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Agent-Based Evolutionary Dynamics for Mixed Autonomy Weaving Ramps
Sheryl Paul, Kexin Wang, Ruolin Li, Jyotirmoy V. Deshmukh
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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 94%
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Summary
This paper develops an agent-based model (ABM) to study mixed-autonomy traffic on weaving ramps, bridging macroscopic equilibrium theory with decentralized vehicle dynamics. The authors use evolutionary game theory, specifically replicator dynamics, to model how individual Connected and Automated Vehicles (CAVs) with varying levels of altruism (Social Value Orientation) adapt their lane choices. The study proves that these decentralized dynamics converge to the unique Wardrop equilibrium predicted by macroscopic models. It further analyzes the impact of finite populations, heterogeneous altruism, imperfect information, and demand changes, showing that the ABM reproduces aggregate predictions while revealing deployment-level phenomena like convergence rates and burden distribution.
Entities (8)
Relation Signals (6)
Agent-based model → appliesto → Weaving Ramps
confidence 98% · We develop an agent-based model of a macroscopic weaving-ramp framework
Agent-based model → models → Connected and Automated Vehicles
confidence 98% · We develop an agent-based model... in which individual vehicles adapt their lane choices... CAVs have θi>0
Agent-based model → convergesto → Wardrop Equilibrium
confidence 97% · We prove convergence of the decentralized dynamics to the unique equilibrium predicted by the macroscopic theory.
Connected and Automated Vehicles → characterizedby → Social Value Orientation
confidence 96% · CAVs can be programmed to account for broader traffic objectives... SVO represents an agent’s relative concern
Human-Driven Vehicles → contrastedwith → Connected and Automated Vehicles
confidence 95% · Unlike human-driven vehicles (HDVs)... CAVs can be programmed to account for broader traffic objectives
Agent-based model → uses → Replicator Dynamics
confidence 95% · individual vehicles adapt their lane choices using an evolutionary game-theoretic update rule... discrete-time replicator rule
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Abstract
Abstract:Existing models of mixed-autonomy weaving ramps characterize how altruistic connected and automated vehicles (CAVs) can improve traffic efficiency at the population level, but provide limited insight into how such behavior emerges from decentralized vehicle interactions or how it is affected by finite populations, heterogeneous preferences, and imperfect information. We develop an agent-based model of a macroscopic weaving-ramp framework in which individual vehicles adapt their lane choices using an evolutionary game-theoretic update rule and altruism-based objectives providing a microscopic interpretation of the original Wardrop model. We prove convergence of the decentralized dynamics to the unique equilibrium predicted by the macroscopic theory. Beyond reproducing aggregate equilibrium behavior, the framework enables the study of deployment-level questions that cannot be addressed by static analysis. Simulation results demonstrate close agreement with the macroscopic predictions while revealing how convergence rates, adaptation to changing traffic conditions, heterogeneous altruism levels among CAVs, and imperfect state information influence system performance and the distribution of altruistic burden across vehicles. These results provide a bridge between equilibrium traffic theory and decentralized mixed-autonomy deployment.
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- Source: https://arxiv.org/abs/2609.36424v1
- Canonical: https://arxiv.org/abs/2609.36424v1
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Agent-Based Evolutionary Dynamics for Mixed Autonomy Weaving Ramps Sheryl Paul Kexin Wang Ruolin Li Jyotirmoy V. Deshmukh Address: University of Southern California, Los Angeles, CA, USA. sherylpa, kwang255, ruolinl, jdeshmuk@usc.edu Abstract Existing models of mixed-autonomy weaving ramps characterize how altruistic connected and automated vehicles (CAVs) can improve traffic efficiency at the population level, but provide limited insight into how such behavior emerges from decentralized vehicle interactions or how it is affected by finite populations, heterogeneous preferences, and imperfect information. We develop an agent-based model of a macroscopic weaving-ramp framework in which individual vehicles adapt their lane choices using an evolutionary game-theoretic update rule and altruism-based objectives providing a microscopic interpretation of the original Wardrop model. We prove convergence of the decentralized dynamics to the unique equilibrium predicted by the macroscopic theory. Beyond reproducing aggregate equilibrium behavior, the framework enables the study of deployment-level questions that cannot be addressed by static analysis. Simulation results demonstrate close agreement with the macroscopic predictions while revealing how convergence rates, adaptation to changing traffic conditions, heterogeneous altruism levels among CAVs, and imperfect state information influence system performance and the distribution of altruistic burden across vehicles. These results provide a bridge between equilibrium traffic theory and decentralized mixed-autonomy deployment. keywords mixed autonomy, weaving ramps, lane choice, evolutionary game theory, agent-based model, social value orientation. 1 Introduction Freeway weaving sections, where entering and exiting flows must cross within a short roadway segment, remain one of the most persistent bottlenecks on highways Cassidy and others (2005). Recent advances in connected and automated vehicles (CAVs) create an opportunity to influence traffic conditions through lane-choice behavior Stern and others (2018). Unlike human-driven vehicles (HDVs), whose decisions emerge from individual preferences, CAVs can be programmed to account for broader traffic objectives while remaining decentralized. Vehicle-level traffic models examine how individual interactions and control decisions affect lane changing, stability, and throughput, while Social Value Orientation (SVO)-based models describe how CAVs balance private and collective costs Talebpour and others (2016); Schwarting and others (2019). A complementary evolutionary-game-theoretic literature studies how strategy shares change over time, from evolutionarily stable strategies and replicator dynamics to traffic-assignment processes that converge to Wardrop equilibria Maynard Smith and others (1973); Taylor and others (1978); Smith (1984). At the macroscopic scale, recent work has developed a game-theoretic model of lane choice at weaving ramps He and others (2025); Wang and others (2026). The model represents HDVs through a Wardrop equilibrium and incorporates SVO into CAV objectives, allowing CAVs to trade off private delay against aggregate congestion. It characterizes when altruistic CAVs improve traffic efficiency, derives closed-form penetration thresholds, and quantifies the achievable reduction in social delay. Thus, the mixed-autonomy weaving problem is analytically characterized at equilibrium. The two modeling perspectives nevertheless leave a complementary gap. The macroscopic theory characterizes equilibrium lane-choice fractions but does not provide a decentralized vehicle-level mechanism for reaching them. Equilibrium existence also does not guarantee convergence under local learning; adjustment may be slow, oscillatory, or unsuccessful Hart and others (2003); Sandholm (2010). Moreover, the static formulation cannot capture adaptation to changing demand, determine whether finite heterogeneous fleets reproduce mean-field predictions, or explain how the altruistic burden is distributed across vehicles. It also assumes exact traffic information, whereas deployed CAVs rely on noisy local observations and imperfect state estimates. On the other hand, the aforementioned vehicle-level models were not designed to reproduce the calibrated cost structure and penetration thresholds of such a weaving-ramp model that incorporates CAVs as well as HDVs. They therefore do not establish whether finite-agent adaptation recovers its aggregate predictions under realistic deployment constraints. These limitations motivate the central question: Can the equilibrium prescriptions of the macroscopic theory be realized by a finite population of decentralized vehicles, and if so, what behavior emerges under adaptation, heterogeneous altruism, and imperfect information? This paper answers the question by treating the calibrated macroscopic model of He and others (2025); Wang and others (2026) as a specification and developing a new agent-based model (ABM) of it. The equilibrium cost structure and penetration benchmarks are inherited from those papers; the vehicle-level dynamics, convergence analysis, and deployment studies below are novel. Contributions: (i) We develop a decentralized realization of the macroscopic weaving-ramp model using replicator-based adaptation from evolutionary game theory Weibull (1995); Sandholm (2010). We show that the resulting lane-choice game is an exact potential game and that the induced dynamics converge globally to the unique Wardrop equilibrium. (i) Using the resulting agent-based model, we characterize deployment phenomena that are absent from the equilibrium theory, including convergence regimes, demand-tracking limits, and the distribution of costs and benefits across altruistic and selfish vehicles. (i) We extend the framework to heterogeneous altruism and partial observability and identify when aggregate behavior is summarized by an effective altruism angle and analyze the impact of noise. (iv) We show that belief sharing improves performance under stationary demand, but can be conservative under abrupt changes. To address this, we introduce a forgetting-factor update that balances accuracy and adaptability. 2 Preliminaries 2.1 Population-game Model We briefly introduce the population-game concepts used in the paper. Consider a population of agents choosing between two pure strategies, S (stay) and B (bypass), and let x∈[0,1]x∈[0,1] denote the fraction of agents selecting S, so that 1−x1-x select B. A two-strategy population game is defined by a pair of cost functions JS,JB:[0,1]→ℝJ^S,J^B:[0,1] , where JS(x)J^S(x) and JB(x)J^B(x) denote the costs incurred by agents choosing strategies S and B, respectively, at population state x. The mean population cost is J¯(x)=xJS(x)+(1−x)JB(x) J(x)=xJ^S(x)+(1-x)J^B(x). A state x⋆∈[0,1]x ∈[0,1] is a Wardrop equilibrium if no agent can reduce its cost by unilaterally switching strategy: x⋆⋅(JS(x⋆)−JB(x⋆))≤0;(1−x⋆)⋅(JB(x⋆)−JS(x⋆))≤0.x · (J^S(x )-J^B(x ) )≤ 0;\ \ (1-x )· (J^B(x )-J^S(x ) )≤ 0. For an interior equilibrium x⋆∈(0,1)x ∈(0,1), this reduces to JS(x⋆)=JB(x⋆)J^S(x )=J^B(x ). Wardrop equilibrium is the user-equilibrium concept from transportation theory and coincides with the symmetric Nash equilibrium of the corresponding population game Wardrop (1952); Sandholm (2010). The continuous-time replicator dynamics associated with a population game (JS,JB)(J^S,J^B) are x˙=x(1−x)(JB(x)−JS(x)). x=x(1-x) (J^B(x)-J^S(x) ). (1) Replicator dynamics model decentralized adaptation: strategies with lower cost increase their population share over time. Any interior stationary point satisfies JS(x⋆)=JB(x⋆)J^S(x )=J^B(x ) and therefore coincides with an interior Wardrop equilibrium. In the agent-based model, vehicles update their lane-choice probabilities using a discrete-time replicator rule; as population size grows and the step size shrinks, the stochastic process converges to Eq. (1) Benaïm and others (2003); Sandholm (2010), so the replicator ODE serves as its analytical approximation. A population game is an exact potential game if there exists a continuously differentiable function Φ:[0,1]→ℝ :[0,1] satisfying dΦdx=JB(x)−JS(x),∀x∈[0,1]. d dx=J^B(x)-J^S(x),\ \ ∀\,x∈[0,1]. In an exact potential game the stationary points of the replicator dynamics coincide with those of the potential, which can be used directly as a Lyapunov function to establish stability Monderer and others (1996); Rosenthal (1973); Sandholm (2010). Let xSx^S and xBx^B denote the fractions of vehicles choosing the stay and bypass strategies, (xS+xB=1x^S+x^B=1). The agent’s private costs are Jself,S=K1SxS+B1S,Jself,B=K1BxB+B1BJ^self,S=K_1^Sx^S+B_1^S,\ \ J^self,B=K_1^Bx^B+B_1^B (2) The K1K_1 and B1B_1 coefficients capture the calibrated traversing and merging costs; their detailed expressions are provided in Section 2.2. The aggregate social cost is the sum of the costs incurred by all traffic classes, weighted by their respective flow fractions and is denoted by JsocJ^soc. Social Value Orientation (SVO) represents an agent’s relative concern for private and collective outcomes by an altruism angle θ∈[0,π/2]θ∈[0,π/2]. For a lane choice a∈S,Ba∈\S,B\, the agent minimizes the effective cost J~a(θ)=cos(θ)Jself,a+sin(θ)(∂Jsoc/∂xa), J^a(θ)= (θ)\,J^self,a+ (θ)\,(∂ J^soc/∂ x^a), (3) where Jself,aJ^self,a is the agent’s private travel cost and ∂Jsoc∂xa ∂ J^soc∂ x^a is the marginal social cost of its lane choice, and the sine–cosine weights are the standard unit-circle parameterization of SVO Van Lange (1999); Murphy and others (2011); Liebrand and others (1986). The boundary cases are θ=0θ=0 (fully selfish) and θ=π/2θ=π/2 (fully altruistic). 2.2 Weaving Ramp Model We now present the macroscopic weaving-ramp model He and others (2025); Wang and others (2026) and state equilibrium predictions that the agent-based model of Sec. 3 will be shown to match. Setup and Parameters. We consider a single highway weaving section with two lanes. Vehicles are partitioned into three movement classes by their origin and destination: entering vehicles, which join the highway at the on-ramp; exiting vehicles, which leave at the off-ramp; and through vehicles, which traverse the section without changing lanes. Each through vehicle in the mainline lane adjacent to the ramps additionally selects one of two lane-choice strategies: stay (S), remaining in lane 1, or bypass (B), shifting to lane 2 to avoid the merge conflict. The model is one-shot at the equilibrium-flow scale: lane choices within the weaving region are resolved into stationary flow ratios, and intra-section spatial dynamics are abstracted into the lane-cost functions below. Let n=(nenter,nexit,n2)n=(n_enter,n_exit,n_2) denote the normalized demand composition, where they denote the fractions of entering, exiting, and through vehicles, respectively. Let x∈[0,1]x∈[0,1] denote the fraction of lane 1 through vehicles choosing S, so that 1−x1-x choose B. Let p∈[0,1]p∈[0,1] denote the CAV penetration rate; the remaining 1−p1-p fraction are HDVs. The cost functions depend on four coefficients (C1t,C1m,C2t,C2m)(C_1^t,C_1^m,C_2^t,C_2^m) from the optimization method in Li et al. (2019) and six coefficients (α,β,γ,δ,ω,ρ)(α,β,γ,δ,ω,ρ), calibrated against microsimulation in He and others (2025).11 1 (C1t,C1m,C2t,C2m)=(1.000, 1.000,1.000, 1.000)(C_1^t,C_1^m,C_2^t,C_2^m)=(1.000,\,1.000,1.000,\,1.000)22 2 (α,β,γ,δ,ω,ρ)=(1.255, 1.138, 2.384, 3.094, 1.000, 1.000).(α,β,γ,δ,ω,ρ)=(1.255,\,1.138,\,2.384,\,3.094,\,1.000,\,1.000). Lane Costs. The travel delay costs of S and B are affine in the lane-choice state x, as seen in (2): with K1S=C1tα+C1m(ωnexit+nenter),B1S=C1t(βnexit+nenter),K1B=C2tγ+C2m(ρn2+δnexit),B1B=C2tn2.K_1^S=C_1^tα+C_1^m(ω\,n_exit+n_enter),B_1^S=C_1^t(β\,n_exit+n_enter),K_1^B=C_2^tγ+C_2^m(ρ\,n_2+δ\,n_exit),B_1^B=C_2^tn_2. The aggregate social cost Jsoc(x,n)J^soc(x,n) is the total social cost experienced by all vehicles in the weaving section, obtained by summing each class’s individual cost weighted by its flow: (i) steadfast and bypassing vehicles in lane 1, (i) through vehicles in lane 2, and (i) the entering and exiting vehicles and can be given as: Jsoc=xS(Jself,s)+xB(Jself,b)+n2(K2SxB+B2S)+nexit(K2exitxS+B2exit)+nenter(K0enterxS+B0enter).J^soc=x^S(J^self,s)+x^B(J^self,b)+n_2(K_2^Sx^B+B_2^S)+n_exit(K_2^exitx^S+B_2^exit)+n_enter(K_0^enterx^S+B_0^enter). The strategy-specific marginal social costs are Ma(x,n):=∂Jsoc/∂xa|(xS,xB)=(x,1−x),a∈S,B.M^a(x,n):= .∂ J^soc/∂ x^a |_(x^S,x^B)=(x,1-x),\ a∈\S,B\. (4) For the calibrated affine lane-cost model, JsocJ^soc is quadratic in x, and MsM^s and MbM^b are affine. Equilibrium Benchmarks. Two benchmark lane-choice states organize the analysis (i) HDV Wardrop point ξ(n)ξ(n), the equilibrium of the pure-HDV population (p=0p=0), obtained by setting J1S=J1BJ_1^S=J_1^B in (2), and (i) the social optimum Γ(n) (n), the steadfast fraction that minimizes the aggregate cost: ξ(n)=K1B+B1B−B1SK1S+K1B;Γ(n)=argminx∈[0,1]Jsoc(x,n).ξ(n)\;=\; K_1^B+B_1^B-B_1^SK_1^S+K_1^B;\ \ \ (n)\;=\; _x∈[0,1]J^soc(x,n). (5) The admissible configurations of interest are those in which the selfish equilibrium and the social optimum are distinct, i.e. 0<ξ(n)<Γ(n)<10<ξ(n)< (n)<1; under this condition altruistic CAVs can steer the population from ξ toward Γ .33 3 This ordering applies to operational regimes in which selfish traffic supplies fewer steadfast vehicles and increasing the weight on marginal social cost moves the equilibrium from ξ toward Γ . Three-Phase Penetration Structure. The aggregate impact of altruistic CAVs as the CAV penetration p varies is established in Wang and others (2026): for any admissible configuration, JsocJ^soc at the population equilibrium follows a three-phase structure in p. (i) At low penetration p∈[0,p1]p∈[0,p_1], JsocJ^soc remains at the HDV-only reference level; (i) over an intermediate band p∈(p1,p2]p∈(p_1,p_2], it decreases strictly in p; and (i) for p∈(p2,1]p∈(p_2,1], it saturates at the social-optimum level. 44 4 The two transition points are: p1=K1B+B1B−B1SK1S+K1B, and p2=(2K1B+B1B−B1S−nexitK2exit−nenterK0enter+n2K2s)(2(K1S+K1B))p_1\;=\; K_1^B+B_1^B-B_1^SK_1^S+K_1^B, and p_2\;=\; (2K_1^B+B_1^B-B_1^S-n_exitK_2^exit-n_enterK_0^enter+n_2K_2^s)(2\,(K_1^S+K_1^B)) where K2exitK_2^exit and K0enterK_0^enter are the xSx^S-coefficients in exiting and entering costs, while K2sK_2^s is the xBx^B-coefficient in Lane-2 through costs. This three-phase structure is the central macroscopic prediction that the ABM of Sec. 3 will be shown to reproduce. SVO Effective Cost. Specializing the definition of SVO (3) to the present setting, a CAV with altruism angle θ∈[0,π/2]θ∈[0,π/2] minimizes the effective cost: J~(⋅)(x,n,θ)=cos(θ)J1(⋅)(x)+sin(θ)M(⋅)(x,n), J^(·)(x,n;θ)\;=\; (θ)\,J_1^(·)(x)\;+\; (θ)\,M^(·)(x,n), (6) for (⋅)∈S,B(·)∈\S,B\, with J1(⋅)J_1^(·) given by (2) and M by (4). The angle θ is a single broadcastable parameter: HDVs use θ=0θ=0, while the CAV fleet adopts a value assigned by the operator. The induced mixed-population equilibrium interpolates between ξ (at θ=0θ=0) and Γ (at θ=π/2θ=π/2), and is the object the ABM of the next section converges to. 3 Agent-Based Model and Theory This section constructs the agent-based model (ABM) that realizes the aggregate statements over population fractions of the macroscopic model at the level of individual vehicles. The ABM uses the lane costs (2), the social-cost machinery, and the SVO effective cost (6) as specified in Sec. 2.2. Agents and Lane-Choice Dynamics. We consider a population of N vehicles indexed by i∈1,…,Ni∈\1,…,N\. Each vehicle is characterized by its (fixed) movement class ci∈enter,exit,throughc_i∈\enter,exit,through\, its (fixed) altruism angle θi∈[0,π/2] _i∈[0,π/2], and its (time-varying) lane-choice probability πi(t)∈[0,1] _i(t)∈[0,1] for selecting the steadfast strategy S. x(t)=1N∑i=1Nπi(t)x(t)\;=\; 1N _i=1^N _i(t) (7) HDVs have θi=0 _i=0; CAVs have θi>0 _i>0, with the value broadcast by the fleet operator. The aggregate steadfast fraction is given by (7). Each vehicle updates its lane-choice probability by a discrete-time replicator rule applied to its own SVO cost: πi(t+1)=πi(t)+ηπi(t)(1−πi(t))[J~iB(x(t))−J~iS(x(t))] _i(t+1)= _i(t)+η\, _i(t) (1- _i(t) ) [ J_i^B (x(t) )- J_i^S (x(t) ) ] (8) where J~i(⋅) J_i^(·) is the effective cost (6) evaluated at the agent’s own angle θi _i, and η>0η>0 is a step size.55 5 We assume that η is small enough that η|J~iB−J~iS|≤1η J_i^B- J_i^S ≤ 1, ensuring that πi(t)∈[0,1] _i(t)∈[0,1] is preserved by the update. The rule is decentralized: each vehicle uses its own altruism angle, the realized lane costs, and the demand composition needed to form the marginal social cost M. The operator’s only intervention is the choice of θ broadcast to the CAV fleet. Mean-Field Approximation and Convergence. We derive a scalar continuous-time approximation using the rescaled time τ=ηtτ=η t. This follows standard stochastic-approximation arguments Benaïm and others (2003), and its agreement with the aggregate agent-based dynamics is assessed numerically. Let Δ0(x):=J1B(x)−J1S(x) _0(x):=J_1^B(x)-J_1^S(x) denote the cost advantage of choosing the steadfast strategy for an HDV. For a CAV with altruism angle θ, the corresponding effective-cost difference is Δθ(x)=cosθΔ0(x)+sinθ[MB(x)−MS(x)]. _θ(x)= θ\, _0(x)+ θ\,[M^B(x)-M^S(x)]. Suppose that a fraction 1−p1-p of the strategic vehicles are HDVs and a fraction p are CAVs with SVO angle θ. Their population-average incentive is then: Δpop(x)=(1−p)Δ0(x)+pΔθ(x) ^pop(x)=(1-p) _0(x)+p _θ(x) (9) From (7) and (8), the exact aggregate change generated by the individual probability updates is x(t+1)−x(t)η=1N∑i=1Nπi(t)(1−πi(t))Δθi(x(t)). x(t+1)-x(t)η= 1N _i=1^N _i(t) (1- _i(t) ) _ _i(x(t)). To obtain a scalar approximation, we assume that the individual probabilities remain concentrated around their population average, giving: (1/N)∑i=1Nπi(1−πi)Δθi(x)≈x(1−x)Δpop(x).(1/N) _i=1^N _i(1- _i) _ _i(x)≈ x(1-x) ^pop(x). x˙=x(1−x)Δpop(x), x=x(1-x)\, ^pop(x), (10) After introducing the rescaled time τ=ηtτ=η t, the resulting continuous-time approximation is (10). Intuitively, x(1−x)x(1-x) measures the population’s responsiveness, whereas Δpop(x) ^pop(x) measures its average incentive to choose the steadfast strategy. Theorem 3.1 (Aggregate potential and convergence) Under the homogenization closure, suppose KS+KB>0K^S+K^B>0 and a>0a>0.66 6 With positive coefficients (α,β,γ,δ,ω,ρ)(α,β,γ,δ,ω,ρ) and nonnegative demand fractions, JsocJ_soc is strictly convex in x with a=4(α+γ+nenter+(ω+δ)nexit+ρn2)>0a=4(α+γ+n_enter+(ω+δ)\,n_exit+ρ\,n_2)>0, identical for the lane-1 and full social-cost forms since the additional movement terms are linear in x. Likewise KS+KB=α+γ+nenter+(ω+δ)nexit+ρn2>0K^S+K^B=α+γ+n_enter+(ω+δ)n_exit+ρ n_2>0. Then, for every p∈[0,1]p∈[0,1] and θ∈[0,π/2]θ∈[0,π/2]: (i) Φpop ^pop is a strictly concave exact potential for the aggregate mean-field game and therefore has a unique maximizer x⋆∈[0,1]x ∈[0,1]. (i) If x⋆∈(0,1)x ∈(0,1), it is the unique interior aggregate Wardrop equilibrium and the unique interior stationary point of (10). It is globally asymptotically stable on (0,1)(0,1) with Lyapunov function V(x)=Φpop(x⋆)−Φpop(x)V(x)= ^pop(x )- ^pop(x). (i) Under u=−Ju=-J, the interior maximizer is the unique symmetric Nash equilibrium of the representative aggregate game and is evolutionarily stable.77 7 The model is written in costs (which agents minimize), whereas potential-game and evolutionary-stability statements are conventionally written in payoffs (which agents maximize). We use the payoff convention u=−Ju=-J. Under this, lower-cost strategy corresponds to higher potential, and convexity of social cost to concavity of the potential. Proof 3.2 (Sketch) By definition, Φ0′=Δ0 _0 = _0 and Φθ′=Δθ _θ = _θ, we have (Φpop)′=Δpop( ^pop) = ^pop, proving the exact potential property. Moreover, from (9) (Φpop)′=(Δpop)′=−(1−p)(K1S+K1B)−p[(K1S+K1B)cosθ+asinθ]<0,( ^pop) =( ^pop) =-(1-p)(K_1^S+K_1^B)-p [(K_1^S+K_1^B) θ+a θ ]<0, so the potential is strictly concave and has a unique maximizer. When this maximizer is interior, (Φpop)′(x⋆)=0( ^pop) (x )=0 is both the aggregate Wardrop condition and the stationary condition for (10). Furthermore, V˙=−x(1−x)[(Φpop)′(x)]2<0 V=-x(1-x)[( ^pop) (x)]^2<0 for every x∈(0,1)∖x⋆x∈(0,1) \x \, proving global asymptotic stability. The Nash and ESS conclusions follow from the strict potential maximum Monderer and others (1996); Sandholm (2010); Weibull (1995). Heterogeneous Fleets. We next allow each CAV’s fixed altruism angle to be drawn from a distribution F on [0,π/2][0,π/2]. Under the same aggregate homogenization closure, the expected cost difference becomes ΔFpop(x)= _F^pop(x)= (1−p)Δ0(x)+pF[cosθ]Δ0(x) (1-p) _0(x)+p\,E_F[ θ]\, _0(x) +pF[sinθ][MB(x)−MS(x)]. +p\,E_F[ θ]\,[M^B(x)-M^S(x)]. (11) Thus, at the aggregate mean-field level, F enters the dynamics only through F[cosθ]E_F[ θ] and F[sinθ]E_F[ θ]. Define the effective angle and coherence factor by θ¯eff=atan2(F[sinθ],F[cosθ]), θ_eff=atan2 (E_F[ θ],E_F[ θ] ), ρF=F[cosθ]2+F[sinθ]2. _F= E_F[ θ]^2+E_F[ θ]^2. Then F[cosθ]=ρFcosθ¯eff,F[sinθ]=ρFsinθ¯eff,E_F[ θ]= _F θ_eff,\ \ E_F[ θ]= _F θ_eff, and (11) can be written as ΔFpop=(1−p)Δ0+pρFΔθ¯eff. _F^pop=(1-p) _0+p _F _ θ_eff. Equivalently, let λF:=1−p+pρF,peff:=pρF1−p+pρF. _F:=1-p+p _F,\ \ p_eff:= p _F1-p+p _F. The heterogeneous drift then factors as ΔFpop=λF[(1−peff)Δ0+peffΔθ¯eff]. _F^pop= _F[(1-p_eff) _0+p_eff _ θ_eff]. Hence its equilibrium and phase portrait coincide with those of the homogeneous aggregate model having angle θ¯eff θ_eff and penetration peffp_eff; the positive factor λF _F changes only the time scale. Because ρF≤1 _F≤ 1, with equality if and only if all CAVs share the same altruism angle, heterogeneity reduces the fleet’s effective aggregate influence. The conclusions of Thm. 3.1 therefore continue to hold for the heterogeneous aggregate mean-field model after replacing (p,θ)(p,θ) by (peff,θ¯eff)(p_eff, θ_eff). This equivalence concerns the expected aggregate dynamics. Finite fleets with persistent agent-specific altruism may differ because lane-choice probabilities can become correlated with behavioral type and because of finite-population sampling variability. Partial Observability and Cooperative Belief Sharing. The SVO effective cost in (6) depends on the demand composition n=(nenter,nexit,n2),n=(n_enter,n_exit,n_2), which determines both the private lane costs and the marginal social costs. The macroscopic model assumes that n is known exactly. In practice, a deployed CAV has access only to noisy local observations of the traffic state. We model the measurement of vehicle i at time t as: yi,t=n+εi,t,εi,t∼(0,σobs2I)y_i,t=n+ _i,t,\ _i,t (0, _obs^2I), with observations independent across vehicles and time. Each vehicle replaces n in (6) by an estimate n^i,t n_i,t formed from available information. We compare two estimators: (i) Memoryless estimator. The vehicle uses only its most recent observation, n^i,tmem=yi,t n_i,t^mem=y_i,t. This models a one-shot lane decision based solely on the current local measurement. (i) Cooperative belief-sharing estimator. Vehicles maintain and communicate beliefs regarding the traffic composition. Let μi,t _i,t denote vehicle i’s belief and let τi,t=1/σi,t2 _i,t=1/ _i,t^2 denote its precision. Whenever a new observation yj,t+1y_j,t+1 is received from another vehicle or roadside infrastructure, the belief is updated according to τi,t+1=τi,t+1σobs2,μi,t+1=μi,t+1/σobs2τi,t+1(yj,t+1−μi,t), _i,t+1= _i,t+ 1 _obs^2,\ \ _i,t+1= _i,t+ 1/ _obs^2 _i,t+1(y_j,t+1- _i,t), and the estimate used in the SVO cost is n^i,tshare=μi,t n_i,t^share= _i,t. This update is the standard Bayesian fusion rule for independent Gaussian measurements and can be interpreted as a simple distributed information-sharing protocol. The next result quantifies the benefit of information-sharing: combining independent observations keeps the demand estimate unbiased while reducing variance. Lemma 3.3 (Estimator bias and variance) Both estimators are unbiased, [n^i,tmem]=[n^i,tshare]=nE[ n_i,t^mem]=E[ n_i,t^share]=n, with Var[n^i,tmem]=σobs2IVar[ n_i,t^mem]= _obs^2I and Var[n^i,tshare]=(σobs2/t)IVar[ n_i,t^share]=( _obs^2/t)\,I. Proof 3.4 The memoryless estimate is n^i,tmem=yi,t=n+εi,t n_i,t^mem=y_i,t=n+ _i,t, giving mean n and variance σobs2I _obs^2I. Bayesian fusion of t independent Gaussian observations has posterior mean equal to the sample average n^i,tshare=1t∑k=1tyk n_i,t^share= 1t _k=1^ty_k, hence mean n and variance σobs2/tI _obs^2/t\,I. Because the effective-cost difference depends linearly on the estimated demand, both unbiased estimators recover the full-information equilibrium on average. They differ only in how much noise they introduce into the updates. Corollary 3.5 (Equilibrium is preserved) The effective-cost difference Δ(x,n)=J~B−J~S (x,n)= J^B- J^S is affine in n, so Δ(x,n)=a(x)⊤n+b(x) (x,n)=a(x) n+b(x) and, by Lemma 3.3, [Δ(x,n^i,t)∣x]=a(x)⊤n+b(x)=Δ(x,n)E[ (x, n_i,t) x]=a(x) n+b(x)= (x,n) for either estimator. The expected replicator vector field and the equilibrium location are thus identical to the full-information case; the estimators differ only in variance, which is O(σobs2)O( _obs^2) for the memoryless rule and decays as 1/t1/t under belief sharing. 4 Experimental Results We evaluate the agent-based realization of Sec. 3 using the parameters of Wang and others (2026), with N=1000N=1000 agents and step size η=0.08η=0.08. Macroscopic-equilibrium validation. For the HDV-only case (p=0p=0), Fig. 1 compares the ABM steady-state bypass fraction with the closed-form Wardrop equilibrium ξ(n)ξ(n) and an independent SUMO simulation. All three agree within 11–2%2\% across the tested demand conditions. Figure 1: HDV-only ABM, closed-form Wardrop equilibrium, and independent SUMO simulation. For mixed traffic, Fig. 2 compares the analytic equilibrium and ABM results over CAV penetration p and altruism angle θ. Both recover the three regimes of p1p_1 and p2p_2: an HDV baseline for p≤p1p≤ p_1, a decreasing transition, and a θ-dependent floor for p≥p2p≥ p_2. Figure 2: Analytic mixed-population equilibrium (left) and ABM results (right), including thresholds p1p_1 and p2p_2. Heterogeneous fleets. We test four homogeneous profiles and five mixtures over θ∈π/6,π/4,π/3,π/2θ∈\π/6,π/4,π/3,π/2\. As shown in Fig. 3, all profiles retain the three-regime structure. Profiles with the same θ¯eff θ_eff produce nearly identical equilibria, while greater heterogeneity primarily affects transient convergence through the coherence factor ρF _F. Figure 3: Social cost for homogeneous and heterogeneous CAV fleets. Profiles with matched θ¯eff θ_eff have similar equilibria. Convergence speed and stability. Fig. 4 reports the settling time TeqT_eq over (p,ηCAV)(p, _CAV) at θ=π/2θ=π/2. Low-penetration fleets require conservative learning rates, whereas higher penetration supports faster adaptation without sustained oscillation. Figure 4: Steps to equilibrium over (p,ηCAV)(p, _CAV) at θ=π/2θ=π/2; dotted lines indicate p1p_1 and p2p_2. Tracking time-varying demand. We vary n2(t)n_2(t) sinusoidally and compare the ABM trajectory with the instantaneous quasi-static equilibrium. Fig. 5 shows that the fleet follows slowly varying demand more accurately: the RMSE is ≈ 0.0070.007 for period 200200 and 0.0120.012 for period 4040. Figure 5: ABM tracking of sinusoidal demand with periods 4040 and 200200 for p>p2p>p_2. Distributional outcomes and free-riding. Aggregate social cost does not reveal how delay is distributed across vehicles. Fig. 6 compares per-agent costs across homogeneous and mixed SVO profiles. The fully altruistic fleet has the highest measured inequity (G≈0.014G≈ 0.014), while the mixed profiles have lower inequity (G≈0.009G≈ 0.009). Thus, full altruism can concentrate the cooperative burden on the vehicles that absorb the conflict-zone delay. Figure 6: Per-agent cost distributions (left) and Gini coefficients (right) across CAV fleet profiles. Partial observability and belief sharing. True demand n is replaced by a noisy estimate yi,t=n+εi,ty_i,t=n+ _i,t, where εi,t∼(0,σobs2I) _i,t (0, _obs^2I). Fig. 7 shows that observation noise delays the transition between penetration regimes but does not change the social-optimum floor. Cooperative belief sharing is closer to the perfect-information result than a memoryless estimator. Figure 7: Social cost under perfect information, memoryless estimation, and cooperative belief sharing. Recovery from demand shocks. We equilibrate the fleet at balanced demand and apply a shock n2:0.33→0.60n_2:0.33→ 0.60 at t=200t=200. As shown in Fig. 8, all fleets reach the same post-shock equilibrium, but recovery is faster at higher penetration: p≥70%p≥ 70\% settles within a few steps, whereas p=40p=40–50%50\% requires approximately 3030–4040 steps. Figure 8: Recovery after a demand-composition shock across CAV penetration rates. Belief sharing under shocks. Bayesian belief sharing reduces estimation variance under stationary demand but can retain stale information after a shock. We therefore discount past observations using τt+1=λτt+σobs−2,λ∈(0,1]. _t+1=λ _t+ _obs^-2,\ \ λ∈(0,1]. Fig. 9 shows that standard fusion (λ=1λ=1) may require hundreds of steps to recover, whereas λ≤0.95λ≤ 0.95 restores convergence within a few steps. Figure 9: Post-shock recovery of social cost under different forgetting factors λ; the shock occurs at t=400t=400. 5 Conclusions We developed an agent-based realization of the macroscopic weaving-ramp altruism framework, proved its convergence to the macroscopic equilibrium, and used it to derive deployment-level findings invisible to the static analysis. These results are limited to a single weaving ramp under a calibrated affine-cost model with homogeneous learning rates. Future work includes networks of coupled weaving sections, and heterogeneous and adaptive learning dynamics. References Benaïm et al. (2003) M. Benaïm et al. Deterministic approximation of stochastic evolution in games. Econometrica 71 (3), p. 873–903. Cited by: §2.1, §3. Cassidy et al. (2005) M.J. Cassidy et al. Increasing the capacity of an isolated merge by metering its on-ramp. Transp. Res. B 39 (10), p. 896–913. Cited by: §1. Hart et al. (2003) S. Hart et al. Uncoupled dynamics do not lead to Nash equilibrium. AER 93 (5), p. 1830–1836. Cited by: §1. He et al. (2025) H. He et al. To stay or to bypass: aggregate strategic decision-making at highway weaving ramps. arXiv:2505.08965. Cited by: §1, §1, §2.2, §2.2. Li et al. (2019) R. Li, N. Mehr, and R. Horowitz An extended game–theoretic model for aggregate lane choice behavior of vehicles at traffic diverges with a bifurcating lane. In 2019 IEEE Intelligent Transportation Systems Conference (ITSC), p. 2226–2231. Cited by: §2.2. Liebrand et al. (1986) W.B.G. Liebrand et al. Value orientation and conformity in social dilemma games. J. Conflict Resolut. 30 (1), p. 77–97. Cited by: §2.1. Maynard Smith et al. (1973) J. Maynard Smith et al. The logic of animal conflict. Nature 246, p. 15–18. External Links: Document Cited by: §1. Monderer et al. (1996) D. Monderer et al. Potential games. Games Econ. Behav. 14 (1), p. 124–143. Cited by: §2.1, Proof 3.2. Murphy et al. (2011) R.O. Murphy et al. Measuring social value orientation. Judgm. Decis. Mak. 6 (8), p. 771–781. Cited by: §2.1. Rosenthal (1973) R.W. Rosenthal A class of games possessing pure-strategy Nash equilibria. Int. J. Game Theory. Cited by: §2.1. Sandholm (2010) W.H. Sandholm Population games and evolutionary dynamics. MIT Press. Cited by: §1, §1, §2.1, §2.1, §2.1, Proof 3.2. Schwarting et al. (2019) W. Schwarting et al. Social behavior for autonomous vehicles. PNAS 116 (50). Cited by: §1. Smith (1984) M.J. Smith The stability of a dynamic model of traffic assignment. Transp. Sci. 18 (3), p. 245–252. Cited by: §1. Stern et al. (2018) R.E. Stern et al. Dissipation of stop-and-go waves via control of autonomous vehicles. Transp. Res. C. Cited by: §1. Talebpour et al. (2016) A. Talebpour et al. Influence of connected and autonomous vehicles on traffic flow stability and throughput. Transp. Res. C. Cited by: §1. Taylor et al. (1978) P.D. Taylor et al. Evolutionary stable strategies and game dynamics. Math. Biosci. 40 (1–2), p. 145–156. Cited by: §1. Van Lange (1999) P.A.M. Van Lange The pursuit of joint outcomes and equality in outcomes. J. Pers. Soc. Psychol. 77 (2), p. 337–349. Cited by: §2.1. Wang et al. (2026) K. Wang et al. When altruism meets autonomy: managing bottleneck congestion with strategic AVs. arXiv:2604.21941. Cited by: §1, §1, §2.2, §2.2, §4. Wardrop (1952) J.G. Wardrop Some theoretical aspects of road traffic research. Proc. Inst. Civ. Eng. 1 (3), p. 325–362. Cited by: §2.1. Weibull (1995) J.W. Weibull Evolutionary game theory. MIT Press. Cited by: §1, Proof 3.2.