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

Anran Hu, Zhexin Wang, Yufei Zhang, Xuan Di

Year: 2026Venue: arXiv preprintArea: math.OCType: PreprintEmbeddings: 108

Intelligence

Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%

Last extracted: 10/1/2026, 7:06:27 AM

Summary

This paper introduces an alpha-potential game framework for decentralized decision-making among heterogeneous autonomous vehicles (AVs). It addresses the computational challenges of finding Nash equilibria in noncooperative multi-vehicle games with asymmetric interactions. The authors construct an alpha-potential function to approximate Nash equilibria, introduce vehicle-specific scaling to reduce interaction asymmetry and tighten equilibrium approximation, and derive social-efficiency guarantees. Numerical experiments validate the framework's ability to handle collision avoidance, lane changing, and intersection crossing.

Entities (12)

autonomous vehicles · subject · 99%alpha-potential game · methodology · 98%Nash Equilibrium · concept · 97%asymmetric interactions · property · 95%Columbia University · organization · 95%Imperial College London · organization · 95%collision avoidance · application · 93%intersection crossing · application · 93%lane changing · application · 93%vehicle-specific scaling · technique · 92%social efficiency · metric · 90%Mean Field Games · related-work · 88%

Relation Signals (9)

alpha-potential game → appliedto → autonomous vehicles

confidence 98% · We study noncooperative multi-vehicle games among heterogeneous autonomous vehicles... We develop an α-potential game framework

Anran Hu → affiliatedwith → Columbia University

confidence 95% · Anran Hu ... Affiliation: Department of Industrial Engineering and Operations Research, Columbia University

Yufei Zhang → affiliatedwith → Imperial College London

confidence 95% · Yufei Zhang ... Affiliation: Department of Mathematics, Imperial College London, London, UK

alpha-potential game → reducescomputationto → Nash Equilibrium

confidence 95% · We develop an α-potential game framework that reduces the computation of an approximate Nash equilibrium (NE) to the minimization of a single auxiliary α-potential function.

vehicle-specific scaling → reduces → asymmetric interactions

confidence 92% · We further introduce vehicle-specific scaling to reduce the effective interaction asymmetry, thereby tightening the equilibrium approximation

alpha-potential game → enables → lane changing

confidence 90% · Numerical experiments demonstrate the flexibility of the framework in capturing ... lane changing and overtaking

alpha-potential game → enables → intersection crossing

confidence 90% · Numerical experiments demonstrate the flexibility of the framework in capturing ... priority-based intersection crossing.

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Abstract

Abstract: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.

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Decentralized Decision-Making among Heterogeneous Autonomous Vehicles: An α-Potential Game Framework Anran Hu Zhexin Wang Affiliation: Department of Industrial Engineering and Operations Research, Columbia University Affiliation: ah4277@columbia.edu, zw3037@columbia.edu Yufei Zhang Affiliation: Department of Mathematics, Imperial College London, London, UK Email: yufei.zhang@imperial.ac.uk Xuan Di* Affiliation: Department of Civil Engineering and Engineering Mechanics, Data Science Institute, Columbia University Email: sharon.di@columbia.edu Abstract 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 α-potential game framework that reduces the computation of an approximate Nash equilibrium (NE) to the minimization of a single auxiliary α-potential function. We explicitly construct this α-potential, establish the existence of its minimizers, and characterize the equilibrium approximation error α 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. Key words. Multi-vehicle systems, α-potential games, decentralized control, Nash equilibrium, asymmetric interactions, price of stability. AMS subject classifications. 91A14, 91A06, 91A15 1 Introduction How can one design scalable control strategies for large populations of autonomous vehicles (AVs) in complex traffic environments? Realistic AV control must account for the heterogeneous behaviors and objectives of interacting vehicles. Different vehicles may compete for limited road space, pursue different desired speeds or routes, and make conflicting decisions when merging, changing lanes, overtaking, or crossing intersections. Capturing such interactions is essential for generating realistic, human-like driving behavior in applications including platooning [15, 51, 58, 32], lane changing [48, 54], merging [46], unsignalized intersections [28], and departure-time choice on transportation networks [47]. These considerations highlight two central challenges: designing decentralized policies that scale to large AV populations and capturing the heterogeneous, self-interested behavior that arises in mixed traffic. Scalable control of large AV systems requires decentralized decision-making. A full-state feedback policy would require each vehicle to continuously observe, communicate, and process information about an increasing number of surrounding vehicles. As the fleet size grows, this creates substantial sensing, communication, and computational burdens, making real-time implementation increasingly difficult. Decentralized policies offer a more scalable alternative by allowing each AV to make decisions primarily from its own state and locally available information [10, 11]. By reducing reliance on global state information and centralized coordination, such policies are better suited to large-scale AV deployment. Meanwhile, heterogeneous and potentially conflicting objectives in mixed traffic motivate a noncooperative game-theoretic formulation of AV decision-making, in contrast to the fully cooperative control paradigm. A substantial part of the AV-control literature assumes that vehicles coordinate toward a common objective using, for example, consensus-based methods [2] or model predictive control [9]. Such an assumption can be restrictive when vehicles have different preferences, risk attitudes, priorities, or incentives. Under a game-theoretic formulation, each AV is modeled as a strategic agent optimizing its own objective while responding to the behavior of others. The resulting behavior is described by a Nash equilibrium (NE), at which no vehicle can improve its objective through a unilateral deviation. Challenges in equilibrium computation. While the game-theoretic formulation better captures self-interested behavior in heterogeneous traffic environments, its main challenge is computational. Computing an NE in a dynamic game is generally more demanding than solving an optimal control problem: standard procedures often rely on iterative best responses, in which each agent repeatedly solves an optimal control problem against the current strategies of the others [1]. The resulting computational burden can grow rapidly with the number of vehicles. Although multi-agent reinforcement learning (MARL) provides a flexible numerical alternative, it generally lacks theoretical guarantees of convergence to an NE in large-scale dynamic games. This motivates the search for structural properties of multi-agent interactions that can render equilibrium computation tractable. Existing work on multi-vehicle games has largely pursued this goal through two theoretical frameworks that exploit particular forms of symmetry. Mean field games (MFGs) [12, 25, 11] assume a large population of statistically similar vehicles interacting weakly and symmetrically through their empirical distribution. (Markov) potential games [36, 52], by contrast, require exact pairwise symmetry in cross-agent interactions: the effect of vehicle j’s control on vehicle i’s objective must be matched by the reciprocal effect of vehicle i’s control on vehicle j’s objective [36]. These structural assumptions facilitate equilibrium analysis, but can be restrictive in heterogeneous traffic environments with asymmetric and strong local interactions. See Section 2.2 for a more detailed discussion of these game-theoretic approaches and their limitations. Our work. We develop an α-potential game framework for decentralized control of heterogeneous AVs, in which each vehicle adopts a decentralized closed-loop policy based on its own state, and its objective depends on the states of other vehicles through heterogeneous interaction weights. Our model accommodates asymmetric and strong local interactions beyond the classical mean field game framework, allowing nearby vehicles to exert nonvanishing influence on one another. This feature is essential for capturing behaviors such as collision avoidance (see Section 3.2). Our main results are as follows. • We develop an α-potential game approach for computing (approximate) NEs in the multi-vehicle game. The framework reduces equilibrium computation to a decentralized control problem with an auxiliary objective, called an α-potential function. It accommodates asymmetric interactions and removes the exact pairwise symmetry required by the (Markov) potential game approach. We explicitly construct the α-potential function, establish the existence of a minimizer, and show that any such minimizer yields an α-NE of the original game. We further characterize the approximation parameter α analytically in terms of the game coefficients, including the length of the decision horizon, the magnitude of the interaction kernel, and the asymmetry of the vehicle interactions (Proposition 4.2). • We further sharpen the equilibrium approximation by applying the α-potential construction to rescaled objectives with vehicle-specific scaling weights (Theorem 4.3). Optimizing these weights to minimize the residual interaction asymmetry yields the tightest approximate-NE guarantee within this family of α-potential functions. We prove that an optimal rescaling exists under a mild connectivity condition (Theorem 4.4). Moreover, we show that when the interaction structure satisfies a Kolmogorov cyclic condition, this rescaling eliminates the approximation error entirely, so that any minimizer of the rescaled potential is an exact NE despite asymmetric interactions (Theorem 4.5). In the general case, we develop an efficient bisection algorithm for computing the optimal rescaling, where each iteration solves a linear feasibility problem and the algorithm converges at a geometric rate (Proposition 4.6). • We derive explicit bounds on the social-efficiency loss of the potential-selected policies, which capture decentralized, self-interested equilibrium behavior, relative to the centralized full-cooperation benchmark (Proposition 4.7 and Corollary 4.8). The bounds make explicit how the interaction structure shapes worst-case social efficiency and thus provide guidance for interaction design. Under pairwise symmetric interactions, the social cost of the selected self-interested equilibrium is at most twice the centralized social optimum. More generally, greater interaction asymmetry can weaken this social-efficiency guarantee through greater dispersion in the rescaling weights, suggesting that limiting excessive asymmetry can improve social-efficiency. • On the computational side, we develop a policy gradient method with decentralized neural network policies to solve the resulting control problems. The algorithm fits naturally within the centralized training and decentralized execution paradigm widely used in MARL [1]. Extensive numerical experiments illustrate the modeling flexibility and the scalability of the framework. We compare weak mean field and strong local interaction regimes, capture heterogeneous vehicle interactions and collision avoidance, and reproduce lane changing, overtaking, and priority based intersection crossing across different traffic configurations. The experiments also confirm that the proposed rescaling procedure can substantially improve the equilibrium approximation. Organization of the paper. The remainder of this paper is organized as follows. Section 2 reviews existing approaches to AV control and compares them with the proposed framework. Section 3 formulates the multi-vehicle games with decentralized policies. Section 4 presents the main theoretical results. It develops the α-potential game framework and establishes equilibrium approximation guarantees, optimal vehicle-specific rescaling, conditions for recovering exact Nash equilibria, and social-efficiency bounds. It also provides an efficient algorithm for computing the optimal rescaling weights. Section 5 presents numerical experiments on longitudinal control, obstacle avoidance, lane changing, and intersection crossing, examining the effects of interaction strength, vehicle heterogeneity, and rescaling. Finally, Section 7 summarizes the main findings and discusses future research directions. 2 Related work We begin with a brief overview of cooperative and noncooperative approaches to AV control, and then discuss existing methods for computing equilibria in noncooperative multi-vehicle games. 2.1 Cooperative and noncooperative frameworks for AV decision-making AV decision-making has been formulated from both cooperative and noncooperative perspectives. In cooperative-control, AVs coordinate toward a common system-level objective, with applications including platooning [21], lane changing [14, 22, 6], merging [13], and intersection management [34]. Decentralized (or distributed) control methods have been developed to reduce the communication and computational burden by relying more heavily on local information [40]. While these approaches decentralize information and computation, they generally retain the premise that vehicles pursue a common or coordinated objective. Noncooperative game-theoretic formulations instead model AVs as self-interested decision makers, and characterize their strategic interactions through Nash equilibria. Such approaches have been applied to lane changing and car-following [43, 48, 57, 53], longitudinal velocity control [25, 49], platooning and convoy control [7, 33, 29], routing [41], merging [50, 56], and intersection interactions [28, 24, 42]. However, computing Nash equilibria in large-scale dynamic multi-vehicle games remains challenging unless additional structure in the game can be exploited. For cooperative and noncooperative settings, MARL provides as a flexible numerical framework for multi-vehicle control in dynamic and potentially partially observed environments [49, 35, 20, 23]. However, ensuring convergence of an MARL algorithm to an NE in a dynamic game requires additional structural assumptions. 2.2 Tractable structures for equilibrium computation in multi-vehicle games The challenges of computing NEs motivate the development of theoretical frameworks that exploit interaction structures to make equilibrium analysis and computation tractable. Two prominent approaches widely used in traffic modeling are mean field games and (Markov) potential games. Although conceptually different, both rely on particular forms of symmetry that can be restrictive in heterogeneous AV settings. The MFG framework considers the large-population limit of similar AVs interacting weakly and symmetrically through their empirical distribution [12, 25, 11]. These assumptions can be restrictive for traffic applications: weak interactions are less suited to capturing strong local effects such as collision avoidance, while homogeneity overlooks heterogeneous objectives and asymmetric interactions that arise in mixed traffic; see Remark 1. (Markov) potential games, by contrast, address the finite-vehicle game directly (see e.g., Liu et al. 36, Jing et al. 27, Yan and Liu 52, Chen et al. 5). Following Monderer and Shapley [38], they posit a global potential function whose change under a player’s unilateral deviation exactly matches the corresponding change in the deviating player’s objective, thereby reducing equilibrium computation to a single optimization problem of the potential function. However, such an exact potential need not exist in dynamic multi-vehicle games. Even when restricting to time-dependent controls without explicit state feedback, potential structure requires exact pairwise symmetry in cross-vehicle interactions [36]. This requirement can be restrictive for heterogeneous AV systems with asymmetric interactions. Recent work relaxes exact potential structure while retaining the computational advantage of optimizing a single auxiliary objective; see, e.g., Candogan et al. 4, Varga 45. In particular, [16] introduces the notion of an α-potential game, in which the change in the potential under a unilateral deviation approximates the corresponding change in a player’s objective up to an error α. The case α=0α=0 recovers an exact potential game, while minimizers of an α-potential yield approximate NEs. This framework has subsequently been developed for broad classes of dynamic games (Guo et al. 17, Guo and Zhang 19, Guo et al. 18). For general stochastic differential games, [17] constructs an α-potential explicitly from the game coefficients and quantifies the approximation error in terms of asymmetry in the interaction structure. We build on this framework for multi-vehicle games with decentralized closed-loop policies and asymmetric interactions. Exploiting the decentralized state dynamics yields a simpler potential construction, while introducing a vehicle-specific scaling further reduces effective interaction asymmetry and sharpens the equilibrium approximation. 3 Methodology This section formulates a dynamic Markov game with decentralized policies for AV control. 3.1 Problem setup Consider a finite-player stochastic differential game with decentralized Markov policies, defined as follows: T>0T>0 is a fixed time horizon, [N]=1,…,N[N]=\1,…,N\, N∈ℕN , is a finite set of players, and for all i∈[N]i∈[N], Ai⊂ℝkA_i⊂R^k is a nonempty compact set representing player i’s action set, and Πi⊂C⁡([0,T]×ℝd,Ai) _i⊂ C([0,T]×R^d;A_i) is the set of player i’s admissible policies that are locally Lipschitz continuous in the spatial variable. We denote by Π=∏i∈[N]Πi = _i∈[N] _i the set of all admissible policy profiles of all players, and by Π−i=∏j≠iΠj _-i= _j =i _j the set of admissible policy profiles of all players except player i. We denote by ϕ=(ϕi)i∈[N]φ=( _i)_i∈[N] and ϕ−i=(ϕj)j∈[N]∖i _-i=( _j)_j∈[N] \i\ a generic element of Π and Π−i _-i, respectively. Each player’s state evolves according to a controlled diffusion, determined by their chosen policy. The player then minimizes a cost functional over the set of admissible policies. More precisely, for any given ϕ=(ϕi)i∈[N]∈Πφ=( _i)_i∈[N]∈ , player i’s state dynamics XiϕiX _i_i is governed by the following dynamics: for all t∈[0,T]t∈[0,T], dXi,t= dX_i,t= bi(Xi,t,ϕi(t,Xi,t))dt+σi(Xi,t,ϕi(t,Xi,t))dWt,Xi,0=ξi, b_i(X_i,t, _i(t,X_i,t))dt+ _i(X_i,t, _i(t,X_i,t))dW_t, X_i,0= _i, (3.1) where ξi∈ℝd _i∈R^d is a given initial state, bi:ℝd×Ai→ℝdb_i:R^d× A_i→R^d and σi:ℝd×Ai→ℝd×m _i:R^d× A_i→R^d× m are Lipschitz continuous functions, and W:Ω×[0,T]→ℝmW: ×[0,T]→R^m is an m-dimensional Brownian motion on a complete probability space (Ω,ℱ,ℙ)( ,F,P). Given ϕ−i∈Π−i _-i∈ _-i, player i determines their optimal policy by minimizing the following objective function over Πi _i: Ji(ϕ) J_i(φ) =[∫0T(fi(Xϕi,t,ϕi(t,Xϕi,t))+∑j≠iλijK(Xϕi,t−Xϕj,t))dt+gi(Xϕi,T)], =E [ _0^T (f_i (X _i_i,t, _i(t,X _i_i,t) )+ _j =i _ijK(X _i_i,t-X _j_j,t) )dt+g_i(X _i_i,T) ], (3.2) where λij≥0 _ij≥ 0 is a given constant, fi:ℝd×Ai→ℝf_i:R^d× A_i→R and K:ℝd→ℝK:R^d→R are given running costs, and gi:ℝd→ℝg_i:R^d→R is a given terminal cost. We assume that fif_i and gig_i are continuous and at most of quadratic growth, and K is a bounded continuous function satisfying K⁡(x)=K⁡(−x)K(x)=K(-x) for all x∈ℝdx∈R^d and K≢0K ≡ 0. We characterize the rational behavior of players in the Markov game (3.1)–(3.2) using the notion of an ϵε-Nash equilibrium given below. Definition 3.1. For any ϵ≥0ε≥ 0, a policy profile ϕ¯=(ϕ¯i)i∈[N]∈Π φ=( φ_i)_i∈[N]∈ an ϵε-Nash equilibrium (NE) of the game (3.1)–(3.2) if Ji(ϕ¯)≤Ji(ϕi,ϕ¯−i)+ϵ,∀ϕi∈Πi,i∈[N].J_i( φ)≤ J_i( _i, φ_-i)+ε, ∀ _i∈ _i,i∈[N]. If ϵ=0ε=0, then ϕ~ φ is called an NE. According to Definition 3.1, a joint policy profile ϕ¯ φ is an ϵε-NE if no player can improve her objective by more than ϵε through any unilateral deviation. Note that although player i’s objective in (3.2) depends on the joint states of all players, player i optimizes (3.2) over decentralized policies that depend only on her own private state XiX_i. Throughout this paper, we impose suitable compactness conditions on the class of admissible policies. Assumption 1. For all i∈[N]i∈[N], Πi _i is compact in the compact-open topology, i.e., Πi _i is equicontinuous on every compact subset of [0,T]×ℝd[0,T]×R^d. Assumption 1 reflects physical constraints on each player’s control. It holds, for example, when Πi _i consists of neural networks with Lipschitz activation functions and bounded weights. Since Πi _i is generally nonconvex, standard equilibrium-existence results do not directly apply to ensure the existence of an (approximate) Nash equilibrium in the game (3.1)-(3.2). 3.2 Specialization to AV control The game (3.1)-(3.2) encompasses various dynamic games in AV control. In this setting, each player represents a vehicle. The dynamics (3.1) can be taken as controlling velocity: dxi,t=ϕi(t,xi,t)dt,t∈[0,T],dx_i,t= _i(t,x_i,t)dt, t∈[0,T], (3.3) where xi,tx_i,t represents the position of the i-th vehicle at time t, and the control vi,t≔ϕi(t,xi,t)v_i,t _i(t,x_i,t) represents the velocity of the i-th vehicle at time t. Alternatively, one can take the dynamics (3.1) as a double integrator with controlled acceleration: for all t∈[0,T]t∈[0,T], dxi,t=vi,tdt,dvi,t=ϕi(t,xi,t,vi,t)dt+σidWt, splitdx_i,t&=v_i,tdt, dv_i,t= _i(t,x_i,t,v_i,t)dt+ _idW_t, split (3.4) where xi,tx_i,t and vi,tv_i,t represent the position and velocity of the i-th vehicle at time t, respectively, and the control ui,t≔ϕi(t,xi,t,vi,t)u_i,t _i(t,x_i,t,v_i,t) represents the acceleration of the i-th vehicle at time t. Vehicle i determines its optimal route by minimizing the objective (3.2), where the terminal cost gig_i specifies vehicle i’s preferred target, the running cost fif_i penalizes control effort and deviations from the target speed, while also enforcing obstacle-avoidance behavior for vehicle i, and the kernel K together with the weights (λij)j≠i( _ij)_j =i characterizes how the spatial distribution of other vehicles influences vehicle i’s preferred route. Typically, the kernel K decreases as the distance between two vehicles increases, modeling congestion-averse behavior. Remark 1 (Local and asymmetric interactions.). Our model accommodates both strong local interactions and asymmetric interactions among AVs, thereby going beyond common structural assumptions in existing game-theoretic approaches. First, unlike the MFG framework [25], interactions need not be only through the empirical distribution. Nearby vehicles may exert nonvanishing influence on one another, which is essential for modeling behaviors such as collision avoidance; see Section 5.1.1. Second, the interaction weights need not be pairwise symmetric, i.e., we allow λij≠λji _ij≠ _ji, in contrast to classical potential-game formulations that rely on symmetric cross-agent interactions. To illustrate the strong-local-interaction feature, following [39], consider λij=Nβ−1,K⁡(z)=ρ⁡(Nβ/d|z|),z∈ℝd, _ij=N^β-1, K(z)=ρ (N^β/d|z| ), z∈R^d, (3.5) where ρ:[0,∞)→[0,∞)ρ:[0,∞)→[0,∞) satisfies lim|z|→∞ρ⁡(z)=0 _|z|→∞ρ(z)=0, and β∈[0,1]β∈[0,1] determines how the interaction strength scales with N. When β=0β=0, each player interacts with all others with strength ⁡(1/N)O(1/N). This is the classical weak-interaction regime underlying mean-field approximations [26, 31]. As N→∞N→∞, the influence of individual players vanishes and interactions are captured through the aggregate population distribution, as in the MFG formulation for traffic flow in [25]. In contrast, when β=1β=1, the decay of ρ implies that each player interacts only with neighbors at distance (N−1/d)O(N^-1/d), but with interaction strength of order one. This strong-local-interaction regime allows nearby vehicles to have a non-negligible effect on individual decisions and is therefore well suited to modeling collision avoidance. Such interactions fall outside the classical mean-field regime, while remaining within the scope of our framework. 4 Main Results 4.1 α-potential function and approximate NEs This section analyzes approximate NEs of the game (3.1)–(3.2) by adapting the α-potential game framework developed in [17] to the present setting with decentralized closed-loop controls. The α-potential game framework reduces the challenging problem of identifying (approximate) NEs into a single optimization problem of the associated α-potential function. This connection is made precise in the following proposition, which follows directly from [17, Proposition 2.1]. Proposition 4.1. Suppose that the game (3.1)-(3.2) is an α-potential game for some α≥0α≥ 0, in the sense that there exists Φ:Π→ℝ : →R, called an α-potential function, such that for all i∈[N]i∈[N], ϕ−i∈Π−i _-i∈ _-i, and ϕi,ϕi′∈Πi _i,φ _i∈ _i, |(Ji(ϕi′,ϕ−i)−Ji(ϕi,ϕ−i))−(Φ⁡(ϕi′,ϕ−i)−Φ⁡(ϕi,ϕ−i))|≤α.|(J_i(φ _i, _-i)-J_i( _i, _-i))-( (φ _i, _-i)- ( _i, _-i))|≤α. (4.1) If ϕ¯∈Π φ∈ satisfies Φ⁡(ϕ¯)∈argminϕ∈Π⁡Φ(ϕ) ( φ)∈ _φ∈ (φ), then ϕ¯ φ is an α-NE of the game (3.1)-(3.2). Intuitively, a function Φ is an α-potential function for the game (3.1)-(3.2) if, whenever a player unilaterally deviates from her strategy, the resulting change in her objective differs from the corresponding change in Φ by at most α. Once constructed, Φ serves as an auxiliary central objective for constructing approximate NEs of the game. In what follows, we analytically construct the α-potential function Φ . Importantly, Φ is not merely an aggregation of the players’ objectives; rather, it preserves the strategic structure of the underlying non-cooperative game and captures the self-interested nature of the players. The α-potential game framework generalizes existing (Markov) potential game approaches to autonomous driving (see, e.g., Liu et al. [36], Yan and Liu [52]) by allowing α>0α>0. This relaxation is particularly important for dynamic games among AVs with asymmetric interactions, which generally do not admit an exact potential function but can be characterized as α-potential games for a suitable α>0α>0. Specifically, the following proposition constructs analytically an α-potential function for the game (3.1)-(3.2) and quantifies the associated α, in terms of the decision horizon, the magnitude of the interaction kernel K and the degree of asymmetry in the interaction weights (λij)i,j∈[N]( _ij)_i,j∈[N]. Proposition 4.2. Define Φ:Π→ℝ : →R such that for all ϕ∈Πφ∈ , Φ⁡(ϕ)≔⁡[∫0TF⁡(Xtϕ,ϕ⁡(t,Xtϕ))t+G⁡(XTϕ)], (φ) E [ _0^TF(X^φ_t,φ(t,X^φ_t))dt+G(X^φ_T) ], (4.2) where Xϕ=(Xiϕi)i∈[N]X^φ=(X _i_i)_i∈[N] is the joint state process, ϕ⁡(t,Xtϕ)=(ϕi(t,Xi,tϕi))i∈[N]φ(t,X^φ_t)=( _i(t,X _i_i,t))_i∈[N] is the joint control process, and F:ℝdN×A→ℝF:R^dN× A→R and G:ℝdN→ℝG:R^dN→R are given by F⁡(x,a)≔∑i=1Nfi(xi,ai)+∑1≤i<j≤Nλij+λji2K(xi−xj),G(x)≔∑i=1Ngi(xi). splitF(x,a)& _i=1^Nf_i(x_i,a_i)+ _1≤ i<j≤ N _ij+ _ji2K(x_i-x_j), G(x) _i=1^Ng_i(x_i). split Then Φ is an α-potential function of the game (3.1)-(3.2) with α≤T‖K‖L∞max⁡∑i≠ji∈[N]⁡|λij−λji|.α≤ T\|K\|_L^∞ _i∈[N] _i≠ j| _ij- _ji|. (4.3) Hence any minimizer of Φ over Π is an α-NE of the game (3.1)-(3.2). Proposition 4.2 highlights the importance of allowing a general α≥0α≥ 0 in (4.1) to accommodate games with asymmetric interactions. Indeed, by Proposition 4.2, α=0α=0 (so that Φ is an exact potential function) if and only if players’ interactions is symmetric, i.e., λij=λji _ij= _ji for all i,j∈[N]i,j∈[N]. Consequently, the (discrete-time) potential game frameworks in Liu et al. [36], Yan and Liu [52] are restricted to AVs with symmetric interactions, whereas our framework accommodates general asymmetric interactions among AVs. We refer to [17, 18] for large-population games with asymmetric interactions and small α. Remark 2 (Construction of α-potential functions). The construction of the α-potential function in Proposition 4.2 differs from those in [17, 18] in two key aspects: (i) Proposition 4.2 considers the Markov game (3.1)–(3.2) with Lipschitz (closed-loop) policies, whereas [17, 18] focus on open-loop controls without explicit state feedback; (i) Proposition 4.2 exploits the independence of state dynamics to construct the α-potential function (4.2) without introducing additional sensitivity equations as auxiliary states, as required in [17, 18]. This more tractable α-potential function enables more efficient computation of approximate NEs. Leveraging Proposition 4.2, the next theorem provides additional flexibility in constructing an α-potential function by appropriately scaling each player’s objective, thereby yielding a tighter approximate NE guarantee for games with asymmetric interactions. Theorem 4.3. Let p=(pi)i∈[N]∈(0,∞)Np=(p_i)_i∈[N]∈(0,∞)^N, and define Φp:Π→ℝ ^p: →R such that for all ϕ∈Πφ∈ , Φp(ϕ)≔⁡[∫0TFp(Xtϕ,ϕ⁡(t,Xtϕ))t+Gp(XTϕ)], ^p(φ) [ _0^TF^p(X^φ_t,φ(t,X^φ_t))dt+G^p(X^φ_T) ], (4.4) where XϕX^φ and ϕ⁡(t,Xtϕ)φ(t,X^φ_t) are defined as in Proposition 4.2, and Fp:ℝdN×A→ℝF^p:R^dN× A→R and Gp:ℝdN→ℝG^p:R^dN→R are given by Fp(x,a)≔∑i=1Npifi(xi,ai)+∑1≤i<j≤Npiλij+pjλji2K(xi−xj),Gp(x)≔∑i=1Npigi(xi). splitF^p(x,a)& _i=1^Np_if_i(x_i,a_i)+ _1≤ i<j≤ N p_i _ij+p_j _ji2K(x_i-x_j), G^p(x) _i=1^Np_ig_i(x_i). split Then Φp ^p admits a minimizer over Π , each of which is an α⁡(p)α(p)-NE of the game (3.1)-(3.2), with α⁡(p)≤T‖K‖L∞maxi∈[N]1pi∑j≠i|piλij−pjλji|.α(p)≤ T\|K\|_L^∞ _i∈[N] 1p_i _j≠ i|p_i _ij-p_j _ji|. Compared with Proposition 4.2, Theorem 4.3 provides a more general construction of the central objective for approximate NEs by introducing a scaling vector p that can mitigate the asymmetry of the interaction matrix. The function Φ in (4.2) corresponds to the special case pi≡1p_i≡ 1 for all i∈[N]i∈[N]. By optimizing p to minimize the residual interaction asymmetry, one can construct Φp ^p whose minimizer achieves a tighter NE gap than that obtained in Proposition 4.2. As will be shown in Section 4.2, this approach can even yield an exact NE for games with asymmetric interactions. 4.2 Optimal scaling weights To obtain the tightest equilibrium approximation guarantee from Theorem 4.3, we minimize the approximation bound over p∈(0,∞)Np∈(0,∞)^N. Define r⋆≔infp∈(0,∞)Nmaxi∈[N]⁡1pi∑j≠i|piλij−pjλji|,r _p∈(0,∞)^N _i∈[N] 1p_i _j≠ i|p_i _ij-p_j _ji|, (4.5) so that the smallest bound obtainable from Theorem 4.3 is α⋆=T‖K‖L∞r⋆.α =T\|K\|_L^∞r . Since the optimization in (4.5) is over the open set (0,∞)N(0,∞)^N, attainment of the infimum is not immediate. We next show that, under a mild connectivity condition on the interaction structure, the problem can be equivalently reformulated over a compact set. Associate with the interaction matrix Λ=(λij)i,j∈[N] =( _ij)_i,j∈[N] the directed interaction graph λ=([N],Eλ),G_λ=([N],E_λ), where there is a directed edge j→ij→ i if and only if λij>0 _ij>0. Thus an edge j→ij→ i means that player j influences the objective of player i. Assumption 2 (Componentwise strong connectivity). The directed graph λG_λ can be decomposed into a disjoint union of strongly connected subgraphs11 1 A directed graph is strongly connected if, for any two vertices i and j, there exists a directed path from i to j. with vertex sets S1,…,SmS_1,…,S_m. That is, [N]=S1∪⋯∪Sm,[N]=S_1∪·s∪ S_m, each subgraph induced by SkS_k is strongly connected, and there are no edges between distinct subgraphs (i.e., λij=λji=0 _ij= _ji=0, for all i∈Sk,j∈Sℓ,k≠ℓi∈ S_k,\ j∈ S_ ,\ k≠ ). Remark 3. Assumption 2 is satisfied, for example, when the entire interaction graph λG_λ is strongly connected. It is also satisfied when the interaction pattern has symmetric support, in the sense that λij>0 _ij>0 if and only if λji>0 _ji>0 for all i≠ji≠ j; in this case, the sets S1,…,SmS_1,…,S_m can be taken as the connected components of the corresponding undirected interaction graph. In particular, the assumption allows interaction matrices with zero entries and disconnected groups of players. This next theorem shows that under Assumption 2, the optimization problem (4.5) admits an equivalent compact formulation, which in particular guarantees the existence of an optimal rescaling vector. Theorem 4.4 (Existence of an optimal rescaling). Suppose Assumption 2 holds. Then (4.5) is equivalent to minp∈[0,∞)N,r∈[0,r0] _p∈[0,∞)^N,\ r∈[0,r_0] r r (P) s.t. .t. ∑j≠i|piλij−pjλji|≤rpi,i∈[N], _j≠ i|p_i _ij-p_j _ji|≤ rp_i, i∈[N], ∑i∈Skpi=|Sk|,k=1,…,m, _i∈ S_kp_i=|S_k|, k=1,…,m, where r0≔max⁡∑j≠i∈[N]⁡|λij−λji|r_0 _i∈[N] _j≠ i| _ij- _ji|. Consequently, the infimum in (4.5) is obtained by some p⋆∈(0,∞)Np ∈(0,∞)^N. We next identify the condition under which the optimal rescaling reduces the approximation error to zero, and hence minimizing (4.4) yields an NE. Theorem 4.5. The following statements are equivalent: (i) r⋆=0r =0; (i) there exists p=(p1,…,pN)∈(0,∞)Np=(p_1,…,p_N)∈(0,∞)^N such that piλij=pjλji,i,j∈[N];p_i _ij=p_j _ji, i,j∈[N]; (4.6) (i) (λij)i,j∈[N]( _ij)_i,j∈[N] satisfies the following Kolmogorov cyclic condition: • The zero pattern is symmetric: for all i≠ji≠ j, λij=0 _ij=0 if and only if λji=0 _ji=0. • For every finite cycle i1,i2,…,im,im+1i_1,i_2,…,i_m,i_m+1 with im+1=i1i_m+1=i_1, ∏k=1mλikik+1=∏k=1mλik+1ik. _k=1^m _i_ki_k+1= _k=1^m _i_k+1i_k. (4.7) If any of the above conditions holds, let p∗=(pi∗)i∈[N]p^*=(p^*_i)_i∈[N] satisfy (4.6). Then any minimizer of Φp⋆ ^p is an NE of the game (3.1)-(3.2). Remark 4 (Rank-one interaction). If λij=γiτj _ij= _i _j for some γi,τi>0 _i, _i>0, then (λij)i,j∈[N]( _ij)_i,j∈[N] satisfies the Kolmogorov cyclic condition. Indeed, the weights p satisfying (4.6) is given by pi=τiγip_i= _i _i, i∈[N]i∈[N], up to a common positive scaling factor. We next turn to the computation of an optimal rescaling vector in the general case. For any fixed r∈[0,r0]r∈[0,r_0], the feasibility problem in (P) is convex in p and can be written as a linear feasibility problem by introducing auxiliary variables for the absolute-value terms. Moreover, feasibility is monotone in r. Therefore, an optimal rescaling parameter can be computed by bisection over r∈[0,r0]r∈[0,r_0]. Algorithm 1 Bisection for Optimizing Rescaling Parameters 1: Interaction matrix (λij)i,j∈[N]( _ij)_i,j∈[N], components S1,…,SmS_1,…,S_m, number of iterations n 2: Set rlow=0,rhigh=r0≔max⁡∑j≠i∈[N]⁡|λij−λji|r_ low=0,\ r_ high=r_0 _i∈[N] _j≠ i| _ij- _ji|. 3: Initialize p=(1,…,1)p=(1,…,1). 4: for k=0,…,n−1k=0,…,n-1 do 5: Set rmid≔rlow+rhigh2r_ mid r_ low+r_ high2. 6: Check feasibility of find p~∈[0,∞)N p∈[0,∞)^N s.t. .t. ∑j≠i|p~iλij−p~jλji|≤rmidp~i,i∈[N], _j≠ i| p_i _ij- p_j _ji|≤ r_ mid p_i, i∈[N], ∑i∈Sℓp~i=|Sℓ|,ℓ=1,…,m. _i∈ S_ p_i=|S_ |, =1,…,m. 7: if the feasibility problem is feasible then 8: Set rhigh=rmidr_ high=r_ mid and p=p~p= p. 9: else 10: Set rlow=rmidr_ low=r_ mid. 11: end if 12: end for 13: return p and r=rhighr=r_ high. The bisection procedure is globally convergent, with an explicit geometric error bound, as stated below. Proposition 4.6 (Convergence of the bisection algorithm). Suppose Assumption 2 holds. Let (pn,rn)(p^n,r^n) denote the output of Algorithm 1 after n bisection iterations. Then 0≤rn−r⋆≤r02n,0≤α⁡(pn)−α⋆≤T‖K‖L∞r02n.0≤ r^n-r ≤ r_02^n, 0≤α(p^n)-α ≤ T\|K\|_L^∞ r_02^n. Consequently, rnr^n converges to r⋆r and α⁡(pn)α(p^n) converges to α⋆α at the rate O⁡(2−n)O(2^-n). Computational complexity. Each iteration of Algorithm 1 requires solving one linear feasibility problem. After introducing auxiliary variables to linearize the absolute-value terms, the problem is a linear program with O⁡(N+Mλ)O(N+M_λ) variables and constraints, where MλM_λ denotes the number of interacting pairs. Hence each iteration can be solved in time polynomial in q≔N+Mλq N+M_λ using standard linear-programming algorithms; see, e.g., Karmarkar 30, Vaidya 44, Cohen et al. 8. In particular, q=O⁡(N2)q=O(N^2) in the dense-interaction case, while q=O⁡(N)q=O(N) when each player interacts with a uniformly bounded number of neighbors. Together with Proposition 4.6, an ε -accurate value of r⋆r requires only O⁡(log⁡(r0/ε))O( (r_0/ )) such linear-programming feasibility solves. 4.3 Social efficiency: game-theoretic versus centralized control After characterizing approximate Nash equilibria, we turn to their social-efficiency implications. The minimizers of the rescaled α-potential functions given in (4.4) capture the self-interested behavior of heterogeneous vehicles in mixed traffic and therefore need not minimize the aggregate social cost achieved under full cooperation. We quantify the resulting efficiency loss by comparing these equilibrium outcomes with the centralized social optimum, thereby measuring how much social welfare is sacrificed to accommodate decentralized, self-interested decision-making. The resulting bounds also provide guidance for the design of vehicle interaction structures. To quantify the efficiency loss relative to full cooperation, define the social cost for any joint policy profile ϕ∈Πφ∈ by SC⁡(ϕ)≔∑i=1NJi(ϕ),SC(φ) _i=1^NJ_i(φ), (4.8) and the optimal social cost achieved in the full-cooperation setting by SCopt≔infϕ∈ΠSC⁡(ϕ).SC opt _φ∈ SC(φ). (4.9) In the sequel, we assume without loss of generality that SCopt>0SC opt>0, and measure the social efficiency of a policy profile ϕφ through the following inefficiency ratio. Definition 4.1 (Social inefficiency). For any ϕ∈Πφ∈ , define InEff⁡(ϕ)≔SC⁡(ϕ)SCopt.InEff(φ) SC(φ)SC opt. (4.10) Thus InEff⁡(ϕ)≥1InEff(φ)≥ 1, with values closer to one indicating greater social efficiency. Remark 5. The inefficiency ratio InEff⁡(ϕ)InEff(φ) extends the Price of Stability (PoS) perspective to arbitrary policy profiles, including approximate Nash equilibria. Indeed, letting ℰE denote the set of NEs, PoS≔infϕ∈ℰSC⁡(ϕ)SCopt=infϕ∈ℰInEff⁡(ϕ),PoS _φ SC(φ)SC opt= _φ InEff(φ), (4.11) so PoS measures the social inefficiency of the most efficient NE. We work with InEff⁡(ϕ)InEff(φ) because, under Assumption 1, an exact NE need not exist when the policy class is nonconvex. Even when exact equilibria exist, identifying the socially most efficient one can be computationally difficult, making the price of stability hard to evaluate. In contrast, InEff⁡(ϕ)InEff(φ) assesses the social performance of any individual policy profile and therefore applies directly to the approximate equilibria constructed in this paper. When α∗=0α^*=0 as in Theorem 4.5, the potential-selected profile is an exact NE, and hence its inefficiency ratio provides an upper bound on PoSPoS. In the sequel, we derive an analytical bound on the inefficiency ratio of the selected equilibrium policy in terms of the asymmetry of the interaction weights (λij)i,j∈[N]( _ij)_i,j∈[N] and the rescaling vector p. The key observation is that both the rescaled α-potential and the social cost decompose into the same player-specific and pairwise interaction cost components, but with different coefficients. This reduces the analysis of social inefficiency in the dynamic game to a comparison of these coefficients and makes explicit how interaction asymmetry and the choice of rescaling weights affect social welfare. To see it, define the non-interaction component of player i’s cost by Ci(ϕ)≔⁡[∫0Tfi(Xi,tϕi,ϕi(t,Xi,tϕi))t+gi(Xi,Tϕi)],C_i(φ) [ _0^Tf_i (X _i_i,t, _i(t,X _i_i,t) )\,dt+g_i(X _i_i,T) ], and the pairwise interaction term by ij(ϕ)≔⁡[∫0TK⁡(Xi,tϕi−Xj,tϕj)t].K_ij(φ) [ _0^TK (X _i_i,t-X _j_j,t )\,dt ]. Then SC⁡(ϕ)=∑i=1NCi(ϕ)+∑1≤i<j≤N(λij+λji)ij(ϕ),SC(φ)= _i=1^NC_i(φ)+ _1≤ i<j≤ N( _ij+ _ji)K_ij(φ), (4.12) whereas the rescaled potential associated with p can be written as Φp(ϕ)=∑i=1NpiCi(ϕ)+12∑1≤i<j≤N(piλij+pjλji)ij(ϕ). ^p(φ)= _i=1^Np_iC_i(φ)+ 12 _1≤ i<j≤ N(p_i _ij+p_j _ji)K_ij(φ). (4.13) The following proposition provides an explicit analytical bound on the inefficiency ratio of the equilibrium policy selected by minimizing the rescaled potential (4.13), in terms of the interaction weights (λij)i,j∈[N]( _ij)_i,j∈[N] and the rescaling vector p. Proposition 4.7 (Social inefficiency guarantee). Suppose that Ci(ϕ)≥0C_i(φ)≥ 0 and ij(ϕ)≥0K_ij(φ)≥ 0 for all admissible ϕφ, i∈[N]i∈[N], and i<ji<j, and SCopt>0SC opt>0. Then for all p∈(0,∞)Np∈(0,∞)^N and ϕp∈arg⁡minϕ∈ΠΦp(ϕ)φ^p∈ _φ∈ ^p(φ),22 2 The bound (4.14) is nontrivial when λij+λji>0 _ij+ _ji>0 for at least one pair i<ji<j. When λij=0 _ij=0 for all i≠ji≠ j, the game reduces to N independent control problems, and is excluded. InEff⁡(ϕp)≤γ¯(p)γ¯(p),InEff(φ^p)≤ γ(p) γ(p), (4.14) where γ¯(p)≔min⁡mini∈[N]⁡pi,min1≤i<j≤Nλij+λji>0⁡piλij+pjλji2(λij+λji), γ(p) \ _i∈[N]p_i,\, _ subarrayc1≤ i<j≤ N\\ _ij+ _ji>0 subarray p_i _ij+p_j _ji2( _ij+ _ji) \, (4.15) and γ¯(p)≔max⁡maxi∈[N]⁡pi,max1≤i<j≤Nλij+λji>0⁡piλij+pjλji2(λij+λji). γ(p) \ _i∈[N]p_i,\, _ subarrayc1≤ i<j≤ N\\ _ij+ _ji>0 subarray p_i _ij+p_j _ji2( _ij+ _ji) \. (4.16) When α∗=0α^*=0 as in Theorem 4.5, the bound in (4.14) can be sharpened to yield a bound on the PoS (see (4.11)). Corollary 4.8. Suppose that there exists p=(p1,…,pN)∈(0,∞)Np=(p_1,…,p_N)∈(0,∞)^N satisfying (4.6). Assume further that the hypotheses of Proposition 4.7 are satisfied. Then PoS≤InEff⁡(ϕp)≤maxi∈[N]⁡pimin1≤i<j≤N⁡pipjpi+pj≤2maxi∈[N]⁡pimini∈[N]⁡pi.PoS (φ^p)≤ _i∈[N]p_i _1≤ i<j≤ N p_ip_jp_i+p_j≤ 2 _i∈[N]p_i _i∈[N]p_i. (4.17) If, in addition, the interaction coefficients are pairwise symmetric, i.e., λij=λji _ij= _ji for all i,j∈[N]i,j∈[N], then PoS≤InEff⁡(ϕ)≤2PoS (φ^1)≤ 2. Remark 6 (Interaction asymmetry and social efficiency). The bound in (4.17) quantifies the worst-case social-efficiency loss associated with decentralized, self-interested behavior relative to the fully cooperative benchmark. In the symmetric-interaction case, the bound InEff⁡(ϕ)≤2InEff(φ^1)≤ 2 implies that, the social cost of the selected self-interested equilibrium is at most twice the centralized social optimum. More generally, under the Kolmogorov cyclic condition, (4.6) implies that, for every interacting pair, pipj=λjiλij. p_ip_j= _ji _ij. Hence asymmetry in the interaction coefficients is reflected directly in the relative magnitudes of the rescaling weights. A larger discrepancy between λij _ij and λji _ji induces greater dispersion in the weights pip_i, which can weaken the worst-case social-efficiency guarantee through the ratio maxi⁡pi/mini⁡pi _ip_i/ _ip_i in (4.17). These bounds also provide a design insight: interaction structures that avoid excessive asymmetry can lead to stronger worst-case social-efficiency guarantees. 5 Numerical Experiments In this section, we aim to evaluate the performance of our proposed model in a variety of traffic settings, summarized in Table 1. Table 1: Numerical simulation scenarios. Scenario (Section) Dim. Control Experiment Goal Interaction regimes (5.1.1) 1D Velocity; acceleration 10 vehicles; weak versus strong interactions (β=0β=0 versus β=1β=1) (Figures 1 and 2) Compare separation and collision avoidance Scalability (5.1.1) 1D Velocity 30 vehicles; strong interactions (β=1β=1) (Figure 3) Assess scalability to a larger number of vehicles Obstacle avoidance (5.1.2) 2D Acceleration 10 vehicles; no obstacle, or a circular obstacle of radius 0.10.1 or 0.50.5 (Figure 4) Assess trajectory responses to obstacle size Rescaling (5.1.3) 1D Velocity; acceleration 9 vehicles with asymmetric interactions (Figure 5); exploitability before and after rescaling (Table 3) Assess size-dependent equilibrium behavior; Evaluate improvement in Nash approximation Lane changing (5.2) 2D Velocity 3 vehicles, 2 lanes; two initial layouts with identical costs and target speeds (Figure 6) Compare lane changes and overtaking behavior Intersection crossing (5.3) 2D Acceleration 8 vehicles; same initial conditions, with versus without major–minor-road differentiation (Figure 7) Assess road-priority effects on yielding and crossing Dim. denotes spatial dimension. Algorithm. To minimize the α-potential function (4.2) and compute an α-NE, we parametrize player i’s policy using a sufficiently expressive neural network33 3 The neural network architecture is feedforward fully connected with one or two hidden layers. ϕiθi:[0,T]×ℝd→Ai _i _i:[0,T]×R^d→ A_i with parameters θi∈ℝLi _i∈R^L_i, and discretize the state dynamics (3.1) on time grid πP=0=t0<⋯<tP=T _P=\0=t_0<…<t_P=T\ for some P∈ℕP∈N: for any i∈[N]i∈[N], Xi,0θi=ξiX _i_i,0= _i, and for any ℓ=0,…,P−1 =0,…,P-1, Xi,tℓ+1θi X_i,t_ +1 _i =Xi,tℓθi+bi(Xi,tℓθi,ϕiθi(tℓ,Xi,tℓθi))Δℓ+σi(Xi,tℓθi,ϕiθi(tℓ,Xi,tℓθi))ΔWℓ, =X_i,t_ _i+b_i(X_i,t_ _i, _i _i(t_ ,X_i,t_ _i)) _ + _i(X_i,t_ _i, _i _i(t_ ,X_i,t_ _i)) W_ , (5.1) where Δℓ=tℓ+1−tℓ _ =t_ +1-t_ , and ΔWℓ=Wtℓ+1−Wtℓ W_ =W_t_ +1-W_t_ . We minimize the following discretized α-potential function Φ⁡(θ)≔⁡[∑ℓ=0P−1F⁡(Xtℓθ,ϕθ(tℓ,Xtℓθ))dt+G⁡(XTθ)] (θ) E [ _ =0^P-1F\! (X_t_ ^θ,φ^θ(t_ ,X_t_ ^θ) )dt+G (X_T^θ ) ] (5.2) over θ=(θi)i∈[N]θ=( _i)_i∈[N]. This is achieved by using the Adam method, where at each iteration, (5.2) is approximated by ΦM(θ) _M(θ) =1M∑m=1M[∑ℓ=0P−1F⁡(Xtℓθ,(m),ϕθ(tℓ,Xtℓθ,(m)))Δℓ+G⁡(XTθ,(m))], = 1M _m=1^M [ _ =0^P-1F(X_t_ ^θ,(m),φ^θ(t_ ,X_t_ ^θ,(m))) _ +G(X_T^θ,(m)) ], (5.3) where (Xθ,(m))m∈[M]=(Xiθi,(m))i∈[N],m∈[M] (X^θ,(m) )_m∈[M]= (X_i _i,(m) )_i∈[N],m∈[M] are M independent trajectories of (5.1). 5.1 Longitudinal control This set of experiments use a fleet of 10 AVs on a long highway (1D) or its 2D extension with a common start position (x0,i=−1x_0,i=-1 in 1D cases and (−1,−1)(-1,-1) in 2D cases) and a target position (zi=1z_i=1 in 1D cases and (1,1)(1,1) in 2D cases). 5.1.1 Interaction regimes in cost function Here we examine the impact of β in (3.5) for both velocity-control and acceleration-control models. Specifically, for each model, we test two choices β=0β=0 and β=1β=1, which characterize the weak-interaction regime and strong-interaction regime, respectively. We consider a 10-player game in AV control and assume that each player has a one-dimensional state, which follows from (3.3) under velocity control and (3.4) under acceleration control with a uniform noise level σi=0.1 _i=0.1. Each vehicle i minimizes its objective function defined in (3.2) with fi(x,ai)=0.1ai2f_i(x,a_i)=0.1a_i^2 and gi(x)=10|x−zi|2g_i(x)=10|x-z_i|^2, where zi=1z_i=1 is the target terminal position for vehicle i. Interaction kernel λij _ij and K are defined in (3.5) with ρ:ℝd→[0,∞)ρ:R^d→[0,∞) given by ρ⁡(z)=(|z|2+1)−1.ρ(z)=(|z|^2+1)^-1. Figure 1: Vehicle trajectories of the control-velocity model for β=0β=0 (left) and β=1β=1 (right) Figure 2: Vehicle trajectories of the control-acceleration model for β=0β=0 (left) and β=1β=1 (right) We observe in Figures 1 and 2 that, in the weak-interaction regime (β=0β=0), both the velocity- and acceleration-control models yield near-identical controls and hence similar trajectories, reflecting the symmetric MFG structure in which agents adopt the same feedback policy. However, in the strong-interaction regime (β=1β=1), vehicles deliberately choose different controls to give each other space, and their paths diverge. This shows that the strong-interaction regime ensures collision avoidance more effectively. In addition, one can see that under velocity-control, vehicles frequently detour (even briefly backtrack) to avoid others yet still reach the target accurately, whereas under acceleration-control, trajectories are smoother but exhibit dispersed terminal positions, indicating greater difficulty in hitting the target precisely. This happens because velocity control gives agents direct authority over position with cheap late corrections, so they can prioritize separation and fix position near the end; acceleration control introduces inertia, making sharp turns and last-minute corrections costly, so the optimizer smooths motion and tolerates small terminal errors. (a) Trajectories of 30 vehicles (b) Potential objective during training Figure 3: Vehicle trajectories and potential objective during training for the 30-vehicle control-velocity model. To further examine the scalability of the proposed approach, we consider a larger-scale experiment with 3030 vehicles under the control-velocity model with β=1β=1. Figure 3 shows both the learned trajectories and the evolution of the potential objective during training. The trajectories remain well structured despite the substantially larger number of interacting vehicles, while the potential objective decreases steadily over the course of optimization. These results indicate that the proposed decentralized policy optimization framework remains effective and computationally tractable in a larger multi-vehicle setting. 5.1.2 Two-dimensional acceleration control with obstacles This section examines vehicle behavior in the presence of circular obstacles on the road. We consider a 10-player game in two-dimensional AV control, i.e. d=2,N=10d=2,N=10. The state dynamics follows (3.4) under acceleration control without noise (σi≡0 _i≡ 0). Each vehicle i minimizes its objective function (3.2), where the terminal cost is gi(x)=2|x−zi|2g_i(x)=2|x-z_i|^2 for zi=(1,1)z_i=(1,1), and the interaction kernel λij _ij and K are defined the same as in Sec. 5.1.1 with β=1β=1. The running cost fif_i depends on the position and the size of the obstacle. Specifically, we consider two circular obstacles centered at (0,0)(0,0) with radii 0.10.1 and 0.50.5, respectively. To capture an obstacle avoidance behavior, we consider the running cost fi(x,ai)=0.02ai2+h⁡(x)f_i(x,a_i)=0.02a_i^2+h(x), with h(x)=1000(1−11+exp⁡[10(1−M|x|2)]),h(x)=1000 (1- 11+ [10(1-M|x|^2)] ), where M=100M=100 for the small obstacle of radius 0.10.1 and M=4M=4 for the large obstacle of radius 0.50.5. The cost h is a differentiable version of forbidding passage through the circular obstacles [3]. For comparison, we also consider the case without obstacles by setting h≡0h≡ 0. Figure 4: Vehicle trajectories under acceleration-control without obstacle (left), with a small obstacle (middle), and with a large obstacle (right) Figure 4 shows that introducing an obstacle causes the vehicle trajectories to adjust accordingly to satisfy obstacle-avoidance constraints. When a large obstacle is present, its wider exclusion region forces the trajectories to deviate earlier and more noticeably from their nominal paths. This results in a global reshaping of the motion patterns. In contrast, a smaller obstacle influences the trajectories only locally: only vehicles passing near its vicinity exhibit visible adjustments, while others remain close to the baseline case. These observations highlight that the trajectories respond adaptively to the obstacle, and the extent of deviation is determined by both the size and placement of the obstacle within the environment. 5.1.3 Impact of vehicle-specific rescaling for equilibrium computation This section examines the effect of asymmetric interactions and confirms that rescaling can improve the equilibrium approximation. Rescaling recovers exact NEs. We first consider a deterministic one-dimensional game with nine vehicles. We conduct this experiment under both control formulations: the vehicle dynamics follow (3.3) under velocity control and (3.4) under acceleration control, with no noise in either case. We adopt the rank-one interaction structure introduced in Remark 4, with λij=γiτj _ij= _i _j for some γi,τi>0 _i, _i>0. As shown in Remark 4, this interaction structure satisfies the Kolmogorov cyclic condition, and the rescaling pi=τi/γip_i= _i/ _i transforms the game into an exact potential game. Consequently, an exact minimizer of the rescaled potential corresponds to a Nash equilibrium. To model heterogeneous vehicle sizes, we divide the nine vehicles into three groups: large, medium, and small, with three vehicles in each group. We set τi=10 _i=10, 33, and 0.50.5 for the large, medium, and small vehicles, respectively. For each vehicle, we sample γiτi∼Unif⁡(0.8,1.2) _i _i (0.8,1.2) and determine γi _i accordingly. Thus, larger vehicles have larger τi _i and smaller γi _i, so they exert stronger influence on other vehicles while being less sensitive to the influence of others. The resulting values of γi _i and τi _i are reported in Table 2. Each vehicle starts from the common initial position x0=−1x_0=-1 and minimizes the objective function in (3.2), with fi(x,ai)=0.02ai2f_i(x,a_i)=0.02a_i^2, gi(x)=2|x−1|2g_i(x)=2|x-1|^2, and interaction kernel K⁡(z)=(N2|z|2+1)−1K(z)=(N^2|z|^2+1)^-1. The running cost penalizes control effort, while the terminal cost encourages the vehicle’s terminal position to remain close to the common target x=1x=1. Table 2: Values of γii∈[9]\ _i\_i∈[9] and τii∈[9]\ _i\_i∈[9]. Vehicle 1 2 3 4 5 6 7 8 9 γi _i 0.115 0.117 0.095 0.395 0.319 0.347 1.805 2.235 2.353 τi _i 10 10 10 3 3 3 0.5 0.5 0.5 Figure 5: Vehicle trajectories of different types under control-velocity (left) and control-acceleration (right) Figure 5 shows a consistent size-dependent ordering of the vehicle trajectories under both velocity and acceleration control. Large vehicles remain near the center of the trajectory bundle, medium vehicles are displaced farther outward, and small vehicles move toward the periphery. This pattern reflects the asymmetric interaction structure: larger vehicles exert a stronger influence on others while being less sensitive to their presence, whereas smaller vehicles are more strongly affected by the larger ones. The same qualitative ordering appears under both control formulations. Rescaling reduces the Nash gap. We further consider settings in which rescaling does not recover an exact potential game and examine how it improves the equilibrium approximation. In this experiment, the vehicle dynamics follow (3.3) under velocity control with no noise. To this end, we introduce a small multiplicative Gaussian perturbation to the rank-one interaction matrix (λij)i,j( _ij)_i,j by replacing each entry with λ~ij=λij(1+εξij) λ_ij= _ij(1+ _ij), where ξij∼i.i.d.(0,1) _ij i.i.d. N(0,1) and ε>0 >0 controls the perturbation magnitude. We choose ε sufficiently small so that all perturbed interaction coefficients remain positive in the realized instance. The perturbation breaks the rank-one structure, and the Kolmogorov cyclic condition no longer holds exactly. We then apply Algorithm 1 to optimize the rescaling parameters p=(p1,…,pN)p=(p_1,…,p_N) and minimize the corresponding upper bound on the equilibrium approximation error. To evaluate the resulting policy profile, for each vehicle i, we hold the strategies of all other vehicles fixed and compute its best-response cost JiBR=infϕi′Ji(ϕi′,ϕ−i)J_i^BR= _ _i J_i( _i , _-i). We define the exploitability as ℰi=Ji(ϕ)−JiBRE_i=J_i(φ)-J_i^BR, which measures the cost reduction that vehicle i could achieve through a unilateral deviation. At a Nash equilibrium, ℰi=0E_i=0 for every vehicle, while smaller exploitability indicates a policy profile closer to a Nash equilibrium in terms of unilateral deviations. We further report the relative exploitability ℰirel=ℰi/Ji(ϕ)×100%E_i^rel=E_i/J_i(φ)× 100\%, which normalizes the exploitability by the vehicle’s cost under the learned policy profile. Table 3: Exploitability before and after scaling. Vehicle Scaling Metric 1 2 3 4 5 6 7 8 9 Before Cost Ji(ϕ)J_i(φ) 5.336 3.774 1.458 4.974 3.830 4.736 11.168 16.183 11.832 Relative exploitability (%) 64.679 49.945 12.646 18.740 22.354 23.210 3.640 0.008 0.009 After Cost Ji(ϕ)J_i(φ) 1.637 1.712 1.642 5.258 4.497 7.148 8.930 13.070 13.290 Relative exploitability (%) 0.180 0.795 0.928 1.598 1.145 0.927 3.717 0.962 2.275 Table 3 compares the costs and relative exploitabilities before and after rescaling. Before rescaling, several vehicles have substantial incentives to deviate from the learned policy, with relative exploitabilities reaching 64.7%64.7\% and 49.9%49.9\% for Vehicles 1 and 2, respectively. After rescaling, the relative exploitabilities are substantially reduced for most vehicles and remain below 4%4\% across all vehicles, with most below 2%2\%. These results show that rescaling substantially reduces exploitability and produces a policy profile closer to a Nash equilibrium. 5.2 Lane changing In this section, we present numerical experiments on lane-changing behavior using the previously established model framework. We consider a three-vehicle lane-changing problem on a two-lane road. The position of vehicle i at time t is denoted by (xi,t,yi,t)(x_i,t,y_i,t), where xi,tx_i,t and yi,ty_i,t are its longitudinal and lateral positions, respectively. The two lanes have lateral center positions c1c_1 and c2c_2, and the road boundaries are denoted by yminy_ and ymaxy_ . Each vehicle controls its longitudinal and lateral velocities, denoted by ϕi,x,t _i,x,t and ϕi,y,t _i,y,t, and seeks to travel close to its prescribed target speed while remaining within the roadway and avoiding collisions with the other vehicles. The objective of vehicle i is Ji(ϕ)=[∫0T(fi(xi,t,yi,t,ϕi,x,t,ϕi,y,t)+∑j≠iK(xi,t−xj,t)T(yi,t−yj,t))dt].J_i(φ)=E [ _0^T (f_i (x_i,t,y_i,t, _i,x,t, _i,y,t)+ _j =iK(x_i,t-x_j,t)T(y_i,t-y_j,t) )dt ]. (5.4) The individual running cost is decomposed as fi=fi,1+fi,2f_i=f_i,1+f_i,2. The first component is fi,1(⋅,⋅,ϕx,ϕy)=q1ϕx2+q2(vides)ϕy2+q3(ϕx−vides)2,f_i,1(·,·, _x, _y)=q_1 _x^2+q_2(v_i^des) _y^2+q_3( _x-v_i^des)^2, where videsv_i^des denotes the target speed of vehicle i. The first and third terms penalize control effort and deviations from the target longitudinal speed, respectively. The second term penalizes lateral motion, with q2(z)=k1/(k2+k3z)q_2(z)=k_1/(k_2+k_3z). Since q2q_2 decreases with the target speed, a vehicle with a higher videsv_i^des incurs a smaller penalty for lateral motion and is therefore more willing to change lanes when doing so helps it maintain its preferred longitudinal speed. The second component regulates the lateral position of the vehicle and is defined by fi,2(x,y,⋅,⋅)= f_i,2(x,y,·,·)= −k4log⁡(∑j=1nexp⁡(−k5(y−cj)2)) -k_4 ( _j=1^n (-k_5(y-c_j)^2 ) ) +k6[(log⁡(1+exp⁡(−k7(y−ymin))))2+(log⁡(1+exp⁡(−k7(y−ymax))))2]. +k_6 [ ( (1+ (-k_7(y-y_ ) ) ) )^2+ ( (1+ (-k_7(y-y_ ) ) ) )^2 ]. Here n=2n=2 is the number of lanes, and cjc_j denotes the lateral center of lane j. The first term smoothly penalizes deviations from the nearest lane center, thereby encouraging vehicles to travel near a lane center during normal driving. The second term penalizes departures from the roadway through soft penalties near the lower and upper road boundaries. The pairwise interaction term in (5.4) penalizes vehicle configurations that may lead to collisions. Specifically, K⁡(z)=(N2|z|2+1)−1K(z)= (N^2|z|^2+1 )^-1 captures the effect of longitudinal separation, with the interaction becoming stronger as two vehicles move closer along the direction of travel, while T⁡(z)=k8[tanh⁡(k9(dlat2−z2))+1]T(z)=k_8 [ (k_9(d_lat^2-z^2) )+1 ] acts as a lateral weighting function that largely suppresses the interaction when the vehicles are separated across different lanes. Here dlatd_lat is chosen slightly below the lane width, so that vehicles in the same lane interact strongly, whereas the interaction rapidly decays as their lateral separation approaches the spacing between adjacent lane centers. Consequently, K⁡(xi,t−xj,t)T(yi,t−yj,t)K(x_i,t-x_j,t)T(y_i,t-y_j,t) induces a strong penalty when two vehicles are longitudinally close and also occupy nearby lateral positions, thereby promoting collision avoidance. At the same time, vehicles traveling in different lanes have negligible interaction even when their longitudinal positions are close. Table 4 summarizes the model parameters, initial vehicle positions, and target velocities used in the experiments. Table 4: Parameter values and initial conditions for the lane-changing experiments in Section 5.2. Parameter k1k_1 k2k_2 k3k_3 k4k_4 k5k_5 k6k_6 k7k_7 k8k_8 k9k_9 q1q_1 q3q_3 c1c_1 c2c_2 dlatd_lat Value 10.0 0.1 1.0 8.0 100.0 20.0 1.0 5.0 10.0 0.1 50 0.5 1.5 0.9 Vehicle 11 22 33 Initial position in Figure 6 (a) (−4.5,0.5)(-4.5,0.5) (−6.5,0.5)(-6.5,0.5) (0.5,1.5)(0.5,1.5) Initial position in Figure 6 (b) (−3.0,0.5)(-3.0,0.5) (−10.5,0.5)(-10.5,0.5) (−3.5,1.5)(-3.5,1.5) Target velocities in Figure 6 (a) and (b) 2.02.0 6.06.0 3.03.0 We use this model to examine how different initial traffic configurations can lead to distinct equilibrium lane-changing behaviors. The two scenarios use the same cost functions, cost parameters, and vehicle-specific target speeds, and differ only in the initial vehicle positions. In both scenarios, the orange vehicle has the highest target speed, followed by the green and blue vehicles. Initially, the orange vehicle travels in the right lane behind the slower blue vehicle, while the green vehicle travels in the left lane. The experiments therefore examine how the faster orange vehicle overtakes the blue vehicle while strategically interacting with the green vehicle in the adjacent lane. The behaviors are shown in Figures 6(6(a)) and 6(6(b)), respectively. (a) Standard overtaking behavior (b) Alternative equilibrium lane-changing behavior Figure 6: Lane changing in velocity control Under the first initial configuration, the relative vehicle positions allow the orange vehicle to perform a standard overtaking maneuver. It first changes from the right lane to the left lane, passes the slower blue vehicle, and then returns to the right lane. The initial spacing leaves sufficient room for the orange vehicle to complete the return maneuver without incurring a large interaction cost. The second initial configuration leads to a qualitatively different equilibrium. The orange vehicle again changes to the left lane to overtake the blue vehicle. However, after passing the blue vehicle, returning to the right lane would place the orange vehicle too close to the blue vehicle, creating a high collision risk and therefore a large interaction cost. The orange vehicle therefore remains in the left lane. Because the green vehicle has a lower target speed than the orange vehicle, the interaction cost of remaining in the left lane increases as the orange vehicle approaches from behind. In equilibrium, this interaction cost exceeds the cost of changing lanes, so the green vehicle moves to the right lane, allowing the two vehicles to maintain sufficient separation. 5.3 Unsignalized intersection crossing This section examines vehicle behavior at an unsignalized intersection. We consider an eight-vehicle setting in which each vehicle follows a prescribed travel direction and passes through the intersection. Each vehicle is encouraged to maintain its prescribed target speed while avoiding collisions with other vehicles. The control variable is the acceleration, and vehicle i minimizes the objective function Ji(ϕ)=⁡[∫0T(fi(xi,t,yi,t,vi,t,ϕi,t)+∑j≠iC⁡(xi,t,yi,t,xj,t,yj,t))t].J_i(φ)=E [ _0^T (f_i(x_i,t,y_i,t,v_i,t, _i,t)+ _j≠ iC(x_i,t,y_i,t,x_j,t,y_j,t) )dt ]. (5.5) Here (xi,t,yi,t)(x_i,t,y_i,t) denotes the position of vehicle i at time t, while vi,tv_i,t and ϕi,t _i,t denote its velocity and acceleration along its assigned travel direction, respectively. The running cost is given by fi(⋅,⋅,v,ϕ)=q4ϕ2+q5i(v−vides)2+q6(log⁡(1+exp⁡(−q7v)))2.f_i(·,·,v,φ)=q_4φ^2+q_5^i(v-v_i^des)^2+q_6 ( (1+ (-q_7v ) ) )^2. The second term penalizes deviations from the target speed videsv_i^des, with a vehicle-dependent coefficient q5iq_5^i. The third term smoothly penalizes negative velocities and thereby discourages reverse motion. The interaction takes the form C(xi,yi,xj,yj)=q8e−rij2/2+q9(log(1+exp(q10(rs−rij))))2,C(x_i,y_i,x_j,y_j)=q_8e^-r_ij^2/2+q_9 ( (1+ (q_10(r_s-r_ij) ) ) )^2, where rij=(xi−xj)2+(yi−yj)2.r_ij= (x_i-x_j)^2+(y_i-y_j)^2. The first term penalizes close vehicle configurations, with the interaction cost increasing smoothly as the distance between two vehicles decreases. The second term is a smooth safety-distance penalty that becomes active when the inter-vehicle distance falls below the prescribed separation threshold rsr_s. This term is particularly important in realistic traffic settings, where vehicles occupy physical space rather than being treated as point masses. Table 5 reports the model parameters, initial conditions, target speeds, and road-dependent speed-tracking coefficients used in the experiments. Table 5: Parameter values and initial conditions for the unsignalized intersection-crossing experiments. Parameter q4q_4 q6q_6 q7q_7 q8q_8 q9q_9 q10q_10 Value 0.1 1.0 10.0 10.0 5.0 10.0 Road type Position Initial position Initial speed Target speed Speed-tracking coefficient q5iq_5^i Major Lead (↑ ) (1.0,−1.5)(1.0,-1.5) 1.5 2.0 300.0 Major Lead (↓ ) (−1.0,1.5)(-1.0,1.5) 1.5 2.0 300.0 Major Follow (↑ ) (1.0,−3.0)(1.0,-3.0) 1.5 2.0 300.0 Major Follow (↓ ) (−1.0,3.0)(-1.0,3.0) 1.5 2.0 300.0 Minor Lead (→) (−1.5,−0.5)(-1.5,-0.5) 1.5 2.0 10.0 Minor Lead (←) (1.5,0.5)(1.5,0.5) 1.5 2.0 10.0 Minor Follow (→) (−3.0,−0.5)(-3.0,-0.5) 1.5 2.0 10.0 Minor Follow (←) (3.0,0.5)(3.0,0.5) 1.5 2.0 10.0 With major–minor-road differentiation (a) t=0.5t=0.5 (b) t=1.0t=1.0 (c) t=1.5t=1.5 (d) t=2.0t=2.0 Without major–minor-road differentiation (e) t=0.5t=0.5 (f) t=1.0t=1.0 (g) t=1.5t=1.5 (h) t=2.0t=2.0 Figure 7: Comparison of the intersection-crossing dynamics at four representative time instants. The top row shows the major–minor-road scenario, while the bottom row shows the scenario without major–minor-road differentiation. Vehicle colors indicate deviations from their target speeds. We compare the equilibrium intersection-crossing behaviors with and without major–minor-road differentiation under identical initial conditions. In the major–minor-road setting, the vertical road is designated as the major road and the horizontal road as the minor road, with vehicles on the minor road expected to yield to those on the major road. This priority structure is encoded through the vehicle-dependent speed-tracking coefficient q5iq_5^i: vehicles on the major road are assigned a larger q5iq_5^i, making deviations from their target speeds more costly, whereas vehicles on the minor road are assigned a smaller q5iq_5^i, allowing them to decelerate more readily when resolving potential conflicts. In the setting without major–minor-road differentiation, the same speed-tracking penalty is used across roads. In both scenarios, the vehicles start from the same prescribed positions, with initial speeds slightly below their respective target speed. Figure 7 shows four representative snapshots from each scenario. Vehicle colors indicate deviations from the corresponding target speeds: blue represents speeds below the target, red represents speeds above the target. Colors closer to white indicate smaller deviations, while darker shades correspond to larger deviations in magnitude. In the major–minor-road scenario, a clear priority-based crossing pattern emerges. Vehicles traveling on the major road exhibit similar behaviors and maintain speeds close to their target values while passing through the intersection. Among the minor-road vehicles, those that have already entered or are approaching the conflict region accelerate to clear the intersection and appear in red, indicating speeds above their target values. In contrast, the following vehicles decelerate and wait for the major-road vehicles to pass, appearing in dark blue as their speeds fall substantially below their targets. Once the major-road vehicles have cleared the intersection, the waiting vehicles accelerate and proceed through the intersection. Without major–minor-road differentiation, no predetermined road hierarchy governs the crossing order. Instead, vehicles resolve potential conflicts through their acceleration and deceleration decisions, with leading vehicles proceeding first and following vehicles yielding as needed. The resulting crossing pattern is therefore determined by the vehicles’ relative positions and strategic interactions rather than by road priority. 6 Proofs We provide in this section the proofs of the main results established throughout the paper. Proof of Proposition 4.2 For all i∈[N]i∈[N], define a new objective of player i: Jis(ϕCLOSE J^s_i(φ )=[∫0T(fi(Xϕi,t,ϕi(t,Xϕi,t))+∑j≠iλij+λji2K(Xϕi,t−Xϕj,t))dt+gi(Xϕi,T)]. )=E [ _0^T (f_i (X _i_i,t, _i(t,X _i_i,t) )+ _j =i _ij+ _ji2K(X _i_i,t-X _j_j,t) )dt+g_i(X _i_i,T) ]. (6.1) By [19, Theorem 3.1], the game with the objectives (Jis)i∈[N](J^s_i)_i∈[N] is a potential game with the potential function Φ defined in (4.2), which implies Φ⁡(ϕi,ϕ−i)−Φ⁡(ϕi′,ϕ−i)=Jis(ϕi,ϕ−i)−Jis(ϕi′,ϕ−i). ( _i, _-i)- ( _i , _-i)=J_i^s( _i, _-i)-J^s_i( _i , _-i). Note that |Jis(ϕ)−Ji(ϕ)|≤ |J_i^s(φ)-J_i(φ)|≤ |∫0T∑j≠i(12(λij+λji)−λij)K(Xi,tϕi−Xj,tϕj)t|≤T‖K‖L∞2∑i≠j|λij−λji|. E | _0^T _j≠ i ( 12( _ij+ _ji)- _ij )K(X _i_i,t-X _j_j,t)dt |≤ T\|K\|_L^∞2 _i≠ j| _ij- _ji|. The desired conclusion then follows from |(Ji(ϕi,ϕ−i)−Ji(ϕi′,ϕ−i))−(Φ⁡(ϕi,ϕ−i)−Φ⁡(ϕi′,ϕ−i))| | (J_i( _i, _-i)-J_i( _i , _-i) )- ( ( _i, _-i)- ( _i , _-i) ) | = = |(Ji(ϕi,ϕ−i)−Ji(ϕi′,ϕ−i))−(Jis(ϕi,ϕ−i)−Jis(ϕi′,ϕ−i))| | (J_i( _i, _-i)-J_i( _i , _-i) )- (J_i^s( _i, _-i)-J^s_i( _i , _-i) ) | ≤ ≤ |Ji(ϕi,ϕ−i)−Jis(ϕi,ϕ−i)|+|Ji(ϕi′,ϕ−i)−Jis(ϕi′,ϕ−i)|≤T‖K‖L∞∑i≠j|λij−λji|. |J_i( _i, _-i)-J_i^s( _i, _-i) |+ |J_i( _i , _-i)-J^s_i( _i , _-i) |≤ T\|K\|_L^∞ _i≠ j| _ij- _ji|. □ Proof of Theorem 4.3 By Proposition 4.2, Φp ^p satisfies for all i∈[N]i∈[N], ϕ−i∈Π−i _-i∈ _-i, and ϕi,ϕi′∈Πi _i,φ _i∈ _i, |(piJi(ϕi′,ϕ−i)−piJi(ϕi,ϕ−i))−(Φp(ϕi′,ϕ−i)−Φp(ϕi,ϕ−i))|≤α~i(p),|(p_iJ_i(φ _i, _-i)-p_iJ_i( _i, _-i))-( ^p(φ _i, _-i)- ^p( _i, _-i))|≤ α_i(p), with α~i(p)=T‖K‖L∞∑j≠i|piλij−pjλji|. α_i(p)=T\|K\|_L^∞ _j≠ i|p_i _ij-p_j _ji|. Hence a minimizer ϕφ of Φp ^p satisfies for all i∈[N]i∈[N] and ϕi′∈Πφ _i∈ , Ji(ϕi′,ϕ−i)−Ji(ϕi,ϕ−i)≥1pi(Φp(ϕi′,ϕ−i)−Φp(ϕi,ϕ−i))−1piα~i(p)≥−1piα~i(p)≥α⁡(p),J_i(φ _i, _-i)-J_i( _i, _-i)≥ 1p_i( ^p(φ _i, _-i)- ^p( _i, _-i))- 1p_i α_i(p)≥- 1p_i α_i(p)≥α(p), which implies that ϕφ is an α⁡(p)α(p)-NE of the game (3.1)-(3.2). We proceed to show that Φp ^p admits a minimizer. Since Πi _i is compact, Π equipped with the product topology is also compact in the compact-open topology. It suffices to show that Φp ^p is continuous. The extreme value theorem then guarantees that Φp ^p admits a minimizer on Π . To show the continuity of Φp ^p, consider a sequence (ϕn)n∈ℕ⊂Π(φ^n)_n∈N⊂ with (ϕin)i∈[N](φ^n_i)_i∈[N] converging to some ϕ∞∈Πφ^∞∈ in the compact-open topology. In the sequel, we write Xin=XiϕinX^n_i=X^φ^n_i_i and ΔXin=Xiϕin−Xiϕi∞ X^n_i=X^φ^n_i_i-X^φ^∞_i_i for all n∈ℕ∪∞n∈N∪\∞\ and i∈[N]i∈[N], and denote by C≥0C≥ 0 a generic constant independent of n∈ℕ∪∞n∈N∪\∞\ and i∈[N]i∈[N], which may take different values in each line. Since A=∏i∈[N]AiA= _i∈[N]A_i is compact, (ϕn)n∈ℕ∪∞(φ^n)_n∈N∪\∞\ is uniformly bounded. The standard moment estimate in [55, Theorem 3.4.3] implies that for all n∈ℕn∈N and i∈[N]i∈[N], ⁡[supt∈[0,T]|Xi,tn|4]≤CE[ _t∈[0,T]|X^n_i,t|^4]≤ C. For each n∈ℕn∈N and R>0R>0, define the stopping time τin,R≔inft∈[0,T]∣|Xtn|>R or |Xt∞|>R∧T. _i^n,R \t∈[0,T] |X_t^n|>R or |X_t^∞|>R\ T. By the stability estimate [55, Theorem 3.2.4] and the locally Lipschitz continuity of ϕ∞φ^∞, ⁡[supt∈[0,T]|ΔXi,t∧τin,Rn|2] E[ _t∈[0,T]| X^n_i,t _i^n,R|^2] ≤C⁡(⁡[∫0τin,R|bi(Xi,tn,ϕin(t,Xi,tn))−bi(Xi,tn,ϕi∞(t,Xi,tn))|2t]CLOSE ≤ C (E [ _0 _i^n,R|b_i(X^n_i,t,φ^n_i(t,X^n_i,t))-b_i(X^n_i,t,φ^∞_i(t,X^n_i,t))|^2dt ] OPEN+⁡[∫0τin,R|σi(Xi,tn,ϕin(t,Xi,tn))−σi(Xi,tn,ϕi∞(t,Xi,tn))|2t]) +E [ _0 _i^n,R| _i(X^n_i,t,φ^n_i(t,X^n_i,t))- _i(X^n_i,t,φ^∞_i(t,X^n_i,t))|^2dt ] ) ≤C[∫0τin,R|ϕin(t,Xi,tn))−ϕi∞(t,Xi,tn)|2dt], ≤ CE [ _0 _i^n,R|φ^n_i(t,X^n_i,t))-φ^∞_i(t,X^n_i,t)|^2dt ], which converges to zero as n→∞n→∞, due to the the convergence of (ϕn)n∈ℕ(φ^n)_n∈N on compact sets. Then by the Cauchy-Schwarz inequality, ⁡[supt∈[0,T]|ΔXi,tn|2] E[ _t∈[0,T]| X^n_i,t|^2] =⁡[supt∈[0,T]|ΔXi,tn|21τin,R=T]+⁡[supt∈[0,T]|ΔXi,tn|21τin,R<T] =E[ _t∈[0,T]| X^n_i,t|^21_ _i^n,R=T]+E[ _t∈[0,T]| X^n_i,t|^21_ _i^n,R<T] ≤⁡[supt∈[0,T]|ΔXi,τin,Rn|2]+[supt∈[0,T]|ΔXi,tn|4]1/2P(τin,R<T)1/2. ≤E[ _t∈[0,T]| X^n_i, _i^n,R|^2]+E[ _t∈[0,T]| X^n_i,t|^4]^1/2P( _i^n,R<T)^1/2. Since τin,R<T⊂supt∈[0,T]|Xtn|≥R∪supt∈[0,T]|Xt∞|≥R\ _i^n,R<T\⊂ \ _t∈[0,T]|X_t^n|≥ R \∪ \ _t∈[0,T]|X_t^∞|≥ R \, by Markov’s inequality and the uniform moment bounds of (Xin)n∈ℕ(X^n_i)_n∈N, we have limn→∞⁡[supt∈[0,T]|Xi,tn−Xi,t∞|2]=0 _n→∞E[ _t∈[0,T]|X^n_i,t-X^∞_i,t|^2]=0. Similar arguments show that limn→∞supt∈[0,T]⁡[|ϕin(t,Xi,tn)−ϕi∞(t,Xi,t∞)|2]=0 _n→∞ _t∈[0,T]E[|φ^n_i(t,X^n_i,t)-φ^∞_i(t,X^∞_i,t)|^2]=0. Hence, possibly after passing to a subsequence, (Xi,tn,ϕin(t,Xi,tn))n∈ℕ(X^n_i,t,φ^n_i(t,X^n_i,t))_n∈N also converges to (Xi,t∞,ϕi∞(t,Xi,t∞))(X^∞_i,t,φ^∞_i(t,X^∞_i,t)) dℙ⊗dtdP dt-a.e., and (Xi,Tn)n∈ℕ(X^n_i,T)_n∈N converges to Xi,T∞X^∞_i,T a.s. Let F:ℝdN×∏i∈[N]Ai→ℝF:R^dN× _i∈[N]A_i→R and G:ℝdN→ℝG:R^dN→R be continuous functions with at most quadratic growth. Then (F⁡(Xtn,ϕn(t,Xtn)))n∈ℕ(F (X_t^n,φ^n(t,X_t^n) ))_n∈N converges to F⁡(Xt∞,ϕ∞(t,Xt∞))F (X_t^∞,φ^∞(t,X_t^∞) ) dℙ⊗dtdP dt-a.e., and (G⁡(XTn))n∈ℕ(G (X_T^n ))_n∈N converges to G⁡(XT∞)G (X_T^∞ ) a.s. The quadratic growth of F and G, the L2L^2 convergence of (Xi,tn,ϕin(t,Xi,tn))n∈ℕ(X^n_i,t,φ^n_i(t,X^n_i,t))_n∈N, and Vitali’s theorem imply limn→∞⁡[∫0TF⁡(Xtn,ϕn(t,Xtn))t+G⁡(XTn)]=⁡[∫0TF⁡(Xt∞,ϕ∞(t,Xt∞))t+G⁡(XT∞)]. _n→∞E [ _0^TF (X_t^n,φ^n(t,X_t^n) )dt+G (X_T^n ) ]=E [ _0^TF (X_t^∞,φ^∞(t,X_t^∞) )dt+G (X_T^∞ ) ]. Applying this to F=FpF=F^p and G=GpG=G^p in (4.4) yields the desired continuity of Φp ^p and finishes the proof. □ Proof of Theorem 4.4 Since there are no interactions between distinct components, the objective in (4.5) is invariant under multiplying all pip_i, i∈Ski∈ S_k, by the same positive constant, independently for each k. Hence, without loss of generality, we may impose ∑i∈Skpi=|Sk|,k=1,…,m. _i∈ S_kp_i=|S_k|, k=1,…,m. Introducing the epigraph variable r then gives (P), initially with p∈(0,∞)Np∈(0,∞)^N and r≥0r≥ 0. We next show that the strict positivity constraint can be relaxed to p∈[0,∞)Np∈[0,∞)^N. Suppose that a feasible solution satisfies pi=0p_i=0 for some i∈Ski∈ S_k. The i-th constraint gives ∑j≠ipjλji=0 _j≠ ip_j _ji=0. Hence pj=0p_j=0 for every j such that λji>0 _ji>0, i.e., for every out-neighbor j of i in λG_λ. Repeating this argument along directed paths starting from i and using the strong connectivity of the subgraph induced by SkS_k, we obtain pj=0p_j=0 for any j∈Skj∈ S_k. This contradicts ∑j∈Skpj=|Sk|>0. _j∈ S_kp_j=|S_k|>0. Thus every feasible solution of (P) satisfies pi>0p_i>0 for all i. Finally, the componentwise normalizations imply 0≤pi≤|Sk|0≤ p_i≤|S_k|, for all i∈Ski∈ S_k. Also, choosing pi=1p_i=1 for all i is feasible with r=r0r=r_0, and hence r⋆≤r0r ≤ r_0. Therefore one may restrict r to [0,r0][0,r_0] without changing the optimal value. The feasible set of (P) is then closed and bounded, and hence compact. The minimum is therefore attained, and, by the preceding argument, any optimizer belongs to (0,∞)N(0,∞)^N. □ Proof of Theorem 4.5 The equivalence between the Kolmogorov cyclic condition and the existence of p∈(0,∞)Np∈(0,∞)^N satisfying (4.6) follows from the standard characterization of diagonally symmetrizable matrices; see, e.g., [37, Proposition 4.15 and Lemma 4.12]. Suppose first that (4.6) holds. Then, for every i∈[N]i∈[N], ∑j≠i|piλij−pjλji|=0, _j≠ i|p_i _ij-p_j _ji|=0, and hence r⋆=0r =0. Conversely, suppose that r⋆=0r =0. Then there exists a sequence (pn)n≥1⊂(0,∞)N(p^n)_n≥ 1⊂(0,∞)^N such that r⁡(pn)≔maxi∈[N]⁡1pin∑j≠i|pinλij−pjnλji|⟶0.r(p^n) _i∈[N] 1p_i^n _j≠ i|p_i^n _ij-p_j^n _ji| 0. For every i≠ji≠ j, |λij−pjnpinλji|≤r⁡(pn). | _ij- p_j^np_i^n _ji |≤ r(p^n). Thus, if λji=0 _ji=0, then λij=0 _ij=0; interchanging i and j shows that the zero pattern is symmetric. Now consider any interacting pair i≠ji≠ j, namely λij+λji>0 _ij+ _ji>0. By the symmetric zero pattern, λij,λji>0 _ij, _ji>0, and therefore limn→∞pjnpin=λijλji. _n→∞ p_j^np_i^n= _ij _ji. Hence, along any cycle i1,…,im,im+1=i1i_1,…,i_m,i_m+1=i_1 with positive interaction weights, 1=∏k=1mpik+1npikn⟶∏k=1mλikik+1λik+1ik,n→∞,1= _k=1^m p_i_k+1^np_i_k^n _k=1^m _i_ki_k+1 _i_k+1i_k, n→∞, which yields (4.7). If a cycle contains a zero interaction coefficient, then the symmetric zero pattern implies that the corresponding reverse coefficient is also zero, so both sides of (4.7) vanish. Thus the Kolmogorov cyclic condition holds. Finally, let p⋆∈(0,∞)Np ∈(0,∞)^N satisfy (4.6). Then maxi∈[N]⁡1pi⋆∑j≠i|pi⋆λij−pj⋆λji|=0. _i∈[N] 1p_i _j≠ i|p_i _ij-p_j _ji|=0. Therefore Theorem 4.3 gives α⁡(p⋆)=0α(p )=0, and every minimizer of Φp⋆ ^p is an NE of (3.1)–(3.2). □ Proof of Proposition 4.6 After n bisection iterations, the interval [rlow,rhigh][r_ low,r_ high] contains r⋆r and has length r0/2nr_0/2^n. Since the algorithm returns rn=rhighr^n=r_ high, 0≤rn−r⋆≤r02n0≤ r^n-r ≤ r_02^n. Moreover, pnp^n is feasible for rnr^n, and hence r⋆≤maxi∈[N]⁡1pin∑j≠i|pinλij−pjnλji|≤rn.r ≤ _i∈[N] 1p_i^n _j≠ i|p_i^n _ij-p_j^n _ji|≤ r^n. Multiplying by T‖K‖L∞T\|K\|_L^∞ gives the bound for α⁡(pn)−α⋆α(p^n)-α . □ Proof of Proposition 4.7 Under Assumption 1, the existence of ϕp∈arg⁡minϕ∈ΠΦp(ϕ)φ^p∈ _φ∈ ^p(φ) follows from Theorem 4.3. By the nonnegativity of the cost components Ci(ϕ)C_i(φ) and ij(ϕ)K_ij(φ), the definitions of γ¯(p) γ(p) and γ¯(p) γ(p) yield γ¯(p)SC⁡(ϕ)≤Φp(ϕ)≤γ¯(p)SC⁡(ϕ),∀ϕ∈Π. γ(p)SC(φ)≤ ^p(φ)≤ γ(p)SC(φ), ∀φ∈ . (6.2) Under Assumption 1, an argument similar to the proof of Theorem 4.3 shows that the social cost admits a minimizer. Let ϕopt∈arg⁡minϕ∈ΠSC⁡(ϕ)φ opt∈ _φ∈ SC(φ). Using (6.2) and the optimality of ϕpφ^p for Φp ^p gives SC⁡(ϕp)≤1γ¯(p)Φp(ϕp)≤1γ¯(p)Φp(ϕopt)≤γ¯(p)γ¯(p)SC⁡(ϕopt).SC(φ^p)≤ 1 γ(p) ^p(φ^p)≤ 1 γ(p) ^p(φ opt)≤ γ(p) γ(p)SC(φ opt). Dividing by SCopt>0SC opt>0 and using Definition 4.1, proves (4.14). □ Proof of Corollary 4.8 By Theorem 4.5, the rescaled potential with rescale parameter p is an exact potential of the rescaled game, and any of its global minimizers is a Nash equilibrium of the original game. Hence PoS≤InEff⁡(ϕp)PoS (φ^p). For every pair satisfying λij+λji>0 _ij+ _ji>0, (4.6) gives piλij+pjλji2(λij+λji)=pipjpi+pj≤min⁡pi,pj. p_i _ij+p_j _ji2( _ij+ _ji)= p_ip_jp_i+p_j≤ \p_i,p_j\. It follows that γ¯(p)=maxi∈[N]⁡pi, γ(p)= _i∈[N]p_i, and γ¯(p)≥min1≤i<j≤N⁡pipjpi+pj. γ(p)≥ _1≤ i<j≤ N p_ip_jp_i+p_j. Therefore, Proposition 4.7 yields the first inequality in (4.17). Furthermore, pipjpi+pj≥12min⁡pi,pj≥12mink∈[N]pk. p_ip_jp_i+p_j≥ 12 \p_i,p_j\≥ 12 _k∈[N]p_k. which gives the second inequality in (4.17). If the interaction coefficients are symmetric, one can take pi=1p_i=1 for all i∈[N]i∈[N] and obtain PoS≤2PoS≤ 2. □ 7 Conclusions and Future Work We develop an α-potential game framework for decentralized decision-making of heterogeneous autonomous vehicles, reducing approximate NE computation to potential minimization. We introduce player-specific rescaling to sharpen the equilibrium approximation and under suitable conditions, recover exact NE despite asymmetric interactions. We further derive social-efficiency guarantees relative to centralized full cooperation. We implement the framework using a neural-network policy-gradient method and demonstrate its performance across collision and obstacle avoidance, lane changing, and intersection crossing. Numerical results highlight the ability of the framework to capture heterogeneous behavior and strong local interactions, and show that rescaling can substantially reduce exploitability in asymmetric games. Future work will address model uncertainty and data-driven learning, particularly how errors in estimated dynamics and interaction parameters affect equilibrium accuracy and social efficiency. Further priorities include convergence analysis of the policy-gradient algorithms and systematic comparisons with HJB-based approaches. Additional directions include equilibrium selection when multiple equilibria exist, extensions to policies using local neighborhood information, and the incorporation of explicit safety constraints. Validation in larger, more realistic traffic environments will further assess scalability and practical applicability. Acknowledgments The corresponding author acknowledges support from the National Science Foundation under the award number CMMI-1943998. Anran Hu acknowledges financial support by InnoHK initiative, The Government of the HKSAR and the AIFT Lab. References [1] S. V. Albrecht, F. Christianos, and L. 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