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Empirically Calibrated Differential Games for Simulating Driver Interactions

Amirmohammad Khakpour, Nachuan Li, Pedram Beigi, Yu, Nie

arXiv:2610.32276v1cs.MAmath.OC

TL;DR

Whether calibrated game-theoretic models can describe human driving better than existing simulator and IRL models remains insufficiently tested. This paper calibrates a differential game of intersection interactions to observed trajectories and evaluates route-only prediction on held-out interactions. Its cross-validation results favor the game over calibrated SUMO, IRL, and independent and cooperative variants.

  • Problem

    Prior calibrated games have not established whether route-only predictions outperform current simulator and IRL models, despite the need to model heterogeneous interaction at urban intersections.

    Source: §I · p. 1

  • Method

    The paper calibrates an N-player differential game to observed vehicle trajectories, where each vehicle pursues its own objectives and anticipates others’ delayed states.

    Source: §I · p. 2, §II · p. 2, §III · p. 2

  • Results

    Cross-validation error is lower than calibrated SUMO by 3.61 m, an IRL model by 0.93 m, and independent and cooperative variants by 0.33 and 0.32 m, respectively.

    Source: §VIII · p. 12

  • Takeaways & Limitations

    For game-theoretic simulators, trajectories identify some parameters, while conflict and steering weights can be set from safety and comfort targets; with a stiffer contact barrier, these settings removed every collision in tight closed-loop conflicts.

    Source: §VIII · p. 13

    Trajectories do not identify the conflict and steering weights, which must instead be set from safety and comfort targets.

    Source: §VIII · p. 13

Abstract

from arXiv · show

Game-theoretic models can resolve vehicle interactions without hand-written priority rules, but whether they describe human drivers once calibrated has rarely been tested. This study formulates a general-sum differential game of vehicle interactions at urban intersections, calibrates it to observed driving, and evaluates it on data not used for calibration. Each vehicle minimizes a cost that combines speed tracking, adherence to a map-derived lane route, control effort, gap keeping based on the intelligent driver model, a predictive conflict term, and a contact barrier, while perceiving the other vehicles with a delay. The game is solved with a feedback iterative linear-quadratic algorithm, and ten behavioral parameters per vehicle are calibrated by Bayesian optimization on trajectories recorded at four intersections in Washington, D.C. The calibrated game reproduces the trajectories of 819 vehicles with a mean position error of 1.00 m. When only each vehicle's route is known, as in a simulator, it is the most accurate of the compared models on 96 held-out vehicles (7.42 m), and ten-fold cross-validation shows that it is significantly more accurate than calibrated SUMO (by 3.61 m), an inverse reinforcement learning model (by 0.93 m), and independent and cooperative variants of the same model (by 0.33 and 0.32 m). The solutions, computed in 2.1 s for a typical interaction, are verified approximate equilibria. Finally, the conflict and steering weights, which the data do not identify, can be set from safety and comfort targets; together with a stiffer contact barrier, this removes the collisions that otherwise occur in 43% of tight closed-loop conflicts.

I. INTRODUCTION

Rule-based interaction models become difficult to extend at urban intersections, while prior calibrated games have not established whether they predict human interactions better than current simulator and IRL models. This paper develops and evaluates a local differential game in which vehicles jointly choose motion while pursuing individual objectives and anticipating others.

I. INTRODUCTION

  • At urban intersections, heterogeneous drivers negotiate right-of-way simultaneously, making rule-based frameworks hard to extend as street-user mixes diversify.Prior calibrated games have not tested whether route-only predictions outperform simulator and IRL models, and often assume drivers perceive others without delay.Source: §I · pp. 1–2
  • The paper calibrates an N-player differential game to observed trajectories and evaluates it on interactions not used for calibration.Each vehicle has its own behavioral parameters, and the Nash equilibrium is a conflict-resolution mechanism rather than a claim that drivers solve games.Source: §I · p. 2

II. PROBLEM SETTING

  • A local interaction is modeled as a finite-horizon game whose equilibrium determines how the vehicles’ conflict unfolds.As vehicles enter and leave, conflicts appear and dissolve, making the simulator a sequence of local games.Source: §II · p. 2

III. MODEL

  • Each vehicle balances preferred speed, lane-route adherence, smooth control, and safe avoidance, with its plans jointly solved as a Nash equilibrium.The model assigns each vehicle ten behavioral parameters; other constants are shared.Source: §III · p. 2

1) Agent costs:

The agent costs combine individual driving goals with terms for interactions, using a map-derived lane route and delayed perceptions of other vehicles.

1) Agent costs:

  • Own-goal terms include preferred-speed tracking, lane-route adherence, control effort, road-boundary repulsion, reverse-motion and speed or lateral-acceleration penalties.The speed reference ramps toward the desired speed and tapers to zero near the destination.Source: 1) Agent costs · p. 3
  • Map-derived lane routes constrain where turns occur, preventing corner-cutting from direct attraction to a destination; observed paths deviate a median 0.30 m, mostly in turns.Fig. 1 depicts routes and observed paths for through, left-turning, and right-turning vehicles.Source: 1) Agent costs · pp. 3–4, Figure 1 · p. 3
  • Interaction costs keep an IDM-based gap to the same-stream leader, penalize predicted close approaches with crossing traffic, and impose a contact barrier against every other vehicle.The conflict term is a smooth surrogate for both vehicles occupying the same space within the next 1.5 s and avoids division by small speeds.Source: 1) Agent costs · p. 4
  • The following and conflict terms use partners’ delayed states, while the agent’s own state and contact barrier are not delayed.The IDM enters as a cost term rather than replacing the vehicle dynamics.Source: 1) Agent costs · pp. 4–5

2) Constraints:

The game starts each agent from its measured state and restricts curvature and acceleration to an admissible set.

2) Constraints:

  • Each agent starts from its measured state, with curvature limited to ±0.25 m^-1 and acceleration limited to −6 to 3.5 m/s^2.These control bounds contain all observed kinematics; state limits are enforced softly by cost terms.Source: 2) Constraints · p. 5

3) Smooth approximations:

The solver uses smooth approximations and a positive-semidefinite Gauss–Newton Hessian approximation to handle costs that may otherwise be nonsmooth or indefinite.

3) Smooth approximations:

  • Nonsmooth operators are replaced with smooth approximations, including softplus, smoothstep, a quadratic hinge, Gaussian kernels, and periodic heading error.For residual costs, the solver keeps the positive-semidefinite Gauss–Newton Hessian component and the exact gradient.Source: 3) Smooth approximations · p. 5

4) Agent’s best response:

Each driver chooses a plan that minimizes its own cost given the other drivers’ plans. The best response is an optimal-control problem in which delayed partner states are treated as given data.

4) Agent’s best response:

B. Nash equilibrium

A Nash equilibrium is a set of plans from which no driver can gain by deviating alone. The algorithm returns a candidate local Nash equilibrium, and nonlinear games can have multiple local equilibria whose selection depends on initialization and parameters.

B. Nash equilibrium

  • At a Nash equilibrium, no driver can gain by deviating alone from the other drivers’ plans.Source: B. Nash equilibrium · p. 5
  • Different passing orders or yielding decisions can define multiple nonlinear-game equilibria, and the reached equilibrium depends on the warm start and parameters.Source: B. Nash equilibrium · p. 5
  • The iterative framework returns a candidate local Nash equilibrium after rolling out, linearizing, solving the local game, and updating the nominal controls.Source: B. Nash equilibrium · p. 5

IV. SOLUTION ALGORITHM

The iterative linear–quadratic game jointly corrects vehicle plans so each accounts for the others’ responses. Its local feedback Nash solver computes coupled feedback strategies through a backward recursion while respecting control bounds and delayed perception.

IV. SOLUTION ALGORITHM

  • Joint corrections make each vehicle’s plan account for how the others respond, unlike separate single-vehicle optimizations.Source: §IV · p. 5

A. Solution of the local feedback LQ Nash game

B. Numerical stabilization and convergence

The solver stabilizes its local game approximations and uses a safeguarded line search, with convergence checked separately from equilibrium verification. Calibration uses time-aligned trajectory error and block-coordinate Bayesian optimization on selected multi-vehicle interaction windows.

B. Numerical stabilization and convergence

  • Nonconvex interaction, contact, and limit costs use Gauss–Newton curvature, with diagonal regularization and an adaptive Levenberg–Marquardt term.Source: B. Numerical stabilization and… · p. 6
  • The final convergence check requires state trajectories and costs to meet tolerances for three consecutive iterations; nonconverged runs may exhaust the budget, stall, or diverge.The best iterate is kept when the run stalls, and equilibrium status is verified separately.Source: B. Numerical stabilization and… · p. 6

V. CALIBRATION

  • The calibration objective is time-aligned RMS position error, so arriving late is penalized roughly by the distance traveled during the delay.Interaction parameters are calibrated only for agents with a partner within 20 m and otherwise remain at their defaults.Source: §V · p. 6

A. Bayesian calibration procedure

  • Cases use an ego-centered window around the junction and include up to five nearby vehicles recorded throughout it; map routes and road boundaries come from the map.Source: A. Bayesian calibration procedure · pp. 6–7
  • Block-coordinate Bayesian optimization updates one agent’s parameters at a time, re-solving the full game and accepting only updates that lower case error.Two sweeps revisit agents after their partners have been updated.Source: A. Bayesian calibration procedure · p. 7

B. Statistical significance of the behavioral mechanisms

The behavioral mechanisms are tested through nested-model comparisons, since strongly autocorrelated trajectory residuals make per-parameter t-statistics uninformative. The study also describes the dataset and calibrates each comparison model under a matched procedure where applicable.

B. Statistical significance of the behavioral mechanisms

  • Median lag-1 autocorrelation of 0.96 motivates nested-model tests comparing full and restricted models on the same cases, budget, and seeds.The tests use case-clustered standard errors and sign measures, and assess interaction terms and perception delay in-sample and on held-out agents.Source: B. Statistical significance of the… · p. 7

A. Data

  • The TGSIM dataset provides 370 training cases with 819 agents and 42 held-out cases with 96 agents across four Washington, D.C., intersections.Cases are groups of vehicles interacting within a common time window; the 412 usable cases have a median junction-box window of 7.9 s.Source: A. Data · p. 7

B. SUMO setup

  • SUMO is calibrated per vehicle using nine vehicle-type parameters, the same cases and optimization budget as the game, and ten random seeds per stochastic run.It uses a network built from the same scene geometry, the Krauß car-following model, and SUMO junction logic.Source: B. SUMO setup · p. 7

C. SMIRL setup

  • SMIRL learns one shared linear reward from the 819 training vehicles and predicts the minimum-cost sampled trajectory, with regularization selected by cross-validation.Its candidate paths follow the game’s lane route, while interaction features extrapolate other agents at constant velocity from their initial states.Source: C. SMIRL setup · p. 7

D. Independent and cooperative variants of the model

  • Independent and cooperative variants share the game’s dynamics, costs, constraints, solver, warm start, and iteration budget, isolating differences in how vehicles’ objectives are coupled.Independent control treats others’ warm-start trajectories as fixed predictions; cooperative control minimizes the team cost, allowing a vehicle to accept a higher individual cost.Source: D. Independent and cooperative… · p. 7, U IOC · pp. 7–8

U IOC

  • Both variants are calibrated separately with the game’s procedure, so accuracy differences reflect coupling rather than parameters tuned only for the game.Source: U IOC · p. 8

E. Comparison setup

The comparison evaluates models under shared information settings and reports trajectory errors, close approaches, and equilibrium checks. Calibration analyses also examine parameter identifiability and how weakly identified weights and contact barriers affect closed-loop conflicts.

E. Comparison setup

  • Held-out comparisons cover 42 cases and 96 agents, while ten-fold cross-validation uses the 370 training cases with group means learned from other folds.Equilibrium checks fix other players’ controls and minimize each player’s cost over its own controls from the solution and two perturbations.Source: E. Comparison setup · p. 8
  • A group-average perception delay does not transfer to held-out drivers, so those predictions use zero delay.Using the group-mean delay increased position errors from 4.33 to 4.55 m with the end location known and from 7.42 to 7.66 m when it was unknown.Source: E. Comparison setup · p. 8, A. Calibration · p. 9
  • Models receive the same initial states, lane routes, and destination information, with no observed positions after t = 0; the route-only setting places the terminal target where the vehicle’s speed reference would carry it.Errors are time-aligned RMS measures for position, speed, acceleration, and jerk, with equal weight per agent; ADE and FDE are also reported.Source: E. Comparison setup · p. 8

A. Calibration

  • Position trajectories identify speed-tracking, acceleration, and terminal-position weights for at least 94% of agents, while the desired speed and delay are identifiable only in prediction.The gap-keeping weight is identified for followers; the conflict and curvature weights are not identified by position because recorded crossings stay far apart and curvature changes smoothness rather than location.Source: A. Calibration · p. 9
  • Raising the conflict weight from 100 to 1000 reduced collisions from 43% to 12%, widened gaps, and softened braking as vehicles adjusted earlier.Collisions fell in 10 of 12 pairs and vanished in six.Source: A. Calibration · p. 9, Table III · p. 9
  • The curvature weight lowered peak lateral acceleration in turns from 4.05 to 1.89 m/s2, while observed turns had a median peak of 1.59 m/s2.The observed-turn interquartile range was 1.15–2.04 m/s2, matching a curvature weight of 100.Source: A. Calibration · p. 10
  • Setting the weakly identified contact and curvature weights from safety and comfort targets reduced collisions from 43% to 12% and improved accuracy from 7.45 to 7.29 m in cross-validation over all 819 agents.This setting also widened gaps, softened braking, and produced human-like peak lateral acceleration of 1.89 m/s2.Source: Table III · p. 9, A. Calibration · p. 10
  • 0 of 93 scenarios collided with the rectangular contact barrier, without deadlock or refitting, though more solves stopped at the 50-iteration limit before converging.The held-out accuracy effects were −0.01 m with the end point given and −0.10 m without it; the confidence intervals included zero.Source: Table III · p. 9, A. Calibration · p. 10

B. Held-out comparison of the five models

With only the route known, the game has the lowest held-out position and speed errors and remains more accurate than the alternatives in cross-validation. Its advantage over independent and cooperative control is concentrated among vehicles that interact.

End location known, arrival time unknown

  • With end location known, mean position errors span 4.09–4.41 m with no significant differences, but the game follows observed speed 21% and acceleration 58% more closely than SUMO.The game also has lower acceleration and jerk errors than SMIRL by 11% and 33%, with equal speed errors.Source: B. Held-out comparison of the five… · p. 10

End location unknown

  • 7.42 m position error and 2.04 m/s speed error make the game best among five models when only the route is known.It is 4.27 m more accurate than SUMO and more accurate than independent OC by 0.37 m, cooperative OC by 0.49 m, and SMIRL by 0.61 m.Source: B. Held-out comparison of the five… · p. 10, Figure 4 · p. 11

Cross-validation

Effect of strategic coupling

  • The game’s cross-validation advantage over independent and cooperative OC is larger for agents with an interaction partner: 0.47 m and 0.42 m versus 0.07 m and 0.15 m.All three models fit training trajectories almost equally well, with errors of 1.00, 1.03, and 1.05 m, respectively.Source: B. Held-out comparison of the five… · p. 10

Effect of driver heterogeneity

  • For a vehicle with unknown parameters, the group mean is the best estimate, while correlated draws better reproduce population variability when the end location is unknown.Group-mean vehicles averaged 3.72 m/s versus 3.20 m/s recorded; correlated draws averaged 3.00 m/s and reproduced 92% of recorded mean-speed spread versus 81%.Source: B. Held-out comparison of the five… · pp. 10, 12, Table VI · p. 12

Interpreting the errors

  • A single prediction from the initial state has 7.42 m error, with about 90% of squared error along the path and some timing error attributable to unrecorded signal phases.The game’s ADE and FDE are 3.02 and 8.41 m at 6 s.Source: B. Held-out comparison of the five… · p. 12

Interaction behaviors and safety

  • For held-out car-following pairs, median RMS gap error is 2.1 m; the model reproduces the passing order in 7 of 8 gap-acceptance pairs with median PET error of 0.25 s.Among 90 held-out agent pairs, the game produced at most one close approach, compared with four for SMIRL.Source: Figure 5 · p. 11, B. Held-out comparison of the five… · p. 12

C. Computation time and convergence

The game solves typical held-out interactions faster than real time, and converged solutions are verified as approximate local equilibria. The conclusion reports predictive advantages, recommends setting unidentified parameters using safety and comfort targets, and identifies further validation and extensions as future work.

C. Computation time and convergence

  • 2.1 s is the median computation time per held-out case with unknown end location, faster than the 8.1 s for SMIRL; SUMO takes 0.7 s.For the game, time rises from 0.4 s for one vehicle to 10.4 s for four; with end location known, the median is 7.4 s.Source: C. Computation time and convergence · p. 12
  • Converged held-out solutions have median relative Nash gap below 10^-6, and 93–94% of players cannot reduce their cost by more than 1% through a unilateral change.The solutions are approximate local equilibria; independent OC plans have gaps 50 to 250 times larger.Source: C. Computation time and convergence · p. 12

VIII. CONCLUSION

  • On route-only prediction, cross-validated position error reductions over SUMO, SMIRL, independent OC, and cooperative OC are 3.61, 0.93, 0.33, and 0.32 m, respectively.The game reproduces the 819 calibrated vehicles’ trajectories with mean error of 1.00 m.Source: §VIII · p. 12
  • For simulators, the conclusion recommends setting unidentifiable conflict and steering weights from safety and comfort targets; a stiffer contact barrier removed collisions in tight closed-loop conflicts.Drawing simulated drivers’ parameters from calibrated drivers in their group reproduces observed traffic variability.Source: §VIII · p. 13
  • Future work includes validation at further sites with signal-phase data, calibration on closer interactions, embedding the game in a full simulator, and extending it to other road users.Source: §VIII · p. 13

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