## 4 Results
### Optimization with the Nash and Stackelberg Approaches  
The Nash equilibrium is based on the assumption of equal average movement times per vehicle during transport operations performed within a transport network. The average travel time through route \(p\) is equal to the total of average travel times through the links that make up this link. The average travel times within the analyzed transport network are expressed by the relationship (25):  
\[\left\{ \begin{array}{c}t_{1,18}(x)=t_{14}(x_{14})+t_{48}(x_{48})\\ t_{2,18}(x)=t_{14}(x_{14})+t_{47}(x_{47})+t_{78}(x_{78})\\ t_{3,28}(x)=t_{25}(x_{25})+t_{54}(x_{54})+t_{48}(x_{48})\\ t_{4,28}(x)=t_{25}(x_{25})+t_{54}(x_{54})+t_{47}(x_{47})+t_{78}(x_{78})\\ t_{5,28}(x)=t_{25}(x_{25})+t_{56}(x_{56})+t_{67}(x_{67})+t_{78}(x_{78})\\ t_{6,38}(x)=t_{36}(x_{36})+t_{67}(x_{67})+t_{78}(x_{78})\end{array} \right.. \tag{25}\]For the travel O-D pair (1,8) there are two routes \(p_{1}\) and \(p_{2}\), while for the O-D pair (2,8), three possible routes \(p_{3}\), \(p_{4}\) and \(p_{5}\) were established. The state of equilibrium is achieved if the following conditions are met (26):  
\[\left\{ \begin{array}{l}t_{1,18}(x)=t_{2,18}(x)\\ t_{3,28}(x)=t_{4,28}(x)=t_{5,28}(x)\end{array} \right.. \tag{26}\]  
The O-D pair (3,8) has only one possible movement route, resulting in the lack of equilibrium for flows moving between nodes 3 and 8. The equilibrium state described by the system of Equations (26) corresponds to a situation in which a change in the route of movement of any participant in the traffic would result in an extended travel time. To perform the optimization using the Stackelberg approach, the marginal times \(m_{ij}(x_{ij})\) for individual links of the road network were calculated. They are presented in the form of relationship (27):  
\[\left\{ \begin{array}{l}m_{14}(x_{14})=\frac{\mathrm{d}[t_{14}(x_{14})\cdot x _{14}]}{\mathrm{d}x_{14}}=5+0.1x_{14}+0.075x_{14}^{2}\\ m_{25}(x_{25})=\frac{\mathrm{d}[t_{25}(x_{25})\cdot x_{25}]}{\mathrm{d}x_{25}}= 10+0.08x_{25}+0.195x_{25}^{2}\\ m_{36}(x_{36})=\frac{\mathrm{d}[t_{36}(x_{36})\cdot x_{36}]}{\mathrm{d}x_{36}} =6+0.06x_{36}+0.075x_{36}^{2}\\ m_{47}(x_{47})=\frac{\mathrm{d}[t_{47}(x_{47})\cdot x_{57}]}{\mathrm{d}x_{47}} =4+0.08x_{47}+0.12x_{47}^{2}\\ m_{48}(x_{48})=\frac{\mathrm{d}[t_{48}(x_{48})\cdot x_{48}]}{\mathrm{d}x_{48}} =5+0.16x_{48}+0.375x_{48}^{2}\\ m_{54}(x_{54})=\frac{\mathrm{d}[t_{54}(x_{54})\cdot x_{54}]}{\mathrm{d}x_{54}} =4+0.07x_{54}+0.06x_{54}^{2}\\ m_{56}(x_{56})=\frac{\mathrm{d}[t_{56}(x_{56})\cdot x_{56}]}{\mathrm{d}x_{56}} =5+0.07x_{56}+0.12x_{56}^{2}\\ m_{67}(x_{67})=\frac{\mathrm{d}[t_{57}(x_{57})\cdot x_{57}]}{\mathrm{d}x_{57}} =1+0.07x_{67}+0.09x_{67}^{2}\\ m_{78}(x_{78})=\frac{\mathrm{d}[t_{78}(x_{78})\cdot x_{78}]}{\mathrm{d}x_{78}} =2+0.05x_{78}+0.105x_{78}^{2}\end{array} \right. \tag{27}\]  
The Stackelberg approach assumes the presence of a minimum total movement time with equal marginal travel times on individual routes in the transport network routes. The marginal travel time through route \(p\) is equal to the total marginal travel times through the links comprising route \(p\). Marginal travel times within the analyzed transport network are expressed by the relationship (28):  
\[\left\{\begin{array}{l}m_{1,18}(x)=m_{14}(x_{14})+m_{48}(x_{48})\\ m_{2,18}(x)=m_{14}(x_{14})+m_{47}(x_{47})+m_{78}(x_{78})\\ m_{3,28}(x)=m_{25}(x_{25})+m_{54}(x_{54})+m_{48}(x_{48})\\ m_{4,28}(x)=m_{25}(x_{25})+m_{54}(x_{54})+m_{47}(x_{47})+m_{78}(x_{Calculations for both optimization approaches were performed with computer assistance using LINGO 18.0 software (Lindo Systems, Chicago, USA). Table 3 lists the results obtained for the individual routes of the transport network. According to the assumptions of the Nash equilibrium, the travel times for each available route \(p\) within each O-D pair (\(a\), \(b\)) are equal.