*3.1. Specification*

It is assumed that every user *n* (*n* is not reported for simplicity) has 2 alternatives available in choice set **I**:

$$\mathbf{I} = \{\text{yes}, \text{no}\} \tag{1}$$

where yes and no refer to the alternative of preferring or not preferring the MaaS scenario in a hypothetical context (stated preference).

Given choice set **I**, the choice of the alternative can be modeled through different models belonging to discrete choice theory, i.e., a random utility model (RUM), a fuzzy UM (FUM), or a quantum UM (QUM). Each user perceives a utility for each alternative belonging to **I** (*Uyes* or *Uno*) and prefers the alternative with the maximum perceived utility. With the reported assumption, the choice probability for the alternatives belonging to I is given by the following:

$$p(\text{yes} \mid \mathbf{I}) = probability(\text{LI}\_{\text{yes}} \ge \mathbf{II}\_{\text{no}}, \text{yes} \in \mathbf{I}, \text{no} \in \mathbf{I}) \tag{2}$$

$$p(\text{no} \mid \mathbf{I}) = 1 - p(\text{yes} \mid \mathbf{I})$$

In this paper, a RUM is adopted in the Logit family (assumed identical and independent Gumbel probability distribution with parameter *θ* for the perceived utilities associated to the 2 alternatives). The preference probabilities for the 2 alternatives are as follows:

$$p(\text{yes} \mid \mathbf{I}) = \mathbf{e}^{V \text{yes} / \theta} / (\mathbf{e}^{V \text{yes} / \theta} + \mathbf{e}^{V \text{no} / \theta}) \tag{3}$$
 
$$p(\text{no} \mid \mathbf{I}) = 1 - p(\text{yes} \mid \mathbf{I})$$

where


The ratio between the expected value of utility (*Vk* with *k* equal to yes or no) and the Gumbel parameter (*θ*) are commonly assumed as a linear combination of attributes (*X1k*, *X2k*,... reported in the vector of attributes *Xk*), with parameters (*β*1, *β*2, ... , reported in the vector of parameters *β*) to be calibrated again during observation:

$$V\_k/\theta = \beta\_1 \bullet X\_1 + \beta\_2 \bullet X\_2 + \dots \dots = \mathfrak{F}' \bullet X\_k \qquad \forall \ k \in \mathbf{I} \tag{4}$$

The choice probability for alternative *k* depends on the vector of the parameters to be calibrated and can be reported as a function *ρ*() of alternative *k* subject to **I** and dependent on the vector of parameters *β* to be calibrated:

$$p(k \mid \mathbf{I}) = \rho(k \mid \mathbf{I}; \mathbf{\mathcal{J}}) \qquad \qquad \forall \ k \in \mathbf{I} \tag{5}$$
