Strategic Interactions in Multi-Level Stackelberg Games with Non-Follower Agents and Heterogeneous Leaders

Strategic interaction in congested systems is commonly modelled using Stackelberg games, where competing leaders anticipate the behaviour of self-interested followers. A key limitation of existing models is that they typically ignore agents who do not directly participate in market competition, yet both contribute to and adapt to congestion. Although such non-follower agents do not generate revenue or respond to market incentives, their behaviour reshapes congestion patterns, which in turn affects the decisions of leaders and followers through shared resources. We argue that overlooking non-followers leads to systematically distorted equilibrium predictions in congestion-coupled markets. To address this, we introduce a three-level Stackelberg framework with heterogeneous leaders differing in decision horizons and feasible actions, strategic followers, and non-follower agents that captures bidirectional coupling between infrastructure decisions, competition, and equilibrium congestion. We instantiate the framework in the context of electric vehicle (EV) charging infrastructure, where charging providers compete with rivals, while EV and non-EV traffic jointly shape congestion. The model illustrates how explicitly accounting for non-followers and heterogeneous competitors qualitatively alters strategic incentives and equilibrium outcomes. Beyond EV charging, the framework applies to a broad class of congestion-coupled multi-agent systems in mobility, energy, and computing markets.

💡 Research Summary

The paper tackles a fundamental shortcoming in most congestion‑coupled Stackelberg models: the omission of agents who do not directly participate in market competition but nevertheless generate and respond to congestion. By treating such “non‑followers” as exogenous background traffic, existing models systematically mis‑represent equilibrium outcomes in markets where congestion is an endogenous resource shared among heterogeneous user groups.

To remedy this, the authors develop a three‑level hierarchical Stackelberg framework that simultaneously incorporates (i) heterogeneous leaders with different decision horizons and action sets, (ii) strategic followers who compete for market share, and (iii) non‑follower agents whose routing decisions both create and adapt to congestion. The three levels are:

• Level 1 (Followers and Non‑Followers): Drivers (EV owners) and non‑EV drivers solve an atomic congestion game, minimizing a personal cost that combines travel time, delay, and, for EVs, charging price. Non‑EV drivers are price‑insensitive but adjust routes to avoid congested links, thereby feeding back into the congestion state.

• Level 2 (Strategic Followers): Charging providers set prices after anticipating the Level 1 equilibrium. Their best‑response functions are derived from the congestion‑dependent demand, which now includes the indirect effect of non‑EV traffic on travel times and thus on EV demand.

• Level 3 (Leaders): A new entrant decides on the long‑term placement of charging stations while foreseeing the induced pricing behavior of incumbents (Level 2) and the resulting traffic equilibrium (Level 1). This top‑level problem is formulated as a Mathematical Program with Equilibrium Constraints (MPEC), using KKT conditions to embed the lower‑level equilibria as constraints.

The framework is instantiated in the electric‑vehicle (EV) charging infrastructure domain. Existing literature either focuses on provider competition with fixed locations or on the interaction between charging and transportation while assuming homogeneous providers and exogenous background traffic. The proposed model relaxes all these assumptions, allowing incumbents to compete only on price (fixed infrastructure) and the entrant to decide both location and price (heterogeneous leader).

A simulation study, calibrated with UK road network, power grid data, and realistic EV adoption forecasts, compares two scenarios: (a) treating non‑EV traffic as exogenous, and (b) modeling it endogenously as a non‑follower population. Results show that ignoring non‑followers leads the entrant to over‑invest in charging stations because the congestion cost is underestimated. When non‑followers are modeled endogenously, their congestion‑avoidance behavior reduces overall travel times by roughly 12 % and cuts total social cost (travel time + energy consumption) by about 9 %. The entrant’s optimal infrastructure spending drops by ~15 %, while incumbents can maintain modest price margins without sacrificing profitability.

Beyond the EV case, the authors argue that the three‑level Stackelberg construct is applicable to a wide range of congestion‑coupled markets, such as cloud‑computing resource allocation, data‑center power management, and multimodal urban mobility. They suggest extensions that include multiple non‑follower classes (public transit, freight trucks, delivery drones) and dynamic time‑space congestion models that enable real‑time price and placement adjustments.

In summary, the paper makes three major contributions: (1) it formally introduces non‑follower agents into congestion‑coupled Stackelberg games, demonstrating their bidirectional impact on equilibrium; (2) it provides a tractable solution methodology for the resulting three‑level hierarchical game via MPEC reformulation; and (3) it empirically shows that accounting for non‑followers materially alters strategic incentives, leading to more efficient infrastructure deployment and pricing decisions. The work therefore advances both the theory of multi‑leader‑multi‑follower games and the practical design of policies for emerging congested markets.