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Decentralized Decision-Making among Heterogeneous Autonomous Vehicles: An $\alpha$-Potential Game Framework

Anran Hu, Zhexin Wang, Yufei Zhang, Xuan Di

arXiv:2609.38731v1math.OCcs.GTcs.MA

TL;DR

Heterogeneous autonomous vehicles need decentralized decision-making that captures self-interested behavior and asymmetric interactions, while equilibrium computation in dynamic games is challenging. The paper constructs an α-potential framework with vehicle-specific rescaling to approximate equilibria and, under a cyclic condition, recover exact equilibria. It also gives social-efficiency guarantees and demonstrates the framework across vehicle-control scenarios.

  • Problem

    Scalable AV control must account for heterogeneous, potentially conflicting objectives, but computing Nash equilibria in dynamic games is computationally challenging.

    Source: §1 · pp. 1–2

  • Method

    The framework constructs an α-potential whose minimization yields approximate equilibria, then uses vehicle-specific rescaling to reduce effective interaction asymmetry.

    Source: §1 · pp. 2–3, §4.1 · p. 7

  • Results

    Under a Kolmogorov cyclic condition rescaling eliminates approximation error, pairwise symmetric interactions give social cost at most twice the centralized optimum, and experiments demonstrate diverse vehicle-control scenarios.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The framework accommodates heterogeneous behavior and strong local interactions in settings including collision avoidance, lane changing, overtaking, and intersection crossing.

    Source: §1 · p. 2, §7 · p. 27

    Future work includes assessing model uncertainty, algorithm convergence, equilibrium selection, explicit safety constraints, and scalability in larger, more realistic traffic environments.

    Source: §7 · p. 27

Abstract

from arXiv · show

We study noncooperative multi-vehicle games among heterogeneous autonomous vehicles, where each vehicle adopts a decentralized closed-loop policy based on its own state, and optimizes an objective that depends on other vehicles through potentially asymmetric interaction weights. We develop an $\alpha$-potential game framework that reduces the computation of an approximate Nash equilibrium (NE) to the minimization of a single auxiliary $\alpha$-potential function. We explicitly construct this $\alpha$-potential, establish the existence of its minimizers, and characterize the equilibrium approximation error $\alpha$ in terms of interaction asymmetry. We further introduce vehicle-specific scaling to reduce the effective interaction asymmetry, thereby tightening the equilibrium approximation and, in important cases, recovering an exact NE despite asymmetric interactions. We also derive social-efficiency guarantees for the potential-selected policies, revealing how the interaction structure shapes worst-case efficiency. Numerical experiments demonstrate the flexibility of the framework in capturing heterogeneous vehicle interactions, collision and obstacle avoidance, lane changing and overtaking under different traffic configurations, and priority-based intersection crossing.

1 Introduction

The paper addresses scalable decision-making for heterogeneous vehicles with potentially conflicting objectives and asymmetric interactions. It develops an α-potential framework for approximate equilibrium computation, sharpens its guarantees through vehicle-specific rescaling, and studies efficiency and traffic-control scenarios.

1 Introduction

  • Decentralized control and noncooperative games address the sensing, communication, and computation burden of large fleets while modeling vehicles with heterogeneous, potentially conflicting objectives.Computing Nash equilibria in dynamic games remains challenging as the number of vehicles grows.Source: §1 · p. 2
  • The α-potential framework reduces approximate Nash equilibrium computation to minimizing an auxiliary function and establishes that its minimizers are approximate equilibria despite asymmetric interactions.The approximation parameter is characterized by game coefficients, including horizon length, interaction magnitude, and asymmetry.Source: §1 · pp. 2–3
  • Vehicle-specific rescaling tightens the equilibrium approximation; under a Kolmogorov cyclic condition it eliminates the error, while optimal weights can be computed by geometrically convergent bisection.An optimal rescaling exists under a mild connectivity condition.Source: §1 · p. 3
  • Under pairwise symmetric interactions, the selected self-interested equilibrium has social cost at most twice the centralized social optimum.Greater interaction asymmetry can weaken the efficiency guarantee through greater dispersion in rescaling weights.Source: §1 · p. 3
  • Policy-gradient experiments demonstrate heterogeneous interactions, collision avoidance, lane changing, overtaking, and priority-based intersection crossing, with rescaling substantially improving equilibrium approximation.The experiments compare weak mean-field and strong local interaction regimes across traffic configurations.Source: §1 · p. 3

2 Related work

Prior AV-control work includes cooperative and noncooperative formulations, while tractable equilibrium methods often rely on symmetry assumptions that can limit heterogeneous traffic models. The paper builds on α-potential games and adds vehicle-specific scaling for decentralized policies and asymmetric interactions.

2.1 Cooperative and noncooperative frameworks for AV decision-making

  • Cooperative AV control pursues shared objectives, whereas noncooperative formulations model self-interested vehicles; computing equilibria in large dynamic games remains challenging.MARL is flexible for dynamic and partially observed settings, but convergence to an NE requires additional structural assumptions.Source: §2.1 · p. 4

2.2 Tractable structures for equilibrium computation in multi-vehicle games

  • Mean field games assume large populations of similar vehicles with weak, symmetric interactions, limiting their fit to strong local effects and heterogeneous objectives.Nearby vehicles can have nonvanishing influence in collision-avoidance settings.Source: §2.2 · p. 4
  • Exact potential games reduce equilibrium computation to one optimization but require exact pairwise symmetry, while α-potential games permit approximate deviation matching and approximate equilibria.The paper builds on this framework for decentralized closed-loop policies, using vehicle-specific scaling to reduce effective interaction asymmetry.Source: §2.2 · pp. 4–5

3 Methodology

The methodology formulates a finite-player stochastic game in which vehicles choose decentralized policies based on their own states while optimizing objectives affected by other players. Its AV specialization permits strong local and asymmetric interactions.

3.1 Problem setup

  • The setup is a finite-player stochastic differential game over a fixed horizon, with compact action sets and admissible decentralized Markov policies.Each player's state follows controlled diffusion dynamics, and the player minimizes a cost against the other players' policies.Source: §3.1 · p. 5
  • A policy profile is an ε-NE when no player can improve its objective by more than ε through unilateral deviation, even though each policy depends only on that player's private state.Compactness of the policy class is assumed; such classes can be nonconvex, including bounded-weight neural networks with Lipschitz activations.Source: §3.1 · p. 6

3.2 Specialization to AV control

  • In the AV specialization, objectives combine target and control costs with obstacle avoidance and weighted interactions that shape route preferences.The interaction kernel and vehicle-specific weights model how other vehicles' spatial distribution influences a vehicle's preferred route.Source: §3.1 · p. 5, §3.2 · p. 6
  • The model allows order-one influence from nearby vehicles and asymmetric pairwise weights, extending beyond weak mean-field interactions and symmetric potential-game assumptions.The strong-local regime is suited to modeling collision avoidance.Source: §3.2 · pp. 6–7

4 Main Results

The paper constructs an α-potential objective whose minimizers provide approximate Nash equilibria for decentralized vehicle games, then uses vehicle-specific rescaling to tighten the guarantee. It also establishes how interaction structure shapes both exact-equilibrium conditions and worst-case social efficiency.

4.1 α-potential function and approximate NEs

  • The construction adapts the α-potential framework to decentralized closed-loop policies and uses independent state dynamics to avoid auxiliary sensitivity equations.The paper presents this as a more tractable construction than prior approaches using open-loop controls.Source: §4.1 · pp. 7–8
  • Minimizers of the constructed α-potential are α-NEs, with the approximation error characterized by the decision horizon, interaction-kernel magnitude, and interaction-weight asymmetry.The construction preserves the game’s strategic structure rather than merely aggregating player objectives.Source: §4.1 · pp. 7–8

4.2 Optimal scaling weights

  • Vehicle-specific objective scaling can reduce residual interaction asymmetry and tighten the approximate-NE guarantee; under the Kolmogorov cyclic condition, it can eliminate the error and yield an NE.The paper identifies finite-cycle conditions and gives rank-one interactions as a case satisfying the cyclic condition.Source: §4.1 · p. 9, §4.2 · p. 10
  • An optimal rescaling vector exists under componentwise strong connectivity, which permits disconnected interaction groups when there are no edges between them.The optimization can be reformulated over a compact set, ensuring attainment.Source: §4.2 · pp. 9–10
  • Optimal rescaling can be computed by bisection over a monotone feasibility problem, with each iteration formulated as a linear program.After n iterations, the error in r is at most r0/2^n; an ε-accurate r⋆ requires O(log(r0/ε)) feasibility solves.Source: §4.2 · pp. 10–11

4.3 Social efficiency: game-theoretic versus centralized control

  • The paper bounds the social inefficiency of potential-selected policies relative to centralized cooperation using interaction weights and rescaling, assuming nonnegative cost components and positive optimal social cost.When α⋆=0, the bound can be sharpened to bound the price of stability.Source: §4.3 · pp. 12–13
  • With pairwise symmetric interactions, the social cost of the selected self-interested equilibrium is at most twice the centralized social optimum.The paper states this as a special-case worst-case efficiency guarantee.Source: §4.3 · p. 13
  • Greater interaction-weight asymmetry can spread the rescaling weights and weaken the worst-case social-efficiency guarantee, while avoiding excessive asymmetry can strengthen it.The guarantee depends in part on the ratio between the largest and smallest rescaling weights.Source: §4.3 · p. 14

5 Numerical Experiments

The numerical experiments evaluate the framework across longitudinal control, obstacle avoidance, rescaling, lane changing, and intersection crossing, examining how interaction strength, vehicle dynamics, environment geometry, and traffic configuration shape policies. At intersections, vehicles yield according to road priority or resolve conflicts according to position and strategic interaction.

5.1.1 Interaction regimes in cost function

  • Under weak interaction (β = 0), velocity- and acceleration-control vehicles use near-identical controls; under strong interaction (β = 1), they differentiate controls and avoid collisions more effectively.With velocity control, vehicles may detour or briefly backtrack while reaching targets accurately; acceleration control yields smoother paths but more dispersed terminal positions. The 10-vehicle cases share common starts and targets.Source: §5.1.1 · pp. 15–16, Figure 1 · p. 15, Figure 2 · p. 15
  • In the 30-vehicle velocity-control case with β = 1, trajectories remain structured and the potential objective decreases steadily during optimization.The authors report that the decentralized policy optimization remains effective and computationally tractable at this larger scale.Source: Figure 3 · p. 16, §5.1.1 · p. 16

5.1.2 Two-dimensional acceleration control with obstacles

  • Obstacle size changes trajectory deviations: a large circular obstacle reshapes paths earlier and more globally, while a small one mainly affects vehicles passing nearby.Without an obstacle, paths provide the baseline for comparison.Source: §5.1.2 · p. 17, Figure 4 · p. 17

5.1.3 Impact of vehicle-specific rescaling for equilibrium computation

  • With rank-one interactions satisfying the Kolmogorov cyclic condition, vehicle-specific rescaling makes the game an exact potential game, so an exact rescaled-potential minimizer is a Nash equilibrium.For heterogeneous vehicle sizes, trajectories show large vehicles near the center, medium vehicles farther outward, and small vehicles at the periphery under both control formulations.Source: §5.1.3 · pp. 17–18, Figure 5 · p. 18
  • After rescaling, relative exploitabilities fall below 4% for every vehicle, with most below 2%, from pre-rescaling highs of 64.7% and 49.9% for Vehicles 1 and 2.The results indicate that rescaling substantially reduces incentives for unilateral deviation and moves the learned profile closer to a Nash equilibrium.Source: §5.1.3 · pp. 18–19, Table 3 · p. 19

5.2 Lane changing

  • Lane-changing equilibria depend on initial positions: the faster orange vehicle overtakes and returns to its lane when space permits, but otherwise stays left as the green vehicle yields right.The two scenarios use the same costs and target speeds; in the second, returning would bring the orange vehicle too close to the blue vehicle, while its approach prompts the slower green vehicle to change lanes.Source: §5.2 · pp. 20–21

5.3 Unsignalized intersection crossing

  • Intersection experiments compare crossing behaviors with and without major–minor-road differentiation under identical initial conditions, encoding priority through vehicle-dependent speed-tracking penalties.Major-road vehicles face greater cost for deviating from target speed, while minor-road vehicles can decelerate more readily to resolve conflicts.Source: §5.3 · pp. 22–23, Figure 7 · p. 22
  • With major–minor-road differentiation, minor-road vehicles clear the conflict region or wait for major-road traffic to pass before proceeding.Vehicles already entering or approaching the conflict region accelerate, while following vehicles decelerate; waiting vehicles proceed after the major-road vehicles clear the intersection.Source: §5.3 · p. 23
  • Without a predetermined road hierarchy, vehicles cross according to relative positions and strategic interactions, with leading vehicles proceeding and followers yielding as needed.Source: §5.3 · p. 23

6 Proofs

The proofs establish existence and equilibrium guarantees for the constructed potentials, then derive results for interaction rescaling, computation, and social efficiency.

Proof of Proposition 4.2

Proof of Theorem 4.3

  • A minimizer of Φ_p exists and is an α(p)-Nash equilibrium, with continuity and compactness establishing existence.The proof obtains the equilibrium property from the potential inequality and the minimizer from continuity on the compact policy space.Source: Proof of Theorem 4.3 · pp. 24–25

Proof of Theorem 4.4

  • Every optimizer of the rescaling problem is strictly positive, and the compact feasible set ensures the minimum is attained.Componentwise normalizations bound the rescaling variables and the objective parameter.Source: Proof of Theorem 4.4 · p. 25

Proof of Theorem 4.5

  • The cyclic condition yields a rescaling with α(p⋆) = 0, so every minimizer of Φ_p⋆ is an exact Nash equilibrium.The proof connects the cyclic condition to vanishing rescaled interaction asymmetry.Source: Proof of Theorem 4.5 · p. 26

Proof of Proposition 4.6

  • After n bisection iterations, the returned r_n satisfies 0 ≤ r_n − r⋆ ≤ r0/2^n.The feasible p_n corresponds to the returned upper bound r_n.Source: Proof of Proposition 4.6 · p. 26

Proof of Proposition 4.7

  • The proof establishes the social-efficiency bound (4.14) by comparing a potential minimizer with a social-cost minimizer.It first establishes that the social cost admits a minimizer.Source: Proof of Proposition 4.7 · p. 26

Proof of Corollary 4.8

7 Conclusions and Future Work

The paper develops an α-potential framework for decentralized heterogeneous-vehicle games, with rescaling that can reduce exploitability and recover exact equilibria under suitable conditions. Its implementation demonstrates applications across several traffic tasks, while uncertainty, convergence, and larger-scale validation remain future work.

7 Conclusions and Future Work

  • The framework reduces approximate Nash-equilibrium computation to potential minimization and uses vehicle-specific rescaling to sharpen equilibrium approximation, sometimes recovering exact equilibria.The paper also derives social-efficiency guarantees relative to centralized full cooperation.Source: §7 · p. 27
  • A neural-network policy-gradient implementation demonstrates heterogeneous behavior and strong local interactions across collision and obstacle avoidance, lane changing, and intersection crossing.The conclusion reports that rescaling can substantially reduce exploitability in asymmetric games.Source: §7 · p. 27
  • Future work includes studying how model and interaction-parameter errors affect equilibrium accuracy and social efficiency, alongside convergence analysis and validation in larger, more realistic traffic environments.Other listed directions include equilibrium selection, local-neighborhood policies, and explicit safety constraints.Source: §7 · p. 27
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