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Choose Your Game Wisely: Measuring Game-Theoretic Structures in Real-World Vehicle Interactions

Yueyuan Li, Rongcheng Nie, Weijie Xi, Mingyang Jiang, Songan Zhang, Hanyang Zhuang, Ming Yang

arXiv:2608.25917v1cs.AIcs.RO

TL;DR

The paper asks how simultaneous, sequential, and asymmetric interaction structures can be measured from real-world vehicle trajectories, addressing limited systematic evidence about behavioral dependence. It develops a trajectory-based framework using behavioral deviations to identify and characterize interaction events, then finds that concurrent, sequential, and persistently ordered structures coexist, with temporal precedence producing a measurable response in only about 41% of assessable directed pairs.

  • Problem

    How behavioral dependence and temporal interaction structure develop within real-world vehicle interaction episodes remains less systematically examined than interaction identification or prediction.

  • Method

    A trajectory-based framework identifies interaction events and quantifies behavioral change onset, temporal organization, response dynamics, and ordering stability using behavioral deviations to verify candidates.

  • Results

    Concurrent and sequential behavioral changes coexist across following, merging, and conflicting interactions, while sequential interactions more often show stable than alternating ordering.

  • Takeaways & Limitations

    Different game-theoretic formulations are complementary abstractions for different interaction regimes rather than a universal structure governing all vehicle interactions.

Abstract

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Game-theoretic models provide principled frameworks for modeling vehicle interactions, but their underlying temporal assumptions have not been systematically examined against real-world driving behavior. In particular, it remains unclear how simultaneous, sequential, and asymmetric interaction structures can be measured from vehicle trajectories. This paper develops a trajectory-based interaction measurement framework to identify interaction events and quantify behavioral change onset, temporal organization, post-onset response dynamics, and ordering stability. The framework uses behavioral deviations to verify candidate interactions. We evaluate the framework on six real-world trajectory datasets, including INTERACTION, highD, inD, rounD, Waymo Open Motion, and nuPlan, covering diverse road geometries, traffic environments, and interaction types. The results show that concurrent and sequential behavioral changes both constitute substantial proportions of observed following, merging, and conflicting interactions. Among sequential interactions, stable ordering is more prevalent than alternating ordering, indicating that persistent asymmetric roles are a common interaction structure. Importantly, temporal precedence does not necessarily coincide with a measurable behavioral response, indicating that temporal ordering alone may not be sufficient to characterize behavioral dependence. These findings show that real-world interactions exhibit concurrent, sequential, and persistently ordered temporal structures. Different game-theoretic formulations are therefore better regarded as complementary modeling abstractions for different interaction regimes rather than as a universal structure governing all vehicle interactions.

I. INTRODUCTION

The paper addresses the limited empirical measurement of how behavioral dependence unfolds between interacting vehicles. It introduces a trajectory-based framework for distinguishing temporal interaction structures and evaluates their prevalence in real-world driving.

  • I. INTRODUCTION: Existing studies identify or model vehicle behavior, but the temporal and directional organization of behavioral responses remains less systematically measured.Proximity or temporal precedence alone does not establish a measurable behavioral response.
  • I. INTRODUCTION: The framework quantifies behavioral change onset, temporal organization, response dynamics, and ordering stability from real-world trajectories.It also measures which vehicle changes first and whether a subsequent measurable response occurs.
  • I. INTRODUCTION: The study establishes an empirical basis for choosing among alternative temporal interaction structures when modeling real-world driving.It positions the contribution as measurement of observed structure rather than another interaction model.
  • I. INTRODUCTION: Concurrent and sequential behavioral changes coexist across following, merging, and conflicting interactions.The framework characterizes these structures across interaction types rather than assuming one universal temporal pattern.
  • I. INTRODUCTION: Sequential interactions are more often characterized by stable than alternating ordering.This finding indicates that persistent asymmetric ordering is common in observed interactions.

II. RELATED WORKS

Prior work uses trajectories to identify, describe, and model vehicle interactions, while game-theoretic studies typically prescribe interaction structure within the model. Consequently, the structure emerging in real driving remains less well understood.

  • II. RELATED WORKS: Trajectory-based studies increasingly describe which vehicles interact and how strongly their behaviors are related, but less often examine behavioral dependence within an interaction episode.Their primary uses include interaction analysis, pattern extraction, and trajectory prediction.
  • II. RELATED WORKS: Game-theoretic driving models represent interdependent vehicle decisions through objectives involving factors such as safety, efficiency, comfort, and social preferences.Both cooperative and non-cooperative formulations have been applied.
  • II. RELATED WORKS: Nash formulations model mutually dependent decisions within the same stage, whereas Stackelberg formulations impose asymmetric sequential leader–follower roles.Hierarchical formulations add further reasoning levels or decision stages.
  • II. RELATED WORKS: Existing game-based methods may update strategies, aggressiveness, or leader–follower relationships as observations become available.Examples use online estimation, Bayesian inference, cognitive hierarchy, or partially observable games.
  • II. RELATED WORKS: Real-world or simulated data are commonly used to calibrate parameters, reproduce behavior, or evaluate models under predefined interaction structures.They are less often used to identify the interaction structure directly.

III. METHOD

The method interprets game-theoretic assumptions through observable temporal behavior in vehicle trajectories. Simultaneous-move interactions correspond to concurrent behavioral changes without measurable temporal precedence.

  • III. METHOD: The framework reviews observable temporal implications of assumptions underlying commonly used game-theoretic interaction models.It uses these implications to connect model structure with trajectory-level behavior.
  • III. METHOD: Simultaneous-move games involve actions selected within the same decision stage without observing other players’ current actions.In vehicle interactions, this corresponds to concurrent behavioral changes.
  • III. METHOD: The Nash equilibrium is a classical solution concept in which each player’s strategy is optimal given the strategies selected by others.The paper uses this formulation as the game-theoretic counterpart of concurrent changes.
  • III. METHOD: Following, merging, and conflicting are illustrated as trajectory-level interaction types in the INTERACTION dataset.The figure provides examples of the three relation categories used in the paper.

2) Sequential-move assumption:

Sequential-move interactions are modeled as ordered behavioral changes, while Stackelberg interactions add persistent asymmetric leader–follower roles. The framework detects and verifies candidate events using trajectory geometry, timing, and speed deviations.

  • 2) Sequential-move assumption:: Sequential-move games describe decisions made at different stages, with one observable action preceding another.Vehicle interactions are represented as temporally ordered sequences of behavioral changes.
  • 2) Sequential-move assumption:: Stackelberg games add asymmetric leader–follower roles in which the follower responds after observing the leader’s action.This persistent ordering can be examined empirically from observed behavioral changes.
  • 2) Sequential-move assumption:: Candidate following, merging, and conflicting events are detected from road topology, spatial compatibility, and temporal compatibility.Reference trajectories and vehicle-occupancy-aware motion corridors support candidate detection.
  • 2) Sequential-move assumption:: Speed deviation is used as additional behavioral evidence because spatiotemporal compatibility alone does not establish a behavioral response.Deviation is normalized against a vehicle-specific baseline computed from control windows.
  • 2) Sequential-move assumption:: A candidate pair is retained when at least one participant exhibits a sufficiently large normalized deviation.The criterion uses τ = 2.0 and retains asymmetric cases where only one participant adjusts markedly.
  • 2) Sequential-move assumption:: Merging and conflicting candidates must encounter the shared spatial region within a temporal gap of at most 1.0 s.This prevents temporally unrelated behaviors from being associated with one event.

C. Temporal Characterization of Interaction

Behavioral change onset is measured relative to a map-constrained reference trajectory, using sustained speed residuals to identify meaningful deviations from expected motion.

  • C. Temporal Characterization of Interaction: The framework identifies behavioral change onset relative to a map-constrained reference trajectory.The reference follows the vehicle’s lane route toward its observed destination while accounting for road geometry and planned deceleration.
  • C. Temporal Characterization of Interaction: A speed residual compares observed vehicle speed with the corresponding map-constrained reference speed.
  • C. Temporal Characterization of Interaction: Behavioral change requires the residual to exceed a threshold continuously for a minimum duration.The reported settings are δ = 0.3 m/s and dmin = 400 ms.

2) Temporal Organization:

Temporal organization is determined from the relative timing of two vehicles’ behavioral-change onsets, distinguishing concurrent from sequential changes and identifying which vehicle changes first.

  • 2) Temporal Organization:: The onset difference ∆tij measures the relative timing of behavioral changes for vehicles i and j.
  • 2) Temporal Organization:: Events are concurrent when |∆tij| ≤ 400 ms and sequential otherwise.For sequential events, the sign of ∆tij identifies which vehicle exhibits an observable behavioral change first.
  • 2) Temporal Organization:: One-sided events have only one detected onset, while unresolved events have neither onset detected.Both categories are counted as non-concurrent in proportion calculations.

3) Role Stability:

The framework extends first-onset comparisons to complete behavioral-change sequences and examines whether a post-onset change occurs in the other vehicle.

  • 3) Role Stability:: Complete sequences are used to capture ordering reversals beyond the initial temporal relationship.
  • 3) Role Stability:: Stable-ordering events have fewer than two ordering reversals, whereas alternating events have two or more.Events with changes detected for only one vehicle are one-sided, and those without detectable changes are unresolved.
  • 3) Role Stability:: For each directed pair i → j, the framework tests whether vehicle j exhibits a new behavioral change after vehicle i’s onset.
  • 3) Role Stability:: The target response threshold combines the fixed residual threshold with the target’s pre-onset baseline mean and variability.The threshold is θj = max(δ, µj,pre + kσj,pre), with k = 2.
  • 3) Role Stability:: Directed pairs are classified as response, preactive, noresponse, or right-censored according to post-onset and pre-onset behavioral changes.A response requires a sustained target residual above θj after the source change; a preactive target changed before the source onset.
  • 3) Role Stability:: Response-lag statistics use only response directions, with post-onset searches limited to 2000 ms.Directions without a qualifying response within that window receive noresponse or right-censored treatment.

D. Statistical Analysis

The study uses clustered bootstrap intervals and a logistic mixed-effects model to compare temporal organization across interaction types and datasets.

  • D. Statistical Analysis: Confidence intervals are estimated with 2,000 scene-level cluster bootstrap replicates and percentile-based 95% intervals.Scenes are sampled with replacement, retaining all interaction events from sampled scenes.
  • D. Statistical Analysis: A logistic mixed-effects model relates concurrent organization to interaction type, event duration, and vehicle count.The model includes a scene-level random intercept and reports adjusted odds ratios with 95% confidence intervals.
  • D. Statistical Analysis: Evaluation covers six trajectory datasets spanning varied road geometries, traffic densities, data sources, and driving environments.The datasets are INTERACTION, highD, inD, rounD, WOMD, and nuPlan, including intersections, roundabouts, highways, and urban streets.
  • D. Statistical Analysis: Table I presents extracted interaction events from different datasets.

B. Temporal Organization of Vehicle Interactions

Vehicle interactions contain both concurrent and sequential behavioral changes, while temporal precedence produces a measurable response in only a subset of directed pairs.

  • Concurrent events comprise 33.81%–58.67% of reported cases, while sequential events comprise 33.69%–54.55%.
  • Temporal organization varies across interaction types, with concurrent and sequential changes both occurring frequently.
  • 41.4% of assessable directed pairs exhibit a measurable behavioral response on average across datasets.
  • About 64% of detected responses occur within the 400 ms concurrent tolerance, ranging from 56.4% to 69.5% across datasets.
  • Temporal precedence does not necessarily produce a measurable behavioral response.

D. Ordering Stability

Once sequential interactions are established, ordering is predominantly stable rather than alternating across datasets and relation types.

  • Stable ordering accounts for 80.46%–100.00% of sequential events across datasets and relation types.
  • Alternating ordering accounts for the remainder of sequential events and is most frequent in Waymo.
  • Overall, temporal ordering is predominantly stable once a sequential interaction is established, while role alternation remains a minority pattern.

1) Simultaneous-move game assumption:

Concurrent, sequential, and stably ordered behavioral patterns all occur in real-world vehicle interactions, supporting complementary rather than universal game-theoretic temporal formulations.

  • Concurrent behavioral changes are common across datasets and interaction types, supporting simultaneous-move formulations.
  • Sequential behavioral changes are prevalent and sometimes exceed concurrent interactions, supporting sequential-move formulations.
  • Stable ordering is substantially more common than alternating ordering among sequential interactions, supporting leader–follower formulations.
  • Concurrent, sequential, and stably ordered patterns coexist across interaction contexts and datasets.
  • Different game-theoretic formulations provide complementary descriptions of real-world vehicle interactions rather than one universal temporal organization.
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