Strategic Traffic Management in Mixed Traffic Road Networks: A Methodological Approach Integrating Game Theory, Bilevel Optimization, and C-ITS

Abstract

The integration of Connected Vehicles into conventional traffic systems presents significant challenges due to the diverse behaviors and objectives of different drivers. Conventional vehicle drivers typically follow User Equilibrium principles, aiming to minimize their individual travel times without considering the overall network impact. In contrast, Connected Vehicle drivers, guided by real-time information from central authorities or private service providers, can adopt System Optimum strategies or Cournot-Nash oligopoly behaviors, respectively. The coexistence of these distinct player classes in mixed-traffic environments complicates the task of achieving optimal traffic flow and network performance. This paper presents a comprehensive framework for optimizing mixed-traffic road networks through a multiclass traffic assignment model. The framework integrates three distinct types of players: conventional vehicle drivers adhering to User Equilibrium principles, Connected Vehicle drivers following System Optimum principles under a central governing authority, and Connected Vehicle drivers operating under Cournot-Nash oligopoly conditions with access to services from private companies. The methodology includes defining a model to achieve optimal mixed equilibria, designing an algorithm for multiclass traffic assignment, formulating strategic games to analyze player interactions, and establishing key performance indicators to evaluate network efficiency and effectiveness. The framework is applied to a real-world road network, validating its practicality and effectiveness through computational results. The extraction and analysis of computational results are used to propose optimal traffic management policies for mixed-traffic environments. The findings provide significant insights into the dynamics of mixed traffic networks and offer practical recommendations for improving traffic management in increasingly complex urban transportation systems.

1. Introduction

Cooperative Intelligent Transport Systems (C-ITS) technologies have led to the introduction of Connected Vehicles (CVs) in road networks equipped with real-time communication capabilities and enabling drivers to access valuable traffic information for informed route selection decisions [1]. The coexistence of conventional cars and CVs is a fact nowadays, as the widespread adoption of Connected and Automated Vehicles (CAVs) is still an ongoing process [2]. The penetration of CVs leads to the generation of challenges and opportunities for existing mobility systems [3], as these technologies can address various network-wide challenges. Another challenging aspect is the diverse driver behaviors and patterns among the drivers of conventional vehicles and CVs. It has been shown that the performance of a network is not necessarily the optimal one in the case of optimized drivers choosing the shortest paths for themselves [4]. The establishment of traffic management policies aligned with this fact seeks to bridge the gap between network-wide optimal routing and travelers’ selfish routing.

The maximization of benefits in mixed networks with drivers of conventional vehicles and Autonomous Vehicles (AVs) that rely on selfish routing is a topic that has been studied from the perspective of multiclass traffic assignment. More specifically, the principles of User Equilibrium [5] have been modeled according to the AVs’ traffic proportion in the networks’ links’ capacity and tailored policies led to the improvement of the network performance and to achieving closer to System Optimum conditions [2,5]. To investigate the network impacts via a multifaceted perspective, where several parameters, such as travel times, emissions, fuel consumption, and safety, are considered, microscopic models have been used to represent drivers’ behavior and integrate Vehicle-to-Everything (V2X) communication capabilities. Such models, however, impede their application in large networks, as they pertain to many parameters and are computationally demanding. Therefore, hybrid approaches that couple macroscopic, mesoscopic, and microscopic processes and result in variable aggregation models have been tested [6].

The concept of central traffic management authorities controlling all vehicles’ routing in the entire network to address various problematic traffic configurations constitutes more of an unrealistic scenario in the case of mixed flows of CAVs and conventional cars. Approaches focusing on influencing the routing of CAVs towards the desired network equilibrium conditions are more feasible. In this essence, routing games, where a central management authority controls the routes of CAVs using System Optimum principles and pricing schemes for conventional cars, have been examined [7]. Exploiting the potential of CVs in a mixed-traffic environment to improve road traffic efficiency and reduce traffic congestion constitutes a great challenge for engineers and researchers. An increasing focus has been directed towards the investigation of intelligent transportation-based traffic congestion strategies. Real-time communication is significant for drivers as it offers valuable traffic information that affects routing decisions. For traffic operators, real-time communication plays a pivotal role in monitoring traffic flow dynamics, hence enabling the timely implementation of appropriate control measures [8].

The scope of this work is to propose a comprehensive framework that integrates game theory principles, bilevel optimization methodologies, and C-ITS technologies’ features to provide a solution for the coordinated traffic management of mixed flows of conventional cars and CVs. The framework’s innovations rely on the simultaneous incorporation of three different driver behaviors expressed through different types of players in strategic games. This concept is then integrated into a multiclass traffic assignment process. The significant contribution of this work is comprised of the development of a concrete framework, consisting of a series of logical steps, and the development of a model that generates outcomes by providing insights into the traffic management policies that should be formulated for overall network efficiency. By integrating various scientific backgrounds, the proposed framework can assist the understanding of mixed traffic dynamics and facilitate the design of effective traffic management policies.

The following chapters present in detail the materials and methods used for the development of the framework, the structure of the framework itself, and finally, the discussion and conclusions.

2. Materials and Methods

Literature Review

A comprehensive literature review was conducted to identify the current state of the art in the domains of traffic management and C-ITS and in the application of bilevel optimization and game theory for traffic management purposes. The purpose was to identify the research gaps and the contributions that the proposed framework could provide.

Table 1. State of the art in C-ITS application for traffic management.

Paper	Main Contribution
Lee and Park [9]	Impact of C-ITS on the effectiveness of Variable Speed Limits (VSL) in freeway bottlenecks using microscopic traffic simulation.
Rivadeneyra et al. [10]	Development of a test algorithm for queue spillback detection and control in congested grids using C-ITS data.
Goodall et al. [11]	Development of an algorithm for controlling traffic signals in a C-ITS environment using VISSIM for microscopic modeling.
Yang et al. [12]	Upgrade of a signal control algorithm for connected vehicles at intersections and use of heuristics for switching signal controls.
Florin and Olariu [13]	Provision of outcomes of a comprehensive survey on traffic signal optimization methods exploiting C-ITS technologies.
L. Li et al. [14]	Review of general traffic control strategies with a focus on transitioning from feedback to feed-forward control using C-ITS technologies.
Priemer and Friedrich [15]	Optimization of signal phases for traffic control.
Hu et al. [16]	Optimization of signal phases and provision of priority to individual cars to optimize departure sequences.
Wu et al. [17]	Optimization of signal phases and provision of priority to individual cars to optimize departure sequences.
Pandit et al. [18]	Optimization of signal phases and provision of priority to individual cars to optimize departure sequences.
He et al. [19]	Estimated arrival information of non-C-ITS vehicles using traffic models.
Guler et al. [20]	Estimated arrival information of non-C-ITS vehicles using traffic models.
Feng et al. [21]	Estimated arrival information of non-C-ITS vehicles using traffic models.
Liang et al. [22]	Extension of a real-time, connected vehicle-based traffic signal control algorithm to balance efficiency and equity at intersections.

,

Table 2. State of the art in bilevel optimization application for traffic management.

Authors	Contribution
Chiou [23]	Continuous Network Design Problem (CNDP) for optimal link capacity expansions and equilibrium flows.
Dempe and Zemkoho [24]	Optimal value reformulation of the road pricing problem, preserving essential data.
Jung et al. [25]	Eco-Traffic Signal System (Eco-TSS) for fuel consumption reduction and improved traffic flow.
Stoilova et al. [26]	Minimization of vehicles queues in front of traffic lights using bilevel optimization.
Stoilova et al. [27]	Bilevel modeling for traffic lights optimization with hierarchical integration of small problems.
Brotcorne et al. [28]	Optimal tolls on network arcs balancing revenue and user travel costs.
Lv et al. [29]	Inexact bilevel programming under stochastic and fuzzy uncertainties for toll scheme design.
Cheng et al. [30]	Lane reservation for special vehicles minimizing impact on regular traffic.
Zhang et al. [31]	Multi-objective bilevel model for lane reservation for hazmat transportation.
Zhao et al. [32]	Centralized integration of intersection control with vehicle trajectory planning.
Stoilova and Stoilov [33]	Hierarchical organization of traffic management problems with bilevel optimization.
Li et al. [34]	Coordinated control of traffic signals considering pedestrian crossing delay.
Hao et al. [35]	Bilevel model for public transport network considering fairness constraints.
Stoilova and Stoilov [36]	Model predictive framework for real-time traffic management in urban networks.
Basciftci and Hentenryck [37]	On-Demand Multimodal Transit System combining network design and rider route choice.
Hong et al. [38]	Bilevel model for lane reduction during Winter Olympics to manage tidal traffic.
Kalashnikov et al. [39]	Optimal toll assignment on an abstract network with quadratic lower-level costs.
Li et al. [40]	Strategy for public recharging infrastructure location and vehicle routing.
Bingfeng et al. [41]	Design of exclusive bus lanes in multimodal traffic networks optimizing total travel cost.

and

Table 3. State of the art in game theory application for traffic management.

Authors	Main Contribution
Li et al. [42]	Balancing User Equilibrium and System Optimum in traffic assignment.
Levy et al. [43]	Stable System Optimum through cooperation in traffic.
Wang and Tang [44]	Relation between User Equilibrium and System Optimum.
Larsson and Patriksson [45]	Traffic system as a noncooperative Stackelberg game.
Dong et al. [46]	Route choice model with traveler information.
Xiao and Liu [47]	New traffic assignment model combining shortest path algorithm and game theory.
He and Fan [48]	Efficient path finding in road networks using game theory.
Lu et al. [49]	Network vulnerability analysis under hurricane evacuation.
Muhuri et al. [50]	Cooperative game theory for disaster traffic management.
Hadas et al. [51]	New centrality measure for network nodes.
Klein et al. [52]	Traffic congestion management through ICT and game theory.
Zhang et al. [53]	Congestion charging optimization using game theory.
Wang et al. [54]	Traffic demand management with game theory.
Portilla et al. [55]	Highway traffic control using distributed model predictive control.
Fisk [56]	Modelling transportation problems with Nash and Stackelberg games.
Han et al. [57]	Online traffic signal coordination using game theory.
Bui et al. [58]	Smart traffic lights control with Cournot and Stackelberg games.
Bui and Jung [59]	Framework for traffic flow improvement in large networks.
Qi et al. [60]	Amber light passing decision-making using double game model.
Astarita et al. [61]	FCD-based adaptive traffic signal control for connected and autonomous vehicles.
Fan et al. [62]	Simulation framework for dynamics in unsignalized intersections.
Daeichian and Hanghani [63]	Fuzzy Q-learning for traffic light adjustment.
Adkins et al. [64]	Traffic management at intersections with correlated equilibrium.
Clempner and Poznyak [65]	Multi-traffic signal-control synchronization using extra proximal method.
Han et al. [66]	Intersection scheduling with multi-agent multi-step game theory.
Du et al. [67]	Online in-vehicle routing mechanism with game theory.
Ahmad et al. [68]	Cooperative Interest-Aware clustering for vehicular smart applications.
Elhenawy et al. [69]	Real-time traffic control for autonomous vehicles at intersections.
Yu et al. [70]	Lane-changing decision-making for autonomous vehicles.
Lin et al. [71]	Lane changing strategy with cooperative game theory.
Kang and Rakha [72]	Decision-making for merging maneuvers using noncooperative game theory.
Kita [73]	Traffic behavior at on-ramp merging sections with game theory.
Louisell [74]	Microsimulation tools for driver behavior assessment in work zones.
Mei et al. [75]	Rule-based Incentive Framework for personalized travel incentives.
Liu and Shen [76]	Optimization of urban traffic jams in China using game theory.
Schönauer et al. [77]	Simulation tool for shared space zones with game theory.
Kokolaki and Stavrakakis [78]	Heuristic parking search strategies with game theory.
Castelli et al. [79]	Game between authorities for freight transport network management.

present the main contributions of studies related to the abovementioned topics, respectively.

Recent studies have extensively explored the impact of C-ITS on traffic management. Lee and Park [9] evaluated the effectiveness of Variable Speed Limits (VSL) in freeway bottlenecks using microscopic traffic simulation. In indicative relevant studies, Rivadeneyra et al. [10] developed an algorithm for queue spillback detection and control in congested grids using C-ITS data. Goodall et al. [11] proposed an algorithm for traffic signal control in a C-ITS environment using VISSIM for microscopic modeling. Yang et al. [12] upgraded a signal control algorithm for connected vehicles at intersections, utilizing heuristics for switching signal controls. Florin and Olariu [13] provided a comprehensive survey on traffic signal optimization methods exploiting C-ITS technologies. Li et al. [14] reviewed general traffic control strategies with a focus on transitioning from feedback to feed-forward control using C-ITS technologies. Additionally, studies by Priemer and Friedrich [15] and Hu et al. [16] have focused on optimizing signal phases for traffic control, providing priority to individual cars to optimize departure sequences. Most of the studies, however, do not analyze the perspective of different stakeholders’ coexistence and interactions, as they investigate traffic at a microscopic level and develop models that do not consider multifaceted objectives for network performance optimization and social welfare. Moreover, the aspect of integrating various types of drivers and the impact of the provision of real-time information to them from different types of service providers has not been investigated in depth.

In the domain of bilevel optimization for traffic management, Chiou [23] addressed the Continuous Network Design Problem (CNDP) for optimal link capacity expansions and equilibrium flows. Dempe and Zemkoho [24] reformulated the road pricing problem while preserving essential data and offering optimal value solutions. Jung et al. [25] introduced an Eco-Traffic Signal System (Eco-TSS) aimed at reducing fuel consumption and improving traffic flow. Stoilova and colleagues [26,27] minimized vehicle queues in front of traffic lights using bilevel optimization and further expanded on bilevel modeling for traffic light optimization by integrating smaller hierarchical problems. Brotcorne et al. [28] balanced revenue and user travel costs through optimal tolls on network arcs. Lv et al. [29] explored inexact bilevel programming under stochastic and fuzzy uncertainties for toll scheme design. The review in the domain of bilevel optimization and traffic management indicates that the works focusing on network design do not integrate traffic information provision and its influence on CVs’ behavior. The perspective of stakeholders’ diversity is once again not highly considered, and neither are the various levels of cooperation or competition among them.

Finally, game theory has also been applied to various aspects of traffic management. Klein and Ben-Elia [52] examined the emergence of cooperation and fair system optimum in road networks through game-theoretic and agent-based modeling approaches. Wang and Tang [44] discussed system optimum and user equilibrium in traffic assignment from a game theory perspective. Portilla et al. [55] proposed a non-linear model of predictive control based on game theory for highway traffic control. Additional studies have focused on the application of game theory in urban traffic congestion management, cooperative road traffic management in disaster scenarios, and the use of game-theoretic approaches for multiple intersections and real-time traffic signal control. From the perspective of game theory integration into traffic management processes, the studies mostly examine isolated aspects of transport systems, relying on specific theoretical models that do not integrate the perspective of multi-stakeholder interactions in real-life networks and are limited to specific scenarios. Furthermore, they do not sufficiently explore the incorporation of C-ITS technologies in the aspects of game theory when applied to traffic management purposes.

While significant progress has been made in these areas, there are notable gaps. In summary, most studies do not comprehensively analyze the interactions and coexistence of different stakeholders, such as conventional vehicles and connected vehicles, under various traffic conditions. The multifaceted objectives of network performance optimization and social welfare are often not fully addressed. Additionally, the integration of real-time information provision and its influence on CV behavior has not been thoroughly investigated. This framework aims to fill these gaps by proposing a robust solution that integrates game theory, bilevel optimization, and C-ITS technologies to manage mixed traffic environments effectively.

3. Framework Design and Structure

The structure of the suggested framework is comprised of a sequence of interrelated steps, beginning with defining the different types of players in the traffic network. Next, a multiclass traffic assignment model is implemented to represent the interactions and the diverse behaviors of the players. These elements are further examined through the design of strategic games targeting the analysis of the potential strategies and decision-making processes of the different players. The assessment of the model outcomes is performed via the definition of Key Performance Indicators (KPIs), which focuses on network efficiency metrics and parameters. The application of the abovementioned steps can take place in a real-life network, generating outcomes that facilitate the definition of optimal traffic management policies. A high-level overview of the steps that comprise the proposed framework is presented in Figure 1. The scope of this work is to elaborate on the first four steps, which constitute the core part of the methodological approach. The last three steps serve as the means to apply the methodological approach, and more information can be found in [80].

The following sections elaborate on the different steps of the framework.

3.1. Definition of Players

The first step includes the definition of the types of players that are integrated into the model. The objective is to map traffic assignment principles with drivers’ behavior. Three different types of players are defined [80]:

• The Un-Cooperative Egoistic (UE) player represents the travelers in the network who can find the route from an origin to a destination with the minimum travel time and, therefore, no longer have the incentive to seek different routes. Every used route for each origin-destination (OD) pair has the same travel time. This type of player is considered the driver of a conventional vehicle with no access to dynamic/real-time information about the road network conditions. These drivers rely on static route choices without benefiting from dynamic updates.
• The System Optimizer (SO) player represents the travelers who choose their routes with the objective of minimizing total travel time in the network. This type of player is considered the driver of a CV, whose dynamic/real-time information provision is catered from a central governing authority, e.g., a traffic management center in the road network aiming for network-wide benefits. The objective of the central governing authority is expressed through System Optimum principles. The services of the central governing authority could include real-time information provision via Variable Message Signs (VMSs) or mobile applications of Mobility as a Service (MaaS) platforms providing C-ITS services.
• The Cournot-Nash (CN) player represents a self-optimizing oligopoly of Cournot travelers whose objective is to minimize the total cost of travelers belonging to the specific team that they belong to as well. This type of driver is subscribed to a private service provider (e.g., Waze, Google). These drivers of CVs have access to the dynamic/real-time information provided by the services of the private provider and make route decisions based on real-time information while following semi-individual objectives.

The three player types (UE, SO, and CN) behaviors directly influence the model’s applicability to real-world mixed traffic networks by reflecting varying levels of information availability, decision-making behaviors, and access to technology among travelers. In real-world scenarios, the proportion of each player type would depend on factors such as the penetration rate of connected vehicle technologies, the extent of centralized traffic management infrastructure, and the market share of private navigation services. This way, an analysis of the influence of different driver behaviors on network performance can be extracted, providing insights into congestion patterns, the effectiveness of real-time information systems, and the potential benefits of cooperative traffic management. However, applying this concept in specific networks would possibly require calibration to account for the actual distribution and behaviors of drivers in each region, as well as variations in technology adoption rates and infrastructure capabilities.

3.2. Players’ Cost Functions

The distinctions among the three players and the benefits they seek to achieve are represented by their respective cost functions, which reflect their strategies and behaviors in selecting routes within the network. These differences in cost functions are captured through the concept of the “perceived cost of each player” and the variations in marginal costs impacting each of them. Marginal cost refers to the effect of adding an extra vehicle to a path from an origin to a destination on the overall network cost [81]. According to the fundamental economic principle of marginal cost pricing, road users on congested roads should pay a toll equal to the difference between the marginal social cost and the marginal private cost to maximize social net benefits [82].

In traffic assignment studies, marginal cost tolls have been employed as a tool to shift a User Equilibrium flow pattern towards a System Optimum in a road network with fixed demand. Specifically, by imposing a congestion fee on flow magnitude for each user traversing a specific link in the network, the resulting traffic flow pattern, originating from the selection of cost-minimizing routes between any origin-destination pair, tends to align with System Optimum, thereby minimizing the total network travel cost. The toll level that achieves this goal corresponds to the additional travel cost inflicted by an extra user on a link upon all users already utilizing that link. The model developed here is grounded in traffic assignment principles, where multiple players simultaneously choose routes within a network. The costs associated with these choices are represented by distinct cost functions that capture the unique strategies, objectives, and rationalities of each player.

• The “perceived cost of each player” reflects the subjective expense each player associates with traveling along a specific link or path in the network based on their strategy. This perceived cost guides their decision-making process when selecting routes. To describe and differentiate the various cost components relevant to each player, the “perceived cost of each player” is defined as follows: Private cost: Often referred to as the “private cost”, it represents an individual’s personal expenditure, such as time or resources, for traveling along a specific route.
• Full marginal cost: This represents the overall change in the total cost of the transport network caused by the addition of one more vehicle. This term highlights the network-wide impact of incremental increases in travel demand.
• Partial marginal cost (oligopoly cost): Refers to the change in total cost borne by a specific subset of travelers (the oligopoly) due to the addition of an extra vehicle. It captures the effects on a particular group of travelers within an oligopolistic context.

Subsequently, the “perceived costs” for the three types of players are expressed as follows:

• UE player: The link cost perceived by the UE player corresponds to the marginal private cost or the actual cost. This strategy focuses solely on the direct impact of the route on the player’s own travel experience, disregarding the broader network effects. The perceived cost for the UE player is simply the travel time on a chosen path, equivalent to the private cost incurred by an individual traveler, emphasizing the self-centered nature of the UE player’s decision-making.
• SO player: The perceived cost for the SO player incorporates the full marginal social cost, reflecting the total change in the network’s cost caused by an additional vehicle. The SO player, aiming for system-wide efficiency, evaluates the broader impact of their route choice on the entire network. The full marginal cost includes a term representing the derivative of travel time with respect to traffic flow, accounting for the systemic effects of individual decisions. This term highlights the SO player’s system-level perspective, considering how their choices influence overall network performance.
• CN player: The perceived cost for the CN player is based on the partial marginal social cost, representing the change in the total cost borne by a subset of travelers within the oligopoly due to an additional vehicle. The CN player’s strategy involves strategic interactions with other players, factoring in both individual travel time and the collective impact on their group. This is captured by incorporating a term representing the derivative of travel time with respect to the flow on paths used by the oligopoly group to which the CN player belongs, reflecting the interdependent nature of their decisions.

Each player’s perceived cost aligns with their unique strategy-individual optimization (UE), system-wide efficiency (SO), or strategic interactions within an oligopoly (CN). These distinct cost perspectives are mathematically represented through specific equations integrated into the model, capturing the strategic differences among the three players.

$$c_{UE}=t\left(v\right)$$

(1)

$$c_{SO}=t\left(v\right)+{\displaystyle \frac{\partial t\left(v\right)}{\partial \left(v\right)}}$$

(2)

$$c_{CN}=t\left(v_{CN}\right)+{\displaystyle \frac{\partial t\left(v\right)}{\partial \left(v\right)}}\times v_{CN}$$

(3)

where:

• $UE$: User Equilibrium player
• $SO$: System Optimum player
• $CN$: Cournot-Nash player
• $v$: the flow on a link
• $t\left(v\right)$: the cost of a link
• $c_{UE}$: perceived link cost of the UE player
• $c_{SO}$: perceived link cost of the SO player
• $c_{CN}$: perceived link cost of the CN player
• $v_{CN}$: flows on the links where CN conditions apply.

Multiclass Traffic Assignment Algorithm

A model was developed in R, a free software environment for statistical computing and graphics [83], to simultaneously perform traffic assignments for the three distinct players. The inputs to the model include the network assignment matrix, the network demand matrix, and a matrix with other network characteristics, i.e., capacity, free flow, or other parameters. The outputs of the model include the network link flows for each player, the total network link flows, and the total network link costs. The network assignment matrix is a binary matrix, where a value of “1” indicates that a link is part of an individual route for a specific OD pair, while a value of “0” indicates that the link is not part of that route. For some OD pairs, more than one route may apply, indicating the alternative paths to reach this destination from its origin (e.g., the a-b OD pair in

Table 4. Simplified network.

OD	Routes	1	2	3
a-b	R1	1	0	0
a-b	R2	0	1	1
a-c	R1	0	1	0
c-b	R1	0	0	1

includes the alternative routes R1 and R2). Figure 2 presents a simplified network where 3 players, namely UE, SO, and CN, coexist. Table 4,

Table 5. Simplified network demand matrix.

OD	UE	SO	CN
a-b	40	0	60
a-c	20	10	30
c-b	0	30	10

and

Table 6. Simplified network characteristics matrix.

Capacity	Free Flow	Congestion Parameter
100	10	2
200	5	3
150	7	2

present indicatively the network assignment matrix, the network demand matrix, and the network characteristics matrix.

An algorithm was developed to implement the model by performing multiclass traffic assignment and flow distribution within the network. The algorithm seeks to optimize the allocation of traffic demand across the various routes in the road network while accounting for the strategies of the three distinct players.

The algorithm iteratively computes route flows, link flows and costs, resulting in balanced traffic distribution across the network for the three different players. The algorithm operates in an iterative manner, with each iteration focusing on refining the assignment of traffic demand to different routes. The algorithm seeks to find a balanced distribution of traffic flows, considering the players’ strategies with their distinct decision-making criteria and overall network efficiency.

3.3. Development of Strategic Games

The development of strategic games, where the model is applied, allows for a comprehensive assessment of its performance and provides outcomes based on which various traffic management policies for a mixed traffic network can be suggested. The approach for the strategic game’s design is comprised of two main types of games:

• Games that represent typical network conditions, evaluating and quantifying the model’s outputs in situations resembling typical daily traffic scenarios.
• Games that represent special events (described in

  Table 7. Indicative special events in the Case Study network.

  Special Event	Description	Topology
  Road closure	Due to road maintenance, construction, or an unforeseen event.	Important roads or intersections.
  Accident	Closure of certain lanes or the entire road due to a collision.	Different locations within the network.
  Public event	Large public events, such as festivals, parades, or sporting events, on the road network, where increased pedestrian and vehicle traffic can lead to congestion.	Segments of the road network close to the public event location.
  Natural disaster	Effects of natural disasters, such as earthquakes or floods, on the road network.	Closure of certain routes or the need for alternate routes.
  Vehicle breakdown	Vehicles break down causing disruptions in traffic.	Key locations in the road network.
  Evacuation scenario	Evacuation in the case of emergencies.	Closure of certain routes for evacuation purposes.

  below) that can significantly impact traffic dynamics and generate outcomes for non-typical traffic conditions.

It should be noted that the objective of this work is not to rely on established theoretical frameworks or prior studies for the development of strategic games but rather on deriving these games empirically by the outcomes of testing the multiclass traffic assignment algorithm under various demand scenarios. The strategic games are designed to capture realistic network behaviors, reflecting practical traffic dynamics that emerge from diverse demand distributions and player interactions.

The design of the strategic games is informed by the need to evaluate the proposed multiclass traffic assignment framework under diverse traffic conditions representative of real-world scenarios. The games are categorized into three main types, each tailored to capture specific network dynamics and player interactions:

• Typical network conditions: These games represent standard traffic scenarios commonly encountered in road networks, such as those dominated by a single-player type (e.g., UE or SO) or scenarios where player types coexist in a balanced distribution. By modeling such scenarios, the framework simulates routine commuting patterns and general road usage dynamics, enabling the assessment of network performance under normal operating conditions.
• Non-typical network conditions: These scenarios are designed to examine the framework’s adaptability and robustness in the face of disruptions, such as accidents, road closures, or large-scale public events. Non-typical conditions often lead to significant deviations from standard traffic patterns, providing an opportunity to evaluate the framework’s ability to manage unexpected changes and maintain network efficiency.
• Strategic interaction scenarios: These games focus on the interactions among CN players, exploring both cooperative and competitive dynamics. Cooperative scenarios simulate collaborative efforts among private service providers to optimize traffic flow and enhance user experience. Conversely, competitive scenarios model rivalry between CN players as they strive for market dominance by offering differentiated services or pricing incentives. These scenarios are reflective of real-world market conditions and allow for the examination of the broader implications of player interactions on traffic dynamics.

This categorization ensures that the framework is rigorously tested across a spectrum of realistic conditions, from everyday traffic flows to exceptional events. The strategic games are empirically derived from the results of the multiclass traffic assignment algorithm applied to various demand scenarios. This approach provides a comprehensive understanding of network performance and informs the development of targeted traffic management policies.

A key element in the strategic games’ design is the sensitivity analysis integrated through the systematic variance in the traffic demands for the different players. This approach of sensitivity analysis has the objective of identifying critical thresholds.

The games are designed under the principle of Nash equilibrium. Each player acts rationally and independently, seeking to optimize their own performance within the network. The players operate simultaneously, making decisions based on their own objectives and information to represent real-world scenarios where different entities, having their own priorities, coexist and interact. The strategy of each player influences the payoffs of the other players.

The allocation of the total network demand for each OD pair among the players in the games requires the design of a distribution mechanism that captures the behavior and the characteristics of each player in each game. The logic for the demand allocation begins with the assignment of a portion of the total demand to the UE player. A portion of the demand is then assigned to the SO player. The remaining demand is allocated to the CN player. It should be noted here that the CN players could be more than one in a network, as each one represents the different private service providers that coexist in the market. The demand allocation to the CN players is considered by factors such as market share and user base. The sensitivity analysis is performed by varying the demand allocation percentages among the players to understand how the different network situations affect the equilibrium conditions and the network performance. The next section elaborates on the various strategic games that were designed.

• Game 0: Distinct UE and SO equilibria in typical network conditions

In this game, the objective is to examine the network dynamics by allocating the total network demand either only to the UE player or only to the SO player under typical traffic conditions. Two scenarios are examined to represent the two distinct equilibria conditions:

• Scenario 0.a represents the total network demand allocation to the UE player. All drivers drive conventional cars and aim to minimize their costs. This leads to a decentralized equilibrium situation, where each user selects the path with the lowest perceived cost to represent the inherent selfish routing behavior of the individual drivers with no access to real-time information. The selection of routes that attract traffic relies on personal optimization and empirical choices.
• Scenario 0.b represents the total network demand allocation to the SO player. All drivers receive real-time information from a central governing authority seeking to minimize the overall system cost, as it considers the collective impact of all users in the network. This form of equilibrium may diverge from individual user preferences to enhance the overall efficiency of the network. The scenario focuses on a centralized planning approach and identifies routes that contribute to the whole network optimization.

The network demand allocation for the two scenarios is presented in

Table 8. Players’ demand allocation in Game 0.

Game 0		Scenario’s Purpose
Scenario 0.a		Inherent selfish routing behavior of the individual drivers with no access to real-time information.
Player	Total Demand Rate
UE	100%
SO	0%
CN1	0%
CN2	0%
Scenario 0.b		Centralized planning approach to minimize the overall system cost
UE	0%
SO	100%
CN1	0%
CN2	0%

.

• Game 1: UE player dominance in typical network conditions

In this strategic game, the network consists mainly of travelers complying with the UE player strategy due to the limited involvement of a central governing authority and the low market penetration of private service providers. The central governing authority, which is responsible for centralized traffic management, has limited resources and capabilities. A traffic management center provides the bare minimum of real-time information and services to travelers. Private service providers have a very low penetration rate in the market. Only a few travelers rely on such services for real-time traffic information and routing guidance. Traveler behavior is primarily driven by conventional commuting patterns and static route choices based on historical preferences. The dominance of the UE is represented through the allocation of a substantial portion of the total demand to the UE player. The allocation of 80% of total network demand to the UE player reflects the dominant presence of conventional vehicles in mixed traffic networks. However, the allocation should not be as high as 90%, which would overestimate the share of conventional vehicles and neglect the gradual integration of connected vehicles into modern road systems. This assumption ensures a more accurate representation of the current state of traffic networks, balancing conventional vehicle dominance with the realistic presence of connected and automated vehicles.

The network demand allocation for this game is presented in

Table 9. Players’ demand allocation in Game 1.

Game 1		Game’s Purpose
Player	Total Demand Rate	Dominance of the UE player—substantial portion of the total network demand.
SO	10%
UE	80%
CN1	5%
CN2	5%
Sum	100%

.

• Game 2: Dominance of the SO player under typical network conditions

In this scenario, the central governing authority, which oversees centralized traffic management and provides real-time information to travelers, enhances its resources and capabilities to deliver improved services. Meanwhile, private service providers maintain a minimal presence in the market, with only a small portion of travelers utilizing their services. The key change in this game is a gradual increase in the number of travelers accessing the real-time information offered by the central governing authority (SO player).

To represent this shift, the model gradually increases the demand allocated to the SO player while reducing the demand allocated to the UE player. This reduction in the UE player’s demand reflects a departure from traditional commuting patterns, as more travelers adopt strategies based on the real-time guidance provided by the SO player. The objective of this game is to analyze how the increased reliance on the SO player influences network equilibrium and overall performance.

The rising demand for the SO player’s services may reflect various conditions within the network that encourage greater reliance on SO behavior. A primary factor is the significant improvement in real-time information provided by the central governing authority. For instance, a citywide deployment of Variable Message Sign (VMS) systems could offer real-time updates on traffic conditions, travel times, congestion, accidents, and recommended routes, enabling travelers to make informed decisions. Additionally, the central governing authority could foster a culture of cooperative optimization by promoting the use of real-time data in route planning. Public awareness campaigns, training programs, and workshops could educate travelers about the benefits of using such information, highlighting the advantages of collective optimization for both individual and network-level efficiency. Incentives could also be introduced to encourage compliance with the SO player’s recommendations. Examples include access to dedicated lanes or preferential parking spots for travelers who follow suggested routes. These measures aim to shift commuter behavior, improve traffic flow, and maximize network performance through greater adoption of SO player strategies.

Table 10. Players’ demand allocation in Game 2.

Game 2			Scenario’s Purpose
Scenario 2.a	Player	Total demand rate	Dominance of the UE player—substantial portion of the total network demand.
	SO	20%
	UE	70%
	CN1	5%
	CN2	5%
Scenario 2.b	SO	30%
	UE	60%
	CN1	5%
	CN2	5%
Scenario 2.c	SO	40%
	UE	50%
	CN1	5%
	CN2	5%
Scenario 2.d	SO	50%	Gradual increase of the demand allocated to the SO player reflecting a shift to driving behaviors complying with real-time guidance provided by the SO player.
	UE	40%
	CN1	5%
	CN2	5%
Scenario 2.e	SO	60%
	UE	30%
	CN1	5%
	CN2	5%
Scenario 2.f	SO	70%
	UE	20%
	CN1	5%
	CN2	5%
Scenario 2.g	SO	80%
	UE	10%
	CN1	5%
	CN2	5%

presents the different schemes of demand allocation, expressed through seven indicative scenarios (Scenario 2.a to Scenario 2.g).

• Game 3: Private service providers’ increased penetration rate in typical network conditions

The objective of this strategic game is to examine the network equilibrium performance when the influence of the CN players has a greater impact factor. The larger share of the total demand allocated to the CN players is justified by the increased market penetration of private service providers, who introduce new features or services in their applications or provide more accurate real-time information. The scenarios reflect a balanced CN influence, representing an equal share of the total demand for both CN players. The logic is to maintain a balanced competitive landscape.

By gradually continuing with the same logic, the percentages of the SO and UE players decrease proportionally as the CN influence grows. These scenarios allow us to examine how the increasing influence of the CN players affects the equilibrium and network performance. The purpose of these scenarios is to explore a range of CN influence levels systematically. By starting with balanced influence, a baseline for comparison can be established, and then a gradual increase of the CN influence will enable the observance of how this affects the network behavior. The balanced CN influence is a modeling simplification designed to examine a specific aspect of network behavior under controlled conditions. The demand allocation in the scenarios is presented in

Table 11. Players’ demand allocation in Game 3.

Game 3			Scenario’s Purpose
Scenario 3.a	Player	Total demand rate	Larger share of total demand allocated to the CN players due to increased market penetration, reflecting a balanced CN influence.
	SO	8%
	UE	60%
	CN1	16%
	CN2	16%
Scenario 3.b	SO	6%
	UE	50%
	CN1	22%
	CN2	22%
Scenario 3.c	SO	4%
	UE	32%
	CN1	32%
	CN2	32%
Scenario 3.d	SO	2%
	UE	18%
	CN1	40%
	CN2	40%

below.

• Game 4: Cooperation of private service providers in typical network conditions

The CN players of this strategic game are considered as willing to collaborate for the benefit of their users (or for other reasons), leading to cooperative behavior. The CN players form a strategic partnership to improve their services and user experience. This partnership could include sharing real-time traffic data, pooling resources, or integrating their services for seamless routing. The primary motivation for the cooperation is to provide better services to the users. The CN players aim to reduce congestion, optimize routes, and enhance the overall travel experience of their users. Real-time information sharing may include traffic updates, road closure notifications, and alternative routes. Indicative cooperation scenarios between the two CN1and CN2 players are described in the following scenarios:

• Scenario 4.a: The objective is cooperative data sharing for congestion management or safety and incident reporting, which represents the cooperation of two CN players in the case of real-time traffic data sharing for effective congestion management. The focus is on network optimization, reduced congestion, and road safety improvement for their users. The two CN players share incident reports, road hazard data, and safety-related information to reduce accidents and improve overall safety for their users. Indicative demand allocation percentages could be UE—70%, SO—15%, CN1—7.5%, and CN2—7.5%. C-ITS services prioritized by the CN players for the achievement of the above-mentioned objectives include Road Works Warning, Road Hazard Warning, In-Vehicle Signage, Mode and Trip Time Advice, and Flexible Infrastructure
• Scenario 4.b represents cooperative user engagement and loyalty programs or cooperative environmental sustainability. The two CN players cooperate to design and implement user engagement and loyalty programs, aiming to improve the overall user experience and loyalty. In the case of the two players cooperating to reduce environmental impacts, they share data and coordinate their efforts to minimize emissions fuel consumption and promote eco-friendly travel options. Indicative demand allocation percentages could be UE—68%, SO—15%, CN1—8.5%, and CN2—8.5%. The C-ITS services that are prioritized by the CN players in this case include Green Light Optimal Speed Advisory (GLOSA) and Green Priority (GP).

• Game 5: Competition of private service providers in typical network conditions

A competitive game represents a situation where the two CN players compete, and their primary focus is on gaining a competitive edge in the market. Such a situation could rely on market competition as the two CN players compete for users and market share, and they prioritize attracting more users to their services rather than cooperating.

Each CN player keeps his/her real-time traffic data exclusive and does not share it with the other. They use this data to provide differentiated services and gain a competitive advantage. The CN players adopt different strategies to outperform each other, such as offering unique features, pricing incentives, or faster route recommendations. In competitive scenarios, it is assumed that the CN players adopt distinct strategies to outcompete each other. For example, one CN player may focus on offering premium services with a higher price point and unique features, while the other CN player may target a broader user base with lower prices and differentiating features. The CN players can engage in price competition to attract users, as they could offer discounts, promotional packages, or other pricing incentives. Each CN player can develop exclusive features or services that are not provided by the competitor. These unique offerings can be a significant driver for user acquisition. They can invest in aggressive advertising and marketing campaigns to gain more visibility and attract users to their platforms. Three indicative scenarios are described for such a case:

• Scenario 5.a represents the case of unique feature differentiation dynamic market behavior or aggressive advertising. The focus is on each CN player introducing unique features to differentiate themselves and attract users, adapting their strategies based on evolving market conditions, or adopting aggressive advertising and marketing campaigns to capture user attention. The demand allocation percentages could be UE—55%, SO—15%, CN1—15%, and CN2—15%.
• Scenario 5.b represents the case of price competition where the CN1 player starts with a higher allocation percentage to emphasize its pricing strategy. He/she competes by offering lower prices or discounts, aiming to attract cost-conscious users. The demand allocation percentages could be UE—55%, SO—10%, CN1—25%, and CN2—10%.
• Scenario 5.c represents the case of user engagement and loyalty, where the CN1 player starts with a slightly higher allocation percentage to emphasize its focus on user engagement and loyalty programs. CN2 competes with a different approach. The demand allocation percentages could be UE—70%, SO—10%, CN1—12%, and CN2—8%.

Concerning the application of strategic games in the case of non-typical network conditions, adjustments should be made to the demand and capacity of the road network to reflect the changed conditions. Indicatively:

• An increase in demand can be achieved for events like festivals, public gatherings, or sports events. This could be accomplished by multiplying the demand on certain links or nodes by a factor that represents the expected increase in traffic.
• If a special event is expected to occur during peak traffic hours, a more significant increase in demand during those hours could be experienced and compared to off-peak times.
• The effect of a road closure or lane reduction event can be represented by a reduction in the capacity of the affected links.

3.4. Definition of KPIs

The evaluation metrics to quantify the model outcomes across the various games were selected to capture metrics that express the parameters of sustainability, efficiency, and user experience in the network under the coexistence of the different players. The KPIs examine metrics of link flows, “perceived cost”, travel time, link delay, link speed, and link CO₂ emissions. More specifically, the primary KPIs used in this framework include:

• Link Flows: This KPI measures the volume of traffic on each link in the network. It helps assess how traffic is distributed across the network and identifies potential congestion points.
• Perceived Cost: This metric evaluates the perceived travel cost from the perspective of different types of drivers. It considers factors such as travel time, fuel consumption, and other costs that influence route choices.
• Travel Time: Average travel time is a critical KPI for understanding the efficiency of the traffic network. It measures the time taken for vehicles to travel between origin and destination pairs.
• Link Delay: This KPI measures the delay experienced on each link due to congestion. It is a key indicator of traffic efficiency and helps identify areas where traffic flow improvements are needed.
• Link Speed: Monitoring the average speed on different links helps evaluate the performance of the network. It is an important metric for understanding how well the traffic management strategies are working.
• CO₂ Emissions: Environmental impact is assessed through the measurement of CO₂ emissions on each link. This KPI is crucial for evaluating the sustainability of traffic management policies.

These KPIs provide a comprehensive view of the network performance under various scenarios and player behaviors. By analyzing these metrics, the framework can generate valuable insights into the effectiveness of different traffic management strategies and their impact on overall network performance. The application of these KPIs ensures that the traffic management policies not only optimize traffic flow but also promote sustainability and enhance user experience.

4. Discussion

The framework provides a comprehensive solution for exploring traffic flow dynamics in mixed networks while considering various players’ strategies, enabling the understanding of how the preferences of different types of drivers influence traffic patterns. The results offer insights into network congestion, route choices, and the impacts of the players’ interactions on overall network efficiency.

Game theory principles are applied to model the decision-making behaviors of different players within the network. These players represent various stakeholders with unique preferences and objectives regarding route choices. Each player in the strategic game seeks to optimize their utility (or minimize their cost) while accounting for the actions of others. The players’ behaviors and their influence on traffic flow mirror the strategic interactions typical of a game. Each player aims to make the best decision based on their individual objectives and their expectations of other players’ actions.

According to Nash equilibrium principles, no player has an incentive to unilaterally deviate from their chosen strategy once equilibrium is reached. The iterative process within the model allows for convergence to this equilibrium, where players’ actions and decisions stabilize based on the interplay of various factors. Route choice is treated as a game where players’ decisions align with their strategies, influencing the overall traffic distribution across the network.

From a multi-player perspective, the proposed model accounts for the preferences and behaviors of three distinct entities:

• UE player: Drivers of conventional vehicles who choose routes based on individual optimization without considering broader network impacts.
• SO player: Drivers who follow real-time information provided by a central governing authority, optimizing for system-wide efficiency.
• CN player: Drivers of connected vehicles (CVs) utilizing Cooperative Intelligent Transport Systems (C-ITS) services from private providers, reflecting strategic behavior within an oligopolistic framework.

The framework optimizes two interdependent problems. At the upper level, the focus is on managing the overall network by optimizing traffic flow distribution to minimize metrics such as total travel time or congestion. This involves adjusting link flows and route choices to improve network performance. At the lower level, the emphasis is on the route choice decisions of individual players, capturing their distinct strategies and preferences within the network.

Each player aims to minimize their individual costs, considering the costs resulting from the upper-level optimization. The bilevel optimization structure of the problem is nested, with the upper-level optimization problem influencing the lower-level optimization problem. The upper level determines the link costs, which then affect the players’ decisions in the lower level.

The model’s capability to analyze the route choices of various types of drivers provides valuable insights into commuter decision-making in a mixed-traffic environment. Traffic managers can better understand how different decision-making criteria influence traffic flow patterns, enabling more informed policy and infrastructure decisions. By examining the behaviors of different players and their impact on traffic flow distribution, the model helps identify bottlenecks, congestion-prone areas, and underutilized routes. This information supports targeted improvements to optimize traffic flow, such as designing more effective route information systems, signage, and incentive programs that encourage congestion-alleviating route choices.

Additionally, the model allows for the evaluation of policy interventions and their effects on traffic flow. From a transport infrastructure planning perspective, analyzing traffic flow patterns under varying penetration rates of different driver types can aid long-term infrastructure planning. For example, it can help identify areas requiring capacity expansion or the development of new roadways specifically for connected vehicles (CVs).

The cooperative behavior of CVs is represented in the model by the SO and CN players. Through real-time information sharing and collaboration with a central governing authority, these players contribute to collective traffic flow optimization, reducing congestion and improving overall mobility efficiency. The SO player’s alignment with a central governing authority reflects the role of centralized traffic management in connected mobility scenarios. This integration enables real-time monitoring, control, and information dissemination, which supports proactive traffic management strategies.

As CV adoption grows, the model serves as a valuable testing platform for integrating real-time CV data into traffic management systems. This integration could optimize key traffic management elements, such as signal timing, route guidance, and incident response, by leveraging the collective intelligence of CVs to enhance network performance and mobility.

The relationship between the players’ strategies and the C-ITS services that can be exploited for traffic management purposes is described in

Table 12. C-ITS integration into the framework.

C-ITS Service	Integration into the Framework
	SO Player	UE Player	CN Player
RWW (Road Works Warning)	CVs receiving real—time information from the TMC. Notifications could be shown in VMSs and refer to ongoing road works, enabling the drivers to make informed decisions and potentially choose alternative routes to avoid congestion.	Without access to such types of real—time information, UE compliant drivers encounter unexpected road works, leading to congestion and delays. Their route choices are based on static information, potentially leading to suboptimal decisions.	CN compliant drivers rely on private services providers, receiving road works warnings via their applications. This dynamic information could influence their route decisions, potentially leading to improved traffic distribution and reduced congestion around road work areas.
RHW (Road Hazard Warning)	By disseminating real—time information about hazards in VMSs, such as accidents or debris on the road, it is presented how SO compliant drivers can receive warnings and adjust their routes to ensure safety.	UE compliant drivers might be unaware of road hazards in advance, causing disruptions in their travel. They might face unpredictable delays due to hazards, affecting their route choices and overall travel time.	CN compliant drivers receive hazard warnings, allowing them to avoid hazardous areas and select safer routes.
FI (Flexible Infrastructure)	The SO compliant drivers have access to dynamic information showcases about lane usage through VMSs.	UE compliant drivers’ route choices might not consider infrastructure adjustments.	CN compliant drivers might receive real—time recommendations to adjust speeds and routes based on changing lanes.
GLOSA (Green Light Optimal Speed Advisory)	SO compliant drivers could face the impact of priority services or speed adjustments made by the CN compliant drivers.	UE compliant drivers cannot receive GLOSA information, missing opportunities to catch green lights. Their route choices potentially include more stops and delays.	CN compliant drivers receive optimal speed advice, enabling them to synchronize their speeds with signal timings. This could lead to reduced stops and smoother traffic progression.
IVS (In-Vehicle Signage)	SO compliant drivers receive via VMSs information about upcoming events for better—informed route decisions.	Without in-vehicle signage information, UE compliant drivers do not consider real—time information, leading to congestion and inefficiencies.	CN compliant drivers receive in—vehicle signage information and adapt their routes. This adaptability leads to more informed and efficient route decisions.
MTTA (Mode and Trip Time Advice)	SO compliant drivers may have access to recommendations about the most suitable mode and estimated trip time through MaaS platforms. Such information may influence for them modal choice and trip planning.	UE compliant drivers lack information about optimal mode choices and trip times. Their decisions do not consider real—time modal availability or variations in travel times, potentially leading to suboptimal choices.	CN compliant drivers benefit from mode and trip time advice. They might receive recommendations to choose the most suitable mode and adjust departure times.

.

5. Conclusions

The objective of this work is to present a novel framework that targets the optimization of mixed traffic flows in road networks through the exploitation of game theory, bilevel optimization, and C-ITS. The framework integrates three different types of players: drivers of conventional cars (UE player), drivers of CVs following the advice of a central governing authority (SO player), and drivers of CVs following the advice provided by C-ITS services of private companies (CN player). The achievement of optimum mixed equilibria conditions for network performance improvement is tested through a multiclass traffic assignment process that aims to model the interactions of the coexistence of various types of drivers in a modern road network. This framework proposes an approach to model and manage the interactions between different stakeholders in a road network, thus facilitating the design of traffic management policies. The equitable distribution of benefits is considered, as a multiclass traffic assignment process is utilized for the achievement of overall network performance. The formulation of strategic games assists in analyzing the interactions and strategies of the different players, helping to understand the potential impacts on traffic management better. The evaluation of the model outcomes relies on KPIs that were defined to cover parameters of traffic efficiency, environment and sustainability. Finally, the application of the model in a real-life road network leads to the generation of results that provide insights for the design of effective traffic management policies.

Future research could examine the integration of additional emerging technologies, such as autonomous vehicles, advanced sensor networks, micromobility, and drones, to advance the framework capabilities and areas of application. From the perspective of the framework’s practical implementation, its scalability could be tested towards employment in large and complex urban and peri-urban networks. The integration of interdisciplinary approaches, including urban planning and environmental science, could provide a basis for coupling transport planning with traffic management operations and facilitate a holistic understanding of traffic dynamics in mixed networks.