## 2 System Model  
In this section, the CROWDMATCH system model is presented. We consider a set of crowdsourcers, such as Amazon Mechanical Turk, Upwork, CrowdFlower, Google Maps, etc., \(\mathcal{M}=\{1,\dots,m,\dots,M\}\), and a set of crowdworkers, denoted as \(\mathcal{N}=\{1,\dots,n,\dots,N\}\). Each crowdsourcer has a total budget \(P_{m}\left[\frac{5}{\mathrm{bits}}\right]\) that is provided as a reward to the crowdworkers in order to incentivize the latter to submit their information to crowdsourcing tasks announced by the crowdsourcer. Each crowdworker is characterized by an amount of information \(I_{n}\) [bits], which can be composed of text, videos, photos, etc., and a corresponding cost \(P_{n}\big{[}\frac{5}{\mathrm{bits}}\big{]}\) to collect the information. The crowdworker's utility is defined as follows:  
\[U_{n}=\frac{\hat{a}_{n}(\hat{P}_{m}-\hat{P}_{n})\hat{I}_{n}}{\sum_{\forall n^{ \prime}\in\mathcal{N}_{m}}\hat{I}_{n^{\prime}}} \tag{1}\]  
where \(a_{n}\in R^{+}\big{[}\frac{\mathrm{bits}}{5}\big{]}\) denotes the evaluation of the crowdworker's profit from uploading \(I_{n}\) amount of information to the crowdsourcer \(m\), and \(\mathcal{N}_{m}\) denotes the set of crowdworkers that selected crowdsourcer \(m\). The values have been normalized for presentation purposes and without loss of generality, as follows: \(\hat{a}_{n}=\frac{a_{n}}{\max\{\{a_{n}\}_{\forall n\in\mathcal{N}}\}}\), \(\hat{I}_{n}=\frac{I_{n}}{\max\{I_{n}\}_{\forall n\in\mathcal{N}}}\), \(\hat{P}_{n}=\frac{P_{n}}{\max\{P_{n}\}_{\forall n\in\mathcal{N}}}\), and \(\hat{P}_{m}=\frac{P_{m}}{\max\{P_{n}\}_{\forall n\in\mathcal{M}}}\). Also, it is noted that the crowdworker's utility depends on the amount of information that all the crowdworkers upload to the selected crowdsourcer; if the crowdsourcer receives a large amount of information from other crowdworkers, then it is less incentivized to provide competitive rewards to the crowdworkers.  
Focusing on the crowdsourcers' benefit from the crowdsourcing process, the corresponding crowdsourcer's utility is formulated as follows:\[U_{m}=\frac{\sum\limits_{\forall n\in N_{m}}U_{n}-\hat{\gamma}_{m}(\hat{P}_{m}- \hat{C}_{m})^{2}}{\delta_{m}\sum\limits_{\forall m^{\prime}\in M}\hat{P}_{m^{ \prime}}} \tag{2}\]  
where \(C_{m}[\frac{\xi}{\text{bits}}]\) denotes the cost incurred by the crowdsourcer for executing its fixed operations, \(\gamma_{m}\)\([\frac{\text{bits}^{2}}{\xi^{2}}]\) signifies the assessment of profit, and \(\delta_{m}\)\([\frac{\xi}{\text{bits}}]\) represents the evaluation of the market impact. Analogously, in consideration of the crowdworker's utility, we normalize the values for presentation purposes and without loss of generality, as follows: \(\hat{C}_{m}=\frac{C_{m}}{\max\{C_{m}\}_{\forall m\in M}}\) and \(\hat{\delta}_{m}=\frac{\delta_{m}}{\max\{\hat{\alpha}_{m}\}_{\forall m\in M}}\). The primary objective for each crowdworker is to select a crowdsourcer to upload its information, with the aim of maximizing its utility. Similarly, each crowdsourcer aims to recruit a group of crowdworkers to perform the designated crowdsourcing task while maximizing its utility. It is noted that each crowdsourcer can accommodate a maximum number of \(N_{m}^{max}\) crowdworkers based on its available budget \(P_{m}\) to provide rewards.