Pricing equilibrium of transportation systems with behavioral commuters

We study Wardrop equilibrium in a transportation system with profit-maximizing firms and heterogeneous commuters. Standard commuters minimize the sum of monetary costs and equilibrium travel time in their route choice, while "oblivious" commuters choose the route with minimal idle time. Three possible scenarios can arise in equilibrium: A pooling scenario where all commuters make the same transport choice; A separating scenario where different types of commuters make different transport choices; A partial pooling scenario where some standard commuters make the same transport choice as the oblivious commuters. We characterize the equilibrium existence condition, derive equilibrium flows, prices and firms' profits in each scenario, and conduct comparative analyses on parameters representing route conditions and heterogeneity of commuters, respectively. The framework nests the standard model in which all commuters are standard as a special case, and also allows for the case in which all commuters are oblivious as the other extreme. Our study shows how the presence of behavioral commuters under different route conditions affects equilibrium behavior of commuters and firms, as well a the equilibrium outcome of the transportation system.


1.
Introduction. Modeling and predicting traffic flows in a transportation system is a practical challenge that warrants consideration of the psychology of commuters' travel choices, as well as the heterogeneity of commuters' reasoning tendencies in the population. Since the application of game theoretic modeling to transportation problems (Wardrop, 1952), a feature of equilbrium analysis is the underlying assumption that all commuters involved are perfectly rational in the game theoretic sense, meaning that they optimally respond to other commuters' rational game theoretic behavior.
However, such assumptions may be considered too idealistic in real world situations, where often at least a subset of commuters may not incorporate strategic reasoning when making their choices. For example, visitors from out of town may know about the local travel route with the shortest distance, but they may not be aware of the typical travel time required on each route under traffic conditions, a factor which is well-taken into consideration by local commuters. Commuters who are especially busy or cognitively burdened, may heuristically know about the route that requires the least travel time based on their past experience, but may not have time to think carefully about how daily weather conditions or other external circumstances will affect the travel choices of other commuters in the population.
These examples share in common that there is a subset of commuters that are equilbrium players in the sense of Wardrop Equilibrium, while another subset of commuters are limited in their strategic considerations (see Camerer, Ho andChong, 2004, Crawford, 2013, and Lien, Zhao and Zheng, 2019). 1 The latter subset of commuters might consider some salient aspects of the travel routes, while neglecting to incorporate the behavior of other commuters who are using the transportation system into their decision. In this paper, we analyze the implication of such behavioral heterogeneity in the population of commuters on traffic flows across different routes, in a market where transportation firms compete with one another by providing substitutable routes to commuters.
In our setup, "oblivious" commuters not only ignore the traffic-contingent costs in the transport system in their decision, but they also do not pay the service price. This scenario corresponds to some plausible real world situations: For example, the oblivious commuters may not be aware of the requirement to pay the service pricethey could be visitors choosing their route for the first time, oblivious to congestion considerations while neglecting to purchase a ticket; As another example, oblivious commuters may be exempt from paying the service price, or receive a deep price discount -for example, the case of children and the elderly, who often may be less strategically sensitive to congestion costs than ordinary commuters. 2 A substantial literature examines the role of boundedly rational commuters in transportation systems. Di and Liu (2016) provides a comprehensive survey of different approaches to bounded rationality utilized in the transportation literature. Our study differs from prior works by modeling a competitive market in the transportation network, whereas most of the prior literature focuses on the 1 The cognitive hierarchy model by Camerer, Ho and Chong (2004) provides a non-equilibrium framework for heterogeneity in strategic thinking, as does the level-k class of models discussed in survey Crawford (2013). In the current paper, we maintain equilibrium concepts while allowing for some commuters to be unaffected by cost factors that are influenced by other commuters. However, one can easily infer that if such commuters took those cost factors into consideration in their utility but effectively ignored it in their decision-making, that equilbrium behavior in our analysis results in suboptimal utility realizations in a scenario where strategically determined payoff components in fact matter. In maintaining equilibrium concepts, our current approach bears resemblance to Lien, Zhao and Zheng (2019) which analyzes equilibria in contest games in which contestants have perception biases. 2 Another application is that some commuters may have priority to use the route for freefor example, certain government vehicles which are exempt from road service charges. From the decision-making standpoint, if such vehicles are exempt from congestion costs, such as the cases of police cars, fire trucks and ambulances, our model also applies without any behavioral labeling (ie. obliviousness). The welfare consequences however, would differ. user (price-free) equilibrium. Our approach follows on that of , Lien, Mazalov, Melnik and Zheng (2016), Kuang, Mazalov, Tang and Zheng (2020), and Kuang, Lian, Lien and Zheng (2020), in analyzing transport systems and traffic flows under duopoly competition between transport providers.
Among studies focusing on user equilibrium with boundedly rational commuters, Mahmassani and Chang (1987) provides an early framework and analysis for boundedly rational user equilibrium. Di, He, Guo and Liu (2014) study the conditions under which the Braess Paradox persists under a boundedly rational user equilibrium. Ye and Yang (2017) incorporate the boundedly rational user equilibrium into the rational behavior adjustment process, and examine the equilibrium existence conditions. Sun, Cheng and Ma (2018) use a boundedly rational user equilbrium approach to modeling travel time reliability. Our study is closely related in approach to Karakostas, Kim, Viglas and Xia (2011), which considers user equilibria with commuters who may be oblivious to congestion costs that are a result of other commuters' choices.
On the topic of boundedly rational commuters, several empirical studies also address the issue. Among them, Jou, Hensher, Liu and Chiu (2010) use a stated choice approach to empirically examine boundedly rational transport mode switching behavior in Taipei. Zhao and Huang (2016) conduct a laboratory experiment to test boundedly rational choice of routes under a satisficing rule. Tang Luo and Liu (2016) conduct numerical simulations to show the increasing effects of driver bounded rationality on costs of travel.
Our study is the first to our knowledge, to consider the role of boundedly rational commuters in the sense of being oblivious to other commuters' influences on traffic flows, in a competitive transport system where prices motivate the equilibrium between all players in the market, firms and commuters. Transport firms compete with one another and set prices strategically, while commuters choose among the provided routes to maximize their utilities, which are influenced by the choices of other commuters through the congestion cost. We show that three possible scenarios can arise in equilibrium: A pooling scenario where all commuters make the same transport choice; A separating scenario where different types of commuters make different transport choices, A partial pooling scenario where some standard commuters make the same transport choice as the oblivious commuters. Our analysis provides conditions under which the flow of commuter traffic varies based on commuter sophistication and route followed, particularly as a function of the total traffic flow intensity and the fraction of oblivious commuters. In particular, we provide conditions under which each of the three scenarios can become an equilibrium.
The paper is organized as follows: Section 2 provides the model setup. Section 3 conducts equilibrium analysis. Section 4 provides a numerical example to illustrate the results. Section 5 concludes and discusses.
2. Model. We analyze a duopoly competition model between two transport service carriers in a transportation route system of parallel structure. Carriers set service prices to maximize their own profits, and commuters choose which carrier's service to use in order to optimize their own objective functions, the form of which depends on the type of commuter, either standard or "oblivious".
The standard commuters minimize their cost which is equal to the service price paid (the monetary cost) plus the actual travel time spent (the waiting cost), while the oblivious commuters merely minimize the idle (unoccupied) travel time of the chosen route.
2.1. Setting. Consider a transport network composed of two parallel routes between origin A and destination B with the following linear BPR (Bureau of Public Roads) latency functions: 3 In the above expressions, z i represents the flow intensity of route i and (t 1 , a 1 , b 1 ); t 2 , a 2 , b 2 ); are latency parameters such that t i indicates the travel time on a completely unoccupied route i, a i denotes the internal effect of route i's own flow on route i, and b i measures the external effect of route −i's flow on route i, where i = 1, 2. 4 Without loss of generality, we suppose that the first route is faster than the second route, that is t 1 ≤ t 2 . As a reasonable setup, we also assume that the internal effects dominate the external effects, that is, Under such a route structure, commuters of flow intensity X travel from origin A to destination C, and each commuter must choose exactly one of the two routes. There are two carriers: Firm 1 takes charge of the first route and sets a service price of p 1 ; Firm 2 takes charge of the second route and sets a service price of p 2 .
Of the coming flow of commuters X, α fraction of them, denoted by x 0 = αX, are oblivious commuters, where 0 ≤ α < 1. They do not pay the transport service price, and ignore congestion merely choosing the fastest route based on the unoccupied travel time (which can also be interpreted as congestion-independent route-specific characteristics such as geographical distance or road length).
The other commuters, of flow intensity (1 − α)X, are standard commuters. They compare the sum of both monetary cost and waiting cost, namely p 1 + f 1 (x) and p 2 + f 2 (x), in using each route, and select the route with minimal total cost.
We use x = (x 0 , x 1 , x 2 ) to denote a profile of subflows among commuters, where the flow of standard consumers is divided into two subflows x 1 and x 2 such that x i represents the flow of standard consumers using route i, i = 1, 2. Thus we have Notice that the oblivious commuters will always choose the first route as long as t 1 < t 2 . In the case where t 1 = t 2 , as a tie breaking rule, we simply assume that route 1 will be chosen by all oblivious commuters. 5 Therefore, the following conditions are satisfied: Both carriers (firms 1 and 2) set their prices strategically and independently to maximize their profits, while their operating costs are simply normalized to zero. To be specific, firm i's profit, H i = x i p i , equals the product of route i flow of standard commuters and firm i's service price.
The setting of the model can be illustrated by Figure 1. 2. Transformation and equivalence. When introducing externalities into the network pricing model, the intuition behind the results can be substantially different from that without externalities. Nonetheless, when the number of routes is 2, the two models are essentially the same through a mathematical transformation, which is due to the overall flow distribution being determined as long as one route's flow is fixed. Such an equivalence between the cases with and without externalities is established in the following proposition. 6 Proposition 1. The 2-route heterogeneous-commuters linear transportation system with externalities, (X; α; x 0 , x 1 , x 2 ; t 1 , t 2 ; a 1 , a 2 ; b 1 , b 2 ), is equivalent to another 2route heterogeneous-commuters linear transportation system without externalities, (X; α; x 0 , x 1 , x 2 ; t 1 , t 2 ; a 1 , a 2 ).
To see why the above proposition holds, note that the latency functions with externalities can be re-written as Using a similar transformation technique, while keeping the feature of externalities, we can also convert the current transportation system with heterogeneous commuters into one with standard commuters only. Note that This gives us the following equivalence proposition.
Proposition 2. The 2-route heterogeneous-commuters linear transportation system with externalities, (X; α; , is equivalent to another 2route standard-commuters linear transportation system with externalities, (X; α; Finally, we can further convert the heterogeneous-commuters linear transportation system with externalities into a standard-commuters system without externalities, through the following transformation: This leads to the third equivalence proposition in which the transformation is achieved from removing both commuters' heterogeneity and cross-route externalities.
Proposition 3. The 2-route heterogeneous-commuters linear transportation system with externalities, (X; α; x 0 , x 1 , x 2 ; t 1 , t 2 ; a 1 , a 2 ; b 1 , b 2 ), is equivalent to another 2route standard-commuters linear transportation system without externalities, (X; α; Comparing the above three equivalence results, it is easy to verify that when there are no oblivious commuters (α = 0), we have t i = t i and a i = a i , i = 1, 2. Also, when there are no externalities 3. Equilibrium analysis. In this section, we characterize the equilibrium of the heterogeneous-commuters 2-carrier 2-route transportation system introduced in Section 2.
Recall that due to the assumption t 1 ≤ t 2 , the oblivious commuters will always follow the route 1. Thus, there are in total three possible scenarios regarding how standard commuters (and oblivious commuters) are allocated between the two routes: 1) Concentration on One Route: In this pooling scenario, all commuters, regardless of whether they are oblivious or standard, travel through the first route.
2) Competition for Standard Commuters: In this partial pooling scenario, oblivious commuters travel through the first route and the standard commuters are distributed between the first and second routes.
3) Separation between Oblivious and Standard Commuters: In this separating scenario, the oblivious commuters travel through the first route and the standard commuters travel through the second route.
In the following subsections, we first discuss each of these scenarios and derive flows, prices and firms' profits, and then characterize the equilibrium depending on the values of model parameters (X; α; t 1 ; t 2 ; a 1 , a 2 ; b 1 , b 2 ).
3.1. Concentration on one route. In general, the oblivious commuters will surely follow the first route since t 1 ≤ t 2 , but the standard commuters make their decision by comparing the sum of both monetary cost and waiting cost in using each route. In order for all standard commuters to choose the first route, it must be that the travel time of the first route with the flow αX of oblivious commuters is no more than the travel time of the unoccupied second route, that is, Furthermore, the first route should also be more attractive for the flow (1 − α)X of standard commuters in the sense that even if the price of the second route is p 2 = 0, these standard commuters will weakly prefer the first route. 8 Thus, the following condition must be satisfied for x = (x 0 , x 1 , x 2 ) = (αX, (1 − α)X, 0): Consequently, the optimal price for firm 1 is given by the expression 2 )X, and the price for firm 2 should be set at p * 2 = 0. 9 We observe that the optimal price for firm 1 is not dependent on α. This is natural, since firm 2's optimal price is determined by the difference between the travel time of the unoccupied second route and the travel time of the fully occupied first route, where all commuters, regardless of whether they are oblivious or standard, have the same effect upon the travel time.
Notice that the price for firm 1 should be non-negative, which requires Inequality (1) presents a necessary condition under this pooling scenario, which suggests that the total flow intensity cannot be too high. However, it does not mean that all commuters will always choose the first route in equilibrium under this condition, which would be considered a sufficient condition. As we confirm in the subsequent analysis, the equilibrium cutoff condition for this scenario is indeed more strict, and essentially depends on the parameter α. Furthermore, when this scenario becomes an equilibrium outcome, the limit of X allowed is an increasing function of α.
Using H (k) i to denote firm i's profit in Scenario k, where i = 1, 2 and k = 1, 2, 3, in this scenario (Scenario 1) the payoffs of the two firms are given by It is obvious that such a fully pooling scenario is good for firm 1 but not for firm 2, as no standard commuters have any incentive to use the second route.

Competition for standard commuters.
In the partial pooling competitive scenario, both firms charge positive prices p 1 > 0, p 2 > 0, and the traffic flow (1 − α)X of standard commuters is split into two subflows, x 1 > 0 and x 2 > 0, which satisfy the following equations according to the indifference condition of the standard commuters: The two firms strategically set their own prices to maximize their profits We now derive the equilibrium by solving the pricing game above. First, we express the subflows (x 1 , x 2 ) in terms of prices (p 1 , p 2 ) by using equations (2), and then substitute them into the firms' profit functions, as shown below.
Then, fixing p 2 we find the best response function of firm 1 through the first order condition of H 1 (p 1 , p 2 ) with respect to p 1 , which yields Similarly, by fixing p 1 , we calculate the best response of firm 2 and obtain From these equations, we find the equilibrium prices Substituting the notation of parameters t , a into expressions (3) and (4) we obtain the equilibrium prices Thus, we can derive the following equilibrium traffic flows Consequently, the equilibrium profits of the firms in this scenario (Scenario 2) are The competitive scenario takes place only if all optimal values for prices and flows, that is x * i , p * i , i = 1, 2, are non-negative, which implies the following condition: (7) A more informative way to rewrite Inequality (7) is as follows: . 10 Inequality (7) represents a necessary condition under the competitive scenario. Again, this is not a sufficient condition. 11 It is worth noting that the presence of oblivious commuters influences the flow, prices and firms' profits of both routes, as exhibited by the parameter α in the cutoff value expressions of Inequality (7). 10 To see whyα The last inequality holds since min{a 1 , a 2 } ≥ max{b 1 , b 2 } and t 2 ≥ t 1 , implying a 2 t 2 > b 1 t 1 .
11 Notice that the lower bound expression of Inequality (7) for X is less than the cutoff expression in Inequality (1), that is, , since a i > b i for i = 1, 2. This implies that there is a potentially overlapping range of parameters such that both scenarios 1 and 2 can be possible. In the subsection of Equilibrium Characterization, we will further consider firms' available pricing strategies and solve for the unique equilibrium which results in one of the possible scenarios.

3.3.
Separation between oblivious and standard commuters. We now consider the last scenario, in which the oblivious commuters of flow αX travel through the first route while all standard commuters of flow (1 − α)X travel through the second route. This means x 1 = 0 and x 2 = (1 − α)X. Following a logical argument similar to that in the first scenario, we can show that such a separating scenario is possible only if and Inequality (8) implies that in order for all standard commuters to choose the second route, the fraction of oblivious commuters must be sufficiently high, and Inequality (9) provides another necessary condition which is that the total flow intensity cannot be too low. Also notice that Inequalities (7) and (9) have an overlapping parameter range so neither of them are sufficient conditions for scenarios 2 and 3, respectively. 12 In this separating scenario, firm 2 can optimally set the price for travel service, which is equal to the difference between the travel time of the first route with all oblivious commuters and the travel time of the second route with all standard commuters, as below (1 − α)X, and the price for firm 1 should be set at p * 1 = 0. Note that, contrary to firm 1's optimal price being independent of α in the first scenario, in the current scenario, firm 2's optimal price increases in α. This result is intuitive since the more oblivious commuters there are, the longer it takes to travel through the first route, and the less attractive the first route is compared the second route, hence the higher the price that firm 2 can set to attract all standard commuters.
Having derived firms' optimal prices, we can easily express the profits of the firms as

Equilibrium characterization.
Having analyzed the optimal pricing strategy under each scenario, we can now completely characterize the equilibrium of the pricing game between the two firms.
In the first scenario, where the total traffic flow X is sufficiently low such that Inequality (1) holds, the situation is fully determined by firm 1. Specifically, firm 1 announces the price p * 1 = t 2 − t 1 − (t 1 a 1 − t 2 b 2 )X and uses this pricing strategy if the profit H  1 . That 12 The upper bound expression of Inequality (7) for X is greater than the cutoff expression in Inequality (9), that is, In the third scenario where the fraction of oblivious commuters α and the total traffic flow X are both sufficiently high such that Inequalities (8) and (9) hold, the situation is fully controlled by firm 2. Specifically, firm 2 sets the price p * is greater than the profit under the competitive scenario H (2) 2 , that is, Based on the previous analyses and the above discussion, we summarize the results by presenting the equilibrium characterization in the following proposition.
There exist the following possible equilibrium scenarios depending on the parameters of the transportation system (X; α; t 1 , t 2 ; a 1 , a 2 ; b 1 , b 2 ): 1) Under Condition (1), if either Condition (7) fails or both Conditions (7) and (10) hold, then all traffic follows the first route: (8) and (9), if either Condition (7) fails or both Conditions (7) and (11) hold, then oblivious commuters follow the first route and standard commuters follow the second route: 3) In all other cases, standard commuters are split between the first route and the second route, where x * 0 = αX, x * i , i = 1, 2 are determined by Equations (5) and (6), and p * i , i = 1, 2 are determined by Equations (3) and (4). Proposition 3.1 confirms that the pooling scenario in which all commuters concentrate on one route (Scenario 1) cannot occur in equilibrium if the total traffic flow is very high. This result is intuitive because with a large X, the route followed by all commuters will become too crowded and eventually some commuters will be attracted by the other route. The proposition also tells us that the separating scenario of different types of commuters using different routes (Scenario 3) cannot occur in equilibrium if either the total traffic flow or the fraction of oblivious commuters is very low. The intuition is that a small X or a small α implies that the traffic flow by oblivious commuters cannot be too large, which means the delay caused by those commuters will not be too large, thus enhancing the comparative advantage of the faster route against the slower route in terms of attracting the standard commuters.
Case 1. All commuters follow the first route. In this case, x 1 = (1 − α)X, x 2 = 0, and optimal prices are given by Since the prices should be non-negative, we obtain the following expression for Inequality (1): The profits of the two firms are H X and H 2 = 0, respectively. Case 2. Both routes are followed by standard commuters. By Equations (3)-(6), the optimal prices and flows in this case are Note that the competitive case takes place only when the optimal values of p 1 , p 2 , x 1 , x 2 are non-negative. This requirement is characterized by Inequality (7), now with the following form The firms' profits are given by Case 3. Standard commuters follow only the second route. In the last case, oblivious commuters follow the first route and standard commuters follow the second route, which means x 1 = 0 and x 2 = (1 − α)X. The optimal prices are p * 1 = 0, p * 2 = −1 + (7α − 4)X. Previous analyses in Section 3. require that Inequalities (8) and (9) must hold in this case: The firms' profits are given by H

TRANSPORTATION SYSTEMS WITH BEHAVIORAL COMMUTERS 347
Equilibrium Characterization. Now we can solve for firms' optimal strategies by comparing their profits in these scenarios.
We first derive the necessary and sufficient condition for Scenario 1 to be an equilibrium outcome. The first scenario becomes an equilibrium outcome if (i) Condition (1) X ≤ 1 3 holds and (ii) whenever the second scenario is possible Condition (10) H 1 has the following specification: The above inequality is equivalent to condition X 1 ≤ X ≤ X 2 where .
Based on the above results, we obtain the condition under which the first scenario is optimal in equilibrium: . (12) Figure 2. The regions of optimal behavior of commuters and firms The area in which Condition (12) holds is filled with vertical blue lines and marked as "Case 1" in Figure 2, where the horizontal axis measures the fraction 5. Conclusion. In this paper we study a transportation system with competing firms, and commuters that vary in terms of their strategic considerations in the congestion problem. The model setup is a reasonable representation of many actual transportation markets in the sense that firms can be reasonably expected to strategically maximize profit, while only some commuters in the population are fully strategic in their transport choices. That is, a subset of commuters may not be expected to be behaving strategically in their transport route choice, due to lack of familiarity with the route, or special permissions given by the government. The absence of strategic considerations may often coincide with a substantially discounted price provided to those commuters compared with the standard strategic commuters, coinciding with our model setup.
In such a model, we show that three scenarios (pooling, partial pooling, and separating) can exist in equilibrium, depending on the parameter values in the model. Our numerical example demonstrates that when overall traffic flows are sufficiently low, regardless of the fraction of oblivious commuters in the population, all traffic will flow through the shorter of the two possible routes. On the other hand, when overall traffic flows are sufficiently high, and the fraction of oblivious commuters is sufficiently high, commuters will separate across the two possible routes, based on their degree of sophistication. That is, oblivious commuters take the shorter route, and standard commuters take the longer route. In the third scenario, oblivious commuters follow the shorter route, while standard commuters travel through both routes. The result may be intuitive to observed scenarios in transportation systems, where a subset of commuters are price sensitive but not so time sensitive. In such situations, when traffic volume is high, those consumers may tend to opt for the route which appears in all aspects less cost incurring, although in fact the real cost when considering congestion factors may be quite high.
There are several potential directions for further development of this framework. Firstly, our analysis has considered only two types of commuters, oblivious and standard. Future work can consider commuters with varying intermediate degrees of obliviousness. Secondly, in the current framework, we have for simplicity modeled the oblivious commuters similarly to the concept of noise traders in the finance literature, in the sense that they do not incur nor pay attention to the monetary price of the transport route, or at least less so compared to the standard commuters. Future work can consider relaxing this assumption to allow oblivious commuters to be more closely incorporated into the pricing interaction in the transport system. Finally, it may be possible to gain an empirical assessment of how prevalent oblivious consumers are in the real world using survey methods, experimental techniques, or transport system electronic data similar to that used in Ford, Lien, Mazalov and Zheng (2019), for example. Such assessments can allow our model to have further policy implications in analyzing transportation systems by simulating the traffic flow consequences of heterogeneous consumer types.