From Fig.
2, we know that both expected detour time
\({\tilde{t}}\) and expected waiting time
w are related to the vehicle fleet size
N and ridesplitting demand
Q (
\(H_v\) can be replaced by
N and
Q). Here
Q is the vector of ridesplitting demand for all OD pairs. Thus, with a slight abuse of notation, we can rewrite detour time and waiting time as
\({\tilde{t}}(Q, N)\) and
w(
Q,
N), respectively. Recall that the expected travel time is the sum of direct trip time and expected detour time, so we can rewrite the travel time as
t(
Q,
N). The utility function for ridesplitting services is then given by
$$\begin{aligned} V_i(Q, N) = \beta _t t_i(Q, N) + \beta _w w_i(Q, N) + \beta _r r_i \end{aligned}$$
(17)
Substituting Equation (
17) into Equation (
9), the ridesplitting passenger demand thus becomes an implicit function of itself.
$$\begin{aligned} Q_i = \frac{D_i e^{V_{i}(Q, N)}}{e^{V_{i}(Q, N)} + \mu _i} \end{aligned}$$
(18)
The ME in the ridesplitting market is the ultimate stable state of the market (the supply-demand interaction eventually damps out), at which the relationships between the system endogenous variables (e.g., passenger demand, average detour time) can be satisfied under a specific operation strategy (e.g., vehicle fleet size, trip fare). Mathematically, the demand-supply equilibrium is established when both demand and supply equations are satisfied simultaneously (Arrow and Debreu
1954). More specifically, under certain operation strategies, an equilibrium in a ridesplitting market is a set of values of
\({\tilde{t}}_i, w_i\) and
\(Q_i\) that satisfies the equations system composed of Equation (
5), (
12) and (
16)–(
18) for all
i in
\({\mathbb {Z}}\). For convenience, we put them below.
$$\begin{aligned} N n_s&= H_v + \sum _{i} Q_i t_i \nonumber \\ {\tilde{t}}_i&= \frac{A_i \sum _j Q_j t_j^d}{N \sum _j Q_j}, \forall i \in {\mathbb {Z}} \nonumber \\ w_i&= \frac{B_i Q_i^{\theta }}{\varOmega _i \sqrt{N n_s - \sum _j Q_j t_j}}, \forall i \in {\mathbb {Z}} \nonumber \\ \varOmega _i&= \frac{n_z\sum _{j\in {\mathbb {Z}}_i} Q_j}{\sum _{k\in {\mathbb {Z}}} \sum _{j\in {\mathbb {Z}}_k} Q_j}, \forall i \in {\mathbb {Z}} \nonumber \\ V_i(Q, N)&= \beta _t t_i(Q, N) + \beta _w w_i(Q, N) + \beta _r r_i, \forall i \in {\mathbb {Z}} \nonumber \\ Q_i&= \frac{D_i e^{V_{i}(Q, N)}}{e^{V_{i}(Q, N)} + \mu _i}, \forall i \in {\mathbb {Z}} \end{aligned}$$
(19)
It is worth pointing out that Equation (
5) and the set of Equation (
18) given different
i describe the supply of and demand for ridesplitting services, respectively. In practice, this equations system can be solved via a hybrid method for nonlinear equations proposed in Powell (
1970). Our numerical experiments indicate that the resultant solutions are always unique under rational operation strategies.