COLLABORATIVE ORGANIZATIONAL LEARNING ALONG A DIRECTION

COLLABORATIVE ORGANIZATIONAL LEARNING ALONG A DIRECTION PI-SHENG DENG Department of Computer Information Systems, California State University Stanislaus, Turlock, CA 95382, United States. E-mail: pdeng@csustan.edu and YANHONG WU Department of Mathematics, California State University Stanislaus, Turlock, CA 95382, United States. E-mail: ywu1@csustan.edu 1

SYNOPTIC ABSTRACT We introduce the effectiveness of agents as vectors relative to the desired direction for improvement, and define a coalition as a collaboration of agents which can improve individual values based on a utility function. We develop algorithms for finding sequentially formed winning coalitions without and with repeated hiring where an agent joins the coalition if it can improve the values for all existing agents. An algorithm for finding the optimal sequentially formed efficient core is also proposed. Key Words and Phrases: coalition, effectiveness, efficient core, value, utility. 2

1. Introduction Traditional research on organizational learning assumes individual mastery and acquisition of the knowledge needed for accomplishing single-stage tasks Cohen and Feigenbaum, 1982). However, organizational learning is a collective learning process Carley and Prietula, 1994). Many studies indicate that knowledge needed for problem solving is often shared by members of communities of practice Badaracco, 1991). In his model of decision making in organization, March 1981) develops the concepts of a loose and shifting coalition, instead of a single decision maker, that selects or accomplishes organizational goals. In a coalition or a community of practice or even a strategic alliance, each member possesses partial but complementary knowledge, so that only the team working together as a whole has the full body of knowledge Badaracco, 1991). Most of organizational tasks are multi-stage tasks which require knowledge agents work collaboratively as a team, and complete the task in a stage-by-stage manner. However, most of the aforementioned studies of organizational learning are conducted in the context of single-stage tasks, and assume knowledge agents having the complete knowledge for solving the task, i.e. assume there is no knowledge gap issue. In addition, we might want to continuously improve learning in an organization and so no specific task quantity is available. Our study here intends to complement the traditional research. We represent each agent by a vector of skills, and an organizational task is represented by a vector of required skills for completing the task, which involves both the magnitude and direction. We consider in this study a coalition of a number of agents which may interact with each other and continuously improve organizational learning along the desired direction the organization wants. Deng and Tsacle 2003, 2006) considered a market-based computational model for collaborative organizational learning. In their model, an organizational task was accomplished through cooperation of a group of experts participating in a coalition and working on the task in a sequential manner. Membership of the coalition was subject to change, through the market mechanism, according to the contribution each member made toward the task accomplishment, and this membership modification over time could be regarded as the change of organizational processes. In the meanwhile, the composition of project teams adapts to the changing needs of the organization. Therefore, we can theorize that an organization 3

learns through the modification of membership of coalitions or communities of practice in achieving organizational goals over time. However, in their model, there was no mechanism for selecting participating agents by considering the degree of closeness of their skill vectors to the task vector. In this work, we establish a dynamic model which selects the agents sequentially and forces organizational learning to proceed along a desired direction with specified task quantities. This direction-guided adaptive process enables the organization to learn about the importance of each skill for task accomplishment. This process also provides knowledge agents the opportunity to earn credit for their contribution to task accomplishment. Our model differs from traditional models in the sense that the organization can continuously improve along the direction and each knowledge agent can be hired multiple times with accumulated capital, which is defined in this paper as its contribution to the coalition along the desired direction. The rest of the paper is organized as follows. In Section 2, we give the definitions of the utility function and coalition, including winning coalitions. In Section 3, we propose a procedure which selects a winning coalition sequentially based on the values an agent can contribute to the coalition. In Section 4, we consider the case when an agent can be hired multiple times, so the existing agent can accumulate its capitals. The infinite time horizon, or the case when the knowledge gap is large is considered in Section 5 where the concept of efficient core is introduced. Several examples are used for illustration in Section 6. The paper ends with Concluding Remarks. 2. Coalition Modeling We define an organizational task to be accomplished by knowledge agents in the organization as T = T 1,..., T n ), where T i is the competency level of skill i needed for accomplishing the task T. A knowledge agent is defined similarly as a vector of skill competency. At the beginning, when we have not chosen any agent yet, the given task T also represents the knowledge gap the organization 4

is facing. By denoting T = T1 2 +... + Tn 2 = T 0 we can define the desired direction as α = α 1,..., α n ) = T1,..., T ) n. T 0 T 0 By using transformation of coordinates, we can assume, without loss of generality, that the desired knowledge gap is defined as T = T 0, 0,..., 0), and α = 1, 0,..., 0). For agent k, we represent the contribution along the desired direction as x k 0, and y k = y 1,k,..., y n 1,k ) is the deviation from the desired direction, for k = 1, 2,..., K. We call z k = x k, y 1,k,..., y n 1,k ), the effectiveness of agent k in making contribution to organizational task accomplishment, for k = 1, 2,..., K. In the following we shall give some notations which are borrowed from cooperative game theory Barron, 2008) but with totally different meanings. Definition 1. A function uz) = ux, y) 0 for z = x, y), x 0, and y R n 1 is called a utility function if i) uz) x and uz) = x if and only if y = 0; ii) ucx, cy) = cux, y); iii) ux, y 1 ) ux, y 2 ) if y 1 y 2 ; iv) uz 1 + z 2 ) uz 1 ) + uz 2 ). The first condition states that if the contribution does not lie on the desired direction there should be a penalty. The second condition means that the utility should increase proportionally as the skills of an agent. The third condition assumes certain kind of symmetry and it means the agent with larger deviation should have smaller utility. The last condition is called super-additivity property, which states that the utility of the cooperation of several agents is at least as high as the sum of individual utilities. The fourth condition is optional and in many situations is too restrictive to be necessary. The following are several typical examples of utility functions used commonly in practice. 5

1) uz) = x ) 1 γ y x for some 0 < γ 1; 2) uz) = x 1 γ y ) z for some 0 < γ 1; 3) uz) = x 1 γ y 2 z 2 ), for some 0 < γ 1. The first two examples do not satisfy the super-additive property in general, while the third example satisfies the super-additivity. Next, we introduce the modeling of coalition. Our model consists of two stages. When facing an organizational task which requires multiple different kinds of skills with a certain level of competency for each) in an organization of K members, each member i can form K 1 i=1 CK 1, i) coalitions with the other members, where Cn, k) denotes the combinations of k out of n. Some of these coalitions will be more capable of performing the task than the other ones. We also assume that a single member may not be capable of completing the entire task alone. Different members have different skill sets. The purpose of forming coalitions is for complementing each other s weakness in tackling the task collaboratively. As a consequence, an agent will form or join a coalition with index set C, if the agent can improve the gains or values) for himself or other agents in the coalition. In this section and next section, we shall assume repeated hiring is not allowed. For simplicity of notation, we shall denote the total effectiveness of a coalition with index set C {1, 2,..., K} as and its utility will be denoted as uz C ). z C = i C z i, Definition 2. i) For a cooperation among agents indexed in C, the value of agent i C is defined as v C i) = x i k C x k uz C ). ii) A cooperation C is called a collaborative coalition if for each i C v C i) uz i ), and x i is called the capital earned by agent i in the coalition for task accomplishment. 6

In other words, an agent in a collaborative coalition should get higher value in the coalition than the utility the agent can obtain by working alone. Theoretically, the solution space consists of 2 K possible coalitions. There are two broad types of coalitions: non-sequentially and sequentially formed, just like the coalitions defined by Banzhaf power index Banzhaf, 1965) and Shapley and Shubik power index Shapley and Shubik, 1954) in the voting systems. There are several approaches available to find the optimal non-sequential coalitions. Dang, Frankovic, and Budinska 2003) considered the optimal creation of agent coalitions for manufacturing and control, and Dang and Jennings 2006) studied the coalition structure generation in task-based settings. Most studies are conducted based on computational algorithms such as the dynamic programming Rahwan, Ramchurn, and Jennings, 2009), simulated annealing and genetic algorithm Aarts and Korst, 2000; Keinanen, 2009). Our work can be treated as the generalization along the second direction and we shall concentrate on the search for a coalition which can be formed sequentially. Definition 3. A coalition is called sequentially formed with agents of index [C] in the sequential order {[1],..., [j],..., [ C ]} where C is the number of agents if for each 1 j C, for all i = 1,..., j. x [i] j i=1 x [i] uz [1] +... + z [j] ) x [i] j+1 k=1 x uz [1] +... + z [j+1] ), [k] That means the addition of agent [j + 1] will increase the values for each of the existing agents[1],..., [j] for j = 1, 2,..., C. Finally, we introduce several different kinds of coalitions. Definition 4. i) A coalition of index C is called a winning coalition if x k T 0, k C and k C,k j x k < T 0, for every j C. ii) A coalition of index C is called efficient if uz C ) = k C x k. 7

iii) A sequentially formed coalition [C] = {[1], [2],..., [ C ]} is called a winning coalition if x [1] +...x [ C ] T 0 and x [1] +... + x [ C 1] < T 0 That means a winning coalition can accomplish the task when its total capital exceeds T 0, and any smaller coalition will no longer be a winning coalition. An efficient coalition enables knowledge learning to proceed in the desired direction, or in other words, the entire coalition s capitals equals the utility. There are some fundamental differences between non-sequential coalitions and sequentially formed coalitions. First, a non-sequential coalition C does not consider the order of entrance of agents and thus only the values of agents in the final coalition is compared to the individual utility. However a sequentially formed coalition [C] not only considers the order of entrance, each added agent must also improve the values for all existing agents. Second, a non-sequential winning coalition may not be a sequentially formed winning coalition no matter how the agents are ordered as they may not satisfy the requirement for a sequentially formed coalition. Third, the ideal situation is that we can find an efficient winning coalition which is optimal in certain sense. In next section, we will propose several procedures to find a sequential winning coalitions. 3. Sequentially Formed Winning Coalitions Deng and Tsacle 2003, 2006) proposed a stochastic sequential selection procedure, where the selection of each agent is based on the strength of each agent, defined as z, but no specific direction of knowledge gap is given. Here we propose a stochastic sequential selection procedure which can be used to find all sequentially-formed winning coalitions. From all the sequentially-formed winning coalitions, we can find the optimal sequential coalition. A stochastic sequential selection procedure: 8

Step 1. Starting from all agents with index denoted by K 1 = {1, 2,..., K} we randomly select one agent based on the utility weight distribution: { } uz 1 ) k K 1 uz k ),..., uz K ). k K 1 uz k ) Denote the selected agent as index [1] and its utility as uz [1] ). Step 2. Given z [1], we define K 2 as the index of all the remaining agents which can improve the value of agent [1], i.e. K 2 = { k [1] : } x [1] uz [1] + z k ) uz [1] ) x [1] + x k Then randomly select an agent with index [2] from the weighted utility distribution based on the values: { } x k uz [1] + z k ), for k K 2. x [1] + x k Step j+1). Generally, given {z [1],..., z [j] }, define 1) K j+1 = { k [i], for i = 1,..., j : x [i] x [1] +... + x [j] + x k uz [1] +...z [j] + z k ) x [i] x [1] +... + x [j] uz [1] +...z [j] ), for all i = 1, 2,...j }, 2) which includes index of all remaining agents which improve the values of those agents in the coalition. An agent with index [j + 1] is randomly selected based on the values for k K j+1. { } x k uz [1] +...z [j] + z k ) x [1] +... + x [j] + x k The procedure is continued until the τ th -agent which satisfies τ = inf{j : x [1] +... + x [j] T 0 }. Thus, [C] = {[1],..., [τ]} will be formed sequentially as a winning coalition. 9

To search for an optimal sequentially formed winning coalition with maximum utility, we can use the following modified procedure: A sequential procedure for selecting coalitions with maximum utility: Step 1. Starting from K 1 = {1, 2,..., K} we select the agent [1] with the maximum utility such that uz [1] ) = max k K 1 uz k ). Step 2. From all agents with index in K 2 defined in 1), we select agent [2] satisfying uz [1] + z [2] ) = max k K 2 uz [1] + z k ). Step j+1). From K j+1 defined in 2), select the agent [j + 1] such that uz [1] + z [2] +... + z [j+1] ) = max k K j+1 uz [1] + z [2] +... + z [j] + z k ). The procedure is continued until τ = inf{j 1 : x [1] +... + x [j] T 0 }. The utility under the sequential coalition will be uz [1] +... + z [τ] ), and the values for each agent will be x [i] τj=1 x [j] uz [1] +... + z [τ] ). for i = 1, 2,..., τ. 4. Sequentially Formed Coalitions with Repeated Hiring In many situations it may be difficult to find a winning coalition if no agents are allowed to be hired repeatedly. If the capital or value of agents can be accumulated, we can hire agents repeatedly. So a winning sequential coalition can always be formed by using any 10

of the agents repeatedly alone. In fact, from Property ii) in the definition for the utility function, we see that the utility is linearly increasing by using a single agent repeatedly. This gives great flexibility to construct a sequential winning coalition. By modifying the procedure given in Section 3, we can use the following procedure: A sequential procedure for forming winning coalitions with repeated hiring: Step 1. From K 1 = {1, 2,..., K}, we randomly select an agent based on their weighted utility distribution: { } uz 1 ) i K 1 uz i ),..., uz K ), i K 1 uz i ) and denote the selected agent as agent [1]. Step 2. Given z [1], randomly select an agent, say [2] from K 2 of all the collection of agents which can improve the value of agent [1]: { K 2 = k : x } [1]1 + I [k=[1]] ) uz [1] + z k ) uz [1] ), 3) x [1] + x k where I [k=[1]] = 1 if k = [1], and 0 if k [1]. Note that when k = [1], the agent [1] is rehired and all the values will be accumulated for this agent with utility u2z [1] ). Since u2z [1] ) uz [1] ), it means that agent [1] will automatically belong to K 2. The second agent [2] can be randomly selected from K 2 based on their weighted utility values: { } 1 + I[k=[1]] )x k uz [1] + z k ), for k K 2. x [1] + x k Step j+1). Generally, given [1],..., [j], denote the weight I i j) as the number of times the agent i being hired up to step j for i {[1],..., [j]}. Define K j+1 = { I [i] j) + I [k=[i]] )x [i] k : for all i = 1, 2,..., j, uz [1] +... + z [j] + z k ) x [1] +... + x [j] + x k } I [i] j)x [i] uz [1] +... + z [j] ). 4) x [1] +... + x [j] That means K j+1 includes all the agents which can improve the accumulated values for the existing agents in the coalition. The agent [j + 1] can be selected based on the weighted 11

utility: { } I k j) + 1)x k uz [1] +... + z [j] + z k ) x [1] +... + x [j] + x k for k K j+1. So for those agents with larger accumulated capitals, the weight will be higher. The procedure continues until τ such that τ = inf{j 1 : x [1] + x [2] +... + x [j] T 0 }. Similarly, we can propose the following optimal sequentially formed winning coalition to maximize the utility. Optimal sequentially formed coalition with repeated hiring: Step 1. From K 1 = {1, 2,..., K}, we select the agent [1] which satisfies uz [1] ) = max k K 1 uz k ). Step 2. Given z [1], we select agent [2] from K 2 defined in 3) such that uz [1] + z [2] ) = max k K 2 uz [1] + z k ). Step j+1). Given z [1],..., z [j], we select agent [j + 1] from K j+1 defined in 4) such that uz [1] +... + z [j] + z [j+1] ) = max k K j+1 uz [1] +... + z [j] + z k ). The procedure stops at the first time τ = inf{j 1 : x [1] +... + x [j] T 0 }. The weighted utility value for agent k can be written as I k τ)x k I 1 τ)x 1 +... + I K τ)x K uz [1] +... + z [ τ] ), where if agent k is never being hired, I k τ) = 0. 5. Sequentially Formed Efficient Core 12

Here we consider the case when T 0, the infinite time horizon case. Our main objective is to select agents to form a coalition along the direction of the organizational task, rather than to find a winning coalition only. Naturally, a repeated use of an efficient coalition will achieve this goal. Note that there may not exist an efficient coalition with or without repeated hiring. For example, if all the agents have deviations y 1,..., y K along one-side of the desired direction, then an efficient coalition will not exist. In the non-sequential case with repeated hiring, let weight I C = {I k : k C} denote the hiring frequencies for agents in cooperation C, and the total effectiveness of C can be written as z C = I k z j. k C The following definition generalizes Definitions 2 and 4. Definition 5. i) A cooperation C with weight I C is called a collaborative coalition, if the value of agent i v C i) = I i x i k C I k x k uz C ) ui i z i ), for all i C. ii) If further k C I k y k = 0, we call the coalition C with weight I C an efficient coalition. iii) If C is no longer efficient with any smaller weight than I C, we call C an efficient core. Note that if in an efficient core C, weight I i = 1 for all i C, it means there is no repeated hiring. The efficient coalitions have the following two properties: i) If C with weight I C is an efficient coalition, then C with weight mi C for any positive integer m is also an efficient coalition; ii) If C 1 with weight I C1 and C 2 with weight I C2 are two efficient coalitions, then C 1 C 2 with the corresponding weight I C1 C 2 is also an efficient coalition. We note that the efficient core defined here is quite different from game theory in Shapley 1953) and Xu and Veinott 2013). Now let us consider how to find a sequentially formed efficient core and an optimal efficient core to maximize the utility. To show the difference from the definition of the efficient core without repeated hiring, we give the following definition of efficient core separately for 13

sequentially formed efficient core with repeated hiring. Definition 6. A sequentially formed coalition [C] = {[1],..., [M]} is called an efficient core if for each j = 1,..., M 1 I [i] j)x [i] x [1] +... + x [j] uz [1] +...z [j] ) I [i]j) + I [[j+1]=[i]] )x [i] x [1] +... + x [j+1] uz [1] +... + z [j+1] ), for all i = 1, 2,..., j, where I k j) is the weight of agent k for k {[1],..., [j]}, y [1] +... + y [j] 0 for j = 1, 2,..., M 1, and y [1] +... + y [M] = 0. Note that there are significant differences between non-sequential efficient cores and sequentially formed efficient cores. A sequentially formed efficient core has order and it may contain smaller efficient core in non-sequential case. On the other hand, a non-sequential efficient core may not be the same as a sequentially formed efficient core. In other words, a sequentially formed efficient core may be the union of several non-sequential efficient core and thus is only a special efficient coalition in non-sequential case. To find sequentially formed efficient core, we can use the similar algorithm developed in the previous section by replacing the stopping rule as to stop at the first time y [1] +... + y [j] = 0. The following is the algorithm to find the sequentially formed efficient core which maximizes utility at each step. A sequential procedure for finding efficient core maximizing the utility: Step 1. From K 1 = {1,..., K}, we select agent [1] satisfying uz [1] ) = max k K 1 uz k ). 14

Step 2. From K 2 defined in 3), we select agent [2] satisfying uz [1] + z [2] ) = max k K 2 uz [1] + z k ). Step j+1). From K j+1 defined in 4), we select agent [j + 1] such that uz [1] +... + z [j] + z [j+1] ) = max k K j+1 uz [1] +... + z [j] + z k ). The procedure continuous until ˆτ = inf{j 1 : y [1] +... + y [j] = 0} Under this efficient core, agent k has the value for k = 1,..., K. I k ˆτ)x k Kk=1 I k ˆτ)x k Note that one can also use this optimal efficient core repeatedly when T 0 is large until the total accumulated capital reaches T 0. In this way, we can guarantee the organization is improving along the desired direction. 6. Illustrations In this section, we give an example of typical utility functions used. Example 1 shows the difference of conditions to form a non-sequential coalition and a sequentially formed coalition under this utility function. Examples 2 and 3 show the difference between the sequentially formed coalitions without and with repeated hirings. Example 1. Let uz) = x 1 γ y ), z for some 0 < γ 1 and z = x 2 + y 2 1 +... + y 2 n 1. Then we can show that i) C K is a coalition iff for all i C y C z C y i z i. 15

ii) C K is a sequentially formed coalition iff for each j = 1, 2,..., C, y [1] +... + y [j] z [1] +... + z [j] y [1] +... + y [j+1] z [1] +... + z [j+1], where C denotes the number of agents in the coalition. The geometrical interpretation of the condition for a sequentially formed coalition is that the angle between the effectiveness of the cooperation and the desired direction gets smaller and smaller. Remark: The same conditions hold for the utility functions ) uz) = x 1 γ y 2, and uz) = x 1 γ y ), z 2 x for 0 < γ 1. Example 2. No repeated hiring) Let z 1 = 7, 1), z 2 = 6, 2), z 3 = 2, 3), z 4 = 4, 4), and z 5 = 2, 1). The utility function is defined as uz) = x 1 ) y, x2 + y 2 and the knowledge gap is T 0 = 14.5 along direction α = 1, 0). 1) Obviously, C 1 = {1, 2, 5}, C 2 = {1, 2, 3, 4}, and C 3 = {2, 3, 4, 5} are all non-sequential efficient coalitions as z 1 + z 2 + z 5 = 15, 0), z 1 + z 2 + z 3 + z 4 = 19, 0), and z 2 + z 3 + z 4 + z 5 = 14, 0). C 1 and C 2 are winning coalitions and C 3 is not a winning coalition. 2) To find the optimal sequentially formed winning coalition, we follow the procedure proposed in Section 3. Step 1. K 1 = {1, 2, 3, 4, 5}. It is easy to check that z 1 has the largest utility at the first step with utility u7, 1)) = 7 1 1 ). 50 16

So [1] = 1 Step 2. To find the second agent, it is easy to check that K 2 = {2} since z 1 + z 2 = 13, 1) and 7 13 13) 1 1 ) > 7 1 1 ). 170 50 So {1, 2} has utility uz 1 + z 2 ) = 131 1/ 170). Therefore [2] = 2 Step 3. To find the third agent [3], we can check that K 3 = {5}. First, if agent 3 is added, z 1 + z 2 + z 3 = 15, 4), the value for agent 1 will be 7 15 15) 1 4 ) < 7 241 13 13) 1 1 ). 170 Second, if agent 4 is added, z 1 + z 2 + z 4 = 17, 3), the value for agent 1 becomes 7 17 17) 1 3 ) < 7 297 13 13) 1 1 ). 170 Third, if agent 5 is added, z 1 + z 2 + z 5 = 15, 0), which is efficient. So it must improve the values for agents [1] and [2]. Since x 1 + x 2 + x 5 = 15 > 14.5, so the sequentially formed winning coalition is [C] = {[1] = 1, [2] = 2, [3] = 5}, which is also efficient. The values for agents 1, 2, and 5 are 7/15, 6/15, and 2/15 respectively. Example 3. With repeated hiring) Let z 1 = 7, 1), z 2 = 6, 2), z 3 = 2, 3), z 4 = 4, 4), and z 5 = 2, 1). The utility function is defined as uz) = x 1 ) y, x2 + y 2 and the knowledge gap is T 0 = 14.5 along direction α = 1, 0). Step 1. To find the optimal sequentially formed winning coalition, the first step will be the same as in Example 2. So [1] = 1. 17

Step 2. We can check that K 2 = {1, 2}. Agents 2 is already checked before. For agent 1, we see that 2z 1 = 14, 2). Obviously, 14 14 14) 1 2 ) > 7 200 1 1 ). 50 Since its utility is 141 2/ 200) = 12.02 which is larger than the utility of z 1 + z 2 given by 131 1/ 170) = 12.0029, so the second agent is still agent 1. Step 3. To select the third agent, we can check that K 3 = {1, 2}. First, for the same reason as in Step 2, we can add agent 1 as 3z 1 = 21, 3) with utility 211 3/ 450) 18.03. Second, we can add agent 2 as 2z 1 + z 2 = 20, 0), which is efficient and has utility 20, and 14 20) > 14 1 1 ). 20 50 Third, we can not add agent 3 since 2z 1 + z 3 = 16, 5) and 14 16 16) 1 5 ) < 14 1 1 ). 251 50 For the similar reason, we can not add agent 4. Finally, we can not add agent 5 as 2z 1 +z 5 = 16, 3) with utility 161 3/ 265) and 14 16 16) 1 3 ) < 14 265 1 1 ). 50 By comparing the utilities, we see agent 2 is the optimal choice. By noting the utility of 2z 1 + z 2 is 20 which is larger than 14.5, so the optimal sequentially formed winning coalition is [C] = {[1] = 1, [2] = 1, [3] = 2}, which is also efficient. The values for agents 1 and 2 are 14/20 and 6/20 respectively. Remark. Since [C] = {[1] = 1, [2] = 1, [3] = 2} is also efficient, from the algorithm given in Section 5, we see that it is the optimal efficient core for the infinite time horizon case. 7. Concluding Remarks In this work, we used vectors to represent the effectiveness of agents and developed algorithms for finding sequentially formed winning coalition based on the improvement of 18

utility in the coalition. In addition, we also proposed the concept of efficient core to improve the organizational learning along the desired direction for task accomplishment. It has to be emphasized that there are some fundamental differences between non-sequential and sequentially formed coalitions. The following several points will be the topics for further consideration. First, we need to build the conditions when a sequentially formed coalition exists with no repeated hiring. Second, conditions for the existence of efficient core in both non-sequential and sequential cases need to be established in order to implement the algorithms. Third, relationships between non-sequential and sequentially formed winning coalitions and efficient cores should be studied. Also, for a large organization, one may consider the random mechanisms for the effectiveness of agents modeled as random vectors for the performance of employees. Acknowledgements The authors are grateful to a referee and the editor s comments which substantially improve the presentation of this article. Reference Aarts, E. and Korst, J. 2000). Simulated annealing and Boltzmann machines. Chichester, Great Britain: John Wiley and Sons. Badaracco, J. L. 1991). The knowledge link: how firms compete through strategic alliance. Boston, MA: Harvard Business School Press. Banzhaf, J. F. 1965). Weighted voting doesn t work: A mathematical analysis. Rutgers Law Review, 192), 317-343. Barron, E. N. 2008). Game theory, an introduction. Hoboken, NJ:John-Wiley and Sons. Carley, K. M. and Prietula, M. J. 1994). ACT theory: extending the model of bounded rationality. In: K.M. Carley and M.J. Prietulaeds), Computational organizational theory. Hillsdale, NJ: Lawrence Erlbaum Associates. 19

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