## 6 Numerical Evaluation
### Scalability Analysis  
In this section, an exhaustive scalability analysis is conducted, considering a progressive increase in the number of crowdworkers and crowdsourcers. A real online platform scenario is considered, consisting of crowdsourcers on the order of tens and crowdworkers on the order of thousands. The objective of this scalability analysis is twofold: first, to showcase the real-time viability of the CROWDMATCH mechanism, and, second, to elucidate the influence of an increasing number of crowdworkers and crowdsourcers on the utility experienced by the crowdworkers.  
Figure 4a,b delineates the execution time and the average achieved utility of crowdworkers, respectively, in relation to the increasing number of crowdworkers involved in the crowdsourcing process. These analyses are conducted under both the Approximate and Accurate CROWDMATCH algorithms. For the sake of clarity in the presentation, we assume that a percentage increase in the number of crowdworkers results in an equivalent percentage decrease in the availability of their information. This approach ensures a consistent overall information pool within the system, irrespective of the number of engaged crowdworkers. The rationale behind this assumption lies in simplifying the presentation and modeling process. By assuming a proportional relationship between the increase in crowdworkers and the decrease in information availability, we aim to create a clear and comprehensible framework for our analysis. This assumption allows us to focus on the core aspects of our model without introducing unnecessary complexity.  
Furthermore, considering that the uploaded information by crowdworkers to crowdsourcers remains constant, a percentage increase in the number of crowdworkers corresponds to an equivalent percentage increase in the number of crowdworkers that each crowdsourcer can accommodate, i.e., \(N^{max}=[3,5,7]\). The results demonstrate that, with an escalating number of crowdworkers participating in the crowdsourcing process, the execution time of both the Approximate and Accurate CROWDMATCH algorithms follows a comparable upward trajectory, as depicted in Figure 4a.  
Turning attention to the average utility achieved by crowdworkers, a discernible downward trend is observed under both the Approximate and Accurate CROWDMATCH algorithms. It should be highlighted that the decreasing trend becomes more pronounced under the Accurate CROWDMATCH algorithm, reflecting heightened competition among crowdworkers competing to be matched with the most rewarding crowdsourcer in terms of incentives.  
Figure 3: Total crowdworkers’ utility per associated crowdsourcer under the (**a**) Approximate, and (**b**) Accurate CROWDMATCH algorithms.  
Focusing our scalability analysis on the scenario of increasing crowdsourcers, Figure 5a,b depicts the execution time and the average utility of the crowdworkers, respectively, in relation to the increasing number of participating crowdsourcers within the crowdsourcing process. The evaluation is conducted under the Approximate and Accurate CROWDMATCH algorithms. The results indicate that, with a growing number of crowdsourcers, the execution times of both the Approximate and Accurate CROWDMATCH algorithms exhibit a similar upward trend (Figure 5a). Nevertheless, the results affirm that, even with a substantial number of participating crowdsourcers, the execution time of the CROWDMATCH mechanism remains within the order of magnitude of sub-milliseconds. Furthermore, the outcomes reveal that an increasing number of crowdsourcers leads to a diminished average utility for crowdworkers. This is attributed to the constant overall number of rewards available in the system, which, when distributed among numerous crowdsourcers, results in smaller portions for individual crowdworkers.