## 5 Discussion  
For the case study analyzed for three scenarios, considerations were made for the traffic flow distribution according to the Nash equilibrium and the Stackelberg approach. Computational analyses resulted in the determination of the optimum size of traffic flows on the links of the transport network. The criteria of equal average times and equal marginal times were met for all O-D pairs in the three scenarios. An increase in traffic flows for each O-D pair in the transport system may result in a reduction in traffic flows on certain links in the road network. For the case study presented, this situation occurred at link (5,6). With an overall increase in traffic flows, it becomes less attractive to travel for vehicles moving from origin node 2 to destination node 8. The selection of node (5,6) for travel in the O-D pair (2,8) results in the need to travel also through node (6,7), which is loaded with a large traffic flow moving in the O-D pair (3,8). Therefore, the \(p_{5}\) route is more attractive when traffic volumes are low in the transportation network. Analysis of road congestion as an increase in travel time through transport links identifies the weakest elements of the road network, which have the greatest impact on the decrease in efficiency of transport processes. In the model for the northern part of Warsaw's road network, the greatest road congestion occurs on links (4,8), (6,7) and (7,8). According to the Nash equilibrium for scenario #3, there was a 269.24% increase in travel time on link (6,7). For the baseline scenario, the average congestion levels for the analyzed network were 66.64% and 60.93% for the Nash equilibrium and the Stackelberg approach, respectively. Relating these results to studies by other authors [24], it was found that the congestion for the user equilibrium condition obtained using the developed model corresponds to the level determined by traffic measurements. Despite focusing on individual benefits for road users, the Nash equilibrium approach turned out to be just slightly less effective compared to minimizing the travel time of the entire traffic flow. The total transport time in the case of the Stackelberg approach was more favourable by approximately 1% compared to the Nash equilibrium approach. It should also be stressed that the time saved can contribute to potential benefits for the environment, due to lower energy consumption and reduced CO2 emissions to the atmosphere. The  
Figure 11: Compared total travel times of the entire traffic flow for the three scenarios. (The difference (%) represents the percentage increase in total travel time between the results of the Stackelberg approach and the Nash equilibrium for each scenario). aim of optimizing traffic flow distribution should be to minimize travel time and energy consumption. Regarding the findings of other authors [35; 36], it should be highlighted that the PoA for the Warsaw case study is at a significantly lower value. Table 12 summarises the results obtained for the Nash and Stackelberg approaches and the results from the literature for the Nguyen-Dupuis network. The graph of the Nguyen-Dupuis network is shown in Appendix D in Figure 1A. Compared to the algorithm proposed by [PERSON] and [PERSON] [67], both the Nash equilibrium and the Stackelberg approach obtained better results. The model proposed by [PERSON] et al. [68] achieved shorter travel times. The approaches considered in the articles [69; 70; 71] usurped significantly higher travel times for the whole network due to additional restrictions introduced by the authors. In contrast, the decision support model for infrastructure investment issues proposed by [PERSON] [72], in the absence of any investment, was less effective in optimising travel times. Table 13 compares the Nash equilibrium and the Stackelberg approach in terms of objective function, constraints and conditions, advantages and disadvantages. The Nash equilibrium aims at minimizing travel time for individual users.