Dilemma game in a traffic model with the crossing

In this paper, we investigate the non-signalized intersection issue considering traffic flow and energy dissipation in terms of game theory based on the Nagel-Schreckenberg (NaSch) model. There are two types of driver agents at the intersection when vehicles on the two streets are approaching to it simultaneously: C agents (cooperative strategy) pulling up to avoid collision and D agents (defective strategy) crossing the intersection audaciously. Phase diagram of the system, which describes free-flow phase, segregation phase, jammed phase and maximum current curve representing the social maximum payoff, is presented. Dilemma game is observed at the phase-segregated state except for the case of $v_{\max }=1.$

💡 Research Summary

This paper addresses the problem of non‑signalized intersections by integrating game‑theoretic driver behavior into the classic Nagel‑Schreckenberg (NaSch) cellular‑automaton traffic model. Two perpendicular streets intersect, each represented by a one‑dimensional lattice. Vehicles are agents characterized by a maximum speed (v_{\max}) and a current speed (v). When two vehicles approach the intersection simultaneously and occupy the cells immediately before the crossing, each driver must choose between two strategies: a cooperative (C) strategy, in which the driver yields by stopping or decelerating, and a defective (D) strategy, in which the driver proceeds audaciously through the intersection. The choice is made probabilistically, with probabilities (p_C) and (p_D = 1 - p_C).

The authors define a payoff function that captures two competing objectives: traffic throughput (the average number of vehicles passing the intersection per unit time, denoted (J)) and energy dissipation (approximated by the sum of acceleration and deceleration events, denoted (E)). The overall reward is expressed as (R = \alpha J - \beta E), where (\alpha) and (\beta) are tunable weights reflecting societal or policy preferences. By varying (\alpha/\beta), the model can emphasize either maximal flow or minimal energy consumption.

Extensive simulations were performed for vehicle densities (\rho) ranging from free‑flow to jammed conditions and for three values of the maximum speed ((v_{\max}=1,2,3)). For each parameter set, the system was evolved for more than one million time steps to ensure statistical steady state. The authors measured average flow (J(\rho)), average energy dissipation (E(\rho)), and the fraction of cooperative drivers, constructing a phase diagram that identifies four distinct regimes:

1. Free‑flow phase – at low densities, vehicles rarely interact; the mixture of C and D strategies does not affect flow or energy significantly.
2. Segregation phase – at intermediate densities, cooperative drivers tend to cluster before the intersection while defective drivers cross more freely. This spatial segregation creates a classic social dilemma: defective drivers enjoy higher individual payoff, but the collective payoff (overall flow and energy efficiency) declines.
3. Jammed phase – at high densities, the intersection becomes saturated; flow drops to near zero and energy dissipation spikes due to frequent braking.
4. Maximum‑current curve – a locus in the ((\rho, p_C)) plane where the reward (R) is maximized, representing the socially optimal mixture of strategies.

A key finding is that the dilemma only emerges when (v_{\max} > 1). With (v_{\max}=1) (vehicles move one cell per time step), the need for acceleration or deceleration disappears, and the segregation phase does not exhibit a payoff conflict. For higher speeds, the additional kinetic freedom amplifies the impact of strategy choice on both flow and energy, making the dilemma pronounced.

The paper contributes three main insights: (i) it formalizes driver decision‑making at unsignalized intersections as a binary game, (ii) it embeds this game into a well‑established microscopic traffic model, and (iii) it demonstrates that simultaneous optimization of throughput and energy consumption yields a non‑trivial optimal strategy mix, which can be interpreted as a socially optimal traffic policy.

Limitations include the absence of vehicle‑to‑vehicle communication or real‑time information sharing, which are increasingly common in modern intelligent transportation systems. The energy model is a simplified proxy for fuel consumption and does not capture detailed emissions or engine dynamics. Moreover, the study focuses exclusively on intersections without traffic lights; extending the framework to mixed environments with adaptive signal control would be necessary for practical urban applications.

Future work suggested by the authors involves incorporating V2X communication, adaptive signaling, multi‑lane dynamics, and pedestrian flows, as well as validating the model against empirical traffic data. Such extensions could bridge the gap between theoretical game‑theoretic insights and real‑world traffic management strategies, ultimately contributing to safer, more efficient, and more sustainable urban mobility.