Motivation

Commuters in big cities have continuously experienced the frustration of congestion and traffic jams [1]. Travel delays, accidents, and road altercations have consistently impacted the economy, society, and the natural environment regarding energy and pollution [2]. One of the pressing challenges of our time is the increasing demand for energy, which requires us to make fundamental transformations in how our societies use and access transportation [3]. Thanks to the technological evolution of mobility (e.g., electrification of vehicles, smart mobility with self-driving cars, and improved vehicle sensor technology [4]) it is highly expected that we will be able to eliminate congestion while significantly increasing mobility efficiency in terms of energy and travel time [5]. Several studies have shown the benefits of emerging mobility systems (EMS) (e.g., ride-hailing, on-demand mobility services, shared vehicles, self-driving cars) in reducing energy and alleviating traffic congestion in a number of different transportation scenarios [6–10]. For a thorough review of models and the possible methods and techniques for smart mobility-on-demand systems, see [11].

The cyber-physical nature (e.g., data and shared information) of EMS is associated with significant control challenges and gives rise to a new level of complexity in modeling and control [12]. Research efforts over the last twenty years have tended to focus on the technological dimension. What is missing is a complementary theoretical study of the broader social implications of smart mobility. The impact of selfish social behavior in routing networks of regular and autonomous vehicles has been studied in [13–15]. Other efforts have addressed “how people learn and make routing decisions” with behavioral dynamics [16, 17]. A game-theoretic framework using sequential games was proposed to study the socioeconomic interactions and the different tradeoffs that emerge between the mobility stakeholders of a mobility “ecosystem” [18]. It seems, though, that the problem of how automation in mobility will affect the tendency to travel and decision-making has not been adequately approached yet. In a recent study [19], it was shown that when daily commuters were offered a convenient and affordable taxi service for their travels, a change of behavior was noticed; the commuters adjusted their travel behavior and activities and used the taxi service considerably more often leading to an 83% overall increase in vehicle miles traveled. Along with other similar studies [20, 21], this shows that EMS most probably may affect people’s tendency to travel and incentivize them to use cars more frequently, which potentially can also lead to a shift away from public transit.

In this paper, we are interested in one open question: Can we devise appropriate mobility prices in a game-theoretic framework for a multimodal transportation system that considers rational or subjective decision-making? Ideally, we want these prices to lead to an equilibrium that ensures an efficient mobility system. To address this question, we first need to understand the behavioral interactions of travelers with different modes of transportation along with the implications to system efficiency. Thus, we study the game-theoretic interactions of travelers seeking to travel in a transportation network comprised of roads used by different modes of transportation (e.g., cars, buses, light rail, and bikes). A key characteristic of our approach is that we adopt the Mobility-as-a-Service (MaaS) concept, i.e., a multimodal mobility system that centrally handles the travelers’ information and provides travel services (e.g., navigation, location, booking, payment). We aim to provide a game-theoretic framework that captures the most significant factors of a traveler’s decision-making in a transportation network under two different behavioral models.

Literature review

One of the standard approaches to alleviate congestion in a transportation system has been through “congestion pricing.” With this approach, we can manage the travelers’ demand while also considering the scarcity of road space or limited public transit supply [22, 23]. Congestion pricing has been instrumental in intelligent and scalable traffic routing, in which the aim is to optimize routing decisions and “guide” travelers via appropriate prices [24–26]. To understand the innate competition over the scarce resources in transportation and the impact of congestion, pricing game theory has been one of the standard tools that can help us investigate the impact of selfishness and traffic efficiency [27, 28]. By adopting a game-theoretic approach, advanced systems have been proposed to assign travelers concrete routes or minimize all travelers’ travel time while studying the system’s Nash equilibrium (NE) under different mechanisms of congestion pricing [29–37]. Key theoretical games used for this approach include routing/congestion games [14, 38–41], which are a generalization of the standard resource-sharing game of an arbitrary number of resources in a network with a finite number of travelers. For example, each traveler may contribute a certain amount of traffic from a source to a destination and affect the overall congestion on a route, thus increasing the travel time for all other travelers.

Another important class of games is potential games, first introduced in [42]. In a potential game, the incentive of all players to change their strategy can be expressed using a single global function called the “potential” function, which depends on the action sets of all players. As it capture the changes of utility as the actions vary, potential games have been used extensively in wide-ranging applications (taxation of public goods [43], economics of shallow lakes [44], and electricity markets [45]. Routing/congestion and potential games have played an instrumental role in understanding competition over shared resources. Both classes of games have been studied in multiple disciplines to model transportation and communication networks [23, 46–48] and common-pool resource games in economics [49]. Thus, for our purposes, it is most appropriate to formulate our problem as a routing game and show that there exists a potential function. This way, we can keep the mathematical analysis tractable and still be able to draw key insights on the impact of the travelers’ decision-making. We deviate from the current state of the art in transportation with game theory by considering negative congestion externalities. Namely, we suppose that if the number of co-travelers that utilize the same route or mode of transportation increases, then a traveler’s utility decreases too [50–52].

So far, most of the existing game-theoretical literature in transportation and routing/congestion games assumes that players’ are rational and intelligent, i.e., each player is a risk-neutral, selfish, utility maximizer. This makes transportation models quite unrealistic, as unexpected travel delays can lead to uncertainty in a traveler’s utility. Irrational decision-making over uncertainties and risks in utility can play a significant role, and its study can help us understand how large-scale systems perform inefficiently. To support this assertion, empirical experiments were conducted that showed how a human’s choices or preferences systematically deviate from the choice or preferences of a rational game-theoretic player [53, 54]. For example, humans compare the outcomes of their choices to a known expected amount of utility (called reference) and decide, based on that reference, whether their utility is a gain or loss. Prospect theory has laid down the theoretical foundations to study such biases and the subjective perception of risk in the utility of humans [55]. Prospect theory has been recognized as a closer-to-reality behavioral model for the decision-making of humans and has been used in a wide range of applications and fields [56], including recent studies in engineering [57–59]. There has also been considerable work at the intersection of transportation studies and prospect theory [60–62]. Thus, in this paper, we offer a game-theoretic formulation for the routing and travel-mode choice problem and propose a pricing mechanism that successfully leads to an efficient equilibrium under the standard assumption of rationality and prospect theory.

Contribution of this paper

In this paper, we formulate a mobility game for the travelers’ routing and travel-mode choices in a multimodal transportation network. We study the existence of a NE and the resulting inefficiencies of the travelers’ decision-making. Our main contribution is to show that although we cannot guarantee equilibrium uniqueness, our pricing mechanism allows us to control the inefficiencies that arise at equilibrium and we can then derive an upper bound for the PoA. In particular, our mobility game considers the impacts of negative congestion externalities and waiting costs in travelers’ decision-making. That way, we offer an improved look at the socioeconomic factors that can affect the efficient and sustainable distribution of travel demand in a transportation network with multiple different modes of transportation (e.g., car, bus, light rail, bike). Moreover, we study the travelers’ decision-making under two behavioral models: (1) rational, where players are selfish and seek to maximize only their own utility; and (2) prospect theory, where the players’ biases and subjectivity are taken into account when decisions are made under risk.

The features that distinguish our work from the state of the art are as follows:

1. we model the interactions between travelers using a mobility game, a combination of an atomic routing and potential game with travel-mode choices and coupled action sets (see Section “Modeling Framework”);
2. we take into account the traffic congestion cost factors using linear cost functions and the waiting time of travelers at different transport hubs; each transport hub allows a traveler to choose any of the available modes of transportation to utilize for their travel needs (see Section “Modeling Framework”);
3. we introduce a mobility pricing mechanism to control travel demand (both regarding the route and mobility service) and study the inefficiencies at a NE by showing that a NE exists (Theorem 1) and deriving a bound that remains small enough as the number of travelers increases (Theorem 2); in addition, we also upper bound the ratio of the maximum social welfare that can be achieved by all travelers to the maximum social welfare that is achieved at a NE (Theorem 3), and
4. we incorporate prospect theory for the travelers’ decision-making under the uncertainty of the transport hubs’ budgets (Theorem 4).

Organization of the paper

The remainder of the paper is structured as follows. First, we present the mathematical formulation of our mobility game, which forms the basis of our theoretical study in this paper. Then, we derive the properties of our mobility game, i.e., we show NE existence, and we bound the Price of Anarchy (PoA) and the Price of Stability (PoS). Then we prove that a NE exists under prospect theory. Finally, we draw conclusions and offer a discussion of future research.

Existence of a nash equilibrium

In this subsection, we prove that for the mobility game , as defined in Definition 12 (please also see Table 1), there exists at least one NE. The key idea of our proof is the use of a potential function, as defined in Definition 13, that captures the changes in utility of an arbitrary traveler that deviates in their action.

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Table 1. A summary of our notation.

https://doi.org/10.1371/journal.pone.0285322.t001

Theorem 1. The mobility game admits a pure-strategy NE.

As it is standard in the existence of NE results for potential games (see Chapter 2 [68]), we start by stating the explicit form of the potential function for the mobility game , i.e., (7) Our goal now is to verify Definition 13, thus showing that is a potential game. Mathematically, for an arbitrary traveler i and for any two actions and , we need to show (8)

Note here that any traveler i that deviates in their action a_i to by changing their route ρ_i to that does not require an additional service j on any new roads , then traveler i’s impact to congestion is negligent. Thus, traveler i can change routes and still travel along an existing service j in road if that service j has not reached its maximum capacity . If traveler i changes their route from ρ_i to and the traveler capacities of all services on that route are not maxed out, then the number of services J_e does not change in the roads that remain the same along both routes (any ). However, the number of services J_e increases by one in any road since we require an additional service j in road e for traveler i. Thus, we have .

If traveler i changes their transport hub v_i to , then their new waiting cost is , where . We make a similar argument for π_i and , and thus, we can write (9) where the first component represents the squared remaining cost that traveler i has to pay for deviating to the alternative transport hub without taking into account their . Thus, it accounts for the base cost of the deviation in the transport hub and mobility payment and the sum of prices paid by other the travelers. The second component represents the squared remaining cost that traveler i has to pay for the alternative transport hub when actually they do consider their , along with the sum of prices paid by other travelers. We note here that the difference between these two terms highlights the impact of traveler i’s price and on their u_i when they deviate and choose an alternative action. If traveler i chooses a higher payment, the remaining cost decreases, and this difference will have a negative impact on their utility. Conversely, if traveler i chooses a lower payment, the remaining cost increases, and this difference will have a positive impact on their utility. Next, we subtract (9) from (4) to get (10) where . Now, we denote all four components of (7) as follows: , , , and . Step by step, we compute the difference of all four different ϕ’s under a_i and as follows (11) where we note that in both v_i and are included, so if we expand the summations that involve the unrelated nodes in , then most terms cancel out (what remains are only the terms that involve the deviations of traveler i, thus we get (11)). Next, we have (12) (13) (14) We define . We take the sum of (11)–(14) and thus, we obtain . This proves that the mobility game is a potential game, and so following from key results (see [68]), we conclude that admits a pure-strategy NE.

Note that this existence result is not straightforward as the action set of any traveler is a coupled set composed of countable and uncountable subsets.

Corollary 1. If the mobility game is played repeatedly, then the travelers’ actions converge to a pure-strategy NE in finite time.

Proof. We aim to show that if the mobility game is played repeatedly, the travelers’ actions will converge to a pure-strategy NE in finite time. This proof relies on Theorem 1 and Theorem 2.6 (pp. 33) from [68]. The mobility game can be classified as a repeated routing game with complete information since each traveler has full knowledge of the mobility system (travel time latencies, network congestion, and other relevant parameters) when making their decisions. We demonstrate convergence to a pure-strategy NE by considering the decision-making process of the travelers; so we outline the iterative process: (i) Traveler chooses their initial action a_i based on the current state of the mobility system. (ii) After observing traveler i’s action, all other travelers choose their actions a_k accordingly, taking into account the updated state of the mobility system. (iii) The mobility system’s state is updated again, reflecting the actions of all travelers in the current round. (iv) Traveler i now evaluates their action a_i and decides whether to deviate to based on the updated state of the mobility system. (v) These steps are repeated for all travelers until no traveler has an incentive to change their action, leading to a convergence to a pure-strategy NE. we observe that travelers compete and as a result, each traveler’s actions are influenced by the actions of others. So, continuous deviations eventually will lead to a pure strategy NE by an iterative process (see Theorem 2.6 from [68]). In conclusion, by showing that the mobility game is a repeated routing game with complete information and that the iterative decision-making process of the travelers leads to a stable equilibrium, we have demonstrated that the travelers’ actions will converge to a pure-strategy NE in finite time.

Price of anarchy and stability analysis

An existence result (Theorem 1) leads to the problem of multiple NE and raises questions about the efficiency of each equilibrium. For example, an important concern is the efficiency of the equilibrium that the travelers will reach (as it is guaranteed by Corollary 1). To address this concern, we provide an analysis based on the Price of Anarchy (PoA) [69], which is one of the most widely-used metrics to measure the inefficiency in a system and provides an understanding of how the travelers’ decision-making affect the overall performance of the system. We provide the formal definition of the PoA.

Definition 15. Let the social welfare of the mobility game be represented by . Then, the PoA is the ratio of the maximum optimal social welfare over the minimum social welfare at a NE, i.e., (15) where is the set of NE, which is guaranteed to be non-empty according to Theorem 1.

Next, we show that, for the mobility game , (15) is as low as possible at an arbitrary NE. Thus, it follows that our PoA result yields an upper bound for the inefficiencies at a NE of the mobility game .

But first, we prove two useful lemmata that are necessary for our work.

Lemma 1. Let denote the optimal action of traveler , define , and at a NE: . Then, we have (16)

Proof. At social optimum, the pricing mechanism is given by (17)

Summing over all travelers now gives (18) So, we use the Cauchy-Schwarz inequality to bound (18), i.e., (19) For any traveler i, it is always true that , , and also . Thus, (19) simplifies to (20) Note that (20) is a second-order polynomial with respect to . Thus, we compute the discriminant , where Δ* denotes the discriminant at the social optimum, so clearly Δ* ≥ 0. So, from (20), we get (21) We need to follow the same steps to obtain a similar relation as (21) for a NE. Hence, we have (22) where . Next, the LHS of (16), if expanded, can be simplified as follows: (23) (24) where denotes traveler i’s payment at an optimal , and thus, (24) can be simplified by noting that (25) Using (25), we impose an upper bound to (24) as follows (26)

We continue by upper bounding the summations in (26): (27) by the Cauchy-Schwartz inequality and relations (21) and (22). Finally, we substitute and into (27) and with (26) we obtain the desired bound.

Lemma 2. We have (28)

Proof. We can show this result by expanding and rearranging (28) to obtain a simplified relation. So, we have (29) We seek to solve for , so we remove the square roots by squaring both sides of (29), i.e., (30) where and . We solve (30) by noting the positivity of the coefficients to obtain (31) We observe that , and so, an upper bound exists for (31). Thus, we have . We substitute back both E and D and get (32) Hence, since , the result follows.

We are ready now to state and prove our PoA result.

Theorem 2. Any inefficiencies of any NE of the mobility game remain low as close to a constant as the number of travelers tends to infinity. Mathematically, we have (33)

Proof. From Definition 14, for some arbitrary traveler , it is clear that , so if we expand the RHS of it, we get (34) where is the budget at an optimal and . Summing over all travelers in (34), and keeping note of yields (35) At this point, we recall that the travel time latency functions are linear. Thus, we can write (36) (37) where and denote the number of services on road e at a NE and social optimum, respectively. Inspired from [70], we impose an upper bound on each component of the RHS of (35), and thus we have the following (38) where we simplified the notation as , used c_e(J_e)≤c_e(J_e + 1) for each , and applied the Cauchy-Schwarz inequality twice. Note that at any . Next, we introduce notation: and . We have (39) Now we introduce the following notation: (40) where is the budget at a , and (41) By Lemma 1, we have a bound for the first two components of (35), thus (42) We combine all relations together (38), (39), and (42) and substitute them into (35) to get (43) where we have used the fact that for any four numbers , k = 1, 2, 3, 4, we have . Then (43) leads to (44) So, we have the following after a simple rearrangement (45) We divide both sides of (45) by J(a^Nash) to obtain (46) By Lemma 2, we reach the desired bound.

Next, we discuss the intuition behind Theorem 2. The travelers of the mobility game are considered selfish and non-cooperative. Thus, one important question is what will be the impact of selfishness on the efficiency of the mobility system. Since the existence of a NE is guaranteed by Theorem 1 and the mobility game converges to at least one NE, we can compare the level of inefficiency at a NE to the social optimum (this is exactly what the PoA does). The bound we have derived in (33) under certain conditions ensures the mobility system’s inefficiency is guaranteed to remain within a constant, and as the number of travelers increases, this bound becomes smaller and smaller. So, our mobility game can ensure in a realistic setting (big city with road infrastructure) with a large number of travelers a sufficiently efficient operation of the mobility system as it could ideally be operated by a central authority “ordering” the travelers how to travel. Our bound in (33) is strong in the sense that it excludes any other possibility of an improvement in efficiency compared to what we can achieve at a NE.

It happens that we can also upper bound using the “potential function method” (as used in [66]) the Price of Stability (PoS) for our mobility game . The PoS is defined as a ratio comparing social optimum and the best possible social welfare at a NE, i.e., (47)

Theorem 3. With linear travel time latency functions and pricing (7), the PoS for the mobility game is upper bounded, i.e., (48)

Proof. Recall that the social welfare of the mobility game is represented by . We aim to compare the J(a) with the potential function Φ(a). So, we have (49) where we have substituted c_e(J_e) = ξ₁J_e + ξ₂. Next, we have (50) We also have the following (51) which leads to (52)

By comparing now (50) and (52), we note that (53) (54) Our next step is to define a NE that happens to maximize the potential function, say to get (55) Therefore, if we divide by , we get the following relation (56)

Theorem 3 helps us understand the mobility game’s NE and provides a metric for how close any NE might be to the social optimum. We can ensure that as the number of travelers increases, the smaller the PoS becomes, guaranteeing that the closer the mobility game’s NE is to the social optimum.

Prospect theory analysis

In this subsection, we introduce prospect theory and its main concepts [56, 71]. We then incorporate prospect theory to our game . One of the main questions prospect theory attempts to answer is how a decision-maker may evaluate different possible actions/outcomes under uncertain and risky circumstances. Thus, prospect theory is a descriptive behavioral model and focuses on three main behavioral factors:

1. Reference dependence: decision makers make decisions based on their utility, which is measured from the “gains” or “losses.” However, the utility is a gain or loss relative to a reference point that may be unique to each decision-maker. It has been shown in experimental studies [56] the reference dependence captures the tendency of a decision-maker to be more affected in their decisions by the changes in attributes than by the absolute magnitudes. For example, the shortest/average travel time between two locations.
2. Diminishing sensitivity: changes in value have a greater impact near the reference point than away from the reference point. For example, an individual is highly likely to discriminate between a 1 and 2 hours travel time but not very likely to notice the difference between 18 and 19 hours travel time.
3. Loss aversion: decision-makers are more conservative in gains and riskier in losses. For example, a traveler may prefer to secure a 45 min commute rather than risking for a 1.5 hours commute.

One way to mathematize the above behavioral factors (1)—(3) is to consider an action by a decision-maker as a “gamble” with objective utility value (e.g., money). We say that this decision maker perceives z subjectively using a value function [54, 72] (57) where z₀ represents a reference point, β₁, β₂ ∈ (0, 1) are parameters that represent the diminishing sensitivity. Both β₁, β₂ shape (57) in a way that the changes in value have a greater impact near the reference point than away from the reference point. We observe that (57) is concave in the domain of gains and convex in the domain of losses. Moreover, λ ≥ 1 reflects the level of loss aversion of decision makers (see Fig 1).

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Fig 1. The value function for three different values of λ.

https://doi.org/10.1371/journal.pone.0285322.g001

Remark 2 To the best of our knowledge, there does not exist a widely-agreed theory that determines and defines the reference dependence [53, 73, 74]. In engineering [62, 75], it is assumed that z₀ = 0 captures a decision maker’s expected status-quo level of the resources.

As we discussed earlier in this subsection, prospect theory models the subjective behavior of decision-makers under uncertainty and risk. Each objective utility is associated with a probabilistic occurrence, say p ∈ [0, 1]. Decision makers, though, are subjective and perceive p in different ways depending on its value. To capture this behavior, we introduce a strictly increasing function with w(0) = 0 and w(1) = 1 called the probability weighting function. This function allows us to model how decision-makers may overestimate small probabilities of objective utilities, i.e., w(p)>p if p is close to 0, or underestimate high probabilities, i.e., w(p)<p if p is close to 1 (see Fig 2). For the purposes of this work, we use the probability weighting function first introduced in [76], (58) where β₃ ∈ (0, 1) represents a rational index, i.e., the distortion of a decision-makers probability perceptions. Mathematically, β₃ controls the curvature of the weighting function (see Fig 2). Although there are many different formulations for the probability weighting function, we use (58) defined in [76] as a single-parameter function and easier computationally to estimate.

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Fig 2. The probability weighting function for three different values of the rational index β₃.

https://doi.org/10.1371/journal.pone.0285322.g002

Next, we define a prospect which is a tuple of the objective utility (gain or loss) and its probability of happening.

Definition 16. Suppose that there are possible outcomes available to a decision-maker and is the kth gain/loss of objective utility. Then a prospect ℓ_k is a tuple of the utilities and their respective probabilities (59) where k = 1, 2, …, K. We denote the kth prospect more compactly as ℓ_k = (z_k, p_k). We have that and ℓ_k is well-ordered, i.e., z₀ ≤ x₁ ≤ ⋯ ≤ z_K. Under prospect theory, the decision-maker evaluates their “subjective utility” as u(ℓ) = ∑_{0≤k ≤ K}v(z_k)w(p_k), where is the profile of prospects of K outcomes.

In the remainder of this subsection, we apply the prospect theory to our game, clearly define the mobility outcomes (objective and subjective utilities), and then show that the prospect-theoretic mobility game admits a NE.

Travelers may be uncertain about the available amount of mobility funds at any transport hub, that is why we define a mobility prospect to represent as a random variable Z with objective utilities z₁, z₂, …, z_K and their probabilities p₁, p₂, …, p_K. Each z_k now represents the uncertain b(v_i). In addition, the reference dependence of each traveler i is represented by . For any traveler i, the probability weighting function is and the value function is , k = 1, 2, …, K. Thus, we have (60) where w_i(p_k) is given by (58), and (61) where β = β₁ = β₂. We can justify β₁ = β₂ in the above definition as it has been verified to produce extremely good results, and the outcomes are consistent with the original data [54]. Next, we explicitly define the reference point for the mobility game as follows where represents the ideal redistribution of wealth to traveler i (since no transport hub v_i should make a profit, i.e., b(v_i) = 0). For the random variable Z, we assume a continuous distribution F with zero mean and a probability density function f, and explicitly have (62) So, by (60), we have , where z(n) represents at each transport hub v_i of an arbitrary traveler i the realization of Z with available mobility funds. The total utility now under prospect theory for a traveler i is (63)

Next, we show that our mobility game under prospect theory is guaranteed to have at least one NE.

Theorem 4. The mobility game under prospect theory admits a pure-strategy NE.

proof. We expand z(n) and subtract and simplify to get (64) where . Substituting (64) into (63) yields (65) where v_i is given by (61). The next step is explicitly defining a new potential function under prospect theory. We have (66) Next, we show that Ψ as given in (66) is an exact potential function. We notice that does not depend on a_−i, i.e., the actions of the other travelers except traveler i. Hence, following similar arguments as in Theorem 1, we obtain . Hence, Ψ is indeed an exact potential function for the mobility game under prospect theory. Therefore, since any action profile that minimizes Ψ results in a NE, the mobility game admits a NE under prospect theory.

Corollary 2. For the mobility game under prospect theory, the sequence of best responses of an arbitrary traveler converges to a NE.

Proof. It is sufficient to note that the action set of any traveler i is compact. Thus, it follows from the results in [42] that the sequence of best responses of any traveler converges to a NE.

Both Theorem 4 and Corollary 2 ensure that the mobility game under the prospect-theoretic behavioral model admits a NE and prospect-based travelers will eventually converge to it. Both results establish that we can still ensure that an equilibrium can be reached under certain conditions for the cost and pricing functions.

Implementation

In this subsection, we outline how our proposed mobility game can be potentially implemented. We consider a major metropolitan area with an extensive road and public transit infrastructure; a good example is Boston. Several key areas in Boston are connected by roads, buses, light rail, and bikes. These areas can serve as transport hubs from which travelers can utilize any of the available modes of transportation. We can apply the MaaS concept and offer on each transport hub travel services (e.g., navigation, location, booking, payment) to all passing travelers. Information can be shared among all travelers via a “mobility app,” which allows travelers to access the services on the transport hubs. Using this app, travelers can pay for their travel needs and, at the same time, receive mobility payments. For example, a traveler who informs the app and uses a bike multiple times (per day or per week) can receive mobility payments. In addition, travelers travel multiple times and interact with each other more than once. So, travelers seek to move from one place to another while competing with many other travelers, use the transport hubs to access their preferred mode of transportation, and pay using a mobility app. Each mode of transportation offers different benefits in utility; for example, a car is more convenient than a bus and is expected to be in high demand. This naturally will lead to inefficiency and congestion.

The mobility game with the utility structure defined in (4) captures the key factors that may play a role in a traveler’s decision-making. It can be seen by Theorem 1 and Corollary 1 that an equilibrium exists and can be reached by the travelers without direct intervention from a central authority. The particular pricing mechanism we have proposed in (1) ensures all travelers with the computational power of their cellphone can quickly derive the NE strategy (route, transport hub, payment). This is important as we can avoid solving a mixed-integer nonlinear program for all the travelers in the mobility system. In addition, by Theorem 2 we can guarantee that the inefficiency of the mobility system stays low as long as the number of travelers remains large (something that is expected in a typical mobility system). We can guarantee that a NE will stay close to the social optimum as the number of travelers increases. Thus, even though we cannot guarantee the uniqueness of a NE, we can ensure that all NE are similarly efficient and nearly as efficient as the social optimum as long as the number of travelers is high.

Using prospect theory, we can also consider how a traveler can feel uncertain about whether they may receive mobility payments for choosing a more sustainable mode of transportation (e.g., bike). Under certain conditions, we show that indeed a NE exists (Theorem 4) and it can be reached by the travelers as they can travel from the hub that is nearest to their home to the hub that is nearest to their work (Corollary 2). Thus, our game framework is proved to lead to a NE under two different behavioral models and capture the impact of the travelers’ decision-making.

Technical discussion and a numerical example

In this subsection, we discuss in more detail the technical implementation of our mobility game. So, we can compute a NE using the potential function, leveraging the fact that our game is a potential game with a finite set of travelers (as shown in Theorem 1). Note that this means that there exists a potential function (given by (7)) that maps each strategy profile to a real value. Intuitively, any change in any traveler’s utility that unilaterally deviates from a strategy is equal to the change in our potential function. Hence, we can find the strategy profiles that simply maximize our potential function. One approach to achieve this for mixed-integer optimization problems is to use the branch-and-bound algorithm [77]. So, by Corollary 1, convergence to a pure strategy NE is guaranteed, thus we can find a NE, compute it, and use for our PoA analysis.

Typically, we solve numerically the optimization problem that arises from the routing game. We can either employ a gradient-based methods or a learning algorithm (e.g., fictitious play). The gradient-based method involves updating the travelers’ strategies by moving in the direction of the gradient of the potential function. However, since the action sets are coupled and include route choices, the optional stop at a transport hub along the route, and the payment for the mobility service, the standard way solve this problem is as a mixed-integer nonlinear program (MINLP) (in such a case use the branch-and-bound or branch-and-cut algorithm). Alternatively, we can first use the specialized algorithm Dijkstra and find the best route and transport hub through the network based on criteria such as shortest distance and least time. Once we have this optimal route, we then use “fmincon” to compute the optimal value of the mobility payments along that route and transport hub.

We now offer a numerical example with a simple transportation network that has one unique origin-destination (OD) pair. In this network, there are three routes, namely route 1, route 2, and route 3. We assume that there are two mobility services: car and bike. The travel time on each route depends on the volume of traffic and the service. We have the following (67) (68) (69) where x_k denotes the fraction of travelers choosing either of routes k = 1, 2, 3 using a car. We also consider that the pricing functions for the two modes: car and bike take the explicit form as τ(car) = 10 and τ(bike) = −2. So, travelers receive a $2 incentive for choosing a bike. Suppose we have 50 travelers in total, with 30 travelers preferring route 1 and route 2 (we call this Group A), while the remaining 20 travelers prefer route 2 and route 3 (we call this Group B). Now, say that Group A chooses to utilize route 1 with a car, and Group B chooses to utilize route 3 using a car. We can now compute the utilities for each traveler: (70) (71) (72) So, for Group A the utility for route 2 (car) is higher than route 1 (car), thus all travelers in Group A will deviate to route 2 (car). Similarly, in Group B the utility for route 3 (bike) is higher than Route 3 (car), and so all travelers will deviate to route 3 (bike). Let us now compute the utilities for any arbitrary traveler, i.e., (73) (74) (75) (76) (77) (78) We continue our equilibrium analysis as follows: for Group A, the utility for route 1 (bike) is higher than Route 2 (car), so travelers will deviate to route 1 (bike). Next, Group B will not deviate as route 3 (bike) (already highest utility). Hence, we have reached the point in which no traveler can deviate and receive better utility; thus we have a NE. We notice that all travelers from Group A and Group B choose route 1 and route 3 utilizing bikes, respectively. We conclude that it is possible as we showed in our theoretical analysis for a NE to exist and it is easily converged to based on our pricing mechanism in this simple multimodal transportation network. Next, at this Ne, all travelers have chosen to utilize the mobility service: bike, which is a sustainable and environmentally-friendly mode of transportation. Our pricing mechanism naturally will favor such modes of transportation and provide incentives to travelers for better utilization. Thus, we can control travel demand and effectively reduce any inefficiencies that may arise from congestion or higher pollution levels caused by car usage. One last note: in this example, we used a similar method to the best-response dynamics approach and found the NE using only a few iterations. Although our example quickly leads to an equilibrium solution, for larger and more complex transportation networks, we cannot draw the same conclusion and thus, it remains future work to adopt more advanced optimization and algorithmic techniques in order to find and compute our mobility game’s NE.

Limitations and future work

One important limitation of our framework is the assumption of complete information. Realistically, we cannot expect travelers to have accurate and complete knowledge of other travelers or the system’s capabilities (network, road capacities). A potential direction for future research should relax this assumption by only allowing travelers to know their own actions and utilities. In the literature, attempts have been made to investigate the emergence of cooperation among selfish travelers and how to bound rationality/irrationality in travel-choice problems [78–81]. A standard technique is Bayesian game-theoretic analyses, and recently, and techniques to learn representations of unknown information from observed data [82–86]. Another interesting direction for future research is to expand the current framework by explicitly designing the socially-efficient pricing functions to achieve the best possible equilibrium in the mobility system using techniques from mechanism design. Furthermore, to showcase the benefits of the proposed game-theoretic approach, a necessary extension of our work is using machine learning techniques [87–91] with real-life data. Finally, future work may explore how to perform a comparative analysis on models focusing on real-time efficacy for similar problems like ours using techniques similar to other fields [92–95].