running time of our method in each case is obtained by testing 100 times and then averaging it (Table 3). We can see from Fig. 4 that even though computational time consumed by our method is more than that by IP-Normal and IPUniform in the beginning, when the number of agents is more than 22, our method is more efficient than any other method. Specifically, the time consumed by our method is 32.7604% of that of IP-Uniform when the number of agents is 25. On the other hand, the curve of our method is not as steep as that of the other methods. To sum up, the method given in this paper is effective and efficient. And finally, we take Table 1 as a specific example and analyze payoff of Retailer 5 in different coalition forms. The payoffs of Retailer 5 are shown in Fig. 5. It is quite remarkable that Fig. 5 shows the case of coalitions that can be formed with Retailer 5. We do not show cases where coalitions cannot be formed. For example, both Retailer 5 and Retailer 6 get lower payoffs in coalition {5, 6} than when they work alone; Retailer 7 gets more than when it

Fig. 4. Comparison of running time of different coalition formation methods.

works alone in coalition {5, 7}, but Retailer 5 gets less than when it works alone. We can see from Fig. 5 that Retailer 5 can get the most payoff in coalition {2, 3, 4, 5} among all coalitions it can possibly join. Similarly, Retailers 2, 3 and 4 have the same case. That is, the proposed coalition formation method can benefit most of the agents in the game. 7. Conclusions and future work This paper proposes a novel method of coalition formation driven by individual agents' benefits. In particular, both the payoff division among agents and the coalition structures are taken into account by the proposed method. A search space pruning approach is first introduced, according to the relationships between marginal contributions and Shapley values. Then, individual agents' transitions among the pruned coalition structures are modeled by the Markov process. Based on both theoretical analysis and empirical evaluation, we show that the proposed method of coalition formation is effective and efficient. Our simulation results indicate that the proposed method is more efficient than other existing coalition formation methods. The practical implication of our work is that the proposed method can be applied to enhance both coalition structures and agents' payoffs which are two primary concerns of group support systems and negotiation support systems.

Table 3 Pruning results of Algorithms 1 and 2 (number shorted for no.). Retailers no.

CS total no.

Superior CS no.

Pruned

3 5 7 ⋮ 23

5 67 1682 ⋮ 4.59 × 1019

1.26 2.27 4.07 ⋮ 3202.84

74.8% 96.6% 99.7% ⋮ N99.9%

Fig. 5. The payoffs (Shapley values) of Retailer 5 in different coalitions.

S.S. Liao et al. / Decision Support Systems 57 (2014) 355–363

One of the novelties of the proposed coalition formation method is that both payoff division and the coalition structure are derived from the perspective of individual agents. Our proposed method has potential applications in many real-world scenarios. The in-depth examination of applying the proposed method to a specific business application will be part of our future research. For example, we will consider the organizational and human issues that may affect coalition formation. Moreover, development of an anytime version of the proposed algorithms will be taken into account in the future. Acknowledgments The authors would like to thank the editor and anonymous reviewers for helpful comments and suggestions. This study was supported by grants from the National Natural Science Foundation of China (71002064 & 71090402), the Fundamental Research Funds for the Central Universities of China (2682013CX075) and the Soft Science Research Plan of Sichuan Province (2013ZR0043). References [1] J.M. Alonso-Meijide, F. Carreras, M.G. Fiestras-Janeiro, G. Owen, A comparative axiomatic characterization of the Banzhaf–Owen coalitional value, Decision Support Systems 43 (2007) 701–712. [2] J.M. Alonso-Meijide, J. Freixas, A new power index based on minimal winning coalitions without any surplus, Decision Support Systems 49 (2010) 70–76. [3] T. Arnold, U. Schwalbe, Dynamic coalition formation and the core, Journal of Economic Behavior & Organization 49 (2002) 363–380. [4] D. Arnott, G. Pervan, A critical analysis of decision support systems research, Journal of Information Technology 20 (2005) 67–87. [5] J.J. Bartholdi, E. Kemahlioglu-Ziya, Using Shapley value to allocate savings in a supply chain, in: J. Geunes, P.M. Pardalos (Eds.), Supply Chain Optimization, Springer, New York, 2005, pp. 169–208. [6] G. Chalkiadakis, C. Boutilier, Sequentially optimal repeated coalition formation under uncertainty, Autonomous Agents and Multi-AgentSystems 24 (2012) 441–484. [7] R. Evans, Coalitional bargaining with competition to make offers, Games and Economic Behavior 19 (1997) 211–220. [8] D.S. Felsenthal, M. Machover, Voting power measurement: a story of misreinvention, Social Choice and Welfare 25 (2005) 485–506. [9] L.A. Guardiola, A. Meca, J. Timmer, Cooperation and profit allocation in distribution chains, Decision Support Systems 44 (2007) 17–27. [10] X. Guo, J. Lim, Decision support for online group negotiation: design, implementation, and efficacy, Decision Support Systems 54 (2012) 362–371. [11] E. Kemahlioglu-Ziya, J.J. Bartholdi, Centralizing inventory in supply chains by using Shapley value to allocate the profits, Manufacturing & Service Operations Management 13 (2011) 146–162. [12] D.M. Kilgour, K.W. Hipel, L. Fang, X. Peng, Coalition analysis in group decision support, Group Decision and Negotiation 10 (2001) 159–175. [13] S. Krichen, A. Laabidi, F.B. Abdelaziz, Single supplier multiple cooperative retailers inventory model with quantity discount and permissible delay in payments, Computers & Industrial Engineering 60 (2011) 164–172. [14] C. Li, K. Sycara, A. Scheller-Wolf, Combinatorial coalition formation for multi-item group-buying with heterogeneous customers, Decision Support Systems 49 (2010) 1–13. [15] L.H. Lim, I. Benbasat, A theoretical perspective of negotiation support systems, Journal of Management Information Systems 9 (1992) 27–44. [16] C. Liu, Stochastic Process, Huazhong University of Science and Technology Press, Wuhan, China, 2006. [17] W.Y. Liu, K. Yue, T.Y. Wu, M.J. Wei, An approach for multi-objective categorization based on the game theory and Markov process, Applied Soft Computing 11 (2011) 4087–4096.

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Dr Stephen Liao is a Professor of Information Systems at the City University of Hong Kong. He earned a Ph.D. from University of Aix-Marseille III and Institute of France Telecom. His articles have been published extensively in various academic journals like MIS Quarterly, Decision Support Systems, IEEE transactions, Communications of the ACM, Information Science, Computer Software and . His current research interests include use of data mining techniques in mobile commerce applications and intelligent business systems, especially intelligent transportation systems. Mr Jia-Dong Zhang received the MSc. degree in computer science and engineering from Yunnan University, China, in 2009. He is currently working toward the Ph.D. degree with the Department of Computer Science, City University of Hong Kong. His research work has been published in journals such as IEEE Transactions on Intelligent Transportation Systems, Transportation Research Part C: Emerging Technologies, and Expert Systems with Applications. His research interests include data mining, machine learning, and business intelligence. Raymond Y.K. Lau is an Assistant Professor in the Department of Information Systems at City University of Hong Kong. He holds a Ph.D. in Information Technology from Queensland University of Technology, a M.S. in Business Systems Analysis and Design from City University, and a M.S. in Information Studies from Charles Sturt University. He has worked at the academia and the ICT industry for over twenty years. He is the author of over one hundred refereed international journals and conference papers. His research work has been published in renowned journals such as ACM Transactions on Information Systems, IEEE Transactions on Knowledge and Data Engineering, IEEE Internet Computing, Journal of Management Information Systems, Decision Support Systems, etc. His research interests include Information Retrieval, Text Mining, and Agent-Mediated e-Commerce. He is the associate editor of the International Journal of Systems and Service-Oriented Engineering. He is a senior member of the IEEE and the ACM respectively. Miss Tianying Wu is a research assistant of Information Systems at the City University of Hong Kong. She received his Master degree in computer science and engineering from the Yunnan University in 20011. Her current research interests are data mining, and game theory.