Empirically Calibrated Differential Games for Simulating Driver Interactions

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

cs.MAOct 23, 2026

Empirically Calibrated Differential Games for Simulating Driver Interactions

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

Organizations: Marco · Department of Civil and Environmental Engineering, Northwestern University, Evanston, IL 60208 USA

Abstract

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.

Figures & tables

Calibrated per vehicle
Symbol	Meaning	Appears in	Search range / value
$w_{v}$	weight of tracking the reference speed	( 5 )	$[1,1000]$ (log scale)
$\bar{v}$	desired cruise speed	( 6 )	$[1,15]$ m/s
$w_{\kappa}$	weight of lateral acceleration (steering effort)	( 9 )	$[0.1,100]$ (log scale)
$w_{a}$	weight of longitudinal acceleration	( 9 )	$[1,500]$ (log scale)
$b_{p}$	weight of reaching the destination position	( 3 )	$[10,2000]$ (log scale)

TABLE I: Model Parameters: The Ten Behavioral Parameters Calibrated per Vehicle, and the Main Fixed Constants

Fig. 1: Map-derived lane routes

\Gamma_{i}

and observed paths of a through, a left-turning, and a right-turning agent, each with the median maximum deviation of its movement.

Fig. 2: Foggy Bottom study area (schematic) and the four analyzed intersections.

Fig. 3: Observed (solid) and calibrated (dashed) trajectories in two training cases; colors distinguish vehicles. (a) Signalized intersection, four vehicles, mean RMS error 0.81 m. (b) Unsignalized intersection, three vehicles, mean RMS error 0.89 m.

Parameter	Agents	Not identifiable	Error range (m)	Error range (m)	Clear minimum	Respond	Clear minimum	Assessment (reason)
		End point given		End point not given		Interaction measures
$w_{v}$	120	0%	1.92	0.96	21%	–	–	identifiable
$\bar{v}$	120	37%	0.29	6.65	32%	–	–	weak in calibration (the end point fixes the arrival); dominant in prediction
$w_{\kappa}$	120	50%	0.15	0.12	2%	–	–	weak (the lane cost fixes the path); matters in simulation
$w_{a}$	120	6%	0.88	0.94	10%	–	–	identifiable
$b_{p}$	120	5%	1.49	1.77	12%	–	–	identifiable

TABLE II: Practical Identifiability of the Calibrated Parameters (120 Training Agents in 60 Cases)

Conflict weight $w_{c}$
Value	Collisions	Min. gap (m)	PET (s)	Peak decel. (m/s 2 )	Peak lat. acc. (m/s 2 )
10	48%	$-0.15$	1.0	2.54	3.04
31.6	47%	$-0.01$	1.0	2.60	3.05
100	43%	$0.39$	1.1	2.51	3.03
316	28%	$0.93$	1.2	2.29	3.06
1000	12%	$2.11$	1.3	1.86	3.13

TABLE III: Closed-Loop Effect of

w_{c}

w_{\kappa}

, and the Contact Barrier in 93 Negotiation Scenarios

Fig. 4: Held-out comparison (42 cases, 96 agents): per-agent RMS errors of time-aligned position, speed, and acceleration, and single-thread computation time per case (right, logarithmic). Nash, independent OC, and cooperative OC are three couplings of our model (blue). Boxes show the quartiles, whiskers extend to 1.5 times the interquartile range, and diamonds mark the means, whose values are printed below the boxes. (a) End location known, arrival time unknown. (b) End location unknown.

Fig. 5: Interaction behaviors captured by the model. Top: paths on the intersection map; bottom: the interaction over time. (a) Car following in a held-out case: the predicted bumper gap to the leader follows the observed one as the follower closes up (RMS gap error 0.63 m). (b) Gap acceptance in a held-out case: the second vehicle approaches slowly and enters the conflict point after the first has cleared it; the observed post-encroachment time of 6.2 s is predicted as 6.1 s. (c) Negotiation in a closed-loop scenario (merging, simultaneous nominal arrival): with a weak conflict term (

w_{c}=10

, dotted) the vehicles come closer than one car length and their outlines overlap; with

w_{c}=1000

(solid) the merging vehicle adjusts early and the minimum distance is 6 m. The outlines mark both vehicles at their closest approach.

					Position		Close
Model	Position	Speed	Accel.	Jerk	partner	no partner	appr.
(1) End location known, arrival time unknown
Nash (our model)	4.325	1.602	0.759	0.868	4.471	4.102	1
Independent (our model)	4.414	1.653	0.778	0.871	4.605	4.121	1
Cooperative (our model)	4.089	1.565	0.761	0.866	4.193	3.931	0
SMIRL	4.161	1.559	0.855	1.304	4.536	3.587	4

TABLE IV: Held-Out Mean RMS Errors of the Five Models (96 Agents) and Close Approaches (90 Agent Pairs)

Model	All (819)	With partner (521)	Without partner (298)
Nash (our model)	7.45	7.62	7.16
Independent (our model)	7.78	8.09	7.23
Cooperative (our model)	7.77	8.04	7.31
SMIRL	8.38	8.55	8.10
SUMO	11.06	10.68	11.72
Nash $-$ Independent	$-0.33$ [ $-0.44$ , $-0.22$ ]	$-0.47$ [ $-0.65$ , $-0.31$ ]	$-0.07$ [ $-0.14$ , $-0.00$ ]

TABLE V: 10-Fold Cross-Validation Over the 370 Calibration Cases, End Location Unknown: Position RMS Error (m) and Paired Difference, Nash Minus the Other Model

(a) RMS error per vehicle
	End point and time given			End point unknown
Parameters	Pos.	Speed	Accel.	Pos.	Speed	Accel.
Group mean	2.47	1.02	0.68	7.43	2.04	0.74
Correlated draw	2.68	1.14	0.79	7.04	2.15	0.89
	Mean	SD	Ratio	Mean	SD	Ratio
Recorded	3.20	1.67	1	3.20	1.67	1

TABLE VI: Held-Out Errors and Speeds of the Game Under Different Parameter Assignments (96 Agents)

	Computation time				Agents by solver outcome
Model	Mean (s)	Median (s)	90th pct. (s)	RTF	Conv.	Budget	Stalled
(1) End location known, arrival time unknown
Nash (our model)	12.20	7.42	30.83	0.49	68	15	13
Independent (our model)	11.69	6.25	20.75	0.42	63	17	16
Cooperative (our model)	11.92	5.61	20.05	0.37	76	14	6
SMIRL	7.56	5.97	14.86	0.71	–	–	–

TABLE VII: Single-Thread Computation Time per Held-Out Case (42 Cases) and Convergence of the Optimization-Based Models (96 Agents)

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Game Theory Agent Trajectory Analysis Autonomous Driving

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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.

Abstract

Human Behavior Simulation

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