## 4 Results
### Optimization with the Nash and Stackelberg Approaches
A transport system reaches an equilibrium state with the traffic flow distribution shown in the column,,Nash\" in Table 3. A movement O-D pair (3.8) may be implemented only through one route \(p_{6}\), therefore the entire traffic flow must be moved through route \(p_{6}\), regardless of the optimization approach. By translating the volume of traffic flow within individual routes of the O-D pairs in question, the authors calculated the volumes of traffic within the links of the road network. The results are listed in Table 4 and presented graphically in Figures 5 and 6. Traffic flow volume and travel time through links (1,4), (2,5) and (3,6) have the same values in the case of the Nash equilibrium and the Stackelberg approach. This results from the transport network structure, which implies the need to send all the traffic flow originating in nodes 1, 2 and 3, respectively, through these three links. The linear element most loaded of the road network is link (7,8), for which the traffic flow within the analyzed system exceeds 10,000 veh./h. On the other hand, the lowest volume is experienced in link (5,6). In the case of the Nash equilibrium, it is more than ten times higher than in the case of the Stackelberg approach traffic distribution. Road congestion values were calculated based on travel time through individual road network links obtained for individual solutions under the assumptions of the Nash and Stackelberg approaches. The level of congestion was calculated as a quotient of the travel time through the link (_i,j_) under specific traffic conditions and the travel time through this link under minimum traffic conditions. The authors calculated the an increase in travel time relative to the conditions of fully free traffic. The least favorable road conditions were observed for link (6,7), in the case of Nash equilibrium, where the transport time was more than 240% longer than under standard conditions. Regardless of the optimization approach, \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Link**} & \multicolumn{2}{c}{**Traffic Flow Volume (\(\times\)10\({}^{3}\) veh.)**} & \multicolumn{2}{c}{**Average Travel Time (Min)**} \\ \cline{2-5}  & **Nash** & **Stackelberg** & **Nash** & **Stackelberg** \\ \hline (1,4) & 4.000 & 4.000 & 5.60 & 5.60 \\ (2,5) & 5.000 & 5.000 & 11.83 & 11.83 \\ (3,6) & 7.000 & 7.000 & 7.44 & 7.44 \\ (4,7) & 1.610 & 3.026 & 4.17 & 4.49 \\ (4,8) & 5.973 & 5.835 & 9.94 & 9.72 \\ (5,4) & 3.583 & 4.861 & 4.38 & 4.64 \\ (5,6) & 1.417 & 0.139 & 5.13 & 5.01 \\ (6,7) & 8.417 & 7.139 & 3.42 & 2.78 \\ (7,8) & 10.027 & 10.165 & 5.77 & 5.87 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of optimization with the Nash and Stackelberg approaches for road network links in scenario #1. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**O-D Pair**} & \multirow{2}{*}{**Route**} & \multicolumn{2}{c}{**Traffic Flow Volume (\(\times\)10\({}^{3}\) veh. )**} & \multicolumn{2}{c}{**Average Travel Time (Min)**} & \multicolumn{2}{c}{**Marginal Travel Time (Min)**} \\ \cline{3-6}  & & **Nash** & **Stackelberg** & **Nash** & **Stackelberg** \\ \hline (1,8) & \(p_{1}\) & 4.000 & 4.000 & 15.54 & 15.32 & 28.30 \\  & \(p_{2}\) & 0.000 & 0.000 & - & - & - \\  & \(p_{3}\) & 1.973 & 1.835 & 26.14 & 26.19 & 39.73 \\ (2,8) & \(p_{4}\) & 1.610 & 3.026 & 26.14 & 26.83 &the lowest level of traffic congestion occurred for link (5,6), where the travel time increased by less than 3%. The results for all links in the road network are summarized in Table 5.