## 6 Numerical Evaluation
### Pure Operation Performance  
In this section, we present a comprehensive analysis of the performance and functionality of the Approximate and Accurate CROWDMATCH algorithms. Specifically, Figure 0(a),b depicts the approximate and accurate utility of crowdworkers, respectively, with respect to their assigned IDs. The results highlight a direct correlation between higher crowdworker IDs, indicative of increased information availability for upload to crowdsourcers, and elevated levels of both approximate (Figure 0(a)) and accurate utility (Figure 0(b)) following the convergence of the Approximate and Accurate CROWDMATCH algorithms. Furthermore, the results reveal that the achieved accurate utility of the crowdworkers is lower compared to the corresponding approximate utility given the consideration of the externalities imposed in the crowdsourcing process, stemming from the other crowdworkers' decisions and the crowdsourcer rewards' availability. Moreover, our findings reveal that the Approximate CROWDMATCH algorithm empowers crowdworkers to judiciously select the most suitable crowdsourcer, thereby maximizing their utility. This selection process takes into account the maximum capacity (\(N_{M}^{max}\)) of each crowdsourcer to accommodate crowdworkers based on the availability of rewards. Figure 1a specifically illustrates the potential approximate utility that the crowdworkers could receive when paired with their preferred crowdsourcer. However, the Approximate CROWDMATCH algorithm ensures a balanced approach by prioritizing crowdworkers characterized by higher information availability, allowing them to be selected first by crowdsourcers to facilitate the crowdsourcing process. Figure 2a,b presents the number of matched crowdworkers and the crowdsourcers' utility as a function of the crowdsourcers' ID under the Approximate and Accurate CROWDMATCH algorithms, respectively. The findings indicate that an increase in crowdsourcers' IDs, signifying greater reward availability, leads to the recruitment of a larger pool of crowdworkers for participation in the crowdsourcing process. This, in turn, results in higher utility, as evidenced in both the Approximate (Figure 2a) and Accurate (Figure 2b) CROWDMATCH algorithms. Furthermore, crowdsourcers with higher IDs, denoting enhanced rewards availability, demonstrate a heightened capacity for accommodating and recruiting crowdworkers. This capacity is initially exhausted, allowing lower-budget crowdsourcers to subsequently engage crowdworkers under the Approximate CROWDMATCH algorithm. On the other hand, the Accurate CROWDMATCH algorithm mitigates the disparity between higher- and lower-budget crowdsourcers, facilitating even the latter to secure a higher share of recruited crowdworkers. Focusing on the utility realized by crowdworkers when paired with crowdsourcers using the Approximate (Figure 3a) and Accurate (Figure 3b) CROWDMATCH algorithms, the results reveal that the utility experienced by the crowdworkers is contingent upon the selection of crowdsourcers. Specifically, within the framework of the Approximate CROWDMATCH algorithm, the crowdworkers possessing abundant information availability are matched with crowdsourcers offering the highest rewards in the crowdsourcing system. Conversely, the crowdworkers with low information availability are paired with  
Figure 1: Crowdworkers’ (**a**) approximate (Equation (3)), and (**b**) accurate utility (Equation (1)). Figure 2: Number of matched crowdworkers to crowdsourcers and crowdsourcers’ utility under the (**a**) Approximate CROWDMATCH algorithm, and (**b**) Accurate CROWDMATCH algorithm. crowdsourcers providing lower rewards. This imbalance leads to substantial variations in the overall utility achieved by the crowdworkers, associated with the crowdsourcers characterized by low versus high reward availability.