## Appendix A  
The domain of \(Z\) implies non-negative constrain because of the logarithmic part. For the given \(r-s\), the Lagrangian function can be written as follows:  
\[\begin{array}{l}L=\sum\limits_{a}\int_{0}^{x_{a}}c_{a}(w)dw+\frac{1}{\theta} \sum\limits_{rs}\sum\limits_{m}\sum\limits_{k}\mu_{m}f_{mk}^{rs}ln\frac{f_{mk }^{rs}}{\alpha_{mki}^{rs}{}^{1/l_{m}}}\\ +\frac{1}{\theta}\sum\limits_{rs}\sum\limits_{m}(1-\mu_{m})\left(\sum\limits_{ k}f_{mk}^{rs}\right)ln(\sum\limits_{k}f_{mk}^{rs})+\sum\limits_{rs}\sum\limits_{k} \lambda^{rs}\big{(}\alpha^{rs}-f_{k}^{rs}\big{)}\end{array} \tag{10}\]  
where \(\lambda^{rs}\) is the Lagrange coefficient for OD pair \(r-s\). Because \(\lambda^{rs}\) is a multiplier associated with equality constrain, it has no sign restriction. With derivatives of \(L\) to zero, we have:  
\[\sum\
olimits_{m}c_{m}\delta_{mk}+\frac{\mu_{m}}{\theta}\left(1+\ln\frac{f_{ mk}}{\alpha_{mki}{}^{1/\mu_{m}}}\right)+\frac{1-\mu_{m}}{\theta}\Big{(}1+ln\sum \
olimits_{k}f_{mk}\Big{)}+\lambda=0. \tag{11}\]  
The relationship cost on particularly path \(k\) is the sum of cost on links of this path and can be expressed as:  
\[c_{k}=\sum\
olimits_{m}c_{m}\delta_{mk}. \tag{12}\]  
Substituting the above formula to Equation (11) and multiplying \(\theta\) for both sides gives:  
\[\theta c_{k}+\mu_{m}ln\frac{f_{mk}}{\alpha_{mki}{}^{1/\mu_{m}}}+1+(1-\mu_{m}) ln\sum\
olimits_{k}f_{mk}-\theta\lambda=0. \tag{13}\]  
Put logarithmic term on left hand side:  
\[\mu_{m}\ln\frac{f_{mk}}{\alpha_{mki}^{\frac{1}{\mu_{m}}}\left(\sum\
olimits_{ k}f_{mk}\right)^{\frac{Em-1}{\mu_{m}}}}=\theta\lambda-\theta c_{k}-1, \tag{14}\]  
and consequently,  
\[\frac{f_{mk}}{\left(\sum\
olimits_{k}f_{mk}\right)^{\frac{Em-1}{\mu_{m}}}}= \alpha_{mki}{}^{1/\mu_{m}}exp\bigg{(}\frac{\theta\lambda-\theta c_{k}-1}{\mu_ {m}}\bigg{)}. \tag{15}\]  
Summing \(f_{mk}\) over \(k\):  
\[\Big{(}\sum\
olimits_{k}f_{mk}\Big{)}^{1/\mu_{m}}=\sum\
olimits_{k}\alpha_{ mki}{}^{1/\mu_{m}}exp\bigg{(}\frac{\theta\lambda-\theta c_{k}-1}{\mu_{m}} \bigg{)}, \tag{16}\]  
and consequently,  
\[\sum\
olimits_{k}f_{mk}=\exp(\theta\lambda-1)\bigg{(}\sum\
olimits_{k}\alpha_ {mki}{}^{1/\mu_{m}}exp\bigg{(}\frac{-\theta c_{k}}{\mu_{m}}\bigg{)}\bigg{)}^{ \mu_{m}}, \tag{17}\]  
and summing \(\sum\
olimits_{k}f_{mk}\) over \(m\):  
\[\sum\
olimits_{m}\sum\
olimits_{k}f_{mk}=exp(\theta\lambda-1)\sum\
olimits_{m }\bigg{(}\sum\
olimits_{k}\alpha_{mki}{}^{1/\mu_{m}}exp\bigg{(}\frac{-\theta c _{k}}{\mu_{m}}\bigg{)}\bigg{)}^{\mu_{m}}=q. \tag{18}\]  
Finally dividing Equation (17) by Equation (18) leads to marginal probability:  
\[p(m)=\frac{\sum\
olimits_{k}f_{mk}}{\sum\
olimits_{m}\sum\
olimits_{k}f_{mk}} =\frac{\left(\sum\
olimits_{k}\alpha_{mki}{}^{1/\mu_{m}}exp(-c_{k}\theta) \right)^{\mu_{m}}}{\sum\
olimits_{m}\left(\sum\
olimits_{k}\alpha_{mki}{}^{1/ \mu_{m}}exp(-c_{k}\theta)\right)^{\mu_{m}}}. \tag{19}\]Notice the Lagrange multiplier is eliminated in the above formulation, and the conditional probability of choosing path k given nest m can be written as:  
\[p(k|m)=\frac{f_{mk}}{\sum_{k}f_{mk}}=\frac{\left(\alpha_{miki}exp(-c_{k}\theta) \right)^{1/\mu_{m}}}{\sum_{k}(\alpha_{miki}exp(-c_{k}\theta))^{1/\mu_{m}}}.\] (A11)  
The probability of a user choosing path k is the product of marginal probability and conditional probability:  
\[p(k)=\sum_{m}p(m)p(k|m)=\frac{\sum_{m}\left(\left(\alpha_{miki}exp(-c_{k} \theta)\right)^{1/\mu_{m}}\left(\sum_{k}(\alpha_{miki}exp(-c_{k}\theta))^{1/\mu _{m}}\right)^{\mu_{m}-1}\right)}{\sum_{m}\left(\sum_{k}(\alpha_{miki}exp(-c_{k} \theta))^{1/\mu_{m}}\right)^{\mu_{m}}}.\] (A12)  
This form of result is identical to the form of [PERSON]'s theory of CNL choice model and [PERSON]'s GNL model as well when \(\mu\)s are different between nests.