Zeroth-Order Stackelberg Control in Combinatorial Congestion Games

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.

We introduce Calibrated Stackelberg Games (CSGs), a generalization of the standard Stackelberg Games (SGs) framework. In CSGs, a principal repeatedly interacts with an agent who (contrary to standard SGs) does not have direct access to the principal's action but instead best-responds to calibrated forecasts about it. This framework provides a powerful and realistic modeling tool that goes beyond assuming that agents use ad hoc and highly specified algorithms for interacting in strategic settings and instead builds on statistical foundations of forecasts and calibration. We show that in CSGs, despite both the principal and the agent having less information than in standard SGs, the principal's optimal utility remains upper and lower bounded by the Stackelberg value of the one-shot game, in both finite and continuous settings. Alongside CSGs, we develop stronger notions of calibration and corresponding algorithms that address two central challenges for calibration in game-theoretic environments. First, achieving point-wise calibration typically incurs an error that scales exponentially with the dimension of the strategy space. Second, the principal's convergence rate in CSGs depends critically on the adaptivity of the agent's calibration algorithm. To address these challenges, we establish a meaningful, efficiently achievable relaxation of calibration based on conditioning on best-response regions. This yields the first notion of calibration in games with a statistical rate that only depends on the number of agents' actions rather than the dimension of the principal's strategy space and that leads to no-swap regret for the agent. We further develop adaptive calibration algorithms for the agents that provide fine-grained, any-time calibration guarantees against adversarial sequences, enabling the principal to achieve faster convergence in CSGs.

We study two-player zero-sum games (TPZSGs) with bandit feedback under fairness constraints requiring every action to be played with probability at least $α/m$. Existing instance-dependent results target $\textit{pure}$ Nash equilibria, while fairness generically produces $\textit{mixed}$ equilibria, a harder learning target. Our key technical tool is a reparametrization: every fair strategy decomposes as $p = (α/m)\mathbf{1} + (1-α)\widetilde{p}$ with $\widetilde{p} \in Δ_m$, and substituting into the payoff form yields $p^{\top}Aq = \widetilde{p}^{\top}\widetilde{A} q$ for a fair payoff matrix $\widetilde{A} := (1-α)A + α\mathbf{1} c^{\top}$, where $c_j = \tfrac{1}{m}\sum_i A(i,j)$ is the column-mean vector. The fair game on $A$ is then equivalent to a standard zero-sum game on $\widetilde{A}$, so equilibrium existence, KKT structure, and LP basis stability reduce to classical results applied to $\widetilde{A}$. We derive the fair minimax value, fair Nash equilibrium, fair regret, and a clean dual representation showing the price of fairness is at most $α(1-1/m)$ and vanishes whenever the unconstrained equilibrium already has full support. Our main result is an $\widetilde{O}(T^{2/3})$ regret bound for an Explore-Then-Commit algorithm, $\texttt{Fair-ETC-TPZSG}$, applicable to general mixed fair equilibria, together with a discussion of why naive action elimination does not readily improve it. When the fair equilibrium has a single dominant action, equivalently when $\widetilde{p}^{\star}$ is a vertex of $Δ_m$, the bound sharpens to instance-dependent $\widetilde{O}(1/\widetildeΔ(α)^{2})$, where $\widetildeΔ(α)$ is the LP-margin gap.