arXiv subjects

Grzegorz Jamróz

Publications and source records attributed to Grzegorz Jamróz.

16 recordsLinked to original sources

arXiv2026

Competitive mediator games and urban CAV routing markets

Inspired by possible future markets of autonomous routing and driving (ARAD), we introduce competitive mediator games and their equilibria which generalize the (coarse) correlated equilibria, which have become a popular research area recently as they not only can be more socially efficient than Nash equilibria but also are limits of algorithmic no-regret multi-agent learning dynamics. We discuss the basic properties of competitive mediator games and prove that in the generic setting of anonymous congestion(routing) games with market-share maximizing mediators all competitive mediator equilibria are monopolies whenever one of the mediators is weakly preferred to other mediators by all users. We apply and interpret these results in the context of new markets of competing ARAD service providers. We also provide a comprehensive overview of these markets and discuss the future mechanism design thereof.

cs.GT↗

arXiv2026

Randomized routing strategies of fleets of CAVs may prove market efficient

In future cities every driver may own a vehicle which could be either independently driven (HDV), or autonomously routed and piloted (CAV). The autonomous operations could be handled by a few competing companies. What is the market structure which would make this market aligned with city goals? In this paper we discuss a variant of the emerging market of collectively routed fleets of CAVs, where revenue for fleet operators is proportional to market share. We provide benchmark scenarios to compare the routing algorithms. We present several routing algorithms and demonstrate that, when the attitudes of human drivers towards CAVs exhibit significant diversity, randomised CAV routing, resulting in unpredictable travel times for HDVs, is more efficient than routing proportional to system optimum/user equilibrium. Based on this, we propose to improve the design of the market by augmenting the market-share objective with mean systemwide travel time in order to limit antisocial randomised strategies of fleet operators and drive the competition towards social welfare oriented cooperation.

cs.MA↗

arXiv2025

Market share maximizing strategies of CAV fleet operators may cause chaos in our cities

We study the dynamics and equilibria of a new kind of routing games, where players - drivers of future autonomous vehicles - may switch between individual (HDV) and collective (CAV) routing. In individual routing, just like today, drivers select routes minimizing expected travel costs, whereas in collective routing an operator centrally assigns vehicles to routes. The utility is then the average experienced travel time discounted with individually perceived attractiveness of automated driving. The market share maximising strategy amounts to offering utility greater than for individual routing to as many drivers as possible. Our theoretical contribution consists in developing a rigorous mathematical framework of individualized collective routing and studying algorithms which fleets of CAVs may use for their market-share optimization. We also define bi-level CAV - HDV equilibria and derive conditions which link the potential marketing behaviour of CAVs to the behavioural profile of the human population. Practically, we find that the fleet operator may often be able to equilibrate at full market share by simply mimicking the choices HDVs would make. In more realistic heterogenous human population settings, however, we discover that the market-share maximizing fleet controller should use highly variable mixed strategies as a means to attract or retain customers. The reason is that in mixed routing the powerful group player can control which vehicles are routed via congested and uncongested alternatives. The congestion pattern generated by CAVs is, however, not known to HDVs before departure and so HDVs cannot select faster routes and face huge uncertainty whichever alternative they choose. Consequently, mixed market-share maximising fleet strategies resulting in unpredictable day-to-day driving conditions may, alarmingly, become pervasive in our future cities.

math.OC↗

arXiv2025

URB -- Urban Routing Benchmark for RL-equipped Connected Autonomous Vehicles

Connected Autonomous Vehicles (CAVs) promise to reduce congestion in future urban networks, potentially by optimizing their routing decisions. Unlike for human drivers, these decisions can be made with collective, data-driven policies, developed using machine learning algorithms. Reinforcement learning (RL) can facilitate the development of such collective routing strategies, yet standardized and realistic benchmarks are missing. To that end, we present URB: Urban Routing Benchmark for RL-equipped Connected Autonomous Vehicles. URB is a comprehensive benchmarking environment that unifies evaluation across 29 real-world traffic networks paired with realistic demand patterns. URB comes with a catalog of predefined tasks, multi-agent RL (MARL) algorithm implementations, three baseline methods, domain-specific performance metrics, and a modular configuration scheme. Our results show that, despite the lengthy and costly training, state-of-the-art MARL algorithms rarely outperformed humans. The experimental results reported in this paper initiate the first leaderboard for MARL in large-scale urban routing optimization. They reveal that current approaches struggle to scale, emphasizing the urgent need for advancements in this domain.

cs.LG↗

arXiv2025

Impact of Collective Behaviors of Autonomous Vehicles on Urban Traffic Dynamics: A Multi-Agent Reinforcement Learning Approach

This study examines the potential impact of reinforcement learning (RL)-enabled autonomous vehicles (AV) on urban traffic flow in a mixed traffic environment. We focus on a simplified day-to-day route choice problem in a multi-agent setting. We consider a city network where human drivers travel through their chosen routes to reach their destinations in minimum travel time. Then, we convert one-third of the population into AVs, which are RL agents employing Deep Q-learning algorithm. We define a set of optimization targets, or as we call them behaviors, namely selfish, collaborative, competitive, social, altruistic, and malicious. We impose a selected behavior on AVs through their rewards. We run our simulations using our in-house developed RL framework PARCOUR. Our simulations reveal that AVs optimize their travel times by up to 5\%, with varying impacts on human drivers' travel times depending on the AV behavior. In all cases where AVs adopt a self-serving behavior, they achieve shorter travel times than human drivers. Our findings highlight the complexity differences in learning tasks of each target behavior. We demonstrate that the multi-agent RL setting is applicable for collective routing on traffic networks, though their impact on coexisting parties greatly varies with the behaviors adopted.

cs.MA↗

arXiv2025

Wardropian Cycles make traffic assignment both optimal and fair by eliminating price-of-anarchy with Cyclical User Equilibrium for compliant connected autonomous vehicles

Connected and Autonomous Vehicles (CAVs) open the possibility for centralised routing with full compliance, making System Optimal traffic assignment attainable. However, as System Optimum makes some drivers better off than others, voluntary acceptance seems dubious. To overcome this issue, we propose a new concept of Wardropian cycles, which, in contrast to previous utopian visions, makes the assignment fair on top of being optimal, which amounts to satisfaction of both Wardrop's principles. Such cycles, represented as sequences of permutations to the daily assignment matrices, always exist and equalise, after a limited number of days, average travel times among travellers (like in User Equilibrium) while preserving everyday optimality of path flows (like in System Optimum). We propose exact methods to compute such cycles and reduce their length and within-cycle inconvenience to the users. As identification of optimal cycles turns out to be NP-hard in many aspects, we introduce a greedy heuristic efficiently approximating the optimal solution. Finally, we introduce and discuss a new paradigm of Cyclical User Equilibrium, which ensures stability of optimal Wardropian Cycles under unilateral deviations. We complement our theoretical study with large-scale simulations. In Barcelona, 670 vehicle-hours of Price-of-Anarchy are eliminated using cycles with a median length of 11 days-though 5% of cycles exceed 90 days. However, in Berlin, just five days of applying the greedy assignment rule significantly reduces initial inequity. In Barcelona, Anaheim, and Sioux Falls, less than 7% of the initial inequity remains after 10 days, demonstrating the effectiveness of this approach in improving traffic performance with more ubiquitous social acceptability.

eess.SY↗

arXiv2025

Detection of coordinated fleet vehicles in route choice urban games. Part I. Inverse fleet assignment theory

Detection of collectively routing fleets of vehicles in future urban systems may become important for the management of traffic, as such routing may destabilize urban networks leading to deterioration of driving conditions. Accordingly, in this paper we discuss the question whether it is possible to determine the flow of fleet vehicles on all routes given the fleet size and behaviour as well as the combined total flow of fleet and non-fleet vehicles on every route. We prove that the answer to this Inverse Fleet Assignment Problem is 'yes' for myopic fleet strategies which are more 'selfish' than 'altruistic', and 'no' otherwise, under mild assumptions on route/link performance functions. To reach these conclusions we introduce the forward fleet assignment operator and study its properties, proving that it is invertible for 'bad' objectives of fleet controllers. We also discuss the challenges of implementing myopic fleet routing in the real world and compare it to Stackelberg and Nash routing. Finally, we show that optimal Stackelberg fleet routing could involve highly variable mixed strategies in some scenarios, which would likely cause chaos in the traffic network.

math.OC↗

arXiv2025

Collaboration Between the City and Machine Learning Community is Crucial to Efficient Autonomous Vehicles Routing

Autonomous vehicles (AVs), possibly using Multi-Agent Reinforcement Learning (MARL) for simultaneous route optimization, may destabilize traffic networks, with human drivers potentially experiencing longer travel times. We study this interaction by simulating human drivers and AVs. Our experiments with standard MARL algorithms reveal that, both in simplified and complex networks, policies often fail to converge to an optimal solution or require long training periods. This problem is amplified by the fact that we cannot rely entirely on simulated training, as there are no accurate models of human routing behavior. At the same time, real-world training in cities risks destabilizing urban traffic systems, increasing externalities, such as $CO_2$ emissions, and introducing non-stationarity as human drivers will adapt unpredictably to AV behaviors. In this position paper, we argue that city authorities must collaborate with the ML community to monitor and critically evaluate the routing algorithms proposed by car companies toward fair and system-efficient routing algorithms and regulatory standards.

cs.MA↗

arXiv2025

RouteRL: Multi-agent reinforcement learning framework for urban route choice with autonomous vehicles

RouteRL is a novel framework that integrates multi-agent reinforcement learning (MARL) with a microscopic traffic simulation, facilitating the testing and development of efficient route choice strategies for autonomous vehicles (AVs). The proposed framework simulates the daily route choices of driver agents in a city, including two types: human drivers, emulated using behavioral route choice models, and AVs, modeled as MARL agents optimizing their policies for a predefined objective. RouteRL aims to advance research in MARL, transport modeling, and human-AI interaction for transportation applications. This study presents a technical report on RouteRL, outlines its potential research contributions, and showcases its impact via illustrative examples.

cs.MA↗

arXiv2024

Social impact of CAVs -- coexistence of machines and humans in the context of route choice

Suppose in a stable urban traffic system populated only by human driven vehicles (HDVs), a given proportion (e.g. 10%) is replaced by a fleet of Connected and Autonomous Vehicles (CAVs), which share information and pursue a collective goal. Suppose these vehicles are centrally coordinated and differ from HDVs only by their collective capacities allowing them to make more efficient routing decisions before the travel on a given day begins. Suppose there is a choice between two routes and every day each driver makes a decision which route to take. Human drivers maximize their utility. CAVs might optimize different goals, such as the total travel time of the fleet. We show that in this plausible futuristic setting, the strategy CAVs are allowed to adopt may result in human drivers either benefitting or being systematically disadvantaged and urban networks becoming more or less optimal. Consequently, some regulatory measures might become indispensable.

cs.MA↗

arXiv2019

On uniqueness of dissipative solutions of the Camassa-Holm equation

We prove that dissipative weak solutions of the Camassa-Holm equation are unique. Thus we complete the global well-posedness theory of this celebrated model of shallow water, initiated by a general proof of existence in [Z. Xin, P. Zhang Comm. Pure Appl. Math. 53 (2000)]. As the dissipative weak solutions, being viscosity solutions, seem to constitute the most physically relevant class of solutions in the wave-breaking regime, the result provides mathematical rationale for the feasibility of the Camassa-Holm equation as a model of water waves encompassing both soliton interactions and wave-breaking.

math.AP↗

arXiv2016

On measures of accretion and dissipation for solutions of the Camassa-Holm equation

We investigate the measures of dissipation and accretion related to the weak solutions of the Camassa-Holm equation. Demonstrating certain properties of nonunique characteristics, we prove a new representation formula for these measures and conclude about their structural features, such us singularity with respect to the Lebesgue measure. We apply these results to gain new insights into the structure of weak solutions, proving in particular that measures of accretion vanish for dissipative solutions of the Camassa-Holm equation.

math.AP↗

arXiv2016

Transfer of energy in Camassa-Holm and related models by use of nonunique characteristics

We study the propagation of energy density in finite-energy weak solutions of the Camassa-Holm and related equations. Developing the methods based on generalized nonunique characteristics, we show that the parts of energy related to positive and negative slopes are one-sided weakly continuous and of bounded variation, which allows us to define certain measures of dissipation of both parts of energy. The result is a step towards the open problem of uniqueness of dissipative solutions of the Camassa-Holm equation.

math.AP↗

arXiv2015

Maximal dissipation in Hunter-Saxton equation for bounded energy initial data

In [Zhang and Zheng, 2005] it was conjectured by Zhang and Zheng that dissipative solutions of the Hunter-Saxton equation, which are known to be unique in the class of weak solutions, dissipate the energy at the highest possible rate. The conjecture of Zhang and Zheng was proven in [Dafermos, 2012] by Dafermos for monotone increasing initial data with bounded energy. In this note we prove the conjecture in [Zhang and Zheng, 2005] in full generality. To this end we examine the evolution of the energy of any weak solution of the Hunter-Saxton equation. Our proof shows in fact that for every time $t>0$ the energy of the dissipative solution is not greater than the energy of any weak solution with the same initial data.

math.AP↗

arXiv2014

Nonnegative measures belonging to $H^{-1}(\mathbb{R}^2)$

Radon measures belonging to the negative Sobolev space $H^{-1}(\mathbb{R}^2)$ are important from the point of view of fluid mechanics as they model vorticity of vortex-sheet solutions of incompressible Euler equations. In this note we discuss regularity conditions sufficient for nonnegative Radon measures supported on a line to be in $H^{-1}(\mathbb{R}^2)$. Applying the obtained results, we derive consequences for measures on $\mathbb{R}^2$ with arbitrary support and prove elementarily, among other things, that measures belonging to $H^{-1}(\mathbb{R}^2)$ may be supported on a set of Hausdorff dimension $0$. We comment on possible numerical applications.

math.FA↗

arXiv2014

Measure-transmission metric and stability of structured population models

In [Gwiazda, Jamróz, Marciniak-Czochra 2012] a framework for studying cell differentiation processes based on measure-valued solutions of transport equations was introduced. Under application of the so-called measure-transmission conditions it enabled to describe processes involving both discrete and continuous transitions. This framework, however, admits solutions which lack continuity with respect to initial data. In this paper, we modify the framework from [Gwiazda, Jamróz, Marciniak-Czochra 2012] by replacing the flat metric, known also as bounded Lipschitz distance, by a new Wasserstein-type metric. We prove, that the new metric provides stability of solutions with respect to perturbations of initial data while preserving their continuity in time. The stability result is important for numerical applications.

math.AP↗