## 3 Approximate CrowbMATCH Based on Matching Theory  
The crowdworkers seek to employ a strategic approach in selecting a crowdsourcer for uploading information, with the objective of maximizing their utilities. Conversely, the crowdsourcers endeavor to identify an optimal group of crowdworkers to enhance their utilities in a distributed fashion. This situation presents a many-to-one matching problem, wherein multiple crowdworkers are paired with a single crowdsourcer, and can be studied based on the matching theory. Definition 1: (Matching) _A set of crowdworkers, denoted as \(\mathcal{N}=\{1,\dots,n,\dots,N\}\) and a set of crowdsourcers, denoted as \(\mathcal{M}=\{1,\dots,M\}\) are regarded as two non-intersecting sets. The matching function \(\Lambda\) represents a mapping of elements from \(\mathcal{N}\) to elements from \(\mathcal{M}\), subject to the following conditions:_  
* _(C_1)_: \(\forall n\in\mathcal{N}\)_,_ \(|\Lambda(n)|\leq 1\)_;_
* _(C_2)_: \(\forall m\in\mathcal{M}\)_,_ \(|\Lambda(m)|\leq N_{m}^{max}\)_;_
* _(C_3)_: \(\Lambda(n)\in\mathcal{M}\iff\Lambda(m)\in\mathcal{N}\)_;_
* _(C_4)_: \(n\in\Lambda(m)\iff\Lambda(n)=m\)_._  
The magnitude of a matching, expressed as \(|\Lambda(\cdot)|\), signifies the cardinality of the matching. \(N_{\max}\) denotes the maximum count of crowdworkers that a crowdsourcer can accommodate, contingent upon its available rewards. In cases where \(\Lambda(n)=\emptyset\), crowdworker \(n\) remains unmatched to any crowdsourcer. Similarly, if \(\Lambda(m)=\emptyset\), crowdsourcer \(m\) is not chosen by any crowdworker. For each crowdworker \(n\in\mathcal{N}\), preference is given to the crowdsourcer that yields the highest utility, as defined in Equation (1). The utility of crowdworker \(n\), as expressed in Equation (1), is intricately tied not only to its individual decision, but also to the decisions of other crowdworkers, captured by the term \(\sum\limits_{m^{\prime}\in\mathcal{N}_{m}}\hat{I}_{m^{\prime}}\). This phenomenon is commonly referred to as externality in matching theory. In the initial phase of our analysis, we consider the absence of externalities in the matching process. Consequently, the reformulation of the crowdworker's approximate utility is articulated as follows:  
\[\hat{U}_{n}=\hat{a}_{n}(\hat{P}_{m}-\hat{P}_{n})\hat{I}_{n} \tag{3}\]  
and the crowdsourcer's approximate utility after omitting the externalities stemming from the rewards offered by the other crowdsourcers is formulated as follows:  
\[\hat{U}_{m}=\sum\limits_{\forall n\in\mathcal{N}_{m}}\hat{U}_{n}-\hat{\gamma} _{m}(\hat{P}_{m}-\hat{C}_{m})^{2} \tag{4}\]Definition 2: **(Preference Relation)** A preference relation \(<\) is a binary relation between elements of the set \(\mathcal{N}\) and elements of the set \(\mathcal{M}\) that is complete, self-referential, and transitive. The preference relation \(<_{n}\) is formally defined as follows: for any crowdworker \(n\in\mathcal{N}\) and any crowdsourcer \(m,m^{\prime}\in\mathcal{M}\), where \(m\
eq m^{\prime}\), the relation is given by  
\[m>_{n}m^{\prime}\iff\hat{\Omega}_{n}(m)>\hat{\Omega}_{n}(m^{\prime}) \tag{5}\]  
Similarly, the preference relation \(<_{m}\) is defined as follows: for any crowdsourcer \(m\in\mathcal{M}\) and any pair of crowdworkers \(n,n^{\prime}\in\mathcal{N}\) where \(n\
eq n^{\prime}\), the relation is given by  
\[n>_{m}n^{\prime}\iff\hat{\Omega}_{m}(n)>\hat{\Omega}_{m}(n^{\prime}) \tag{6}\]  
With Definition 2, preference lists for crowdsourcers and crowdworkers are established. Subsequently, based on these preference lists, crowdsourcers and crowdworkers are matched with their preferred partners through the Approximate CROWDMATCH algorithm, outlined in Algorithm 1.