## 5 Model Sample Validation
### Validity and Sensitivity Analysis
#### 5.2.1 Comparative Analyses of Validity  
1. Fairness  
In terms of the model performance, overall, the fairness model performed the best, followed by the efficiency-fairness model, and the efficiency model performed the worst. Furthermore, the difference between the efficiency-fairness model and the fairness model was significantly smaller than that between the efficiency-fairness model and the efficiency model, with mean fairness evaluation index values of 1.37, 1.32, and 1.21, respectively, as shown in Figure 6.  
Regarding saturation, under low saturation conditions, the efficiency-fairness model showed better performance. The mean improvement in the fairness evaluation index for saturation levels between 0.1 and 0.4 was 0.16, while for saturation levels between 0.5 and 0.8, it was only 0.075. In terms of trend, higher saturation levels correlated with higher fairness evaluation index values. At saturation levels of 0.1 and 0.8, the mean fairness evaluation index values were 1.27 and 1.33, respectively. Furthermore, the higher the saturation level, the smaller the differences between the models. At saturation levels of 0.1 and 0.8, the mean differences between the models were 0.115 and 0.03, respectively, especially notable between the efficiency-fairness model and the efficiency model, with differences of 0.16 and 0.09 at saturation levels of 0.1 and 0.8, respectively. The following figure illustrates the fairness evaluation index curves for each model.  
1. Efficiency  
In terms of the models, overall, the efficiency model performed the best, followed by the efficiency-fairness model, and the fairness model performed the worst. Furthermore, the difference between the efficiency-fairness model and the efficiency model was significantly smaller than the difference between the efficiency-fairness model and the fairness model. The mean vehicle delay values were 13.72, 14.08, and 35.12, respectively, which are shown in Figure 7.  
Regarding saturation, the saturation level had little impact on the efficiency-fairness model. As for the trend, higher saturation levels correlated with greater vehicle delays. At saturation levels of 0.1 and 0.8, the mean vehicle delays were 18.4 and 24.98, respectively.  
Figure 6: Three-model fairness evaluation index.  
Additionally, as saturation levels increased, the differences between the models did not change significantly, with a mean difference of 10.7. At saturation levels of 0.1 and 0.8, the differences were 10.16 and 11.12, respectively. The following figure illustrates the delay curves for each model.  
(3) Conversion rate  
In terms of the models, overall, the efficiency-fairness model significantly outperformed the fairness model, with mean conversion rates of 0.09 and 9.6, respectively, a difference exceeding 100 times. Furthermore, the efficiency-fairness model ranged from a minimum of 2.37 to a maximum of 0.14 for the fairness model, representing a difference of over 16 times, which is shown in Figure 8.  
Regarding saturation, under low saturation conditions, the efficiency-fairness model exhibited better performance. Excluding the outlier at saturation level 0.8, the mean conversion rates for saturation levels 0.1 to 0.4 were 4.0, greater than the rates for saturation levels 0.5 to 0.7, which were 2.76. With changes in saturation levels, the two models showed different trends. While the fairness model exhibited a decreasing conversion rate with increasing saturation levels, the efficiency-fairness model showed fluctuations in its conversion rate with changes in saturation levels without a clear trend. The figure below illustrates the conversion rate curves