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Probabilistic Game-Theoretic Traffic Routing

Emilio Benenati, Sergio Grammatico

TL;DR

The paper addresses routing for self-interested vehicles under stochastic decision strategies by casting the problem as an aggregative generalized Nash equilibrium with a first-order latency approximation. It develops a mild monotonicity condition and employs an semi-decentralized Inertial Forward-Reflected-Backward (I-FoRB) algorithm to compute a v-GNE, while a receding-horizon MPC formulation is proposed for potential games to enable online, scalable optimization. Theoretical results show the approximation error vanishes with increasing total traffic, monotonicity ensures convergence, and the RHNE controller is asymptotically stable under a suitable terminal cost. Numerical experiments on a 12-node network demonstrate reduced congestion and travel time relative to shortest-path routing, validating practical applicability for real-time, information-driven traffic management.

Abstract

We examine the routing problem for self-interested vehicles using stochastic decision strategies. By approximating the road latency functions and a non-linear variable transformation, we frame the problem as an aggregative game. We characterize the approximation error and we derive a new monotonicity condition for a broad category of games that encompasses the problem under consideration. Next, we propose a semi-decentralized algorithm to calculate the routing as a variational generalized Nash equilibrium and demonstrate the solution's benefits with numerical simulations. In the particular case of potential games, which emerges for linear latency functions, we explore a receding-horizon formulation of the routing problem, showing asymptotic convergence to destinations and analysing closed-loop performance dependence on horizon length through numerical simulations.

Probabilistic Game-Theoretic Traffic Routing

TL;DR

The paper addresses routing for self-interested vehicles under stochastic decision strategies by casting the problem as an aggregative generalized Nash equilibrium with a first-order latency approximation. It develops a mild monotonicity condition and employs an semi-decentralized Inertial Forward-Reflected-Backward (I-FoRB) algorithm to compute a v-GNE, while a receding-horizon MPC formulation is proposed for potential games to enable online, scalable optimization. Theoretical results show the approximation error vanishes with increasing total traffic, monotonicity ensures convergence, and the RHNE controller is asymptotically stable under a suitable terminal cost. Numerical experiments on a 12-node network demonstrate reduced congestion and travel time relative to shortest-path routing, validating practical applicability for real-time, information-driven traffic management.

Abstract

We examine the routing problem for self-interested vehicles using stochastic decision strategies. By approximating the road latency functions and a non-linear variable transformation, we frame the problem as an aggregative game. We characterize the approximation error and we derive a new monotonicity condition for a broad category of games that encompasses the problem under consideration. Next, we propose a semi-decentralized algorithm to calculate the routing as a variational generalized Nash equilibrium and demonstrate the solution's benefits with numerical simulations. In the particular case of potential games, which emerges for linear latency functions, we explore a receding-horizon formulation of the routing problem, showing asymptotic convergence to destinations and analysing closed-loop performance dependence on horizon length through numerical simulations.
Paper Structure (17 sections, 14 theorems, 96 equations, 3 figures, 2 algorithms)

This paper contains 17 sections, 14 theorems, 96 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Traffic routing as a Generalized Nash Equilibrium problem
  4. Affine formulation of the system dynamics
  5. Control objective and constraints
  6. Generalized Nash equilibrium problem
  7. Generalized Nash Equilibrium seeking
  8. Monotonicity of the game
  9. Semi-decentralized equilibrium seeking
  10. Receding horizon game formulation
  11. Equivalent finite horizon optimal control problem
  12. Stability of receding horizon Nash equilibrium control
  13. Numerical study$^1$
  14. Conclusion
  15. Proofs of Section \ref{['sec:traffic_GNE']}
  16. ...and 2 more sections

Key Result

Lemma 1

Let $\omega_i$ in eq:def_x satisfy: Then, $\omega_i\in (\Delta^{|\mathcal{E}|})^T \times (\Delta^{|\mathcal{N}|})^{(T+1)}$ and a choice of $(\Pi^i_t)_{i\in\mathcal{I}, t\in \mathcal{T}}$ such that $\rho_t^i$ follows the dynamics in eq:nonlin_dyn is: for all ${ (a, b) \in\mathcal{E}, t \in \mathcal{T}, i\in\mathcal{I}}$.

Figures (3)

  • Figure 1: $\max_{t} \sigma_{(a,b)}^{t} / \overline{c}_{(a,b)}$, compared to the congestion obtained by the shortest path routing. The dotted line denotes $c_{(a,b)} / \overline{c}_{(a,b)}$. The dots show the median values. The shaded area highlights the 95% confidence interval. We show in red the performance of the shortest path solution (SP).
  • Figure 2: Difference between approximated and empirical travel time with respect to $V$, the number of vehicles per population.
  • Figure 3: Comparison of the total cost incurred by the agents, with respect to the shortest path without traffic information.

Theorems & Definitions (21)

  • Remark 1
  • Lemma 1
  • Proposition 1
  • Definition 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Remark 2
  • Remark 3
  • Lemma 5
  • ...and 11 more