# Outsmarting Human Design in Airline Revenue Management
## 5 Experiments
### Estimating the Price Sensitivity and Arrival Rate under Constrained Capacity  
For our second experimental setting, we consider that the systems must estimate both demand model parameters, i.e., the arrival rate \(\
u\) and the price sensitivity \(\phi\), from historical bookings. The true arrival rate can assume any value in the range \(\
u_{*}\in[55/22,80/22]\), making the flights' capacity \(C=50\) finite in practice. As in the previous experiment, we consider that the true price sensitivity can assume any value in the range \(F_{5}^{*}\in[2.1,3.8]\). For the same reasons as in the previous experimental setting, we train the RL agent  
Figure 8: Comparison between heuristic and reinforcement learning (RL) roll-out policies. For convenience, we represent the revenue-maximizing policy. over a clipped interval that is larger than the evaluation interval (\(\
u\in[50/22,85/22]\) and \(F_{5}\in[1.5,4.3]\)). As in the previous section, we first analyze how the RMS and the RL agent perform in the evaluation interval with respect to revenue performance and model quality. Then, we compare how the two policies differ for a specific value of true demand behavior parameters. In Figure 8(a), we can see how the average normalized revenue and the MSE for the demand model parameters respond to different values of true price sensitivity within the evaluation interval. From the left chart, we can see that RL outperforms RMS in the entire interval, with a larger advantage occurring with higher values of frat5. In the center and right charts, we see that the MSE errors for both parameters are close to each other, but RL consistently demonstrates a worse MSE performance. The same behavior is also present when the true customer price sensitivity is fixed, and we only vary the arrival rate, as in Figure 8(b). This may seem paradoxical: how can RL perform better from the revenue maximization perspective while having a worse estimation of the demand model parameters? Unfortunately, we do not have a complete understanding of RL strategy to precisely answer this question, thus we can only speculate about its strategy. As we observe in the following analysis, instead of reducing model uncertainty, the agent can learn to select the \"safe price\" which it is certain will not lose too much revenue, especially when the estimated demand model parameters are found to be too extreme (i.e., close to the clipping limits). For the second analysis, in Figure 10, we display the RMS and the RL agent pricing policies when the true arrival rate and price sensitivity are fixed at \(\
u_{*}=70/22\) and \(F_{5}^{*}=2.9\). We choose this point because the flights' average load factor is 70%, which brings a natural price variability and a high amount of booking data to estimate the demand model parameters (the historical data contains an average of 1078 bookings). Given these two properties, obtaining a good estimation of the demand model parameters is relatively easy for the RMS, which puts in question the value of price experimentation. The RMS achieves a normalized revenue performance of 80.6%, with an MSE for a price sensitivity and an arrival rate of 0.007 and 0.20, respectively. The agent is better for revenue maximization,  
Figure 9: The comparison between the average performance of the revenue management system, reinforcement learning, and revenue-maximizing policies (99% confidence level). with a normalized revenue performance of 89.9%, but it has a worse model quality with an MSE for a price sensitivity and an arrival rate of 0.008 and 0.26, respectively. When comparing the two policies, we see that the major difference is that the agent prefers to price center fares $130, $150, and $170 more frequently (72.1%) than the RMS (65.3%). This strategy has a negative impact on the overall model quality, because it decreases the price variability in the historical data, but it turns out that center fares are better for revenue maximization than higher fares such as $210 and $230, because they are a less extreme given the true demand behavior.