## 4 Accurate CROWDMATCH Based on Coalition Games  
Given the presence of an externality, the outcome derived from the matching outlined in Section 3 using the Approximate CROWDMATCH algorithm ensures an approximate optimality. The outcome of the stable matching derived from the Approximate CROWDMATCH algorithm acts as an initial input to the Accurate CROWDMATCH algorithm. Specifically, a coalition game has been formulated, along with the corresponding Accurate CROWDMATCH algorithm, to enhance the matching outcome. This approach is devised to address the externalities and achieve an improved result by building upon the findings of the initial matching outcome. Definition 3: (Coalition Game) The coalition game is formally characterized by the triple (\(\mathcal{N},\mathcal{M},\{U_{m}\}_{\forall m\in\mathcal{M}}\)), where \(\mathcal{N}\) denotes the set of players (crowdworkers), \(\mathcal{M}\) represents the set of coalitions (crowdsources), and \(U_{m}\) signifies the utility function associated with each coalition (i.e., crowdsourcer), as defined in Equation (2). Definition 4: (Switching Rules) The following switching rules must be followed in the coalition game:  
**Rule 1:** For a crowdworker \(n\) who has not selected a crowdsourcer, \(n\) will join a crowdsourcer \(m\), if \(\exists m=\arg\max_{m^{\prime}\in\mathcal{M}}\{U_{m^{\prime}}(m^{*}\cup\{n\})- U_{m^{\prime}}(m^{*})|U_{m^{\prime}}(m^{*}\cup\{n\})-U_{m^{\prime}}(m^{*})>0\}\). The new set of coalitions is \(\mathcal{M}=\{\mathcal{M}\setminus\{m\}\}\cup\{m\cup\{n\}\}\);  
**Rule 2:** For \(n\in m\), \(n\) will leave crowdsourcer \(m\) if \(U_{m}(m\setminus\{n\})>U_{m}(m)\). The new set of coalitions is \(\mathcal{M}=\{\mathcal{M}\setminus\{n\}\}\cup\{m\setminus\{n\}\}\);  
**Rule 3:** For \(n\in\mathcal{N}_{m}\) and crowdsourcer \(m^{\prime}\), \(n\) will leave the original crowdsourcer \(m\) (where \(m\
eq m^{\prime}\)) and join another crowdsourcer \(m^{\prime}\) if and only if \(U_{m}(m\setminus\{n\})+U_{m^{\prime}}(m^{\prime}\cup\{n\})>U_{m}(m)+U_{m^{ \prime}}(m^{\prime})\). The new set of coalitions is \(\mathcal{M}=\{\mathcal{M}\setminus\{m,m^{\prime}\}\}\cup\{m\setminus\{n\}\} \cup\{m^{\prime}\cup\{n\}\}\);  
**Rule 4:** For \(n\in m\) and \(n^{\prime}\in m^{\prime}\), where \(n\
eq n^{\prime}\), \(n\) and \(n^{\prime}\) switch crowdsourcers if \(U_{m}((m\setminus\{n\})\cup\{n^{\prime}\})+U_{m^{\prime}}((m^{\prime}\setminus \{n^{\prime}\})\cup\{n\})>U_{m}(m)+U_{m^{\prime}}(m^{\prime})\). The new set of coalitions is \(\mathcal{M}=\{\mathcal{M}\setminus\{m,m^{\prime}\}\}\cup\{(m\setminus\{n\}) \cup\{n^{\prime}\}\}\cup\{(m^{\prime}\setminus\{n^{\prime}\})\cup\{n\}\}\). In accordance with the switching rules articulated in Definition 4, we have formulated the Accurate CROWDMATCH Algorithm, presented in Algorithm 2. This algorithm is strategically devised to streamline the formation of robust coalitions between crowdsourkers and crowdsourcers within the context of the crowdsourcing process. Its core aim is to systematically enhance the utility for the crowdworkers, optimizing their contributions to crowdsourcers and facilitating the efficient execution of the crowdsourcing tasks. Simultaneously, it endeavors to improve the utility for the crowdsourcers, thereby establishing a mutually beneficial framework that supports their collaborative involvement in the crowdsourcing process. Theorem 1: (Nash-Individually Stable Matching) The concept of Nash-Individually stable matching, represented as \(\Lambda^{*}\), pertains to the allocation of crowdworkers to crowdsourcers. This allocation is considered stable when no individual crowdworker can improve their utility by switching to a different crowdsourcer. The Accurate CROWDMATCH algorithm is specifically crafted to ensure the existence of at least one Nash-Individually stable partition, denoted as \(\Lambda^{*}\).