Supplementary Material for Wang, S. et al.: "The Dynamical Relation Between Individual Needs and Group Performance: A Simulation of the Self-Organising Task Allocation Process"

1Institute of Systems Engineering, Dalian University of Technology, No.2 Linggong Road, Ganjingzi District, Dalian 116024, China 2Institute of Business Studies, Hanze University of Applied Science, Zernikeplein 23, Groningen 9747 AS, Netherlands 3Groningen Center for Social Complexity Studies, University of Groningen, Hoendiepskade 23/24, Groningen 9718 BG, Netherlands Correspondence should be addressed tow.jager@rug.nl

There are two types of individual-level entities. One is the tasks, each task is characterised by identity number (ID(t j )), required knowledge (T K(t j )), and duration (T D(t j )). The required knowledge (T K(t j )) of a task t j indicates the minimum mastery of knowledge needed to complete t j ,tk l j ∈ [0, 1], it represents the amount of one type of specialised knowledge or a specific skill. The task duration (T D(t j )) shows the time needed to execute tasks, td j represents the minimal execution time of the task t j under the maximal motivation and knowledge, the actual execution time relates to the factual motivation and knowledge of agents.
The other is the individual. Supposed that a team consists of m individuals, each individual is denoted as a i . Individuals (agents) are characterised by a set of state variables: identity number (ID(a i )), competence (AK(a i )), and motivation (AM (a i )). Competence (AK(a i )) refers to one's ability to complete a task, mainly related to the knowledge and skills possessed by the individual. Each agent a i has certain knowledge and specific skills as prerequisites to complete tasks, di erent agents have di erent knowledge types and amounts. Assume that agent a i has several types of knowledge, its competence is expressed as ak l i ∈ [0, 1], indicating how well a i masters the lth dimension knowledge, ak l i = 0 represents a i not know the lth dimension knowledge, while ak l i = 1 means a i masters it. Additional to competence, the members' motivation is also a fundamental determinant of human behaviour (Latham & Pinder ), referring to the member's willingness to perform a task. The SDT (Ryan & Deci ) identifies three fundamental psychological needs underlying this motivation: competence, autonomy, and belonging (relatedness). Therefore, the motivation of agent a i comprises of the satisfaction of these three needs, they are, the satisfaction of motivational need for competence (M nCom(a i )), need for autonomy (M nAut(a i )), and need for belonging (M nBel(a i )). Learning of agents is an important process for adapting and developing team capabilities, besides, contrary to learning, forgetting is also an important method of adapting and responding to the organisational environment. A list of parameters for individuals is provided in Table . Parameter As a social-level entity, the connection describes the influence relationship among individuals. Here, we utilise connection strength between agents to represent the amount of influence. The social interaction which is involved in the process of task allocation relies on the connection between individuals, and then it reacts to this connection. Every agent has a erent (incoming) connections from other agents and e erent (outgoing) connections to other agents.

Process overview and scheduling
In a team, when tasks need to be assigned, how do members allocate tasks through self-organisation instead of being interfered with by other managers? Here, based on the WORKMATE (Zoethout et al. ), we extend and describe the team task-performing process as the following seven steps: • The project arrivals at the team.
• Make an initial choice. Whether an agent is capable and willing to do a particular task is related to four elements, they are, individual motivation, competence, and the threshold of motivation and competence. If the agent's knowledge is over the task required knowledge, which means the agent is competent to perform the task, and if motivation exceeds the threshold (T a m), which indicates agents are willing to use this knowledge to execute the task. When both individual competence and motivation are higher than their threshold, the agent will make an I Do (I-DO, I want to do it) choice, on the contrary, it will make the You DO (Y-Do, I don't want to do it) choice. There are two following rules to determine whether the agent takes the task or not.
Rule : If AK(a i ) ≥ T ak and AM (a i ) ≥ T am Then I-Do Rule : If AK(a i ) < T ak and AM (a i ) < T am Then Y-Do Where T am is the motivation threshold, and T ak is the knowledge requirement of tasks, which equals tk l j . The algorithm pseudocode of making an initial choice is shown in Algorithm at the page .
Algorithm : Making an initial choice Input : Individual competence(AK) and motivation (AM ), the motivation threshold (T am ) and knowledge threshold (T ak equals task requirement), the parameter λ to balance the knowledge and motivation Output: The initial choice of individuals for tasks IC mn for the agent a i ← 0 to m do for the task t j ← 0 to n do if AK < T ak then if AM < T am then Calculate the V I−node and V y−node of a i with equation and // indicating the Initial choice of ai for all tj end for return IC mn • Interact between agents. Based on step ( ), every agent will have its initial choice, the initial choice of all agents for each task could be categorised into three situations (see Figure ): In the first situation (a), there is no competition for tasks within the team, it is a complementary situation, which means only one member wants to do a particular task, and no interaction will occur. In the second situation (b) more members want to do a particular task, and competition for the task arises. In this situation, the members will interact with each other until one of them is influenced to change their choice. In the third situation (c) none of the members wants to do the particular task. Because the task needs to be completed for the project to finish, all members interact to influence others to do the task until one of them changes their choice and accepts the task. On the basis of this initial choice, the agents start influencing each other. The influencing process aims to reach a complementary situation in which the agent can actually do the task that it wants while influencing the other agents to do the tasks it doesn't want to do. This process applies to all tasks and all agents involved in a project.

Figure :
The initial choice of all members for a task.
• Reach the final allocation. the interaction process will continue until the complementary allocation outcome is reached, which shows each task is assigned and only assigned to one agent. The algorithm pseudo-code of reaching a final allocation is shown in Algorithm . Interact between agents whose IC ( a i , t j ) = 1 until one of them is influenced to change its choice

Algorithm : Reaching a final allocation
// none of the agents wants to do the particular task for tj Interact between agents whose IC ( a i , t j ) = 0 until one of them is influenced to change its choice • Perform the allocated task a er finalising the interaction process.
• Agents adjust their competence and motivation, the former resulting from executing certain tasks or using particular knowledge, the latter is a ected by the allocated tasks and previous processing experience.
• The next project arrives, repeating the self-organising process from step ( ).

Design concepts
Basic principles: this model builds on the ability of individuals to make decisions whether or not to perform certain tasks based on their competence and motivation (Wilke & Meertens ). According to Self-Determination Theory (SDT), three fundamental psychological needs are underlying this motivation: competence, autonomy, and belonging. When team members make decisions whether or not to perform certain tasks, their competence and motivation to perform di erent tasks are key factors in making this decision (Wilke & Meertens ).
Emergence: the emergent e ect we are interested in when using the model for the purpose described in the paper is the performance and satisfaction of the team under di erent individual needs, i.e., when three types of motivational needs have distinct importance in the team. Currently, performance is measured by recording the performance time, including coordination time and execution time.
Heterogeneity: the agents of the team are heterogeneous, their motivation and competence are dynamic. The learning and forgetting experience of agents, as well as the interaction between agents, will cause agents to continue to undergo adaptive changes.
Adaptation: motivation and competence are di erent when facing di erent tasks. Also, individual needs may be fulfilled to a varying extent a er dealing with distinguished tasks, and accordingly, their motivations will change adaptively.
Objectives: for a task-performing system, the competence and motivation of each agent are constantly adjusted according to the outcome of allocation and execution. Under di erent conditions, every agent within the team will develop di erently through continuous processing of the project, and the team's performance and satisfaction will also change accordingly.
Interaction: pairs of individuals interact if they cannot reach the complementary situation in which every task is selected and only by one person. In the course of task allocation, the social interaction involved will not only a ect the extent to which agents want to do and do not want to do tasks but also change the connections strength between agents.
Learning and forgetting: the process of handling tasks and using knowledge is a process of the learning experience for individuals, this acquired experience could be attributable to increased knowledge about the task being performed (Jarkas & Horner ). The more frequently someone performs a task, the better they get at it. The knowledge of the agent will improve a er the task has been executed because an agent will learn the particular knowledge by doing tasks, the learning amount relates to the times of uses of certain knowledge. Similarly, all of the knowledge not being used will be gradually forgotten with time. Stochasticity: this model deliberately avoids introducing stochasticity to minimise the impact of randomness on task allocation. There is only one potential source of randomness may exist, when the initial choice of agents is the same, and the willingness and unwillingness calculated according to their motivation and competence are the same or the di erence of the values between individuals is negligible, in this case, the task would be randomly allocated to one of those agents. Observation: for model analysis, the group-level data were recorded with time to answer how three individual needs a ect team performance. Observation includes metrics measuring the team performance (coordination time and performance time), and metrics that reflect the group satisfaction.

Details Initialisation
The model is aimed to explore the dynamical relationship between three individual needs and group performance. The model is by default initialised by creating an artificial team composed of three individuals, each individual has three di erent kinds of knowledge. For simplifying the experiment, all parameters of individuals within the same team are identical, parameters are set as the following Table . As a proof of concept in our first experimental design, we start with a team composed of members that have a small di erence concerning their knowledge. Having di erences in knowledge is more realistic, and it provides a better start condition for having meaningful interactions than a situation where the agents would be perfectly identical. The following Table shows the knowledge of the three agents. Averaged over this knowledge of the agents have a similar knowledge level.
The three agents have to complete a project. These projects can be more or less di icult, depending on the minimum level of knowledge that is required for an agent to be capable of performing a task. In our experimental design, we implemented a moderate project (Table ). The tasks of the moderate project can be performed by at least two agents.

Parameter
Value To observe the performance and development of the team, these three agents have to complete projects continuously. In our experimental design, the same project will be handled by the same team times in succession, each time step in the experimentation represents a complete process of executing a project.

Input data
In the application of the model, further inputs from external sources such as data files are not required once the model is initialised.

Sub-models
The motivational need for the competence Competence is the ability of an individual to do a job appropriately, competence is o en considered to be contextual. If the task is too easy for a worker, the worker would be bored, if the task is too di icult, the worker would be anxious, an agent would be most motivated if the knowledge of itself close to the task requirement (Schiepe-Tiska ), this suggestion is consistent with the concept of flow, which describes when people are in a state of balance between one's skill and perceived di iculty (task demand), they will be in the zone and highly committed to the task. Hence, we need an inverted U-shape curve for explaining the relation between the need for competence and individual knowledge concerning task requirements. The satisfaction of the need for the competence of an agent a i for a task t j could be denoted as M nCom(a i , t j ) ∈ (0, 1] (Equation ), when ak l i equals to tk l j , M nCom(a i , t j ) = 1, where τ i ∈ (0, 1], indicates to what extent agent a i can tolerant or accept the di erence in knowledge, the bigger τ i is, the more discrepancy a i can accept.
Where τ i ∈ (0, 1], indicates to what extent agent a i can tolerant or accept the di erence in knowledge, the bigger τ i is, the more discrepancy a i can accept.

The motivational need for the autonomy
Autonomy is the capacity of an agent to act following its preference and willingness rather than under the influence of others. Agents want to be autonomous and have individual liking to use a certain skill, expertise, or knowledge, the more agents are capable of doing the tasks they like, the more perceived sense of choice ability, the more satisfied they are with the need for autonomy.
Generally, the sense of choice occurs when a certain activity is executed, which has a certain impact on the subsequent perception of autonomy. Therefore, the satisfaction of autonomy is influenced by both momentto-moment and previous experience. Here we choose a simple solution, the satisfaction of autonomy is being increased if the initial choice of team members is consistent with the final allocation. If not, the satisfaction of autonomy would be decreased. This implies that for each task, the satisfaction of autonomy can be a ected even when the agent does not want to perform it, nor performs it. Hence, we denote the need for autonomy of an agent for a particular task as M nAut(a i , t j ) ∈ (0, 1) (Equation ).

The autonomy of the agent is being defined to what extent can an agent get what it wants. As the di erence between the initial choice (what an agent wants) and the final allocation (what an agent gets) is larger, the need
for autonomy is less fulfilled. We take the di erence between the initial choice and the final allocation as the instantaneously perceived autonomy by agents. This instantaneous perception of autonomy of agent a i for the task t j can be denoted as Equation , Where σ i shows the reactivity of each agent's perceived autonomy of the di erence between final choice and initial choice, Dif f t (F inalallocation, Initialchoice) = 0or1, when Dif f t = 1, it indicates the final allocation of a i is equal to its initial choice when facing t j at time t, thus a i gets what it wants, which implies a i perceived autonomy at time t, when Dif f t = 0, it shows the final allocation of a i is distinguished from its initial choice, and a i didn't perceive autonomy at this moment. The instantly perceived autonomy will gradually weaken over time, for a i , a er a while T , the p t Aut(a i , t j ) will decay into (as Equation ), Where µ i represents the decay rate, the value of µ i for each member may be di erent, and the rate of decay is also di erent. The larger the µ i , the faster the decay, and vice versa.
The satisfaction of autonomy is continuously accumulated, and the accumulation process goes through a process that grows from fast to slow, for this, the motivation of agent a i for the task t j driven by the need for autonomy at time t is expressed as, indicates the initial motivation of autonomy, Inf (pAut(a i , t j ) implies the influence of autonomy perceived in the past on the current motivation of autonomy, p i t−i Aut(a i , t j ) represents the perceived autonomy of a i on t j at the time i a er time (t−i). Inf (pAut(a i , t j ) ∈ [0, 1], generally, an agent will not be a ected by the past unrestrictedly, the upper limit for this impact is in this paper, indicating that when the past impact accumulates to , the agent will consider itself autonomous so far, and the cumulative equation for the satisfaction of autonomy at this moment is as Equation .
Where i shows to what extent the autonomy is influenced by the previous experience, the larger the i , the more autonomy is a ected by history. It is precisely because each agent has di erent reactivity, decay, and accumulation for autonomy, which reflects the heterogeneity of agents.

The motivational need for the belonging
The need for belonging refers to the need the agent has of being connected and as a part of a larger whole. In a setting in which a team performs a project, this logically implies that the agent wants to be a part of the team that performs the project. As we addressed that the sense of belonging of each individual depends on its feelings of connectedness. In this paper, we determined to adopt social connectivity as a proxy to represent members' belonging. Basically, this social connectivity refers to connectivity to the other agents, it is defined as an experience of belonging and relatedness, based on quantitative and qualitative social relationships (Walton et al. ). Social connectivity is intended to capture the social experience derived from a recent interaction. It is the connectivity between individuals and groups from the overall level that pertains to one's social network (Jose et al. ). Table : Representation of the connection strength between members within the team.
From the perspective of social connectivity, the need for belonging is related to connectivity between individuals. In this paper, we utilise connection strength to measure such connectivity, which is dynamic, depending on the collaboration and interaction between agents. The connection strength between a i and other agents at time t, can be expressed as CS i→j t (a i ), i = j, where, j = 1, 2, · · · , m. The status of connection strength of the entire team is shown in Table . CS indicated the ratio of the collaboration of a i and a j having been working on the same project at time t, α i ∈ (0, 1] indicates to what extent the collaboration rate a ects connection strength, the collaboration status between agents can be expressed as the symmetric matrix M C (Equation ), since the collaboration is mutual, so the mutual connections between agents are enhanced at the same time as the collaboration. In M C, C i→j t indicates whether or not a i and a j collaborate at time t, C i→j t = C j→i t = 0, 1, when C i→j t = 1, it shows a i and a j collaborate, they work on the same project at time RoA i→j t ∈ [−1, 1] refers to the ratio at which a i should be accountable to a j on the outcome of the interaction between them, (1 − α i ) ∈ (0, 1] implies that how much the accountability a ects connection strength. The accountability between agents of the team can be expressed as the matrix M A (Equation ), A i→j t = −1, 1, when A i→j t = 1, it indicates that a i cause the final allocation of a j to be inconsistent with its initial choice, for a i , the connection between a j would be strengthened due to the positive interaction outcome with a j , however, for a j , the connection would be weakened because of the negative interaction causing the inconsistency of a j 's choice occurs, accordingly, A j→i t = −1. Since the accountability between any two agents is likely to be distinct, M A is an asymmetric matrix, when I i→j = 0,RoA i→j t = t A i→j t I i→j , I i→j = 0,RoA i→j t = 0, I i→j represents the number of interactions between a i and a j , the matrix M I represents the interaction among agents of the entire team (Equation ).
Based on the above description, the belonging of an agent is related to its overall connectivity, which is denoted by Equation , M nBel a (a i ) = β i log 2 (1 + CS(a i )) ( ) shows the average connection strength between a i and other agents, β i ∈ (0, 1], which indicates to what extent the need for belonging of a i a ected by its overall connectivity. The overall expected motivation of agent a i (M K(a i )). In the workplace, it is feasible for workers to make choices based on a combination of three motivations. In this paper, this combination is called overall expected motivation, which is a weighted average related to the di erent importance of three needs, as Equation . M respectively represent the importance of motivation for competence, autonomy, belonging of individuals, they may be distinguished, which implies that agents believe that the three needs of the overall motivation are of di erent importance.

The initial choice of an agent
Specifically, when every agent decides whether or not to perform a task is determined by the value of two conceptual nodes, an I-node and a Y-node, which respectively means 'I will do it' and 'I will not do it, I want you to do it. Both I-node and Y-node have values varying from to , indicating the extent to which the agent wants and does not want to do a particular task, and the values are related to four elements: agent knowledge, knowledge requirement of tasks, agent motivation, and motivation threshold. The values of I-node (V I−node ) and Y-node (V Y −node ) represent di erent extents. The former indicates to what extent I want to do a certain task, and the latter indicates to what extent I do not want to do a certain task. If the value of I-node is not lower than Y-node (V I−node ≥ V Y −node ), then I do, and vice versa. Based on the relationship between the four elements aforementioned, the following four zones can be divided, shown in Figure   Zone : when there is insu icient knowledge and motivation, AK < T a k and AM < T a m, which will consequently result in a Y-Do choice. The value of I-node and Y-node can be simply described as Equation and Equation : where the motivation is su icient, and the knowledge is insu icient, AK < T ak and AM >= T am (as Equation and Equation ).
where the knowledge and motivation of an agent are both exceeding their threshold, AK >= T ak and AM >= T am which leads to an I-Do choice. In this situation, the value of the I-node is a function of the knowledge (AK), motivation (AM ), and their threshold (T ak , T am ), and a parameter λ ∈ [0, 1] which implies the balance between knowledge and motivation (ad Equation and Equation ).
In this situation, where the knowledge is su icient, but motivation is insu icient,AK >= T ak and AM < T am . We define the value of the I-node as a function of its knowledge, and the Y-node as a function of its motivation (as Equation and Equation ).

The individual learning and forgetting
The learning phenomenon has proved applicable in various organisations, people can benefit from previous experience and therefore they would "learn" to improve productivity. Learning is an important method of adapting and responding to the organisational environment and an important process for adapting and developing team capabilities. The learning process stems from individuals repeating the same task and gaining skill or e iciency from their own experience and practice (Jarkas & Horner ). The process of handling tasks and using knowledge is also a process of the learning experience for individuals, this acquired experience could be attributable to increased knowledge about the task being performed (Jarkas & Horner ). The more frequently someone performs a task, the better they get at it.
The knowledge of the agent will improve a er the task has been executed because an agent will learn the particular knowledge by doing tasks, the learning amount relates to the times of uses of certain knowledge. The learning speed will change from slow to fast, then to slow. The learning curve is asymptotic, and the idealized general form of learning curve follows the Sigmoid function (Leibowitz et al. ; Murre ) (as Equation ).
Where T p (ak l i ) represents the period of a i continuously performing a particular knowledge, AK t (a i,l ) shows the lth knowledge of a i at time t. Besides, γ i ∈ (0, 1] illustrates a learning ability of a i , the larger the γ i , the higher the learning ability. Besides, there is enough empirical evidence that knowledge depreciation (forgetting) occurs in organisations (Argote ). All of the knowledge not being used will be gradually forgotten with time. The behaviour of knowledge forgetting is usually assumed to have the same form as knowledge learning, except that the forgetting rate is negative and the learning rate is positive (Jaber ). This widely recognised assumption is consistent with the suggestion that the learning and forgetting e ects are considered as mirror images of each other (Globerson et al. ). As we elaborated on the learning curve, forgetting curves also comply with an asymptotic curve, see the following equation .

Metrics and indicators
In this paper, we measure how fast projects are performed by the team and their satisfaction a er handling the projects. Simply, indicators are designed from two aspects: performance time and Group satisfaction. Performance Time(T perf ) consists of two parts: allocation time and completion time (as Equation ). The former refers to the time used during the allocation process, also known as coordination time, denoted as T coor (as Equation ), T coor (a i , a j ) shows the coordination time spent on the task t j between the agent a i and a j , the latter part demonstrates the time used to execute the whole project, equalling to the time spent by the person who completed the task last.
T perf = T coor + M ax(t exec (a i )) ( ) T coor = M ax(T coor (a i , a j )) ( ) Where t exec (a i ) represents the time of a i spending on executing the tasks which are allocated to it, as Equation .
t exec (a i ) = T D(t j ) λ × AK(a i ) + (1 − λ) × AM (a i ( ) Group Overall Satisfaction shows the overall satisfaction of the team as a whole, here, we adopt the average satisfaction of all members as the group overall satisfaction, which is denoted as Equation , Where m represents the number of team members. The implications of W (a i ),I c (a i ),I a (a i ),I b (a i ) are consistent with their implications in individual motivation. For every agent, they have three types of motivational needs, for a particular need, its satisfaction level shows to what extent it has been fulfilled, we define individual need satisfaction as the average of agent's satisfaction for every task, denoted as SnCom(a i ), SnAut(a i ), SnBel(a i )) (Equation ( -)), they respectively represent the satisfaction of need for competence, autonomy, and belonging.

Discussion
There are several variables and parameters in the model, which are not directly relevant to achieving our purpose, may influence the results. Such variables include the team size, project size, types of knowledge, and other individual parameters. Hence, we fixed those parameters, make the team processes projects in the context created by the experimental settings.