## 3 Route Choice Modeling
### Model Description  
RRM models are developed as a counterpart method to RUM models to avoid IIA property in the most widely used multinomial logit (MNL) model. As a discrete choice paradigm, the RRM model with its non-linear formulation provides us a new perspective to analyze the decision-makers' (i.e., metro passengers' in our research) routing behaviors under the delays caused by metro emergencies [15]. The core of RRM models lies in the postulation that when choosing from a set of alternatives, passengers are more concerned about reducing their regret, which derives from their chosen alternative perform worse than the nonchosen alternatives in terms of one or several attributes. In order to achieve this goal, passengers have to compare on the attribute level between every two alternatives in the choice set, rather than just comparing the utility of the alternatives, which is the choice mechanism of RUM models. Following this particular principle, the formula of random regret can be given by Equation (4):  
\[RR_{i}=R_{i}+\varepsilon_{i}=\sum_{j\
eq i}\sum_{M}\text{ln}\Big{(}1+\exp \Big{(}\beta_{m}(x_{jm}-x_{im})\Big{)}\Big{)}+\varepsilon_{i} \tag{4}\]  
where \(RR_{i}\) is the random regret generated from choosing alternative \(i\); \(\varepsilon_{i}\) is the stochastic term referring to the unobserved factors that influence passengers' choices; \(R_{i}\) is the random regret value calculated by the aforementioned explanatory variables. \(\beta_{m}\) is the attribute \(x_{m}\)'s parameter that needs to be estimated; \(x_{im}\), \(x_{jm}\) are the values of attribute \(x_{m}\) of alternative \(i\) and alternative \(j\), \(i\), \(j\) belongs to the choice set \(C\); \(M\) is the number of attributes in an alternative, \(\text{m}\in M\).  
The probability of each alternative in the RRM model can be specified by Equation (5):  
\[\begin{split} P(i\|C)&=P(RR_{i}<RR_{j},\forall i, j\in C,i\
eq j)\\ &=P(-R_{i}>-R_{j})=\frac{\exp(-R_{i})}{\exp(-R_{i})+\frac{1}{C \
eq j}\exp(-R_{j})}\end{split} \tag{5}\]  
The non-linear characteristic of the RRM model can be seen in Equation (4), and it is capable of describing the change of random regret. When \(\beta_{m}\) and \(x_{jm}-x_{im}\) are synchronously positive or negative, which means that if the attribute has a positive (negative) effect, but the chosen alternative \(i\) performs worse (better) than alternative \(j\) in this attribute, the regret will rise rapidly. Conversely, when the product of \(\beta_{m}\) and \(x_{jm}-x_{im}\) is negative, the random regret is pretty close to 0. It is because that if the attribute has a positive effect (\(\beta_{m}>0\)) and alternative \(i\) outperforms alternative \(j\) (\(x_{jm}<x_{im}\)) in attribute \(x_{m}\), passengers have already made a good choice and there is no need to regret anymore. Comparing with the linear-additive RUM model, the non-linear characteristic result in another two unique properties of RRM models, semi-compensate property and compromise effect. Semi-compensate property refers to the asymmetry on the changes of attributes' values, in detail, the influence on the random regret of the increase (or decrease) of an attribute's value could not be fully compensated by the changes of another attribute. That is, if the route has the shortest travel time but the highest travel cost, the advantage of travel time and the disadvantage of travel cost cannot offset each other. This property leads to the fact that alternative with more balanced attribute values rather than alternatives with obvious good and bad attribute values can achieve the smallest random regret, and that is the phenomenon of compromise effect.