Chance-Constrained Correlated Equilibria for Robust Noncooperative Coordination

Abstract

Correlated equilibria enable a coordinator to influence the self-interested agents by recommending actions that no player has an incentive to deviate from. However, the effectiveness of this mechanism relies on accurate knowledge of the agents' cost structures. When cost parameters are uncertain, the recommended actions may no longer be incentive compatible, allowing agents to benefit from deviating from them. We study a chance-constrained correlated equilibrium problem formulation that accounts for uncertainty in agents' costs and guarantees incentive compatibility with a prescribed confidence level. We derive sensitivity results that quantify how uncertainty in individual incentive constraints affects the expected coordination outcome. In particular, the analysis characterizes the value of information by relating the marginal benefit of reducing uncertainty to the dual sensitivities of the incentive constraints, providing guidance on which sources of uncertainty should be prioritized for information acquisition. The results further reveal that increasing the confidence level is not always beneficial and can introduce a tradeoff between robustness and system efficiency. Numerical experiments demonstrate that the proposed framework maintains coordination performance in uncertain environments and are consistent with the theoretical insights developed in the analysis.

Figures (4)

• Figure 1: Illustration of the vertiport occupancy coordination scenario. Aircraft arrive through multiple approach queues, where each queue represents a self-interested agent that dispatches aircraft and determines which combination of vertiports to occupy or yield. A central coordinator broadcasts correlated recommendations that suggest vertiport usage while preserving agents' autonomy. The interaction forms a noncooperative coordination game in which agents may follow or deviate from the recommended actions.
• Figure 2: Illustration of CC-CE feasible-set contraction in the decision space $z \in \Delta(\mathcal{X})$ for two constraints $c \in \{1,2\}$. Uncertainty shifts the nominal constraints $m_c(z)$ inward by $q(\alpha)\sigma_{i(c)}$, yielding the chance-constrained boundaries $g_c(z;\alpha,\sigma_{i(c)})$. Constraint $c=1$ (blue) has small uncertainty and represents a structural bottleneck, while constraint $c=2$ (red) corresponds to an information bottleneck. In this illustration, the optimal solution $z^\ast$ lies at the intersection where both constraints are active, with dual sensitivities $\lambda_1$ and $\lambda_2$. This highlights that the value of reducing uncertainty depends jointly on the dual sensitivity and the uncertainty level.
• Figure 3: Sensitivity of coordination performance to the confidence level $\alpha$. $\gamma$ controls the severity of congestion penalties. For visualization, scores are standardized and normalized by the $\alpha=0.75$ case. When congestion penalties are large ($\gamma=1.5$), increasing $\alpha$ reduces performance. When penalties are smaller ($\gamma=1.02$), performance is not monotonic in $\alpha$. The lower panel zooms in on the smaller $\gamma$ values for clarity.
• Figure 4: Performance comparison between information acquisition strategies. System cost is normalized by the baseline CC-CE solution. The Infogain ($\lambda\sigma$) strategy consistently achieves the lowest expected system cost, showing that effective information acquisition must account for both constraint sensitivity and uncertainty magnitude. Average values are marked with $\circ$.

Theorems & Definitions (12)

• Definition 1: Correlated equilibrium
• Definition 2: Chance-constrained correlated equilibrium
• Theorem 1: $\sigma$--sensitivity
• proof
• Corollary 1: Constraint-level sensitivity
• proof
• Theorem 2: $\alpha$ sensitivity
• proof
• Corollary 2: $\alpha$ sensitivity as information gain
• proof