# Utility Distribution Strategy of the Task Agents in Coalition Skill Games
## 3 Definition of the Problem Model  
The formal definition of the problem model, the multiple self-interested agent coalition skill game, is given in Definition 1. **Definition 1**. : _A multiple self-interested agent coalition skill game includes three sets: service agent set \(R\): = {\(r_{1}\), \(r_{2}\), \(\cdot\), \(r_{n}\)}, skill set \(S\): = {\(s_{1}\), \(s_{2}\),\(\cdot\), \(s_{l}\)} and task agent set \(T\): = {\(t_{1}\), \(t_{2}\),\(\cdot\), \(t_{m}\)}, where \(n\) = \(l\)\(R\)\(|\)\(,\)\(l\) = \(l\)\(S\)\(|\)\(,\) and \(m\) = \(|\)\(T\)\(|\)\(,\)\(|\)\(|\)\(|\) denotes the cardinality of the set *. For \(i\) {\(\in\) {1, 2,\(\cdot\), \(\cdot\), \(n\)}, \(j\) {1, 2,\(\cdot\), \(\cdot\), \(l\)}, \(RS_{i,j}\) = (\(RS_{i,j}\) = 0) indicates service agent \(r_{i}\) possesses (does not possess) skill \(s_{j}\). For \(j\) {1, 2,\(\cdot\), \(\cdot\), \(l\)}, \(k\) {1, 2,\(\cdot\), \(m\)}, \(ST_{j,k}\) = 1 (\(ST_{j,k}\) = 0) indicates task agent \(t_{k}\) needs (does not need) skill \(s_{j}\). Utilities of all task agents are denoted by a vector of \(U\): = {\(u_{1}\),\(u_{2}\),\(\cdot\),\(\cdot\),\(u_{m}\)}._  
Utility distribution schemes of all the task agents are denoted with \(TS\). For \(k\) {1, 2,\(\cdot\), \(m\)}, \(j\) {1, 2,\(\cdot\), \(l\)}, \(TS_{k,j}\)(\(\tau\)) denotes the share distributed to \(s_{j}\) by \(t_{k}\) at time \(\tau\), if \(ST_{j,k}\) = 0, \(TS_{k,j}\)(\(\tau\)) = \(-1\). \(RTS_{i,0}\)(\(\tau\)) denotes the serial number of the task agent selected by \(r_{i}\)\(\in\)\(R\) at time \(\tau\). If \(r_{i}\)\(\in\)\(R\) does not select any task agent, \(RTS_{i,0}\)(\(\tau\)) = 0. For \(I\) {1, 2,\(\cdot\), \(n\)} and \(j\) {1, 2,\(\cdot\), \(l\)}, \(RTS_{i,j}\)(\(\tau\)) = 1 (\(RTS_{i,j}\)(\(\tau\)) = 0) denotes \(r_{i}\) provides (does not provide) the selected task agent \(t_{RTS_{i,0}\)(\(\tau\))} with skill \(s_{j}\) at time \(\tau\). If \(RTS_{i,0}\)(\(\tau\)) = 0, for \(j\){1, 2,\(\cdot\), \(l\)}, \(RTS_{i,j}\)(\(\tau\)) = 0. \(RTS_{i}\)(\(\tau\)) denotes the allocation of task agents to service agents at time \(\tau\). Task agent \(t_{k}\) can be completed if all needed skills are provided. Given \(RTS(\tau)\), the system revenue is defined as the sum of the utilities of the tasks that can be completed. The optimal task allocation to a multiple self-interested agent coalition skill game is an allocation of task agents to service agents which maximizes its system revenue. In this paper, it is assumed that: (1) each service agent is allowed to select at most one task at any time, but it can use multiple skills when performing a task. (2) A service agent is willing to abandon the currently selected task and select a new task only if the new one can bring it a higher individual revenue. The states of service agents and task agents are defined in Definitions 2 and 3 respectively. **Definition 2**. : _(State of service agent) The state of service agent \(r_{i}\)\(\in\)\(R\) at time \(\tau\) is denoted by \(r_{i}\)(\(\tau\)): = \(<\)\(RS_{i}\), \(RTS_{i,\cdot}\)(\(\tau\))\(>\), where \(RS_{i,\cdot}\) is the \(i\)th row of \(RS\) and \(RTS_{i,\cdot}\)(\(\tau\)): = {\(RTS_{i,0}\)(\(\tau\)), \(RTS_{i,1}\)(\(\tau\)),\(\cdot\), \(RTS_{i,l}\)(\(\tau\))}._  
**Definition 3**. : _(State of task agent): The state of \(t_{k}\)\(\in\)\(T\) at time \(\tau\) is denoted by \(t_{k}\)(\(\tau\)): = \(<\)\(ST_{\cdot,k}\), \(TSN_{k}\), \(u_{k}\), \(TS_{k,\cdot}\)(\(\tau\))\(>\), where \(ST_{\cdot,k}\) is the \(k\)th column of \(ST_{k}\)\(\in\)\(Z^{+}\) denotes the number of skills needed by \(t_{k}\) (\(lt\) is assumed that at most one unit of each skill is needed). \(u_{k}\) corresponds to the utility of \(t_{k}\) representing the value that completing the task is worth. \(TS_{k,\cdot}\)(\(\tau\)) is the \(k\)th row of \(TS(\tau)\)._  
The Task Selection Strategies of Service Agents and the Utility Distribution Strategies of Task Agents  
Section 4.1 describes the task selection strategies of the service agents. Section 4.2 describes the utility distribution strategies of the task agents.