## Abstract

With the development of the Internet and mobile phone technology, it is much easier to access ridesharing information via mobile applications. In this paper, the relationship between the demand of ridesharing passengers (RPs), ridesharing drivers (RDs) and solo drivers (SDs) in a ridesharing compensation scheme is studied by a stochastic ridesharing user equilibrium (SRUE), which contains a mode choice model and a route choice model. The mode choice model and the route choice model influence each other. The SRUE is first expressed as a fixed-point problem mathematically. Six possible states of OD pairs are discussed. Then the existence of SRUE is proved. The method of successive weighted averages is adopted to solve the problem. It is found that there will be a higher demand of ridesharing passengers for journeys with longer travel time. Moreover, with the increase of the ridesharing compensation, the demand of ridesharing passengers is not always decreasing, and the demand of ridesharing drivers is not always increasing.

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## Acknowledgements

This work was supported by the Key Research and Development Program (No. 2016YFC0802508), and the National Natural Science Foundation of China (Nos.11672289, 71621001, 71631002).

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## Appendix

### Appendix

### Theorem 1

If the driving cost function \( {c}_v^w\left({t}_e^w\right) \) is a continuous and monotone increasing function of the equilibrium travel time \( {t}_e^w \), then the mapping function *q =**MC*(*t*_{e}) is continuous.

### Proof

Firstly, we prove that \( {q}_{rp}^w \), \( {q}_{rd}^w \) and \( {q}_{sd}^w \) are unique for any \( {t}_e^w \).

If the driving cost function \( {c}_v^w={c}_v^w\left({t}_e^w\right) \) is a monotone increasing function of the equilibrium travel time \( {t}_e^w \), its inverse function \( {t}_e^w={c_v^w}^{-1}\left({c}_v^w\right) \) is also a monotone increasing function.

Tables 1 and 2 show that each OD has six possible states according to the different value of *δ* and \( {c}_v^w \). Let’s firstly consider the first one of the six states: State 1 where *δ* > 0 and \( {c}_v^w<{c_v^w}^{\ast } \), which means \( {t}_e^w<{c_v^w}^{-1}\left({c_v^w}^{\ast}\right) \), \( {c}_{sd}^w<{\overline{c}}_{rd}^w<{\overline{c}}_{rp}^w \), \( {c}_{sd}^w<{c}_{rd}^w<{c}_{rp}^w \), \( {c}_{sd}^w<{c}_{rd}^w<{\overline{c}}_{rd}^w \) and \( {q}_{rp}^w<{q}_{rd}^w \). Let’s denote a function \( f\left(c,t\right)=c-\left({e}^{\theta c-{\theta c}_{rp}^w}\right){\overline{c}}_{rd}^w-\left(1-{e}^{\theta c-{\theta c}_{rp}^w}\right){c}_{sd}^w \). The partial derivative of *f*(*c*, *t*) on *c* is \( \frac{\partial f}{\partial c}=1-\theta \left({\overline{c}}_{rd}^w-{c}_{sd}^w\right){e}^{\theta c-{\theta c}_{rp}^w} \). For \( {c}_{sd}^w<{\overline{c}}_{rp}^w \), so \( \frac{\partial f}{\partial c}>0 \), which means it’s monotone increasing on *c* in State 1 if *t* is given. We can get that \( f\left({c}_{sd}^w,t\right)={c}_{sd}^w-\left({e}^{\theta {c}_{sd}^w-{\theta c}_{rp}^w}\right){\overline{c}}_{rd}^w-\left(1-{e}^{\theta {c}_{sd}^w-{\theta c}_{rp}^w}\right){c}_{sd}^w={e}^{\theta {c}_{sd}^w-{\theta c}_{rp}^w}\left({c}_{sd}^w-{\overline{c}}_{rd}^w\right)<0 \) and \( f\left({\overline{c}}_{rd}^w,t\right)={\overline{c}}_{rd}^w-\left({e}^{\theta {\overline{c}}_{rd}^w-{\theta c}_{rp}^w}\right){\overline{c}}_{rd}^w-\left(1-{e}^{\theta {\overline{c}}_{rd}^w-{\theta c}_{rp}^w}\right){c}_{sd}^w=\left(1-{e}^{\theta {\overline{c}}_{rd}^w-{\theta c}_{rp}^w}\right)\left({\overline{c}}_{rd}^w-{c}_{sd}^w\right)>0 \). We have known that the function \( f\left({c}_{sd}^w,{t}_e^w\right) \) is monotone increasing when the value of *c* is between \( {c}_{sd}^w \) and \( {\overline{c}}_{rd}^w \), and\( f\left({c}_{sd}^w\right)<0 \), \( f\left({\overline{c}}_{rd}^w\right)>0 \). As a result, there is one and only one \( c={c}_{rd}^w \) that \( f\left({c}_{rd}^w,{t}_e^w\right)=0 \). \( {c}_{rp}^w \) and \( {c}_{sd}^w \) can be obtained from Eqs. (4) and (16). As a result, the value of \( {c}_{rp}^w \), \( {c}_{rd}^w \) and \( {c}_{sd}^w \) is unique when \( {t}_e^w<{c_v^w}^{-1}\left({c_v^w}^{\ast}\right) \). By Eq. (13), \( {q}_{rp}^w \), \( {q}_{rd}^w \) and \( {q}_{sd}^w \) are unique when \( {t}_e^w<{c_v^w}^{-1}\left({c_v^w}^{\ast}\right) \).

For a similar reason, we can get that \( {q}_{rp}^w \), \( {q}_{rd}^w \) and \( {q}_{sd}^w \) are unique for the other five states. As a result, \( {q}_{rp}^w \), \( {q}_{rd}^w \) and \( {q}_{sd}^w \) are unique for any \( {t}_e^w \).

Then we prove that \( {q}_{rp}^w \), \( {q}_{rd}^w \) and \( {q}_{sd}^w \) are continuous at \( {t}_e^w \).

We still consider State 1 as an example. \( {c}_{rp}^w \) and \( {c}_{sd}^w \) are obtained from Eqs. (2) and (4). Obviously, \( {c}_{rp}^w \) and \( {c}_{sd}^w \) are continuous at \( {t}_e^w \). \( {c}_{rd}^w \) is obtained by solving Eq. (16). We have proved that \( {c}_{rd}^w \) is unique for any \( {t}_e^w \). Therefore, \( {c}_{rd}^w \) can be viewed as a function of \( {t}_e^w \). Denote the function is \( {c}_{rd}^w=h\left({t}_e^w\right) \). If *c*_{0} and *t*_{0} satisfy *c*_{0} = *h*(*t*_{0}), we want to prove *h*(*t*) is continuous at *t*_{0}. For \( \frac{\partial f}{\partial c}>0 \) in State 1, *f*(*c*_{0} + *ε*, *t*_{0}) > 0 and *f*(*c*_{0} − *ε*, *t*_{0}) < 0. For *f*(*c*, *t*) is continuous at *t*, ∃*t* ∈ (*t*_{0} − *δ*, *t*_{0} + *δ* ), *f*(*c*_{0} + *ε*, *t*) > 0 and *f*(*c*_{0} − *ε*, *t*) < 0. Therefore *c*_{0} + *ε* < *h*(*t*) < *c*_{0} − *ε*. As a result, \( {c}_{rd}^w=h\left({t}_e^w\right) \) is continuous at *t*_{0}. Therefore, \( {c}_{rp}^w \), \( {c}_{rd}^w \) and \( {c}_{sd}^w \) are continuous at \( {t}_e^w \). By Eq. (13), \( {q}_{rp}^w \), \( {q}_{rd}^w \) and \( {q}_{sd}^w \) are obtained by \( {c}_{rp}^w \), \( {c}_{rd}^w \) and \( {c}_{sd}^w \). In another word, \( {q}_{rp}^w \), \( {q}_{rd}^w \) and \( {q}_{sd}^w \) are continuous at \( {t}_e^w \) in this case.

For the other five states, we can get the same conclusion for similar reasons. Therefore, \( {q}_{rp}^w \), \( {q}_{rd}^w \) and \( {q}_{sd}^w \) are continuous at \( {t}_e^w \).

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Yan, C., Hu, M., Jiang, R. *et al.* Stochastic Ridesharing User Equilibrium in Transport Networks.
*Netw Spat Econ* **19, **1007–1030 (2019). https://doi.org/10.1007/s11067-019-9442-5

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### Keywords

- Transportation
- Network equilibrium
- Ridesharing
- Demand
- Expected cost