\section{Introduction}
Access to transportation is a crucial factor in determining quality of life. In highly populated locations, public transit can provide mobility services on a large scale, connecting individuals to jobs, educational opportunities, nutritious food, and medical visits \cite{sanchez2004transit}. In low-density locations, however, public transit agencies find it financially difficult to maintain an acceptable level of service, resulting in restricted coverage or transit gaps \cite{borjesson2020rural}. In these underserved locations, rich inhabitants can still use their own vehicles to go around, but disadvantaged groups (e.g., low-income individuals) are more likely to rely on public transportation and have fewer options to meet their fundamental travel needs.
This transit equity gap hurts us all: when our neighbors are isolated and struggling, it undermines the local economy and weakens the social cohesion of the city. Today, these negative impacts are further compounded by the COVID-19 pandemic \cite{meredith2021relationship}. Closing this equity gap is more important than before.

However, transportation is on the verge of dramatic upheaval.  Autonomous vehicle (AV) technology is being tested and deployed in numerous cities around the world. While level-5 fully autonomous driving could be decades away, ride-hailing services with level-4 AVs, offered by transportation network companies (TNCs) in urban and suburban areas, are expected to arrive much earlier \cite{Waymo2022SF}. Automation will bring radical changes to TNC business models: (a) compared to human drivers who selfishly reposition themselves to pursue individual earning opportunities, AVs can be centrally dispatched to achieve system-level objectives \cite{zhang2016control}; (b) compared to human drivers who share a significant portion of the trip fare, AVs can remove the costs associated with human drivers and reduce the trip fare substantially \cite{chen2016operations}. These favorable factors have profound implications on transport equity: with better control of their fleets and lower operational costs, TNCs can improve mobility services in underserved areas and enhance transportation access for disadvantaged populations while preserving financial sustainability.

Despite its potential in mitigating inequities, the actual equity impact of AV deployment in the TNC market remains uncertain. Unlike public transportation, AV technology is being led by multi-billion dollar companies in the private sector. The primary goal of these companies is to maximize the profits, which is naturally contradicted with the equity objectives of the social planner. As a result, the deployment of AV through TNC platforms may negatively affect spatial equity\footnote{ Spatial equity, also referred to as horizontal equity, requires that people in different regions of the city should
have access to the same quality of mobility services.} and social equity\footnote{Social equity, also referred to as vertical equity, requires that people that are disadvantaged in socioeconomic status (e.g., income, education, etc) should be favored by transport policies.} within the multimodal transportation network. On the spatial side, for-profit ride-hailing companies (e.g., Uber, Waymo, Baidu, etc.) tend to focus their services in regions with high demand \cite{hughes2016transportation}. These high-demand areas are often congested communities in the city core, which already have robust public transit services. The geographic concentration of TNC services will further enlarge the accessibility gap between wellserved and underserved areas, which exacerbates spatial inequity. On the social side, as for-hire AVs flock into these areas, they will compete with public transit, degrade transit service qualities, and disproportionately affect low-income individuals that are more transit-dependent \cite{hall2018uber,erhardt2022transportation}. This will enlarge the accessibility gap between passengers with distinct socioeconomic status (e.g., level of income),  which exacerbates social inequity. These issues can be further reinforced as AV technology enables TNCs to have lower costs and stronger control over their fleet. Without regulatory interventions, market forces may drive us to a transport equity catastrophe: public transit ridership will decline, traffic congestion will increase, people who are already transportation-disadvantaged will be left further behind, and the benefits of AVs will be inaccessible to those who need them most.

This paper aims to assess the equity impacts of for-hire AVs in the ride-hailing network and investigate regulatory policies that ensure AV deployment can benefit underserved areas and disadvantaged groups. We consider a TNC platform that deploys a fleet of AVs to offer mobility-on-demand services to maximize its profit, a public transit agency that offers transit services to maximize transit ridership under a fixed budget, and a group of multiclass passengers with different income levels who can choose among ride-hailing, public transit, and bundled services that combines ride-hailing and public transit to reach their destinations. To maintain an equitable multimodal transportation network, it is crucial to investigate: (a) whether AV deployment will exaggerate spatial and social inequity in an unregulated environment; (b) whether it is necessary to impose regulatory intervention; and (c) how to design regulatory policies to mitigate the negative impacts of AVs and improve transport equity. To this end, we will develop a game-theoretic network equilibrium model to examine how travelers with heterogeneous socioeconomic attributes make mode choices in the multimodal transportation system, how the TNC platform and the transit agency interact with each other to reach their own objectives, and how these interactions are affected by the imposed regulations. The major contribution of this paper are summarized below:
\begin{itemize}
    \item We develop a game-theoretic model to characterize the strategic interactions between the TNC platform and the public transit agency over the transport network. The model captures the essential economic aspects  of the multimodal transportation network, including the spatial prices of the TNC platform, the service fare and frequencies of public transit, the spatial distribution of idle AVs, waiting times for ride-hailing and transit services, and the modal choice of multiclass passengers with distinct income groups. We use the model to evaluate the spatial and social equity in transport access among distinct population groups. {\em To our best knowledge, this is the first work that studies spatial and social equity for multimodal transportation networks considering the strategic interaction between TNC platforms and public transit agencies.}
    \item The proposed game-theoretic model consists of a profit-maximizing subproblem for the TNC platform and a ridership maximization subproblem for the public transit agency, which are interdependent and highly non-concave. We use the best response algorithm to derive a candidate equilibrium solution, and evaluate the quality of these solutions by exploring the special structures of each subproblem. In particular, for TNC's profit-maximization subproblem, we use primal decomposition to compute the globally optimal solution to a relaxed reformulation of the problem, which establishes a tight upper bound to evaluate the performance of TNC's decisions. For the transit ridership maximization subproblem, we reformulate the problem and find conditions under which the reformulated problem is concave and its globally optimal solution can be efficiently obtained. Overall, this enable us to assess the quality of the derived candidate equilibrium solution by showing that it is at least as good as an $\epsilon$-Nash equilibrium \cite{daskalakis2006note, li2019connections}, where $\epsilon$ can be determined numerically through the above procedure.
    \item We use Theil index to quantify both spatial and social inequity in transport accessibility and investigate the equity impacts of AV deployment in the absence of regulations. We find that although the proliferation of AVs improves overall accessibility, the benefits are not fairly distributed across different geographic locations and among distinct population groups, and the deployment of AVs will exacerbate both spatial and social inequity gaps at the same time. We point out that the increase in spatial inequity arises from the geographic concentration of ride-hailing services in high-demand areas due to the for-profit nature of the TNC platform, and the increased social inequity gap is due to the fact that the benefits of AV-enabled ride-hailing services are primarily enjoyed by  individuals with higher income, whereas those with lower income  are disproportionately transit-dependent.
    \item We evaluate the equity impacts of two regulatory policies, including (a) a minimum service-level requirement on ride-hailing service, which improves the spatial equity in the transport network; (b) a subsidy on ride-hailing trips that serve as first/last-mile connection to public transit, which promotes the use of public transit and improves the social equity of the transport network.  We show that the minimum service-level requirement entails a trade-off: as a higher minimum service level is imposed, the spatial inequity reduces, but the social inequity will be exacerbated. In contrast, subsidies on bundled trips can enhance accessibility, increase transit ridership,  improve the profit of the ride-hailing platform, and more crucially, simultaneously bridge spatial and social inequity gaps. The reasons for the above results are identified and carefully discussed in Section \ref{service} and \ref{subsidies}, respectively.
\end{itemize}

The remainder of this article will proceed as follows.  Related works in the existing literature will be surveyed in Section \ref{related works}. A game-theoretic model will be developed for the competition between TNC and transit in Section \ref{game-theoretic model}. A solution method will be proposed in Section \ref{solution method}, and numerical studies will be presented in Section \ref{market outcomes}-\ref{regulations}. Finally, concluding remarks are offered in Section \ref{conclusion}.

\section{Related Works} \label{related works}

There is a large body of literature that investigates the operational strategies of for-hire AV fleets, examines the interaction between TNC and public transit, quantifies their impacts on transport equity, and identifies measures to mitigate their negative externalities. Below we summarize the relevant literature from four aspects: (a) autonomous mobility-on-demand services; (b) interactions between TNC and public transit; (c) impacts of AVs and TNCs on transport equity; and (d) regulations on AVs and TNCs.

\subsection{Autonomous Mobility-on-Demand Services}

A recurrent topic of interest in the literature is the planning and operation of for-hire AV fleets to provide autonomous mobility-on-demand (AMoD) services. \cite{zhang2016control} presented a queuing-theoretic model to investigate the optimal rebalancing strategies of AMoD fleets. Their case study of New York showed that the current taxi demand in Manhattan can be met with about 8,000 AVs (roughly 70\% of the size of the current taxi fleet operating in Manhattan). \cite{wen2019value} investigated the value of demand information in AMoD systems and showed that aggregate demand information can lead to better service performance. \cite{zhang2016model} developed a model predictive control (MPC) approach to optimize vehicle scheduling and routing in AMoD systems. \cite{tsao2018stochastic} utilized short-term probabilistic forecasts for dispatching and rebalancing of AMoD systems and developed a stochastic MPC algorithm to efficiently solve the problem. \cite{alonso2017predictive} proposed a predictive method for the vehicle routing and assignment for an AMoD system with ridesharing. \cite{iglesias2019bcmp} formulated AMoD systems as a closed, multi-class Baskett-Chandy-Muntz-Palacios (BCMP) queuing network model to characterize the passenger arrival process, traffic, the state-of-charge of electric vehicles, and the availability of vehicles at the stations and derive the routing and charging policies. \cite{hyland2018dynamic} used agent-based simulations to evaluate the performance of AV-traveler assignment strategies. Their results showed that the spatial distribution of traveler requests significantly impacts the empty fleet miles generated by shared AVs. \cite{gueriau2018samod,guo2020deep} investigated the ride-sharing vehicle dispatching as well as the request-vehicle assignment problem for AMoD systems using reinforcement learning. \cite{chen2016operations} examined the operation of shared electric AV fleets under distinct vehicle range and charging infrastructure settings. \cite{turan2020dynamic} investigated the joint routing, battery charging, and pricing for electrified AMoD systems and used deep reinforcement learning to develop a near-optimal control policy. For a comprehensive literature review on the operation of AMoD systems, please refer to \cite{zardini2022analysis}.
{\em Note that all the aforementioned works primarily focus on the planning and operation of AMoD systems, while the interaction between AMoD and other modes is not explicitly considered.}

\subsection{Interactions Between AMoD and Public Transit}

An increasing body of literature revealed the interaction between AMoD/ridesourcing services and public transit using both data-driven and model-based methods. Based on real-world data, \cite{hall2018uber} estimated the effect of Uber on public transit ridership using a difference-in-differences design. Their results showed that Uber is a complement to the average transit agency, increasing ridership by five percent after two years. On the other hand, \cite{meredith2021relationship} analyzed detailed individual trip records at a spatially and temporally granular level, and found that only approximately 2\% of TNC trips complement public transit, while 45\% to 50\% of TNC trips substitute transit and this percentage drops during COVID-19 shutdowns. Based on realistic TNC data and public transit data, \cite{erhardt2022transportation} found that TNCs are responsible for a net ridership decline of about 10\%, offsetting net gains from other factors such as service increases and population growth. In modeling-wise, \cite{ke2021equilibrium} proposed a user equilibrium based mathematical model to explore the complement and substitution of ride-sourcing to public transit. \cite{zhu2020analysis} established a network equilibrium model to analyze the multimodal commute behavior with ridesplitting programs as both feeders and competitors to public transit. \cite{zhu2021competition} developed a bilevel game-theoretic approach to model the cooperative and competitive relationship between the TNC and the government. They highlighted that a carefully designed subsidy can benefits both the TNC and society, especially in areas with low public transit accessibility. \cite{salazar2019intermodal} and \cite{gurumurthy2020first} considered AMoD services for first/last-mile connection to public transport. \cite{salazar2018interaction} presented a network flow model to capture the interaction between AMoD and public and transit, and designed a pricing and tolling scheme that achieves the social optimum. \cite{pinto2020joint}, \cite{sieber2020improved} and \cite{kumar2022planning} studied the joint planning and operation of shared AV fleet and public transit to improve mobility services in low-density areas.

This paper significantly differs from the aforementioned works in two aspects: (a) the aforementioned studies primarily consider the multimodal transportation system at the aggregate level, and in contrast, our work considers a multimodal transportation network with multiclass travelers; (b) all aforementioned studies neglect the equity impacts of ride-hailing services in the multimodal transportation systems. {\em To our best knowledge, our paper is the first work that studies spatial and social equity for multimodal transportation networks considering the strategic interaction between TNC platforms and public transit agencies.}

\subsection{Impacts of AVs and TNCs on Transport Equity}

Emerging technologies, such as AVs and TNCs, have profoundly disrupted the transportation sector, but their impacts on transport equity remain unclear \cite{dianin2021implications}. On the one hand, AVs and TNCs have great potential to close existing inequity gaps \cite{shaheen2017travel}. \cite{brown2018ridehail} revealed that black riders were 73 percent more likely than white riders to have a taxi trip canceled and waited for 6-15 minutes longer than white riders, while TNC services nearly eliminate the racial-ethnic discrimination in service quality. \cite{palm2021equity} found significant academic evidence that new mobility technologies like ride-hailing have the potential to meaningfully address some disadvantaged travelers’ transportation problems or close existing gaps in transit services like the first/last mile. \cite{wen2018transit} simulated and evaluated integrated AV and public transportation systems and showed that encouraging ride-sharing, allowing in-advance requests, and combining fare with transit help enable service integration and encourage sustainable travel. \cite{chen2017connecting} compared two different relative spatial position (RSP) designs in an integrated e-hailing/fixed-route transit system and showed the great potential of e-hailing in filling transit gaps. \cite{cohn2019examining} and \cite{nahmias2021benefits} examined the equity impacts of AVs and showed that AMoD services could expand transportation access to car-free and underserved populations and bridge the mobility gaps between demographic groups. \cite{ahmed2020quantifying} revealed that shared AV mobility services can significantly increase job accessibility and low-income workers in low-density areas enjoy a higher benefit. On the other hand, there are growing concerns that TNCs and AVs may widen existing inequity gaps. \cite{ge2016racial} and \cite{yang2021equitable} investigated racial discrimination in TNCs and reported that customers with color experience more trip cancellations and longer waiting times. \cite{hughes2016transportation} studied the spatial distribution of TNC services and found that waiting time is lower in densely populated areas. In addition, the deployment of AV technology could also raise new issues such as induced demand \cite{cohn2019examining} and exacerbated geographic concentration \cite{jiao2021shared}. Given these concerns, it is widely agreed that regulatory policies should be imposed to ensure that AV technology promotes equity in future mobility systems \cite{milakis2017policy}.

{\em Distinct from most aforementioned works, we aim to reveal the fundamental mechanisms by which market incentives lead to systemic transport inequity under the interaction between TNC and public transit.} We will develop a mathematical framework that characterizes how market forces drive TNC services and public transit in the absence of regulatory intervention and quantifies both spatial and social inequity in the multimodal transportation system.

\subsection{Regulations on AVs and TNCs}

Regulations on TNCs and AVs are emerging topics that receive increasing attention from the transportation research community recently. In the existing literature, policy studies on human-driver TNCs are primarily focused on driver minimum wage \cite{parrott2018earnings,li2019regulating, shetty2022analysis}, congestion charges \cite{li2021impact,li2021spatial}, and cap on the platform commission rate \cite{vignon2021regulating,vignon2023regulating}. \cite{parrott2018earnings} examined the effects of the proposed minimum wage policy in New York City. \cite{li2019regulating} developed a queueing theoretic equilibrium model to assess the impacts of distinct regulations on TNCs including (1) A minimum wage for drivers, (2) a cap on the number of drivers or vehicles, and (3) a per-trip congestion tax. \cite{li2021impact} further compared the impacts of two types of congestion charge: (a) a charge per TNC trip, and (b) a charge per vehicle operating hour, on TNCs. \cite{li2021spatial} studied the spatial pricing for ride-sourcing platforms under congestion charge and evaluated the impacts of three forms of congestion charge: (a) a one-directional cordon charge on ride-sourcing vehicles that enter the congestion area; (b) a bi-directional cordon charge on ride-sourcing vehicles that enter or exit the congestion area; (c) a trip-based congestion charge on all ride-sourcing trips. \cite{vignon2021regulating} showed that a commission cap and a toll are effective in regulating the market and achieving a socially desirable outcome. \cite{vignon2023regulating} regulated the competitive market with ride-hailing services and street-hailing services and found that commission cap regulation can replicate the second-best in the market and it is effective even in a congested setting.

On the other hand, regulations on AMoD services are relatively under-examined, and we only find a handful of relevant works. For instance,  \cite{freemark2020policies} surveyed municipal officials throughout the U.S. to investigate how they support various AV-related regulatory policies. \cite{simoni2019congestion} used agent-based simulations to investigate how toll strategies on AMoD systems impact traffic congestion and social welfare. \cite{salazar2019intermodal} considered an intermodal AMoD system that coordinates with public transit, and designed a tolling scheme to achieve the social optimum in a perfect market with selfish agents. \cite{dandl2021regulating} developed a tri-level framework to capture the interactions among regulatory agencies, AMoD service providers, and travelers.

{\em Although TNC regulations have garnered a growing amount of attention in recent years, neither the equity aspects of TNC regulation nor the interactions between distinct modes under TNC regulation have been investigated.  Our paper differs from all aforementioned works as we investigate how to design policies that interact with the multimodal transportation system to mitigate spatial and social inequity. This has not been studied in the existing literature.}

\section{The Game-Theoretic Model} \label{game-theoretic model}

Consider a city divided into $M$ geographic zones. These zones are connected by a multimodal transport network which consists of a road network $\mathcal{G}_r(\mathcal{V},\mathcal{E}_r)$ and a transit network $\mathcal{G}_t(\mathcal{V},\mathcal{E}_t)$, where $\mathcal{V}$ denotes the set of zones, $\mathcal{E}_r$ denotes the set of roads, and $\mathcal{E}_t$ denotes the set of transit lines. We consider passenger trips at the zonal granularity and assign each trip with an origin zone $i\in\mathcal{V}$ and a destination zone $j\in\mathcal{V}$. The travel demand is subdivided into $K$ classes based on income levels in each zone. In the multimodal transport system, the TNC platform operates a fleet of AVs to provide mobility-on-demand services on the road network, and the public transit agency provides transit services through the transit network, and passengers in different classes make mode choices over the multimodal transport network. The decision-making of passengers, the TNC platform, and the public transit agency interact with each other and constitute the market equilibrium. This section formulates a mathematical model to capture the competition between the TNC platform and the public transit agency over a transportation network. The details of the model are presented below.

\subsection{Incentives of Passengers}

We consider a discrete choice model for passengers who choose among: (1) direct ride-hailing/AMoD services, indexed by $a$; (2) public transit, indexed by $p$; (3) bundled services combining AMoD and transit, indexed by $b$; and (4) outside option, indexed by $o$, to reach their destinations at the minimum cost. Using AMoD for the first-mile or/and last-mile connection, bundled services can be further categorized into three possible scenarios: (1) the bundle of first-mile AMoD service and public transit, indexed by $b_1$; (2) the bundle of last-mile AMoD service and public transit, indexed by $b_2$; and (3) the bundle of both first-mile and last-mile AMoD service and public transit, indexed by $b_3$.  The costs of different mobility modes collectively determine passenger demand through the following relation:
\begin{equation} \label{general_demand_function}
\begin{cases}
    \lambda_{ij,k}^a = \lambda_{ij,k}^0 F^a\left(c_{ij,k}^a,c_{ij,k}^p,c_{ij,k}^{b_1},c_{ij,k}^{b_2},c_{ij,k}^{b_3},c_{ij,k}^{o}\right) \\
    \lambda_{ij,k}^p = \lambda_{ij,k}^0 F^p\left(c_{ij,k}^a,c_{ij,k}^p,c_{ij,k}^{b_1},c_{ij,k}^{b_2},c_{ij,k}^{b_3},c_{ij,k}^{o}\right) \\
    \lambda_{ij,k}^{b_1} = \lambda_{ij,k}^0 F^{b_1}\left(c_{ij,k}^a,c_{ij,k}^p,c_{ij,k}^{b_1},c_{ij,k}^{b_2},c_{ij,k}^{b_3},c_{ij,k}^{o}\right) \\
    \lambda_{ij,k}^{b_2} = \lambda_{ij,k}^0 F^{b_2}\left(c_{ij,k}^a,c_{ij,k}^p,c_{ij,k}^{b_1},c_{ij,k}^{b_2},c_{ij,k}^{b_3},c_{ij,k}^{o}\right) \\
    \lambda_{ij,k}^{b_3} = \lambda_{ij,k}^0 F^{b_3}\left(c_{ij,k}^a,c_{ij,k}^p,c_{ij,k}^{b_1},c_{ij,k}^{b_2},c_{ij,k}^{b_3},c_{ij,k}^{o}\right) \\
    \lambda_{ij,k}^{o} = \lambda_{ij,k}^0 F^{o}\left(c_{ij,k}^a,c_{ij,k}^p,c_{ij,k}^{b_1},c_{ij,k}^{b_2},c_{ij,k}^{b_3},c_{ij,k}^{o}\right)
\end{cases},
\end{equation}
where $\lambda_{ij,k}^{t},t\in \mathcal{T}=\{a,p,b_1,b_2,b_3,o\}$ represents the arrival rate of passengers of mode $t$ from origin zone $i$ to destination zone $j$ in income class $k$; $\lambda_{ij,k}^0$ is the arrival rate of potential passengers from zone $i$ to zone $j$ in  income class $k$; $c_{ij,k}^{t},t\in \mathcal{T}$ denotes the average generalized travel costs for passengers from origin $i$ to destination $j$ in income class $k$ by choosing mode $t$; $F^{t}(\cdot),t\in \mathcal{T}$ is an aggregate function which captures the proportion of passengers choosing mode $t$. We assume that $F^t(\cdot)$ is a continuously differentiable function with $\frac{\partial F^t}{\partial c_{ij,k}^{t'}}<0$ if $t=t'$ and $\frac{\partial F^t}{\partial c_{ij,k}^{t'}}\geq 0$ otherwise. This indicates that the demand for a specific mode is negatively dependent on the generalized cost of this mode, but positively dependent on the generalized cost of its alternative modes. Under the well-established logit model, the passenger demand over distinct modes can be characterized as:
\begin{equation} \label{logit_demand_function}
    \lambda_{ij,k}^t = \lambda_{ij,k}^0 \frac{\exp \left(-\epsilon c_{ij,k}^t\right)}{\sum_{{t'} \in \mathcal{T}} \exp \left(-\epsilon c_{ij,k}^{t'}\right)} \quad t \in \mathcal{T} ,
\end{equation}
where $\epsilon$ is the scaling parameter in the logit model. Note that (\ref{general_demand_function}) includes the logit model (\ref{logit_demand_function}) as a special case, and the logit model is a classical discrete choice model which has been extensively applied in traffic demand modeling \cite{ahmed2020quantifying,ke2021equilibrium}. For analytical tractability, we use the logit model (\ref{logit_demand_function}) as the passenger demand functions throughout rest of this paper.

{The generalized travel costs of trips by different mobility modes are defined as the weighted sum of waiting time, in-vehicle time, trip fare, and access/walking time (if any).} They satisfy:
\begin{equation} \label{generalized_travel_cost}
    \begin{cases}
    c_{ij,k}^a = \alpha_k w_i^a + \beta_k \frac{l_{ij}^a}{v_a} + \gamma_k (b + r_i^a l_{ij}^a) \\
    c_{ij,k}^p = \alpha_k w_{ij}^p + \beta_k \frac{l_{ij}^p}{v_p} + \gamma_k r^p l_{ij}^p + \theta_k \left(\frac{d_i}{v_w} + \frac{d_j}{v_w}\right) \\
    c_{ij,k}^{b_1} = \alpha_k \left(w_i^a + w_{ij}^p \right) + \beta_k \left(\frac{d_i}{v_a} +  \frac{l_{ij}^p}{v_p}\right) + \gamma_k \left(b + r_i^a d_i + r^p l_{ij}^p\right) + \theta_k \frac{d_j}{v_w} \\
    c_{ij,k}^{b_2} = \alpha_k \left( w_{ij}^p + w_j^a \right) + \beta_k \left( \frac{l_{ij}^p}{v_p} + \frac{d_j}{v_a}\right) + \gamma_k \left( r^p l_{ij}^p + b + r_j^a d_j\right) + \theta_k \frac{d_i}{v_w} \\
    c_{ij,k}^{b_3} = \alpha_k \left(w_i^a + w_{ij}^p + w_j^a\right) + \beta_k \left(\frac{d_i}{v_a} +  \frac{l_{ij}^p}{v_p} + \frac{d_j}{v_a}\right) + \gamma_k \left(b + r_i^a d_i + r^p l_{ij}^p + b + r_j^a d_j\right) \\
    \end{cases} ,
\end{equation}
where $w_i^a$ is the average waiting time for AMoD services in zone $i$; $w_{ij}^p$ is the average waiting time for transit services from zone $i$ to zone $j$; $b$ is the average base fare of AMoD rides; $r_i^a$ is the average per-distance rate of AMoD rides originating from origin zone $i$; $r^p$ is the average per-distance transit fare; $l_{ij}^a$ and $l_{ij}^p$ denote the average trip distances from zone $i$ to zone $j$ using transit and AMoD, respectively; $d_i$ is the average first/last-mile distance (i.e., the average distance from the exact origin/destination to the nearest transit station) in zone $i$; $v_a$, $v_p$ and $v_w$ represent the average speed of TNC autonomous vehicles, the average operating speed of public transit, and the average walking speed, respectively; $\alpha_k$, $\beta_k$, $\gamma_k$ and $\theta_k$ are the average weights associated with waiting time, in-vehicle time and trip fare, and walking time for passengers in income class $k$, which correspond to passengers' valuation on waiting time, trip time, monetary cost and walking time, respectively.

\begin{remark}
The structure of the TNC platform's pricing scheme $b+r_i^a l_{ij}^a$ is consistent with the industry practice. For instance, both Uber and Lyft have a base fare, a fixed per-time fare and a per-distance fare. They first calculate the variable trip fare based on the trip time (time-based charge) and trip distance (distance-based charge), then the total trip fare is calculated as the sum of the fixed base fare and the variable trip fare multiplied by a surge multiplier that reflects the real-time imbalance between supply and demand in each zone $i$. Considering a uniform average speed of TNC vehicles on roads, the time-based charge is proportional to the distance-based charge. In this case, the pricing policy $b+r_i^a l_{ij}^a$ can be viewed as an approximation of the sum of the base fare, the time-based charge and the distance-based charge multiplied by the zone-specific surge multiplier.
\end{remark}

The average passenger waiting time for AMoD services and transit services depends on the operational decisions of the TNC platform and the public transit agency. Below we characterize these endogenous relations for AMoD services and public transit separately.

{\em Waiting time for AMoD services}: The passenger waiting time for AMoD services $w_i^a$ depends on the passenger demand and the vehicle supply in the TNC market. To characterize $w_i^a$, note that each trip is initiated by a passenger who requests a pickup from zone $i$ through the user app (for either direct AMoD service or first-mile/last-mile service). Upon receiving the request, the platform immediately matches the passenger to the closest idle vehicle in the same zone. The matched idle vehicle then travels to pick up the waiting passenger. In this case, the TNC platform does not prioritize either type of AMoD services in the vehicle-passenger matching process, and passengers are served on a first-come-first-served basis. Consequently, the direct AMoD rides and  AMoD rides as first/last mile connections are indifferent in passenger waiting time. Typically, the ride confirmation time (from the ride request being submitted to a vehicle being assigned) is negligible compared to the pickup time (from a vehicle being assigned to passenger pickup) in the matching process. Therefore, we ignore the ride confirmation time and approximate the passenger waiting time as the passenger pickup time. The pickup time depends on the distance of the nearest idle vehicle to the passenger, and the passenger waiting time $w_i^a$ can be characterized as a monotone function of the average number of idle vehicles $N_i^I$ in zone $i$. With slight abuse of notation, we use $w_i^a\left(N_i^I\right)$ to denote the passenger waiting time function of AMoD services. In this paper, we assume the passenger waiting time follows the well-established "square root law", which has been widely used in street-hailing taxi market \cite{douglas1972price},  radio dispatching taxi market \cite{arnott1996taxi},  and online ride-hailing market \cite{li2021spatial}, among others. This leads to the following passenger waiting time function:
\begin{equation} \label{waiting_time_AMoD}
    w_i^a\left(N_i^I\right) = \frac{A_i}{\sqrt{N_i^I}},
\end{equation}
where $A_i$ is the scaling parameter that captures possible factors in the matching between idle TNC vehicles and passengers, such as the size of the zone, the average traffic speed on roads, and the demand/supply distribution, etc. Equation (\ref{waiting_time_AMoD}) indicates that passengers' average waiting time is inversely proportional to the square root of the number of idle AVs. The intuition is that if both waiting passengers and idle vehicles are uniformly and independently distributed in each zone, the distance between a passenger and her/his closest idle vehicle is inversely proportional to the square root of the total number of idle vehicles, which further determines the average waiting time. A detailed justification of (\ref{waiting_time_AMoD}) can be found in \cite{li2019regulating}.

{\em Waiting time for transit services}: The passenger waiting time for transit services $w_{ij}^p$ from zone $i$ to zone $j$ depends on the service frequencies of transit lines traversed by the passenger. Note that each zone probably contains multiple transit stations, thus passengers may board the transit service at different stations in the origin zone, or exist the transit service at distinct stations in the destination zone. This leads to multiple transit routes that connect two zones, each of which may incur distinct waiting times for passengers. To characterize the {\em average} passenger waiting time of public transit from zone $i$ to zone $j$, we assume that passengers choose the nearest station to board the public transit, and get off at the station which is closest to the destination. In this case, if we further assume passenger origins and destinations are uniformly distributed in each zone, then passenger demand for public transit from zone $i$ to zone $j$ is uniformly distributed over different transit routes between $i$ and $j$. Without loss of generality, let $\mathcal{R}_{ij}$ denote the set of all possible transit routes from zone $i$ to zone $j$. A transit route may traverse multiple transit lines and thus each transit route $r\in \mathcal{R}_{ij}$ is a set of transit lines. We assume that:
\begin{assumption} \label{assump_waiting_time_transit_2}
(i) vehicles arrive exponentially with a mean of the transit headway; and (ii) passengers can be served by the first arriving vehicle.
\end{assumption}
Assumption \ref{assump_waiting_time_transit_2} states that vehicle arrival at stations follows the exponential distribution, and upon the vehicle arrives at the station, passengers can get on the vehicle immediately without waiting. Under the assumption, the mean waiting time of a specific transit line $l$ is the inverse of its service frequency \cite{spiess1989optimal}, i.e., $\frac{1}{f_l}$, which has been widely used in transit-related studies \cite{liu2018remote,kumar2021designing,du2022sensitivity,li2023strategy}. Finally, the {\em average} waiting time for public transit $w_{ij}^p$ from zone $i$ to zone $j$ can be approximated by the average of the waiting time along different transit routes connecting zone $i$ and zone $j$:
\begin{equation} \label{waiting_time_PT}
    w_{ij}^p = \frac{1}{|\mathcal{R}_{ij}|} \sum_{r\in \mathcal{R}_{ij}} \sum_{l \in r} \frac{1}{f_l} ,
\end{equation}
where $\mathcal{R}_{ij}$ is the set of transit routes connecting zone $i$ and zone $j$ due to multiplicity of transit stations in the same zone; $|\mathcal{R}_{ij}|$ is the total number of transit routes from zone $i$ to zone $j$; and $f_l$ is the service frequency of transit line $l$.

\begin{remark}
We assume that origins/destinations of passengers, idle vehicles, and transit stations are uniformly distributed in each zone when deriving (\ref{waiting_time_AMoD}) and (\ref{waiting_time_PT}). However, for first-mile/last-mile AMoD services, the spatial distribution of waiting passengers and idle vehicles is affected by the locations of transit stations. We argue that in practice, each zone is usually traversed by multiple transit lines from distinct directions. The spatial distribution of transit stations along these transit lines can be reasonably approximated by the uniform distribution. In this case, passengers either waiting at transit stations or their origins and idle vehicles can be approximately regarded as uniformly distributed across the zone. However, we conjecture that even if they are not uniformly distributed, (\ref{waiting_time_AMoD}) should still hold with a different value of $A_i$ and (\ref{waiting_time_PT}) can be still approximated under differentiated weights of the waiting time of distinct transit routes. We leave it as future work to validate this conjecture.
\end{remark}

\subsection{Incentives of the TNC Platform}

Consider a TNC platform that hires a fleet of $N$ autonomous vehicles to provide mobility-on-demand services. Each TNC vehicle is in one of the following statuses: (a) cruising on the street and waiting for the passenger; (b) on the way to pickup a passenger; and (c) carrying a passenger. At the stationary state, the total number of vehicle hours $N$ should satisfy the following conservation law:
\begin{equation} \label{vehicle_hour_conservation}
\begin{split}
    N = \sum_{i=1}^M N_i^I + & \sum_{i=1}^M \sum_{j=1}^M \sum_{k=1}^K \left( \lambda_{ij,k}^a w_i^a + \lambda_{ij,k}^{b_1} w_i^a + \lambda_{ij,k}^{b_2} w_j^a + \lambda_{ij,k}^{b_3} \left(w_i^a + w_j^a\right)\right) + \\ & \sum_{i=1}^M \sum_{j=1}^M \sum_{k=1}^K \left(\lambda_{ij,k}^a \frac{l_{ij}}{v_a} + \lambda_{ij,k}^{b_1} \frac{d_i}{v_a} + \lambda_{ij,k}^{b_2} \frac{d_j}{v_a} + \lambda_{ij,k}^{b_3} \left(\frac{d_i}{v_a} + \frac{d_j}{v_a}\right)\right) .
\end{split}
\end{equation}
Based on Little's law, the first term in the right-hand side of (\ref{vehicle_hour_conservation}) represents the operating hours of idle vehicles cruising on the street; the second term accounts for the operating hours of vehicles that are on the way to pick up passengers; and the third term sums the operating hours of vehicles that are occupied with a passenger. The total operating hours of picking up vehicles and occupied vehicles can be further decomposed into operating hours of vehicles that serve four different types of AMoD services (e.g., direct service, first-mile service, last-mile service, and first-mile and last-mile service), respectively.

The AMoD platform determines the base fare $b$, the per-distance rates $r_i^a$, the spatial distribution of idle AVs $N_i^I$, and the fleet size $N$ to maximize its profit subject to the equilibrium conditions. The profit maximization for the TNC platform can be formulated as follows:\footnote{We acknowledge that since our primary goal is to characterize competition between distinct modes, for simplicity, the repositioning  of idle TNC vehicles is not explicitly considered in this model. We leave it as future work. }
\begin{equation} \label{Incentives_AMoD}
    \begin{split}
    \max_{b, \mathbf{r^{a}},\mathbf{N^I}, N} \quad & \sum_{i=1}^M \sum_{j=1}^M \sum_{k=1}^K \lambda_{ij,k}^a \left(b+r_i^a l_{ij}^a\right) + \lambda_{ij,k}^{b_1}\left(b+r_i^a d_i\right) + \lambda_{ij,k}^{b_2} \left(b+r_j^a d_j\right) + \lambda_{ij,k}^{b_3} \left(2b + r_i^a d_i + r_j^a d_j\right) - N C_{av} \\
    \mathrm{s.t.} \quad & (\ref{logit_demand_function}), (\ref{generalized_travel_cost}), (\ref{waiting_time_AMoD}), (\ref{vehicle_hour_conservation}) \\
    \end{split}
\end{equation}
where $C_{av}$ is the hourly operating cost of an AV. The objective in (\ref{Incentives_AMoD}) defines the platform profit as the difference between the revenue and the total operating cost of AVs. The total revenue is the ride fares collected from passengers choosing four distinct types of AMoD services. The platform decisions are subject to the passenger demand model (\ref{logit_demand_function}) and (\ref{generalized_travel_cost}), the waiting time function (\ref{waiting_time_AMoD}), and vehicle conservation  (\ref{vehicle_hour_conservation}).

\subsection{Incentives of the Public Transit Agency}

Consider a public transit agency that operates the transit system. Generally, a maximum per-distance fare and maximum acceptable waiting time are imposed by the regulatory agency for transit services connecting distinct areas of the city to guarantee an acceptable service cost and service quality of public transit \cite{SanFrancisco2022Muni}. Let $r_{max}^p$ and $w_{max}^p$ be the maximum allowable per-distance fare and waiting time for transit services, respectively, then we have the following upper bounds on $r^p$ and $w_{ij}^p$:
\begin{equation} \label{waiting_time_constraint}
\begin{cases}
    r^p \leq r_{max}^p \\
    w_{ij}^p \leq w_{max}^p \quad \forall i,j=1,\dots,M
\end{cases}.
\end{equation}
In the meanwhile, a profit constraint is imposed to guarantee the economic sustainability of the transit agency, which requires that the difference between the revenue and the operating cost be larger than a reservation level $\pi_0$:
\begin{equation} \label{profit_constraint}
    \sum_{i=1}^M \sum_{j=1}^M \sum_{k=1}^K r^p l_{ij}^p (\lambda_{ij,k}^p + \lambda_{ij,k}^{b_1} + \lambda_{ij,k}^{b_2} + \lambda_{ij,k}^{b_3}) - \sum_{l=1}^L f_l C_l \geq \pi_0 ,
\end{equation}
where $C_l$ is the per-vehicle hourly operating cost of transit line $l$.

The public transit agency determines the transit fare $r^p$, and the service frequencies $f_l$ to maximize the public transit ridership subject to the passenger demand model (\ref{logit_demand_function}) and (\ref{generalized_travel_cost}), the waiting time function (\ref{waiting_time_PT}), the transit fare and waiting time constraints (\ref{waiting_time_constraint}), and the profit constraint (\ref{profit_constraint}). The ridership maximization for public transit can be cast as:
\begin{equation} \label{Incentives_PT}
    \begin{aligned}
    \max_{r^p,\mathbf{f}} \quad & \sum_{i=1}^M \sum_{j=1}^M \sum_{k=1}^K \lambda_{ij,k}^p + \lambda_{ij,k}^{b_1} + \lambda_{ij,k}^{b_2} + \lambda_{ij,k}^{b_3} \\
    \mathrm{s.t.} \quad & (\ref{logit_demand_function}), (\ref{generalized_travel_cost}), (\ref{waiting_time_PT}), (\ref{waiting_time_constraint}), (\ref{profit_constraint})
    \end{aligned}
\end{equation}

\begin{remark}
We postulate that the public transit agency aims to maximize the total ridership. This is consistent with the nature of public transport agencies (e.g. the government) aiming to provide reliable and sustainable mobility services to minimize negative externalities of vehicle travel (e.g. emissions and congestion). Although a large number of existing studies model the objective of transit agencies as (1) user benefit maximization (or equivalently, user cost minimization) \cite{hasselstrom1982public,ouyang2014continuum}, (2) total welfare maximization \cite{chien2002optimization,zhu2021competition}, (3) waiting time minimization \cite{chakroborty2003genetic,verbas2015integrated}, and (4) total cost minimization \cite{fan2008tabu,daganzo2010structure}, etc, we argue that ridership maximization \cite{yoo2010frequency,verbas2013optimal} is a good proxy for the maximization of social welfare or user benefit under a fixed budget. Maximizing total social welfare or passengers' surplus would complicate the formulation without providing extra insights.
\end{remark}

\subsection{Equilibrium of the Game}

The profit maximization for the TNC platform (\ref{Incentives_AMoD}) and the ridership maximization for the public transit (\ref{Incentives_PT}) constitute the game problem. Note that although the operational decision of the TNC platform $(b,\mathbf{r^a},\mathbf{N^I})$ and the public transit agency $(r^p,\mathbf{f})$ does not explicitly appear in the objective of the other, they interact with each other by affecting passengers' mode choices in the passenger demand model (\ref{logit_demand_function}) and (\ref{generalized_travel_cost}). Therefore, the platform profit of the TNC and the ridership of public transit not only depend on its own operational decisions, but also on the operational strategy of the counterpart. We assume that the TNC platform and the public transit agency interact simultaneously and they have perfect information on each other's strategy. The game is at equilibrium if the TNC platform cannot increase its profit and the public transit agency cannot increase the transit ridership by unilaterally changing their operational strategies. Without loss of generality, let $\xi^{a}=(b,\mathbf{r^a},\mathbf{N^I})$ and $\xi^{p}=(r^p,\mathbf{f})$ be the operational strategy of the TNC platform and the public transit agency, respectively, and denote $\pi^a(\xi^a,\xi^p)$ and $\pi^p(\xi^a,\xi^p)$ as the TNC platform profit and the transit ridership, respectively. We can formally define the equilibria $\xi^*=\left(\xi^{a^*},\xi^{p^*}\right)$ that satisfies:
\begin{equation} \label{equilibrium_condition}
    \begin{cases}
    \pi^a(\xi^{a^*},\xi^{p^*}) \geq \pi^a(\xi^a,\xi^{p^*}) \\
    \pi^p(\xi^{a^*},\xi^{p^*}) \geq \pi^p(\xi^{a^*},\xi^p)
    \end{cases}, \quad \forall \xi^a, \xi^p \geq 0 .
\end{equation}
which is consistent with the general definition of Nash equilibrium.

In practice, each player in the game may be indifferent to a small change in its objective function (e.g., the change only accounts for a negligible portion of the objective value). This motivates a more relaxed solution concept that encapsulates a broader range of equilibrium solutions, where players will not unilaterally change their strategies as far as the current solution is approximately optimal. More rigorously, we can define such relaxed solution as the $\epsilon$-Nash equilibrium  \cite{daskalakis2006note, li2019connections}, denoted as $\xi^*=\left(\xi^{a^*},\xi^{p^*}\right)$, which satisfies:
\begin{equation} \label{equilibrium_condition}
    \begin{cases}
    \pi^a(\xi^{a^*},\xi^{p^*}) \geq \pi^a(\xi^a,\xi^{p^*})-\epsilon \\
    \pi^p(\xi^{a^*},\xi^{p^*}) \geq \pi^p(\xi^{a^*},\xi^p)-\epsilon
    \end{cases}, \quad \forall \xi^a, \xi^p \geq 0 .
\end{equation}
At an $\epsilon$-Nash equilibrium, each player can improve its objective value by at most $\epsilon$ via deviating from the equilibrium strategy $\xi^*=\left(\xi^{a^*},\xi^{p^*}\right)$, given that the other player follows the equilibrium strategy. Therefore, it indicates that the players are motivated to play the equilibrium strategy if they are indifferent to a change of $\epsilon$. Note that Nash equilibrium can be viewed as the special case of the $\epsilon$-Nash equilibrium, correpsonding to the case of $\epsilon=0$.

\section{The Solution Method} \label{solution method}

This section solves the game problem by computing the Nash equilibrium of the game using the best response method and conducting an ex-post evaluation of the performance of the obtained solution. Specifically, we will first derive a candidate Nash equilibrium solution by the best response algorithm, which iteratively solves the profit maximization problem of the TNC platform and the ridership maximization problem of the transit agency using standard interior point methods until convergence. We comment that this step is not difficult because both the best response algorithm and interior point methods are well-established approaches. However, since each of these subproblems is non-convex, the derived candidate solution is only locally optimal, and it is quite challenging to validate how good the derived solution is compared to the globally optimal solution for each decision maker. To address this concern, we treat each subproblem separately: (a) we first focus on the profit-maximization problem of the TNC platform, and relax the consistency of its decisions to derive a tight upper bound for characterizing the optimality gap of the derived solution; (b) we then  reformulate the ridership maximization problem and derive the conditions under which the reformulated problem is concave and thus its global optimal solution can be efficiently computed. Overall, the proposed methods enable us to conduct an ex-post evaluation of the performance of the obtained solution by showing that the derived solution is at least as good as an $\epsilon$-Nash equilibrium, where the value of $\epsilon$ can be derived numerically. Below we investigate the profit-maximization problem for the TNC platform and the ridership-maximization problem for the transit agency to develop a numerical framework to compute   $\epsilon$.

\subsection{Profit Maximization of the TNC Platform}\label{sectionTNC}
In the profit maximization problem (\ref{Incentives_AMoD}), the involved constraints are all equality constraints. Note that given $\xi^p$, (\ref{logit_demand_function}), (\ref{generalized_travel_cost}) and (\ref{waiting_time_AMoD}) jointly determine $\lambda_{ij,k}^t,t\in\left\{a,b_1,b_2,b_3\right\}$ as a function of $b$, $r_i^a$, $r_j^a$, $N_i^I$, and $N_j^I$. With a slight abuse of notation, we denote it as $\lambda_{ij,k}^t=\lambda_{ij,k}^t\left(b,r_i^a,r_j^a,N_i^I,N_j^I\right), t\in\{a,b_1,b_2,b_3\}$. By further substituting (\ref{vehicle_hour_conservation}) into the objective function, the original problem  can be transformed into the following unconstrained optimization:
\begin{equation} \label{Incentives_AMoD_unconstrained}
    \begin{split}
    \max_{b, \mathbf{r^{a}},\mathbf{N^I}} \quad & \sum_{i=1}^M \sum_{j=1}^M \sum_{k=1}^K  \lambda_{ij,k}^a\left(b,r_i^a,r_j^a,N_i^I,N_j^I\right) \left[b+\left(r_i^a-\frac{C_{av}}{v_a}\right) l_{ij}^a - C_{av} \cdot w_i^a\left(N_i^I\right)\right] \\ + & \lambda_{ij,k}^{b_1}\left(b,r_i^a,r_j^a,N_i^I,N_j^I\right)\left[b+\left(r_i^a-\frac{C_{av}}{v_a}\right) d_i - C_{av} \cdot w_i^a\left(N_i^I\right)\right] \\ + & \lambda_{ij,k}^{b_2}\left(b,r_i^a,r_j^a,N_i^I,N_j^I\right) \left[b+\left(r_j^a-\frac{C_{av}}{v_a}\right) d_j - C_{av} \cdot w_j^a\left(N_j^I\right)\right] \\ + & \lambda_{ij,k}^{b_3}\left(b,r_i^a,r_j^a,N_i^I,N_j^I\right) \left[2b + \left(r_i^a-\frac{C_{av}}{v_a}\right) d_i + \left(r_j^a-\frac{C_{av}}{v_a}\right) d_j - C_{av} \cdot w_i^a\left(N_i^I\right) - C_{av} \cdot w_j^a\left(N_j^I\right)\right] \\ - & \frac{C_{av}}{M} \sum_{i=1}^M \sum_{j=1}^M N_i^I ,
    \end{split}
\end{equation}
where we equivalently rewrite $C_{av}\sum_{i=1}^M N_i^I$ as $\frac{C_{av}}{M}\sum_{i=1}^M \sum_{j=1}^M N_i^I$.
Based on (\ref{Incentives_AMoD_unconstrained}), we introduce auxiliary variables $b_j$, $r_{ij}^a$, and $N_{ij}^a$, where $i,j=1,\dots,M$ to formulate the equivalent problem. The intuition behind the auxiliary variables is that the platform determines the base fare $b_j$, the ride fares $r_{ij}^a,i=1,\dots,M$, and the number of idle vehicles $N_{ij}^I,i=1,\dots,M$ for the specific destination zone $j$. Without loss of generality, let $\xi_j^a=\left(b_j,r_{1j}^a,N_{1j}^I,r_{2j}^a,N_{2j}^I,\dots,r_{Mj}^a,N_{Mj}^I\right)$ be the operational strategy exclusively for destination zone $j$, and define $\pi_j^a$ as the corresponding profit from destination zone $j$:
\begin{equation}
    \begin{split}
        & \pi_j^a\left(\xi_j^a\right) = \sum_{i=1}^M \sum_{k=1}^K \lambda_{ij,k}^a\left(b_j,r_{ij}^a,r_{jj}^a,N_{ij}^I,N_{jj}^I\right) \left[b_j+\left(r_{ij}^a-\frac{C_{av}}{v_a}\right) l_{ij}^a - C_{av} \cdot w_i^a\left(N_{ij}^I\right)\right] \\ + & \lambda_{ij,k}^{b_1}\left(b_j,r_{ij}^a,r_{jj}^a,N_{ij}^I,N_{jj}^I\right)\left[b_j+\left(r_{ij}^a-\frac{C_{av}}{v_a}\right) d_i - C_{av} \cdot w_i^a\left(N_{ij}^I\right)\right] \\ + &\lambda_{ij,k}^{b_2}\left(b_j,r_{ij}^a,r_{jj}^a,N_{ij}^I,N_{jj}^I\right) \left[b_j+\left(r_{jj}^a-\frac{C_{av}}{v_a}\right) d_j - C_{av} \cdot w_j^a\left(N_{jj}^I\right)\right] \\ + &\lambda_{ij,k}^{b_3}\left(b_j,r_{ij}^a,r_{jj}^a,N_{ij}^I,N_{jj}^I\right) \left[2b_j + \left(r_{ij}^a-\frac{C_{av}}{v_a}\right) d_i + \left(r_{jj}^a-\frac{C_{av}}{v_a}\right) d_j - C_{av} \cdot w_i^a\left(N_{ij}^I\right) - C_{av} \cdot w_j^a\left(N_{jj}^I\right)\right] \\ - & \frac{C_{av}}{M} \sum_{i=1}^M N_{ij}^I, \quad j=1,\dots,M.
    \end{split}
\end{equation}
Note that the platform actually deploys a uniform operational strategy for distinct destination zones, therefore the destination-specific operational strategy $\xi_j^a$ should be consistent across different destination zones. In this case, the original problem (\ref{Incentives_AMoD_unconstrained}) is equivalent to the following optimization:
\begin{equation} \label{Incentives_AMoD_equivalent}
    \begin{split}
        \max_{\xi_1^a,\xi_2^a,\dots,\xi_M^a} \quad & \sum_{j=1}^M \pi_j^a(\xi_j^a) \\
        \mathrm{s.t.} \quad & \xi_1^a = \xi_2^a = \dots =\xi_M^a
    \end{split}
\end{equation}

Based on the equivalent formulation (\ref{Incentives_AMoD_equivalent}), we will slightly modify the decision consistency constraint $\xi_1^a=\xi_2^a=\dots=\xi_M^a$ to derive an upper bound. In particular, let $\overline{\mathcal{V}}=\left\{\mathcal{V}_1,\mathcal{V}_2,\dots,\mathcal{V}_m\right\}$ be a partition of $\mathcal{V}$, i.e., $\bigcup_{x=1,\dots,m} \mathcal{V}_x = \mathcal{V}$ and $\mathcal{V}_x \cap \mathcal{V}_y = \emptyset, \forall \mathcal{V}_x,\mathcal{V}_y\in \overline{\mathcal{V}}, x \neq y$, and define the relaxed problem as
\begin{equation} \label{Incentives_AMoD_relaxed}
    \begin{split}
        \max_{\xi_1^a,\xi_2^a,\dots,\xi_M^a} \quad & \sum_{j=1}^M \pi_j^a(\xi_j^a) \\
        \mathrm{s.t.} \quad & \xi_j^a = \xi_{j'}^a \quad \forall j,j' \in \mathcal{V}_1 \\
        & \xi_j^a = \xi_{j'}^a \quad \forall j,j' \in \mathcal{V}_2 \\
        & \dots \\
        & \xi_j^a = \xi_{j'}^a \quad \forall j,j' \in \mathcal{V}_m
    \end{split} ,
\end{equation}
where we break the complete decision consistency and only require that the operational strategies are consistent across a subset of destination zones.
Note that (\ref{Incentives_AMoD_relaxed}) can be further decomposed over the partition $\overline{\mathcal{V}}$:
\begin{equation} \label{Incentives_AMoD_relaxed_subproblem}
    \begin{split}
        \max_{\xi_j^a,j \in \mathcal{V}_x} \quad & \sum_{j \in \mathcal{V}_x} \pi_j^a(\xi_j^a) \\
        \mathrm{s.t.} \quad & \xi_j^a = \xi_{j'}^a \quad \forall j,j' \in \mathcal{V}_x
    \end{split} ,
    \quad x=1,\dots,m .
\end{equation}
The optimal solution to (\ref{Incentives_AMoD_relaxed_subproblem}), if obtained, provides an upper bound for the optimal value of the original problem, thus we have the following result:
\begin{proposition} \label{proposition_upper_lower_bound}
Suppose that $\xi^*=\left(\xi^{a^*},\xi^{p^*}\right)$  is a candidate Nash equilibrium. Let $\xi_j^{a^*},j\in\mathcal{V}_x$ be the optimal solution to (\ref{Incentives_AMoD_relaxed_subproblem}) given $\xi^{p^*}$, and let $\pi_{\mathcal{V}_x}^{a^*}$ be the corresponding optimal value. We have $\sum_{x=1}^m \pi_{\mathcal{V}_x}^{a^*} \geq \pi^a\left(\xi^{a^*},\xi^{p^*}\right) \geq \pi^a \left(\frac{\sum_{x=1}^m \pi_{\mathcal{V}_x}^{a^*} \xi_{j,j \in \mathcal{V}_x}^{a^*}}{\sum_{x=1}^m \pi_{\mathcal{V}_x}^{a^*}}, \xi^{p^*} \right)$ .
\end{proposition}
\begin{proof}
(1) $\sum_{x=1}^m \pi_{\mathcal{V}_x}^{a^*}$ is the optimal value of the relaxed problem (\ref{Incentives_AMoD_relaxed}) given $\xi^{p^*}$. Since (\ref{Incentives_AMoD_relaxed}) is a relaxed problem of (\ref{Incentives_AMoD_equivalent}), and $\xi^{a^*}$ is the optimal solution to (\ref{Incentives_AMoD_equivalent}) given $\xi^{p^*}$, then we have $\sum_{x=1}^m \pi_{\mathcal{V}_x}^{a^*} \geq \sum_{j=1}^M \pi_j^a \left(\xi^{a^*}\right) = \pi^a\left(\xi^{a^*},\xi^{p^*}\right)$.
(2) Since $\xi_{j,j\in\mathcal{V}_x}^{a^*}, \forall x=1,\dots,m$, is a feasible solution to (\ref{Incentives_AMoD_equivalent}), a non-negative linear combination of $\xi_{j,j\in\mathcal{V}_x}^{a^*},x=1,\dots,m$ is also a feasible solution to (\ref{Incentives_AMoD_equivalent}). Therefore, $\sum_{j=1}^M \pi_j^a \left(\frac{\sum_{x=1}^m \pi_{\mathcal{V}_x}^{a^*} \xi_{j,j \in \mathcal{V}_x}^{a^*}}{\sum_{x=1}^m \pi_{\mathcal{V}_x}^{a^*}}\right) \leq \sum_{j=1}^M \pi_j^a \left(\xi^{a^*}\right) = \pi^a\left(\xi^{a^*},\xi^{p^*}\right) $.
\end{proof}
Proposition \ref{proposition_upper_lower_bound} establishes an upper bound and a lower bound for the platform profit at Nash equilibrium. The intuition behind the upper bound is that the relaxation of the consistency of platform decisions provides the platform a higher degree of freedom to manage the AMoD service and thus leads to higher profitability. In the meanwhile, we use a weighted sum of optimal solutions to (\ref{Incentives_AMoD_relaxed_subproblem}) to generate a feasible solution and a tight lower bound of the original problem (\ref{Incentives_AMoD_equivalent}). These weights are determined based on the intuition that the platform combines distinct operational strategies $\xi_{j,j\in\mathcal{V}_x}^{a^*}$ for distinct sets of destination zones $\mathcal{V}_x$ to generate a uniform strategy, and the operational strategy on the set of destination zones with a higher potential profit $\pi_{\mathcal{V}_x}^{a^*}$ is assigned with larger weight.

The upper bound derived by Proposition \ref{proposition_upper_lower_bound} is very useful for us because whenever a local solution to (\ref{Incentives_AMoD_equivalent}) is given, we can compare the optimal value with respect to the upper bound, which offers an upper bound on the distance between the optimal value of the local solution and the optimal value of the unknown globally optimal solution. This enables us to assert how good a solution is compared to the globally optimal one, and it will be an important intermediate step for assessing the overall quality of the Nash equilibrium. However, to use this result, we do need to numerically compute the globally optimal solution to the nonconvex program (\ref{Incentives_AMoD_relaxed_subproblem}). Fortunately, despite its non-convexity, (\ref{Incentives_AMoD_relaxed_subproblem}) has a decomposable structure, which can be addressed by primal decomposition. To this end, let $\xi_{ij}^a=\left(r_{ij}^a,N_{ij}^I\right)$ be the operational decision on the specific OD pair $ij$ and define $\pi_{ij}^a$ as the corresponding profit from OD pair $ij$:
\begin{equation}
    \begin{split}
        & \pi_{ij}^a\left(b_j,\xi_{jj}^a,\xi_{ij}^a\right) = \sum_{k=1}^K \lambda_{ij,k}^a\left(b_j,r_{ij}^a,r_{jj}^a,N_{ij}^I,N_{jj}^I\right) \left[b_j+\left(r_{ij}^a-\frac{C_{av}}{v_a}\right) l_{ij}^a - C_{av} \cdot w_i^a\left(N_{ij}^I\right)\right] \\ + & \lambda_{ij,k}^{b_1}\left(b_j,r_{ij}^a,r_{jj}^a,N_{ij}^I,N_{jj}^I\right)\left[b_j+\left(r_{ij}^a-\frac{C_{av}}{v_a}\right) d_i - C_{av} \cdot w_i^a\left(N_{ij}^I\right)\right] \\ + &\lambda_{ij,k}^{b_2}\left(b_j,r_{ij}^a,r_{jj}^a,N_{ij}^I,N_{jj}^I\right) \left[b_j+\left(r_{jj}^a-\frac{C_{av}}{v_a}\right) d_j - C_{av} \cdot w_j^a\left(N_{jj}^I\right)\right] \\ + &\lambda_{ij,k}^{b_3}\left(b_j,r_{ij}^a,r_{jj}^a,N_{ij}^I,N_{jj}^I\right) \left[2b_j + \left(r_{ij}^a-\frac{C_{av}}{v_a}\right) d_i + \left(r_{jj}^a-\frac{C_{av}}{v_a}\right) d_j - C_{av} \cdot w_i^a\left(N_{ij}^I\right) - C_{av} \cdot w_j^a\left(N_{jj}^I\right)\right] \\ - & \frac{C_{av}}{M} N_{ij}^I, \quad i,j=1,\dots,M.
    \end{split}
\end{equation}
Note that $\xi_j^a = \left(b_j,\xi_{1j}^a,\xi_{2j}^a,\dots,\xi_{Mj}^a\right)$ and $\pi_{j}^a\left(\xi_j^a\right)=\sum_{i=1}^M \pi_{ij}^a\left(b_j,\xi_{jj}^a,\xi_{ij}^a\right)$. Therefore, the optimization (\ref{Incentives_AMoD_relaxed_subproblem}) is equivalent to:
\begin{equation} \label{Incentives_AMoD_relaxed_equivalent}
    \begin{split}
        \max_{b_j,\xi_{ij}^a,j \in \mathcal{V}_x,i=1,\dots,M} \quad & \sum_{j \in \mathcal{V}_x} \sum_{i=1}^M \pi_{ij}^a (b_j,\xi_{jj}^a,\xi_{ij}^a) \\
        \mathrm{s.t.} \quad & b_j = b_{j'} \quad \forall j,j' \in \mathcal{V}_x \\
        & \xi_{ij}^a = \xi_{ij'}^a \quad \forall j,j' \in \mathcal{V}_x, i=1,\dots,M
    \end{split} ,
    \quad x = 1,\dots,m .
\end{equation}
For given $b_j,\xi_{jj}^a,j\in \mathcal{V}_x$, we define the subproblems:
\begin{equation} \label{Incentives_AMoD_primal_subproblem}
    \begin{split}
        \max_{\xi_{ij}^a, j \in \mathcal{V}_x} \quad & \sum_{j \in \mathcal{V}_x} \pi_{ij}^a(b_j,\xi_{jj}^a,\xi_{ij}^a) \\
        \mathrm{s.t.} \quad & \xi_{ij}^a = \xi_{ij'}^a \quad \forall j,j' \in \mathcal{V}_x
    \end{split} ,
    \quad i = 1,\dots,M .
\end{equation}
with optimal values $\pi_{i,\mathcal{V}_x}^{a^*},i=1,\dots,M$. In this case, $\pi_{i,\mathcal{V}_x}^{a^*},i=1,\dots,M$ is a function of $b_j,\xi_{jj}^a,j\in \mathcal{V}_x$, and (\ref{Incentives_AMoD_relaxed_equivalent}) is equivalent to the master problem:
\begin{equation} \label{Incentives_AMoD_primal_master}
\begin{split}
    \max_{b_j,\xi_{jj}^a,j\in\mathcal{V}_x} \quad & \sum_{i=1}^M \pi_{i,\mathcal{V}_x}^{a^*} \\
\mathrm{s.t.} \quad & b_j = b_{j'} \quad \forall j,j' \in \mathcal{V}_x
\end{split} .
\end{equation}
Note that (\ref{Incentives_AMoD_primal_subproblem}) only involves two decision variables $\xi_{ij}^a=\left(r_{ij}^a,N_{ij}^I\right)$ if we absorb the equality constraint into the objective. Similarly, (\ref{Incentives_AMoD_primal_master}) has $1+2\cdot |\mathcal{V}_x|$ decision variables if we absorb the equality constraint, where $|\mathcal{V}_x|$ is the number of zones in $\mathcal{V}_x$. Since both (\ref{Incentives_AMoD_primal_subproblem}) and (\ref{Incentives_AMoD_primal_master}) are small-scale problems, we can use grid search method to find the globally optimal solutions in parallel.
\begin{remark}
The partition $\bar{\mathcal{V}}$ is flexible and we can carefully choose the composition of $\bar{\mathcal{V}}$ to address the trade-off between computational complexity and the quality of the upper bound. In particular, if we partition the space $\mathcal{V}$ into a large number of smaller sets, then the decomposed problem (\ref{Incentives_AMoD_primal_master}) has a smaller dimension, which can be more efficiently solved. However, this induces more relaxation for the consistency of decisions, which leads to a worse upper bound. Our numerical simulation indicates that dividing all the zones $\mathcal{V}$ into pairwise groups (i.e., $|\mathcal{V}_x|=2$) will lead to  fast computation and  tight upper bound at the same time. More details will be reported in case studies.
\end{remark}

\subsection{Ridership Maximization of Transit Agency} \label{sectiontransit}
We next discuss how to solve the globally optimal solution to the ridership maximization problem for the public transit agency. Note that given $\xi^a$, (\ref{logit_demand_function}) and (\ref{generalized_travel_cost}) jointly determine $\lambda_{ij,k}^t,t\in\left\{p,b_1,b_2,b_3\right\}$ as a function of $r^p$ and $w_{ij}^p$. With a slight abuse of notation, we denote it as $\lambda_{ij,k}^t=\lambda_{ij,k}^t\left(r^p,w_{ij}^p\right),t\in\left\{p,b_1,b_2,b_3\right\}$. The ridership maximization problem (\ref{Incentives_PT}) is equivalent to
\begin{equation} \label{Incentives_PT_equivalent}
    \begin{split}
    \max_{r^p,\mathbf{f},\mathbf{w^p}} \quad & \sum_{i=1}^M \sum_{j=1}^M \sum_{k=1}^K \lambda_{ij,k}^p \left(r^p,w_{ij}^p\right) + \lambda_{ij,k}^{b_1} \left(r^p,w_{ij}^p \right) + \lambda_{ij,k}^{b_2} \left(r^p,w_{ij}^p\right) + \lambda_{ij,k}^{b_3} \left(r^p,w_{ij}^p \right) \\
    \mathrm{s.t.} \quad & w_{ij}^p = \Phi_{ij}^T \left[\frac{1}{f_1},\frac{1}{f_2},\dots,\frac{1}{f_L}\right]^T \quad \forall i,j=1,\dots,M \\
    & r^p \leq r_{max}^p,  \quad w_{ij}^p \leq w_{max}^p \quad \forall i,j=1,\dots,M \\
    & \sum_{i=1}^M \sum_{j=1}^M \sum_{k=1}^K r^p l_{ij}^p \left[\lambda_{ij,k}^p \left(r^p,w_{ij}^p\right) + \lambda_{ij,k}^{b_1} \left(r^p,w_{ij}^p\right) + \lambda_{ij,k}^{b_2} \left(r^p,w_{ij}^p\right) + \lambda_{ij,k}^{b_3} \left(r^p,w_{ij}^p\right)\right] - \sum_{l=1}^L C_l f_l \geq \pi_0
    \end{split}
\end{equation}
where $\mathbf{w^p}=(w_{11}^p,w_{12}^p,\dots,w_{MM}^p)^T$ is the vector of average waiting times for public transit of different OD pairs and $\Phi_{ij}$ is the vector that relates $w_{ij}^p$ to $\mathbf{f}$ based on (\ref{waiting_time_PT}). Specifically, $\Phi_{ij}$ is a $L$-dimensional vector in which each element accounts for the number of occurrences of a specific transit line in the set of transit routes $\mathcal{R}_{ij}$ connecting zone $i$ and zone $j$ divided by the total number of transit routes $|\mathcal{R}_{ij}|$. We note that the key challenge for (\ref{Incentives_PT_equivalent}) is that both the objective function and constraints are non-convex. To address this challenge, we first introduce a change of variable and reformulate (\ref{Incentives_PT_equivalent}) into an equivalent form as follows:
\begin{lemma} \label{lemma_reformulation}
Define $N_{ij}^p = \frac{1}{w_{ij}^p}$. The optimization (\ref{Incentives_PT_equivalent}) can be equivalently reformulated as:
\begin{equation} \label{Incentives_PT_reformulation}
    \begin{aligned}
    \max_{r^p,\mathbf{f},\mathbf{N^p}} \quad & \sum_{i=1}^M \sum_{j=1}^M \sum_{k=1}^K \lambda_{ij,k}^p \left(r^p,\frac{1}{N_{ij}^p}\right) + \lambda_{ij,k}^{b_1} (r^p,\frac{1}{N_{ij}^p} ) + \lambda_{ij,k}^{b_2} \left(r^p,\frac{1}{N_{ij}^p}\right) + \lambda_{ij,k}^{b_3} \left(r^p,\frac{1}{N_{ij}^p}\right) \\
    \mathrm{s.t.} \quad & N_{ij}^p \leq \dfrac{1}{\Phi_{ij}^T \left[\dfrac{1}{f_1},\ldots,\dfrac{1}{f_L}\right]^T} \quad \forall i,j=1,\dots,M \\
    & r^p \leq r_{max}^p,  \quad N_{ij}^p \geq \frac{1}{w_{max}^p} \quad \forall i,j=1,\dots,M \\
    & \sum_{i=1}^M \sum_{j=1}^M \sum_{k=1}^K r^p l_{ij}^p \left[\lambda_{ij,k}^p \left(r^p,\frac{1}{N_{ij}^p}\right) + \lambda_{ij,k}^{b_1} \left(r^p,\frac{1}{N_{ij}^p}\right) + \lambda_{ij,k}^{b_2} \left(r^p,\frac{1}{N_{ij}^p}\right) + \lambda_{ij,k}^{b_3} \left(r^p,\frac{1}{N_{ij}^p}\right)\right] - \sum_{l=1}^L C_lf_l \geq \pi_0
    \end{aligned}
\end{equation}
\end{lemma}
The proof of Lemma \ref{lemma_reformulation} is trivial since it only involves a change of variable. Intuitively, $N_{ij}^p$ is virtually similar to the number of idle vehicles $N_{ij}^I$ in TNC's AMoD services. Equivalent to adjusting the average waiting time $w_{ij}^p$ of distinct OD pairs, the public transit agency determines the number of vehicles $N_{ij}^p$ for distinct OD pairs to maximize the ridership. Note that the reformulated problem (\ref{Incentives_PT_reformulation}) is non-convex, which is difficult to address. However, we find that under certain mild conditions, if we fix $r_p$, then both the objective function and constraints of (\ref{Incentives_PT_equivalent}) can be concave with respect to $\mathbf{f}$ and $\mathbf{N^p}$. To make this more clear,  we first introduce the following proposition as a preliminary result:
\begin{proposition} \label{proposition_concavity}
Given $\xi^a$ and $r^p$, (\ref{Incentives_PT_reformulation}) is concave with respect to $\mathbf{f}$ and $\mathbf{N^p}$ if
\begin{equation} \label{condition_concavity}
    N_{ij}^p \geq \frac{\epsilon}{2} \frac{\sum_{k=1}^K \alpha_k^2 \lambda_{ij,k}^0 P_{ij,k} \left(1-P_{ij,k}\right) \left(1-2P_{ij,k}\right)}{\sum_{k=1}^K \alpha_k \lambda_{ij,k}^0 P_{ij,k} \left(1-P_{ij,k}\right)}, \quad \forall i,j=1,\dots,M,
\end{equation}
where $P_{ij,k}$ is the percentage of passengers choosing public transit ($p$) and bundled services ($b_1$, $b_2$ and $b_3$) from origin $i$ to destination $j$ in income class $k$.
\end{proposition}
The proof of Proposition \ref{proposition_concavity} is deferred to Appendix A. Proposition \ref{proposition_concavity} provides an important condition (\ref{condition_concavity}) under which the concavity of (\ref{Incentives_PT_reformulation}) with respect to $\mathbf{f}$ and $\mathbf{N^p}$ can be established. However, both sides of  (\ref{condition_concavity}) involve decision/endogenous variables, which cannot be used directly without knowing the values of these variables. To address this concern, we will simplify (\ref{condition_concavity})  by reformulating it as a  constraint on $N_{ij}^p$. Note that (\ref{condition_concavity}) indeed defines a constraint on $N_{ij}^p$, which is determined by the intersection of its left-hand side and the right-hand side. The left-hand side of  (\ref{condition_concavity}) is a linear function of $N_{ij}^p$, whereas the right-hand side of  (\ref{condition_concavity}) is a function of $r^p$ and $N_{ij}^p$. For notation convenience, we denote the right-hand side of (\ref{condition_concavity}) as $RHS(r^p,N_{ij}^p)$, and examine the two boundaries of $RHS(r^p,N_{ij}^p)$ when $N_{ij}^p$ approaches $0$ or $\infty$.
In particular, for $r^p \geq 0$, based on the definition of generalized travel cost (\ref{generalized_travel_cost}), we can write $c_{ij,k}^t$ as functions of $r^p$ and $N_{ij}^p$:
\begin{equation} \label{cost_given_AMoD_1}
    \begin{cases}
    c_{ij,k}^a = \text{const}_{ij,k}^a \\
    c_{ij,k}^p = \gamma_k l_{ij}^p r^p + \frac{\alpha_k}{N_{ij}^p} + \text{const}_{ij,k}^p \\
    c_{ij,k}^{b_1} = \gamma_k l_{ij}^p r^p + \frac{\alpha_k}{N_{ij}^p} + \text{const}_{ij,k}^{b_1} \\
    c_{ij,k}^{b_2} = \gamma_k l_{ij}^p r^p + \frac{\alpha_k}{N_{ij}^p} + \text{const}_{ij,k}^{b_2} \\
    c_{ij,k}^{b_3} =  \gamma_k l_{ij}^p r^p + \frac{\alpha_k}{N_{ij}^p} + \text{const}_{ij,k}^{b_3} \\
    c_{ij,k}^o = \text{const}_{ij,k}^{o} \\
    \end{cases},
\end{equation}
Based on (\ref{cost_given_AMoD_1}) and (\ref{logit_demand_function}), it can be easily verified that $\lim_{N_{ij}^p \to 0} P_{ij,k} = 0$. In this case, by Bernoulli's rule, we have
\begin{equation} \label{RHS_0}
    \lim_{N_{ij}^p \to 0} RHS(r^p,N_{ij}^p) = \lim_{P_{ij,k}\to 0} \frac{\epsilon}{2} \dfrac{\sum_{k=1}^K \alpha_k^2 \lambda_{ij,k}^0 (1-6P_{ij,k}+6P_{ij,k}^2)}{\sum_{k=1}^K \alpha_k \lambda_{ij,k}^0 (1-2P_{ij,k})}=\frac{\epsilon}{2} \frac{\sum_{k=1}^K \alpha_k^2 \lambda_{ij,k}^0}{\sum_{k=1}^K \alpha_k \lambda_{ij,k}^0}.
\end{equation}
Similarly, based on (\ref{cost_given_AMoD_1}) and (\ref{logit_demand_function}), when ${N_{ij}^p}$ approaches infinity, we have
\begin{equation} \label{Np_infty}
    \lim_{N_{ij}^p \to \infty} P_{ij,k} =\dfrac{\sum_{t\in\{p,b_1,b_2,b_3\}}\exp(-\epsilon \cdot (\gamma_k l_{ij}^p r^p + \text{const}_{ij,k}^t))}{\sum_{t\in\{p,b_1,b_2,b_3\}}\exp(-\epsilon \cdot (\gamma_k l_{ij}^p r^p + \text{const}_{ij,k}^t)) + \sum_{t\in\{a,o\}} \exp(-\epsilon \cdot \text{const}_{ij,k}^t)} .
\end{equation}
The right-hand side of (\ref{Np_infty}) is a constant within the range $(0,1)$ and we denote it as $\hat{P}_{ij,k}\in(0,1)$. Given $\hat{P}_{ij,k}$, we have
\begin{equation}
 \label{RHS_infty}
    \lim_{N_{ij}^p \to \infty} RHS(r^p,N_{ij}^p) = \frac{\epsilon}{2} \frac{\sum_{k=1}^K \alpha_k^2 \lambda_{ij,k}^0 \hat{P}_{ij,k} \left(1-\hat{P}_{ij,k}\right) \left(1-2\hat{P}_{ij,k}\right)}{\sum_{k=1}^K \alpha_k \lambda_{ij,k}^0 \hat{P}_{ij,k} \left(1-\hat{P}_{ij,k}\right)} .
\end{equation}
Combining (\ref{RHS_0}) and (\ref{RHS_infty}), we conclude that the right-hand side of (\ref{condition_concavity}) is bounded, whereas the left-hand side of (\ref{condition_concavity}) is a linearly increasing function. Therefore, by continuity of $RHS(r^p,N_{ij}^p)$, there must exist a $\hat{N}_{ij}^p$ such that as far as $N_{ij}^p \geq \hat{N}_{ij}^p$, then $N_{ij}^p \geq RHS(r^p,N_{ij}^p)$. For this purpose, $\hat{N}_{ij}^p$ can be defined as the maximum solution to the following fixed-point equation
\begin{equation} \label{fixed_point_equation}
    N_{ij}^p = RHS(r^p,N_{ij}^p) = \frac{\epsilon}{2} \frac{\sum_{k=1}^K \alpha_k^2 \lambda_{ij,k}^0 P_{ij,k} \left(1-P_{ij,k}\right) \left(1-2P_{ij,k}\right)}{\sum_{k=1}^K \alpha_k \lambda_{ij,k}^0 P_{ij,k} \left(1-P_{ij,k}\right)} .
\end{equation}
Clearly $\hat{N}_{ij}^p$ depends on $r^p$, and without loss of generality, we denote it as $\hat{N}_{ij}^p (r^p)$. Furthermore, let $\bar{N}_{ij}^p=\max_{0\leq r_p\leq r_{max}^p} \hat{N}_{ij}^p (r^p)$, we are then ready to present the major theorem regarding the concavity of  ridership maximization problem (\ref{Incentives_PT_reformulation}):
\begin{theorem} \label{themrem_concavity}
Assume that $\xi^a$ is fixed, the ridership maximization problem (\ref{Incentives_PT_reformulation}) is concave with respect to $\mathbf{f}$ and $\mathbf{N^p}$ for any given $r^p\in\left[ 0,r_{max}^p \right]$, as far as $\bar{N}_{ij}^p\leq \dfrac{1}{w_{max}^p}$ $\forall i,j=1,\dots,M.$
\end{theorem}
Theorem \ref{themrem_concavity} is a direct consequence of Proposition \ref{proposition_concavity} and the aforementioned analysis: since the upper bound on transit waiting time requires that $N_{ij}^p\geq 1/w_{max}^p$, we can conclude that as long as  $\bar{N}_{ij}^p\leq 1/w_{max}^p$, then we have $N_{ij}^p\geq \bar{N}_{ij}^p$, which satisfies the concavity conditions (\ref{condition_concavity}). This provides a condition based on which we can determine the concavity of (\ref{Incentives_PT_reformulation}) with respect to $\mathbf{f}$ and $\mathbf{N^p}$ without knowing the value of decision variables or endogenous variables. Our case study suggests that the premise of Theorem~\ref{themrem_concavity} holds under a large range of realistic model parameters.

It is important to note that in Theorem \ref{themrem_concavity}, the concavity only holds when $r^p$ is excluded from the decisions. In fact, the ridership maximization problem (\ref{Incentives_PT_reformulation}) is not jointly concave with respect to $r^p$ and other decision variables. However, Theorem \ref{themrem_concavity} is still very useful to compute the globally optimal solution to (\ref{Incentives_PT_reformulation}) because it enables us to consider a hierarchical algorithm, where we first fix $r^p$ and solve the remaining decision variables which are concave, then we repeat this process for distinct values of $r^p$ to search for the globally optimal solution. In summary, we consider a hierarchical algorithm in the following two steps: first we fix $r^p$ and compute the optimal value $\pi^{p^*}(\xi^a,r^p)$ to (\ref{Incentives_PT_reformulation}) by optimizing other decision variables aside from $r^p$; then we vary $r^p$ and enumerate all possible $r^p$ to find the optimal ${r}^{p^*}$ that maximizes $\pi^{p^*}(\xi^a,r^p)$. Since $r^p$ is only one-dimensional, and the problem is concave with respect to the rest of the decision variables, this algorithm can efficiently find the globally optimal solution to (\ref{Incentives_PT_reformulation}).  Let us denote the optimal solution corresponding to ${r}^{p^*}$ as $\xi^{p^*}$, then we have the following corollary:
\begin{corollary} \label{proposition_lower_bound_PT}
Assume that $\xi^a$ is fixed, and suppose $\bar{N}_{ij}^p\leq \dfrac{1}{w_{max}^p}$ $\forall i,j=1,\dots,M$, then $\xi^{p^*}$ is the globally optimal solution to the ridership maximization problem (\ref{Incentives_PT_reformulation}).
\end{corollary}

To summarize, we run the best response algorithm to iteratively compute the profit maximization problem (\ref{Incentives_AMoD}) and the ridership maximization problem (\ref{Incentives_PT}), using the standard interior-point method \cite{mehrotra1992implementation}. After the algorithm converges, we obtain a candidate Nash equilibrium, which might be the local solution to each player's optimization problem. To validate the quality of these candidate solutions, we use the result of Proposition \ref{proposition_upper_lower_bound}, Proposition \ref{proposition_concavity}, and Theorem \ref{themrem_concavity}, so that the profit maximization problem (\ref{Incentives_AMoD}) can be solved approximately with a theoretical upper bound, whereas the   ridership maximization problem (\ref{Incentives_PT}) can be solved exactly under certain mild conditions.   Details of the algorithm are summarized in Algorithm \ref{algorithm1}.

\begin{algorithm}
\caption{Best response algorithm and the ex-post evaluation on Nash equilibrium} \label{algorithm1}
\begin{algorithmic}[1]
\REQUIRE Initial guess of operational strategy $\xi_{(0)}^a=\left(b,\mathbf{r^a},\mathbf{N^I}\right)$ and $\xi_{(0)}^p=\left(r^p,\mathbf{f}\right)$, the convergence tolerance $\sigma$, the maximum number of iterations $\hat{n}$, a partition $\overline{\mathcal{V}}=\left\{\mathcal{V}_1,\mathcal{V}_2,\dots,\mathcal{V}_m\right\}$ of $\mathcal{V}$.
\STATE Setup stopping criterion: $||\xi_{(n)}^a-\xi_{(n-1)}^a||_2 \leq \sigma$ and $||\xi_{(n)}^p-\xi_{(n-1)}^p||_2 \leq \sigma$
\FOR {$n=1,\dots,\hat{n}$}
\STATE Fix $\xi^p=\xi_{(n-1)}^p$, solve (\ref{Incentives_AMoD}) using interior-point algorithm, and obtain the optimal solution $\xi_{(n)}^a$.
\STATE Fix $\xi^a=\xi_{(n)}^a$, solve (\ref{Incentives_PT}) using interior-point algorithm, and obtain the optimal solution $\xi_{(n)}^p$.
\IF {stopping criterion satisfied}
\STATE Obtain the equilibrium strategy $\xi^{a^*}=\xi_{(n)}^a$ and $\xi^{p^*}=\xi_{(n)}^p$.
\STATE \algorithmicbreak
\ENDIF
\ENDFOR
\FOR {$x=1,\dots,m$}
\STATE Solve (\ref{Incentives_AMoD_relaxed_subproblem}) given $\xi^p=\xi^{p^*}$ using primal decomposition algorithm in Section \ref{sectionTNC}.
\STATE Obtain the optimal solution $\xi_{j,j\in\mathcal{V}_x}^{a^*}$ and the corresponding optimal value $\pi_{\mathcal{V}_x}^{a^*}$.
\ENDFOR
\STATE Calculate the upper bound $\overline{\pi}^a=\sum_{x=1}^m \pi_{\mathcal{V}_x}^{a^*}$ and the lower bound $\underline{\pi}^a=\pi^a \left(\frac{\sum_{x=1}^m \pi_{\mathcal{V}_x}^{a^*} \xi_{j,j \in \mathcal{V}_x}^{a^*}}{\sum_{x=1}^m \pi_{\mathcal{V}_x}^{a^*}}, \xi^{p^*} \right)$
\STATE Solve (\ref{Incentives_PT_reformulation}) given $\xi^a=\xi^{a^*}$ using the proposed hierarchical algorithm in Section \ref{sectiontransit} and obtain the optimal solution ${\xi}^{p^*}$ and corresponding optimal value $\pi^{p}\left(\xi^{a^*},{\xi}^{p^*}\right)$.
\ENSURE the equilibrium strategy $\xi^*=(\xi^{a^*},\xi^{p^*})$, the platform profit $\pi^a(\xi^{a^*},\xi^{p^*})$ and its upper bound $\overline{\pi}^a$ and lower bound $\underline{\pi}^a$, the transit ridership $\pi^p(\xi^{a^*},\xi^{p^*})$, the global optimality of $\xi^{p^*}$.
\end{algorithmic}
\end{algorithm}

The aforementioned numerical framework is very useful for us to evaluate the quality of the derived Nash equilibrium. This is because it indicates that the candidate Nash equilibrium is at least as good as an $\epsilon$-Nash equilibrium, where the value of $\epsilon$ can be given by the difference between the profit of the ride-hailing platform at the Nash equilibrium $\pi^a(\xi^{a^*},\xi^{p^*})$ and its upper bound $\bar{\pi}^a$. In particular, we summarize the above discussion as the following proposition:
\begin{proposition} \label{proposition_epsilon_nash}
Suppose the best response process (line 2-9) in Algorithm \ref{algorithm1} converges to a candidate Nash equilibrium $(\xi^{a^*},\xi^{p^*})$, which satisfies $\bar{N}_{ij}^p\leq \dfrac{1}{w_{max}^p}$ for $\forall i,j=1,\dots,M$, then $(\xi^{a^*},\xi^{p^*})$ is an $\epsilon$-Nash equilibrium of the game problem defined by (\ref{Incentives_AMoD}) and (\ref{Incentives_PT}), and we have:
\begin{equation}
\label{value_of_epsilon}
\epsilon=\bar{\pi}^a-\pi^a(\xi^{a^*},\xi^{p^*}).
\end{equation}
\end{proposition}
The above proposition can be easily derived based on Proposition \ref{proposition_upper_lower_bound}, Theorem \ref{themrem_concavity}, and the definition of the $\epsilon$-Nash equilibrium, thus its proof is omitted.

\section{Market Outcomes in the Absence of Regulation} \label{market outcomes}

This section investigates the equity impacts of AVs in the unregulated environment. To this end, we first propose evaluation metrics to quantify both spatial and social equity in the multimodal transportation system. Second, we demonstrate how AVs impact market outcomes and transport equity through numerical studies for San Francisco.

\subsection{Evaluation of Equity}

To characterize spatial and social equity, we first define $u_{ij,k}^t=-c_{ij,k}^t, t\in\mathcal{T}$ as the average utility of passengers from origin zone $i$ to destination zone $j$ in income class $k$ by choosing mode $t$. We further define the accessibility measure $A_{ij,k}$ as the expected maximum utility, which evaluates the expected received utility of passengers in income class $k$ from origin $i$ to destination $j$ in the multimodal transportation system. Based on the multinomial logit (MNL) model (\ref{logit_demand_function}), the maximum expected utility $A_{ij,k}$ can be written as the logsum formula \cite{ho2006combined,van2021evaluating}:
\begin{equation} \label{logsum accessibility}
    A_{ij,k} = \frac{1}{\epsilon} \log \sum_{t\in\mathcal{T}} \exp\left(\epsilon u_{ij,k}^t\right) .
\end{equation}

Given the accessibility measure $A_{ij,k}$, we will use Theil coefficient to quantify the transport equity in the multimodal transport system. To this end, we first denote $\lambda_{i,k}$, $\lambda_{k}$, $\overline{\lambda}$ as the passenger demand from zone $i$ in income class $k$, the passenger demand in income class $k$, and the total passenger demand, respectively. They satisfy:
\begin{equation}
\begin{cases}
    \lambda_{i,k} = \sum_{j=1}^M \sum_{t\in\mathcal{T}} \lambda_{ij,k}^t \\
    \lambda_{k} = \sum_{i=1}^M \sum_{j=1}^M \sum_{t\in\mathcal{T}} \lambda_{ij,k}^t \\
    \overline{\lambda} = \sum_{i=1}^M \sum_{j=1}^M \sum_{k=1}^K \sum_{t\in\mathcal{T}} \lambda_{ij,k}^t
\end{cases} .
\end{equation}
Correspondingly, we denote $A_{i,k}$, $A_k$, $\bar{A}$ as the average accessibility of passengers originating from zone $i$ in income class $k$, the average accessibility of passengers in income class $k$, and the average accessibility of all passengers, respectively. They are calculated as:
\begin{equation}
\begin{cases}
A_{i,k} = \frac{\sum_{j=1}^M \sum_{t\in\mathcal{T}} \lambda_{ij,k}^t A_{ij,k}}{\lambda_{i,k}} \\
A_k = \frac{\sum_{i=1}^M \sum_{j=1}^M \sum_{t\in\mathcal{T}} \lambda_{ij,k}^t A_{ij,k}}{\lambda_{k}} \\
\overline{A} = \frac{\sum_{i=1}^M \sum_{j=1}^M \sum_{k=1}^K \sum_{t\in\mathcal{T}} \lambda_{ij,k}^t A_{ij,k}}{\bar{\lambda}}
\end{cases} .
\end{equation}

Finally, we define the Theil (T) coefficient of the accessibility distribution in the multimodal transportation system as:
\begin{equation} \label{Theil_coefficient}
    T = \text{WITHIN} + \text{BETWEEN} = \underbrace{\sum_{k=1}^K \sum_{i=1}^M \left(\frac{\lambda_{i,k}}{\overline{\lambda}} \frac{A_{i,k}}{A_k}\right) \ln \left(\frac{A_{i,k}}{A_k}\right)}_\text{spatial equity} + \underbrace{\sum_{k=1}^K \left(\frac{\lambda_k}{\bar{\lambda}} \frac{A_k}{\overline{A}}\right) \ln \left(\frac{A_k}{\bar{A}}\right)}_\text{social equity} .
\end{equation}
As shown in (\ref{Theil_coefficient}), the Theil coefficient can be decomposed into two components. The "WITHIN" component calculates the inequality in the distribution of passenger accessibility across distinct geographic zones {\em within} different groups. Consequently, it characterizes the spatial inequity due to the differentiated mobility service across distinct zones. The "BETWEEN" component evaluates the difference in the distribution of average passenger accessibility {\em between} different income classes, which captures the social inequity arising from the unequal socioeconomic status. Note that a larger Theil coefficient indicates a more inequitable distribution of benefits across different zones or socioeconomic groups. It can be easily verified that if all zones have the same level of accessibility, then the ``WITHIN" component of (\ref{Theil_coefficient}) is zero. Similarly, if all classes of passengers have the same level of accessibility, then the ``BETWEEN" component of (\ref{Theil_coefficient})  is zero.

\subsection{Case Studies} \label{case study}

Consider a case study for San Francisco, where a TNC platform operates autonomous vehicles for mobility-on-demand services and the public transit agency manages a transit network to provide public transit services. We conduct a numerical study using realistic synthetic data for San Francisco. The data consists of the origin and destination of TNC trips at the zip-code granularity, which is synthesized based on the historic total TNC pickups and dropoffs in each zone \cite{SanFrancisco2022TNC} together with a choice model calibrated with survey data. The zip code zones of San Francisco are shown in Figure \ref{fig:SF_multimodal}. Based on the TNC data, we remove zip code zones 94123, 94127, 94129, 94130, 94134 from our analysis since they have negligible trip volumes. We also merge zip code zones 94104, 94105, and 94111 into a single zone, and merge zip code zones 94108 and 94133 into a single zone since each of these individual zones is very small. We synthesize the public transit network based on the Muni system in San Francisco, which is a network of light-rail metro trains, rapid buses, regular buses, cable cars, etc \cite{SanFrancisco2022Muni}. The synthetic transit network consists of the five light rail lines ('F', 'J', 'KT', 'M', and 'N') and the three rapid bus lines ('5R', '9R', '38R') since they form the major skeleton of the Muni network and undertake the majority of trip volumes. The route information of these transit lines can be found in \cite{SanFrancisco2022Muniroutes} and the synthetic transit network is shown in Figure~\ref{fig:SF_multimodal}. To guarantee that passengers are at least accessible to public transit in each zone, we further aggregate zip code zones 94123 and 94115 since they are geographically adjacent\footnote{In practice, each zone can be covered by public transit although the density of stations and service frequency are spatially unequal.}.
We further define the underserved area $\mathcal{U}$ as consisting of zip code zones 94112, 94114, 94116, 94117, 94118, 94121, 94122, 94124, 94131, 94132 based on the density of transit stations in different zones. Note that the underserved area is generally the remote area in the city.

\begin{figure}[t]
    \centering
    \includegraphics[width=0.58\linewidth]{figures/SF_multimodal_network.pdf}
    \caption{Zip code zones of San Francisco County and the synthetic public transit network.}
    \label{fig:SF_multimodal}
\end{figure}

We consider heterogeneous passengers with different income levels. In particular, passengers are categorized into three distinct classes, namely, low-income class (class 1), medium-income class (class 2), and high-income class (class 3). Generally, low-income people have a lower valuation of time but a higher valuation of money compared to high-income people, and passengers have a higher valuation of waiting/walking time than that of in-vehicle time. Therefore, the following conditions hold:
\begin{equation}
    \begin{cases}
    \alpha_k,\theta_k > \beta_k \quad \forall k=1,2,3 \\
    \alpha_1 < \alpha_2 < \alpha_3 \\
    \beta_1 < \beta_2 < \beta_3 \\
    \gamma_1 > \gamma_2 > \gamma_3
    \end{cases} .
\end{equation}

In summary, the model parameters involved in the multimodal transportation system are
\begin{equation}
    \Theta = \left\{\lambda_{ij,k}^0, \alpha_k, \beta_k, \gamma_k, \epsilon, c_{ij,k}^o, A_i, l_{ij}^a, l_{ij}^p, d_i, v_a, v_p, v_w, r_{max}^p, w_{max}^p, C_{av}, C_{l}, \pi_0  \right\} .
\end{equation}
Due to the lack of real data on AMoD, we calibrate the values of these model parameters based on human-driver-based TNC data and Muni data in San Francisco and map it to the future scenario. In particular, $\lambda_{ij}^0$ is set to satisfy $0.15\lambda_{ij}^0=\lambda_{ij}$, where $\lambda_{ij}$ is the current trip distribution of ride-sourcing trips in San Francisco from zone $i$ to zone $j$, such that 15\% of potential passengers choose to take AMoD services. We further split the potential passenger demand $\lambda_{ij}^0$ among three income classes to obtain $\lambda_{ij,k}^0$. To characterize the spatial difference in socioeconomic characteristics, we assume population distributions of low-income, middle-income, and high-income individuals as 0.4, 0.5, and 0.1 in remote areas and 0.2, 0.5, and 0.3 in core areas\footnote{For simplicity, we define the remote area as the underserved area $\mathcal{U}$. The core area consists of the zip code zones except those in $\mathcal{U}$.}. The potential passenger demand $\lambda_{ij,k}^0$ is then calculated proportionally. The trip distances $l_{ij}^a$ and $l_{ij}^p$ are obtained from Google map estimates. The average speed of TNC vehicles $v_a=17.937$ mph and the average operating speed of public transit $v_p=14.349$ mph are estimated based on the current TNC data and Muni data. For the first-mile/last-mile distance $d_i$, we assume that it follows the "square root law", which indicates that $d_i$ is inversely proportional to the number of transit stations $N_i^T$ in each zone:
\begin{equation}
    d_i = \frac{B_i}{\sqrt{N_i^T}} ,
\end{equation}
where $B_i$ is the scaling parameter that captures the area of the zone and the distribution of transit stations. The average walking speed is $v_w$ is set as 3.48 mph. The travel cost of the outside option $c_{ij,k}^o$ is assumed to be proportional to $l_{ij}^a$. To account for both socioeconomic and spatial heterogeneity in $c_{ij,k}^o$, we assume that $c_{ij,1}^o=1.2c_{ij,2}^0$ and $c_{ij,2}^o=1.25c_{ij,3}^o$ such that the travel cost of the outside option of low-income passengers is 20\% higher than that of medium-income passengers and 50\% higher than that of high-income passengers. We further assume that the per-distance cost of the outside transport mode in the remote area is 50\% higher than that in the urban core. Based on data from the Muni system \cite{SanFrancisco2022Munifares,SanFrancisco2022Munifrequencies}, the per-vehicle operating costs of different transit lines $C_{l}$ are calibrated and we set the maximum acceptable per-distance fare $r_{max}^p$ and waiting time $w_{max}^p$ for transit services as \$3/\text{mile} and 20 minutes, respectively.

The rest of the model parameters are set as:
\begin{gather*}
    \alpha_1 = 0.5; \quad \alpha_2 = 1.0; \quad \alpha_3 = 2.0; \quad \beta_1 = 0.15; \quad \beta_2=0.30; \quad \beta_3 = 0.65; \\ \gamma_1 = 3.0; \quad \gamma_2 = 1.5; \quad \gamma_3 = 0.75; \quad \theta_1 = 0.5; \quad \theta_2 = 1.0; \quad \theta_3 = 2.0; \\ \epsilon = 0.10; \quad A_i = 7.894 \times \text{Area of zone $i$}; \quad B_i = 1.609 \times \text{Area of zone $i$}; \quad \pi_0 =  \$1\mathrm{e}4 \text{/hour} .
\end{gather*}
We adjust the values of these parameters such that when the hourly operating cost of AVs is roughly equivalent to the average driver wage of TNCs, the corresponding market outcomes at Nash equilibrium are close to real-world data of San Francisco (e.g., modal share, trip volumes, average trip fare and frequencies of transit lines). In particular, when $C_{av}=\$30\text{/hour}$, we execute Algorithm \ref{algorithm1} to compute the Nash equilibrium and conduct the ex-post evaluation on the obtained equilibrium. The best response algorithm converges to the same equilibrium $\xi^*=(\xi^{a^*},\xi^{p^*})$ in a few iterations under distinct initial guesses. At the equilibrium, the TNC profit $\pi^a(\xi^{a^*},\xi^{p^*})$ is \$95,613/hour. Under a pairwise partition $\bar{\mathcal{V}}$\footnote{$\mathcal{V}$ is partitioned into subsets, each of which contains two zones, e.g., $\mathcal{V}_1=\{1,2\}$, $\mathcal{V}_2=\{3,4\}$, $\dots$, and $\mathcal{V}_9=\{17,18\}$.}, the established upper bound $\overline{\pi}^a$ is \$97,526/hour. Based on Proposition \ref{proposition_epsilon_nash}, this indicates that $\epsilon$ only accounts for 2.0\% compared with the globally optimal solution to (\ref{Incentives_AMoD}), which is a very tight approximate Nash equilibrium. In the meanwhile, the corresponding lower bound $\underline{\pi}^a$ is \$95,146/hour, which is very close to the equilibrium profit $\pi^a(\xi^{a^*},\xi^{p^*})$. This validates the nature of TNC's decision-making: geographic zones with higher potential profits are given higher priority. {Besides, our numerical experiments showed that the concavity conditions $\bar{N}_{ij}^p\leq \frac{1}{w_{max}^p},\forall i,j=1,\dots,M$, hold and have fairly safe gaps under a large range of model parameters, which verifies the global optimality of $\xi^{p^*}$.} For the market outcomes under $C_{av}=\$30/\text{hour}$, the TNC base fare is \$9.6/trip and the average per-distance rate of AMoD services is \$3.61/mile. The calculated trip fare $9.6+3.61\times2.6$ is close to the fare estimates \cite{SanFrancisco2022rate} for a 2.6-mile trip \cite{castiglione2016tncs}. The per-distance fare of public transit is \$1.14/mile and the service frequencies of the transit lines range from 9.26 per hour to 20.10 per hour, which is consistent with the current fare and frequency setting of Muni system \cite{SanFrancisco2022Muniroutes,SanFrancisco2022Munifares}. Moreover, the modal share of public transit is 22\%, which also coincides with the reported data in the survey \cite{SanFrancisco2022modalshare}.

To investigate the impacts of AVs on the multimodal transportation system, we gradually reduce the  cost of AVs, i.e., $C_{av}$, to simulate the scenario in which technological progress makes AVs less expensive over time, resulting in the ride-hailing industry's gradual acceptance of AVs. To this end, we fix all other model parameters, solve the game problem under distinct $C_{av}$ based on Algorithm \ref{algorithm1}, examine how the evolution of AV technology affects the market outcomes in the absence of regulations, and evaluate transport equity using Theil coefficient based on (\ref{Theil_coefficient}). The results are presented in Figure \ref{fig:TNC_no_reg}-\ref{fig:modal_share_no_reg}. Figure \ref{fig:TNC_no_reg} shows TNC's operational strategy and corresponding outcomes under distinct $C_{av}$. As the cost of AV reduces, the ride fare of TNC trips decreases (Figure \ref{fig:base_fare_no_reg}), the number of vehicles on the ride-hailing platform increases (Figure \ref{fig:total_number_vehicles_no_reg}), and the profit of the platform improves (Figure \ref{fig:TNC_profit_no_reg}). These results are intuitive since the reduction in AV cost lowers the marginal cost of AMoD services and thus both the TNC platform and passengers are direct beneficiaries of AV deployment. However, a less intuitive insight is that the spatial distribution of AMoD services is unbalanced:
\begin{itemize}
    \item Idle AVs are highly concentrated in the urban core (solid lines in Figure \ref{fig:number_idle_vehicle_no_reg}), indicating that the platform prefers to deploy idle vehicles in high-demand areas.  This is intuitive because TNC has higher utilization of AVs and higher potential profits in high-demand areas, which promotes it to deploy more idle vehicles to attract passengers from urban centers. However, the geographical concentration of ride-hailing vehicles will not only exacerbate traffic congestion in the urban core, but also leads to spatial inequality in the waiting time/service quality across distinct zones.
    \item As AV cost reduces, the number of idle vehicles increases in both high-demand and low-demand areas of the city, but the gap of service quality between  high-demand and low-demand areas is further enlarged (i.e., compare solid and dashed lines in Figure \ref{fig:number_idle_vehicle_no_reg}). This indicates that without regulatory intervention, lowering the cost of AV will promote the proliferation of AVs, but at the same time exacerbate the existing spatial inequality gaps in the multimodal transportation network.
\end{itemize}

\begin{figure}[t]
     \centering
     \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/TNC_no_regulations/average_trip_fare.pdf}
         \caption{{Average TNC trip fare under distinct $C_{av}$.}}
         \label{fig:base_fare_no_reg}
     \end{subfigure}
     \hfill
     \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/TNC_no_regulations/total_number_AVs.pdf}
         \caption{Total number of AVs under distinct $C_{av}$.}
         \label{fig:total_number_vehicles_no_reg}
     \end{subfigure}

     \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/TNC_no_regulations/TNC_profit.pdf}
         \caption{Hourly TNC profit under distinct $C_{av}$.}
         \label{fig:TNC_profit_no_reg}
     \end{subfigure}
     \hfill
     \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/TNC_no_regulations/number_idle_AVs.pdf}
         \caption{Number of idle vehicles in different zones under distinct $C_{av}$.}
         \label{fig:number_idle_vehicle_no_reg}
     \end{subfigure}
    \caption{TNC's operational decisions and outcomes under distinct $C_{av}$.}
    \label{fig:TNC_no_reg}
\end{figure}

To understand how the deployment of AVs affects social equity, we shall compare the impacts of AVs on distinct passenger income classes. Note that low-income populations are more transit-dependent, thus to characterize social equity, we will first examine the operational decisions of the transit agency.
Figure \ref{fig:PT_no_reg} presents the public transit agency's operational decisions and the corresponding outcomes under distinct $C_{av}$. Based on the simulation results, the proliferation of AVs has a moderate impact on the transit operational decisions:  as $C_{av}$ reduces, the fare of public transit monotonically increases (Figure \ref{fig:transit_fare_no_reg}),  the service frequencies of most transit lines decreases (Figure \ref{fig:transit_frequency_no_reg}), the revenue and operating cost of the transit network slightly decrease (Figure \ref{fig:transit_revenue_no_reg}), and the total ridership of public transit under distinct $C_{av}$ exhibits two regimes: when $C_{av}>\$36/\text{hour}$, the transit ridership increases as $C_{av}$ reduces; when $C_{av}\leq \$36/\text{hour}$, the transit ridership decreases as $C_{av}$ drops (Figure \ref{fig:transit_ridership_no_reg}).  The above results reveal some interesting competitive and complementary effects of AMoD services on public transit. Note that the TNC provides on-demand door-to-door mobility services (direct AMoD services, i.e., mode $a$), which is a substitution for public transit. On the contrary, TNC trips serving as first/last-mile connections to public transit (bundled services, i.e., mode $b_1$, $b_2$, and $b_3$) could be a complement/feeder to public transit. The TNC and public transit agency carefully determine their operational strategies to balance the competitiveness and complementarity between AMoD services and public transit. When $C_{av}$ is relatively high, the competitiveness/substitution of AMoD services to public transit is weak, and the complementarity dominates. To strengthen the complementary effect and fully utilize the benefits of reduced AV cost, the public transit agency improves the service frequencies of some transit lines to attract more passengers to choose bundled services (Figure \ref{fig:transit_frequency_no_reg}). In this case, although the number of passengers choosing direct public transit drops, the passenger arrival rates of bundled services increase, which leads to increased total ridership of public transit (Figure \ref{fig:transit_ridership_no_reg}). The total operating cost of the transit network increases (Figure \ref{fig:transit_revenue_no_reg}), and therefore the agency raises the transit fare to maintain a balance of revenue and expenditure (Figure \ref{fig:transit_fare_no_reg}). On the other hand, when $C_{av}$ is relatively low, the competitiveness/substitution of direct AMoD services to public transit increases as $C_{av}$ reduces and gradually becomes the dominant effect. The competition between the TNC and public transit deteriorates transit services and procures passengers to deviate from public transit (Figure \ref{fig:transit_frequency_no_reg}). The total transit ridership significantly decreases as $C_{av}$ reduces (Figure \ref{fig:transit_ridership_no_reg}), and both the revenue and operating cost of the transit network decrease (Figure \ref{fig:transit_revenue_no_reg}). However, we acknowledge that the above impacts are insignificant because the AV cost only influences the transit agency indirectly through competition.

\begin{figure}[t]
     \centering
     \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/PT_no_regulation/transit_fare.pdf}
         \caption{Per-distance transit fare under distinct $C_{av}$.}
         \label{fig:transit_fare_no_reg}
     \end{subfigure}
     \hfill
    \begin{subfigure}[b]{0.48\textwidth}
     \centering
     \includegraphics[width=\textwidth]{figures/PT_no_regulation/transit_frequency.pdf}
     \caption{Frequencies of different transit lines under distinct $C_{av}$.}
     \label{fig:transit_frequency_no_reg}
    \end{subfigure}
    \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/PT_no_regulation/PT_revenue_cost.pdf}
         \caption{Transit revenue and operating cost under distinct $C_{av}$.}
         \label{fig:transit_revenue_no_reg}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/PT_no_regulation/PT_ridership.pdf}
         \caption{Total public transit ridership under distinct $C_{av}$.}
         \label{fig:transit_ridership_no_reg}
    \end{subfigure}
    \caption{Public transit's operational decisions and outcomes under distinct $C_{av}$.}
    \label{fig:PT_no_reg}
\end{figure}

To further investigate the impact of AVs on social equity, we compare the mode choices of different income classes of the multimodal transportation system in Figure \ref{fig:modal_share_no_reg}.  As AV cost reduces, the modal shares of direct AMoD services (mode $a$) and bundled services (mode $b_1$, $b_2$, and $b_3$) increase, and the proportion of passengers choosing public transit (mode $p$) decreases for distinct income classes (Figure \ref{fig:modal_share_low_no_reg}-\ref{fig:modal_share_high_no_reg}). Overall, passengers shift from the outside option (e.g., walking, bicycling, and driving) and public transit to direct AMoD services and bundled services as AV cost reduces (Figure \ref{fig:modal_share_overall_no_reg}). These results are intuitively straightforward: the decrease in AV cost lowers the marginal cost of AMoD services, which promotes cheaper and improved AMoD services and instead disrupts transit services (in the second regime). However, the impacts of AVs on the modal choices of distinct classes of travelers are different, indicating that the distribution of benefits of AV is unequal:
\begin{itemize}
    \item Low-income individuals are highly transit-dependent. More than half of low-income individuals take public transit and over a third of them choose the outside option (e.g., walking or bicycling) (Figure \ref{fig:modal_share_low_no_reg}). As AV cost reduces, the increase in the modal share of AMoD services and bundled services is insignificant (Figure \ref{fig:modal_share_low_no_reg}). To understand this, note that low-income people have a relatively high valuation of monetary cost and a low valuation of time. As AV cost reduces, although ride fares and the waiting time of AMoD services decrease, the TNC deploys a small number of idle AVs and set relatively high ride fares (compared to transit) in remote areas with a high proportion of low-income individuals, who have to stick to public transit although its travel cost increases.
    \item The evolution of AV technology significantly transforms medium-income individuals' mode choices. As AV cost reduces, the number of medium-income individuals taking transit decreases while the passenger arrival rate of direct AMoD services gradually increases and finally surpasses that of public transit (Figure \ref{fig:modal_share_medium_no_reg}). In the meanwhile, an increasing number of medium-income individuals choose bundled services (Figure \ref{fig:modal_share_medium_no_reg}), although they prefer to use AMoD as the first-mile or last-mile connection rather than for both the first-mile and last-mile connection (Figure \ref{fig:modal_share_medium_no_reg}).
    \item High-income individuals prefer convenient and fast mobility options. As AV cost reduces, the majority of high-income individuals shift from the outside option (e.g., driving) to direct AMoD services and only a small proportion of individuals choose public transit and bundled services (Figure \ref{fig:modal_share_high_no_reg}). High-income people have a high valuation of time and therefore they are unwilling to experience a long trip time and prefer driving or direct AMoD services.
\end{itemize}

To summarize, the above analysis indicates that the proliferation of AVs will increase both spatial and social inequity gaps in the multimodal transportation network. This is consistent with the simulation results presented in  Figure \ref{fig:Theil_no_reg}, where both the 'WITHIN' component (i.e, spatial inequity) and the 'BETWEEN' component (i.e., social inequity) increase and the Theil coefficient significantly increases as $C_{av}$ reduces. We comment that the reason for the expansion of both the spatial inequity gap and social inequity gap is twofold. On the one hand, the spatial inequality in accessibility across different zones is mainly caused by the geographic concentration of AMoD services. As mentioned above, due to the for-profit nature of the TNC platform, idle AVs are highly concentrated in the urban core (Figure \ref{fig:number_idle_vehicle_no_reg}), and passengers in the urban core enjoy more convenient services compared to those in remote areas. The geographical imbalance in AMoD services across distinct zones causes spatial inequity in accessibility. Besides, such geographic imbalance in AMoD services exacerbates as AV cost reduces (Figure \ref{fig:number_idle_vehicle_no_reg}), which enlarges the spatial inequality in accessibility and reinforces the spatial inequity gap (Figure \ref{fig:Theil_no_reg}). On the other hand, the increase in social inequity is not only due to the moderate disruptiveness of AMoD services to public transit, but also because the benefits of reduced AV costs are primarily enjoyed by medium-income and high-income individuals, and the increase in the accessibility of low-income individuals is less significant compared to that of medium-income and high-income individuals (Figure \ref{fig:accessibility_groups}). Therefore, the social inequity gap exacerbates as AV cost reduces (Figure \ref{fig:Theil_no_reg}).

\begin{figure}[h!]
     \centering
     \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/passengers_no_regulation/modal_share_low.pdf}
         \caption{Modal share in low-income class under distinct $C_{av}$.}
         \label{fig:modal_share_low_no_reg}
     \end{subfigure}
     \hfill
    \begin{subfigure}[b]{0.48\textwidth}
     \centering
     \includegraphics[width=\textwidth]{figures/passengers_no_regulation/modal_share_medium.pdf}
     \caption{Modal share in medium-income class under distinct $C_{av}$.}
     \label{fig:modal_share_medium_no_reg}
    \end{subfigure}
    \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/passengers_no_regulation/modal_share_high.pdf}
         \caption{Modal share in high-income class under distinct $C_{av}$.}
         \label{fig:modal_share_high_no_reg}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/passengers_no_regulation/modal_share_overall.pdf}
         \caption{Overall modal share under distinct $C_{av}$.}
         \label{fig:modal_share_overall_no_reg}
    \end{subfigure}
    \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/equity_no_regulation/accessibility_groups.pdf}
         \caption{Average accessibility of different income groups under distinct $C_{av}$.}
         \label{fig:accessibility_groups}
     \end{subfigure}
     \hfill
     \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/equity_no_regulation/Theil.pdf}
         \caption{Theil coefficient, WITHIN component, and BETWEEN component under distinct $C_{av}$}
         \label{fig:Theil_no_reg}
     \end{subfigure}
    \caption{Modal share and transport equity in the multimodal transportation system under distinct $C_{av}$.}
    \label{fig:modal_share_no_reg}
\end{figure}

\section{Impacts of Regulatory Policies} \label{regulations}

Our numerical study so far suggests that spatial inequality and social inequality gap enlarges as AV cost reduces. This section proposes two regulatory policies, i.e., a minimum service-level requirement and a subsidy on bundled services, to mitigate the negative equity impacts of AVs in the multimodal transportation network.

\subsection{A Minimum Service-Level Requirement} \label{service}

To improve spatial equity, we consider a service-level requirement that mandates TNCs to guarantee an acceptable waiting time for AMoD services in all areas of the city. Such policy has already been implemented in practice, for instance,  New York City requires that 90\% of wheelchair-accessible vehicle requests be fulfilled within 15 minutes \cite{NYCWAV}. Under the proposed model, this policy can be easily modeled as:
\begin{equation} \label{minimum_serivce}
    w_i^a\left( N_i^I \right) \leq w_{max}^a, \forall i=1,2,\dots,M ,
\end{equation}
where $w_i^a(\cdot)$ is the average waiting time for AMoD trips starting in zone $i$, and $w_{max}^i$ is the maximum allowable waiting time set by the regulatory agency to ensure reasonable and equitable service quality for travelers across distinct zones. In this case, the TNC determines its operational strategy to maximize its profit subject to the market equilibrium conditions (\ref{logit_demand_function}), (\ref{generalized_travel_cost}), (\ref{waiting_time_AMoD}) and (\ref{vehicle_hour_conservation}) and the minimum service-level requirement (\ref{minimum_serivce}). The public transit agency strategically adjusts its operational decisions to maximize transit ridership subject to the equilibrium constraints (\ref{logit_demand_function}), (\ref{generalized_travel_cost}), (\ref{waiting_time_PT}), (\ref{waiting_time_constraint}) and (\ref{profit_constraint}). Note that the minimum service-level requirement does not change the structure of the profit maximization problem for the TNC and the ridership maximization problem for public transit. We can use Algorithm \ref{algorithm1} to solve the game problem and conduct an ex-post evaluation on the computed Nash equilibrium.

To investigate the impacts of the minimum service-level requirement, we fix $C_{av}$ as 20\$/hour and remain other model parameters unchanged, solve the game problem under distinct $w_{max}^a$, and present how it impacts the market outcomes and transport equity in Figure \ref{fig:TNC_service}-\ref{fig:equity_service}.

\begin{figure}[t!]
     \centering
     \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/TNC_service/average_trip_fare.pdf}
         \caption{{Average trip fare of the ride-hailing services under distinct values of $w_{max}^a$.}}
         \label{fig:base_fare_service}
     \end{subfigure}
     \hfill
     \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/TNC_service/number_idle_AVs.pdf}
         \caption{Number of idle vehicles in different zones of the transport network under distinct values of $w_{max}^a$.}
         \label{fig:number_idle_vehicle_service}
     \end{subfigure}

     \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/TNC_service/TNC_profit.pdf}
         \caption{Hourly profit of the TNC platform under distinct values of $w_{max}^a$.}
         \label{fig:TNC_profit_service}
     \end{subfigure}
     \hfill
     \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/PT_service/transit_fare.pdf}
         \caption{ Trip fare (per-distance) of transit services under different values of $w_{max}^a$.}
         \label{fig:transit_fare_service}
     \end{subfigure}

    \begin{subfigure}[b]{0.48\textwidth}
     \centering
     \includegraphics[width=\textwidth]{figures/PT_service/transit_frequency.pdf}
     \caption{Frequencies of different transit lines under distinct values of $w_{max}^a$.}
     \label{fig:transit_frequency_service}
    \end{subfigure}
     \hfill
    \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/PT_service/PT_ridership.pdf}
         \caption{Total public transit ridership per minute under distinct values of $w_{max}^a$.}
         \label{fig:transit_ridership_service}
    \end{subfigure}
    \caption{Operational decisions of the TNC platform and transit agency under distinct $w_{max}^a$.}
    \label{fig:TNC_service}
\end{figure}

Figure \ref{fig:TNC_service} shows the operational strategies of TNC platform and the public transit agency under distinct $w_{max}^a$. Under a stricter minimum service-level requirement,  the number of idle vehicles in distinct zones remains unchanged initially and then increases, especially in the low-demand areas (Figure \ref{fig:number_idle_vehicle_service}). In this case, the TNC earns a reduced profit under a stricter minimum service-level requirement (Figure \ref{fig:TNC_profit_service}) and increases the average ride fare to improve profitability (Figure \ref{fig:base_fare_service})). Clearly, this lower bound enforces the TNC to improve the service quality in low-demand areas where the TNC is reluctant to place its idle AVs in the absence of regulations. This improves spatial equity while hurting the profitability of TNC's AMoD business.
Figure \ref{fig:transit_fare_service}-\ref{fig:transit_ridership_service} presents the transit agency's operational strategies and the corresponding outcomes under distinct $w_{max}^a$. As $w_{max}^a$ reduces, the agency slightly lowers the transit fare (Figure \ref{fig:transit_fare_service}) and increases the service frequencies of its transit lines (Figure \ref{fig:transit_frequency_service}). This leads to reduced waiting time for transit services and increased total transit ridership (Figure \ref{fig:transit_ridership_service}). Similarly, this can be explained as a consequence of the complementarity between AMoD services and public transit. Imposing a minimum service-level requirement improves the service quality of AMoD services, especially in underserved/remote areas (Figure \ref{fig:number_idle_vehicle_service}). It promotes the agency to improve the transit service and reinforce the AMoD-transit complementarity to attract more passengers to choose bundled services and stimulate a higher transit ridership.

Figure \ref{fig:equity_service} demonstrates the impacts of the minimum service-level requirement on accessibility and transport equity in the multimodal transportation system. As $w_{max}^a$ reduces,  the inequality in accessibility across distinct zones reduces (Figure \ref{fig:Theil_service}), while the inequality gap between the accessibility of different income groups keeps widening (Figure \ref{fig:accessibility_groups_service}-\ref{fig:Theil_service}). Consequently, as $w_{max}^a$ reduces, the spatial inequity ('WITHIN' component) decreases, whereas the social inequity ('BETWEEN' component) keeps increasing, and the overall inequity gap (Theil coefficient) first reduces and then enlarges (Figure \ref{fig:Theil_service}). This indicates that imposing a minimum service-level requirement is double-edged: on the one hand, it reduces the geographic concentration of idle vehicles and improves spatial equity; on the other hand, enforcing more idle AVs in less lucrative areas primarily benefits high-income individuals in underserved areas, who rely on AMoD much more than low-income individuals in the same zone. In this case, the accessibility improvement of low-income individuals is insignificant due to the increased TNC ride fares and high-income individuals are the primary beneficiaries of the minimum service-level requirement, which further reinforces the existing social inequity gap.

\begin{figure}[t]
     \centering
     \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/equity_service/accessibility_groups.pdf}
         \caption{Average accessibility of different income groups under distinct valuse of $w_{max}^a$.}
         \label{fig:accessibility_groups_service}
     \end{subfigure}
     \hfill
     \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/equity_service/Theil.pdf}
         \caption{Theil coefficient, WITHIN component, and BETWEEN component under distinct $w_{max}^a$}
         \label{fig:Theil_service}
     \end{subfigure}
    \caption{Accessibility and transport equity under distinct $w_{max}^a$.}
    \label{fig:equity_service}
\end{figure}

\begin{figure}[!t]
     \centering
     \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/TNC_subsidy/average_trip_fare.pdf}
         \caption{{Average trip fare of the ride-hailing services under distinct values of subsidy $s$.}}
         \label{fig:base_fare_subsidy}
     \end{subfigure}
     \hfill
     \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/TNC_subsidy/number_idle_AVs.pdf}
         \caption{Number of idle vehicles in different zones of the transport network under distinct values of subsidy $s$.}
         \label{fig:number_idle_vehicle_subsidy}
     \end{subfigure}
    \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/TNC_subsidy/TNC_profit.pdf}
         \caption{Hourly profit of the TNC platform under distinct values of subsidy $s$.}
         \label{fig:TNC_profit_subsidy}
     \end{subfigure}
     \hfill
     \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/PT_subsidy/transit_fare.pdf}
         \caption{Trip fare (per-distance) of transit services under different values of  subsidy $s$.}
         \label{fig:transit_fare_subsidy}
     \end{subfigure}
     \begin{subfigure}[b]{0.48\textwidth}
     \centering
     \includegraphics[width=\textwidth]{figures/PT_subsidy/transit_frequency.pdf}
     \caption{Frequencies of different transit lines under distinct values of subsidy $s$.}
     \label{fig:transit_frequency_subsidy}
    \end{subfigure}
    \hfill
    \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/PT_subsidy/PT_ridership.pdf}
         \caption{Total public transit ridership under distinct values of subsidy $s$.}
         \label{fig:transit_ridership_subsidy}
    \end{subfigure}
    \caption{Operational decisions of the TNC platform and the public transit agency under distinct $s$.}
    \label{fig:TNC_subsidy}
\end{figure}

\subsection{Subsidies on Bundled Services} \label{subsidies}

To strengthen the synergy between TNC and transit, and to improve spatial and social equity at the same time, we consider a subsidy on TNC trips that connect passengers to public transit. We focus on the underserved/remote areas $\mathcal{U}$ where there is a large population of low-income transit-dependent households who are poorly served by public transit and AMoD services. A subsidy $s$ is provided for TNC trips that start from or end near a transit stop in zone $i\in \mathcal{U}$. Under the subsidy, the generalized travel costs of distinct mobility options can be captured by:
\begin{equation} \label{generalized_travel_cost_subsidy}
    \begin{cases}
    c_{ij,k}^a = \alpha_k w_i^a + \beta_k \frac{l_{ij}^a}{v_a} + \gamma_k (b + r_i^a l_{ij}^a) \\
    c_{ij,k}^p = \alpha_k w_{ij}^p + \beta_k \frac{l_{ij}^p}{v_p} + \gamma_k r^p l_{ij}^p + \theta_k \left(\frac{d_i}{v_w}+\frac{d_j}{v_w}\right) \\
    c_{ij,k}^{b_1} = \alpha_k \left(w_i^a + w_{ij}^p \right) + \beta_k \left(\frac{d_i}{v_a} +  \frac{l_{ij}^p}{v_p}\right) + \gamma_k \left(b + r_i^a d_i - s \mathbbm{1}_i^{\mathcal{U}} + r^p l_{ij}^p\right) + \theta_k \frac{d_j}{v_w} \\
    c_{ij,k}^{b_2} = \alpha_k \left( w_{ij}^p + w_j^a \right) + \beta_k \left( \frac{l_{ij}^p}{v_p} + \frac{d_j}{v_a}\right) + \gamma_k \left( r^p l_{ij}^p + b + r_j^a d_j - s \mathbbm{1}_j^{\mathcal{U}}\right) + \theta_k \frac{d_i}{v_w} \\
    c_{ij,k}^{b_3} = \alpha_k \left(w_i^a + w_{ij}^p + w_j^a\right) + \beta_k \left(\frac{d_i}{v_a} +  \frac{l_{ij}^p}{v_p} + \frac{d_j}{v_a}\right) + \gamma_k \left(b + r_i^a d_i - s \mathbbm{1}_i^{\mathcal{U}} + r^p l_{ij}^p + b + r_j^a d_j - s \mathbbm{1}_j^{\mathcal{U}} \right) \\
    \end{cases}
\end{equation}
where $\mathbbm{1}_i^{\mathcal{U}}$ is an indicator function that equals 1 if $i\in \mathcal{U}$ and 0 otherwise. In this case, the TNC deploys its operational strategy to maximize its profit subject to the modified generalized travel costs (\ref{generalized_travel_cost_subsidy}) and other unchanged equilibrium conditions (\ref{logit_demand_function}), (\ref{waiting_time_AMoD}) and (\ref{vehicle_hour_conservation}). Similarly, the public transit agency determines the operational strategy to maximize the transit ridership subject to the modified generalized travel costs (\ref{generalized_travel_cost_subsidy}) and other unchanged equilibrium constraints (\ref{logit_demand_function}), (\ref{waiting_time_PT}), (\ref{waiting_time_constraint}) and (\ref{profit_constraint}). Note that the subsidy $s$ is a constant which does not alter the structure of the profit maximization problem for the TNC and the ridership maximization problem for public transit. Therefore, we can still use Algorithm \ref{algorithm1} to compute the Nash equilibrium and perform an ex-post evaluation on the obtained equilibrium.

\begin{figure}[!t]
     \centering
     \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/equity_subsidy/accessibility_groups.pdf}
         \caption{Average accessibility of different income groups under distinct values of $s$.}
         \label{fig:accessibility_groups_subsidy}
     \end{subfigure}
     \hfill
     \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/equity_subsidy/Theil.pdf}
         \caption{Theil coefficient, WITHIN component, and BETWEEN component under distinct values of $s$}
         \label{fig:Theil_subsidy}
     \end{subfigure}
    \caption{Accessibility and transport equity under distinct $s$.}
    \label{fig:equity_subsidy}
\end{figure}

To investigate the impacts of subsidies for bundled services, we focus on the nominal case with $C_{av}=20\$/\text{hour}$ and remain other model parameters unchanged as in Section \ref{case study}, solve the game problem under distinct $s$, and demonstrate how it affects the market outcomes and transport equity in Figure \ref{fig:TNC_subsidy}-\ref{fig:equity_subsidy}.

Figure \ref{fig:base_fare_subsidy}-\ref{fig:TNC_profit_subsidy} shows TNC's operational strategy and the corresponding market outcomes under distinct $s$. As the subsidy increases, the TNC raises the trip fare (Figure \ref{fig:base_fare_subsidy}), increases the number of idle vehicles in distinct zones (Figure \ref{fig:number_idle_vehicle_subsidy}), and earns an increased profit under a higher subsidy level (Figure \ref{fig:TNC_profit_subsidy}). Interestingly, the majority of increased idle AVs are deployed in remote zones (dashed lines in Figure \ref{fig:number_idle_vehicle_subsidy}). This is because subsidizing TNC trips that connect passengers to public transit directly reduces the monetary costs of bundled trips in underserved/remote areas and attracts more passengers to choose bundled services, which enables the TNC to reduce the base fare and raise its per-distance fares in distinct zones to earn increased trip fares. To accommodate the increased demand for first/last-mile AMoD services, the TNC primarily deploys more idle vehicles in underserved areas, thus improving spatial equity.
Figure \ref{fig:transit_fare_subsidy}-\ref{fig:transit_ridership_subsidy} demonstrates the public transit agency's operational decisions and the corresponding outcomes under distinct $s$. As the subsidy level improves, the transit agency decreases its fare (Figure \ref{fig:transit_fare_subsidy}) and increases the frequencies of distinct transit lines (Figure \ref{fig:transit_frequency_subsidy}), which reduces passengers' waiting time for transit in distinct zones and leads to an increased total public transit ridership (Figure \ref{fig:transit_ridership_subsidy}). We argue that this is again attributed to the complementary effect of AMoD services on public transit: subsidizing TNC trips as first/last-mile connections strengthens the complementarity of AMoD services to public transit, thus the public transit agency reduces the transit fare and improves the service frequencies to attract more passengers to choose bundled services, which increases the total transit ridership.

Figure \ref{fig:equity_subsidy} shows the impacts of subsidies on accessibility and transport equity in the multimodal transportation system.  As the subsidy increases, the accessibility of all different income groups increases at the same time (Figure  \ref{fig:equity_subsidy}). More importantly, the accessibility increase in underserved/remote areas is more significant than that in core areas and the accessibility increase of low-income and medium-income individuals are much higher than that of high-income individuals, and the inequality in accessibility across distinct geographic zones and between distinct income groups reduce with the subsidy. Consequently, both spatial inequity ('WITHIN' component) and social inequity ('BETWEEN' component) reduce and the overall transport inequity (Theil coefficient) alleviates (Figure \ref{fig:Theil_subsidy}). The equity improvement is natural since subsidies are devoted to AMoD trips for first/last-mile connections in underserved areas with a large proportion of low-income transit-dependent individuals. It incentivizes the TNC to deploy more idle AVs in remote areas and reduces the geographic concentration of AMoD services (Figure \ref{fig:number_idle_vehicle_subsidy}). In the meanwhile, it promotes the transit agency to provide improved transit services (Figure \ref{fig:transit_fare_subsidy}-\ref{fig:transit_frequency_subsidy}). In this case, the subsidies reduce the travel costs of both public transit and bundled services, which improves accessibility and especially benefits low-income transit-dependent individuals. Consequently, subsidies on bundled services can bridge the existing spatial and social inequity gaps at the same time.

\section{Conclusion} \label{conclusion}

This paper assesses the equity impacts of AVs and investigates regulatory policies that guarantee the benefits of AV deployment reach underserved areas and transportation-disadvantaged groups. A network game-theoretic model is formulated to characterize the intimate interactions among TNC's location-differentiated price, the transit fare, services frequencies of distinct transit lines, waiting times for AMoD services, waiting times for transit services, TNC fleet size, and the modal share of different income groups in the multimodal transportation system, by which the incentives of multiclass passengers, the TNC, and the public transit agency are captured. An algorithm is developed to compute the Nash equilibrium and conduct an ex-post evaluation of the performance of the obtained Nash equilibrium. Theil coefficient is utilized to quantify both the spatial and social inequality in individuals' accessibility in the multimodal transportation system.

Based on the developed framework, we reveal the spatial and social inequity gaps without regulatory interventions and investigate regulatory policies that improve transport equity. Through numerical study, we showed that although individuals' accessibility improves, spatial and social inequity gaps enlarge as AV technology evolves. The for-profit nature of TNC leads to the geographic concentration of AMoD services and exacerbates spatial inequity. In the meanwhile, the competition between AMoD and public transit disrupts transit services and the benefits of reduced costs for AMoD services are primarily distributed to higher-income individuals who are less transit-dependent, which expands the social inequity gap. The impacts of two regulatory policies are evaluated: (a) a minimum service-level requirement on AMoD services; and (b) subsidies on AMoD trips for first/last-mile connections. We showed that a minimum service-level requirement incurs a trade-off: as a higher minimum service level is required, the spatial inequity decreases while the social inequity keeps increasing. Therefore, the regulatory agency should evaluate the trade-off between spatial equity and social equity and carefully control the minimum service-level requirement. On the other hand, subsidizing bundled services will benefit all stakeholders: higher subsidy yields increased TNC profit, transit ridership, and accessibility, and more importantly, improved spatial and social equity at the same time.

This paper delivers a comprehensive socioeconomic analysis of AV-enabled mobility future, with a special focus on transport equity. One future extension is extending the model to capture a mixture of AVs and human drivers in a TNC market and consider equity from both passenger and driver sides. Another extension would be incorporating other socio-economic factors (other than income levels) into the study of social equity.

\section*{Acknowledgments}  {This research was supported by Hong Kong Research Grants Council under project 26200420, 16202922, and National Science Foundation of China under project 72201225.}

\bibliographystyle{unsrt}