Capital Support under Strategic Equilibrium Uncertainty: Convex Safety, Endogenous Incentives, and Exact Network Computation

Research Description Can financial intervention make a banking system safer while simultaneously changing the strategic incentives that determine its stability? This study addresses a fundamental challenge in financial economics: capital support does not merely alter financial resources—it can reshape creditor behavior, transform Nash equilibria, and fundamentally change the geometry of systemic-risk prevention. Integrating financial network theory, game-theoretic equilibrium analysis, robust optimization, and exact computation, the research establishes three complementary mathematical results. Theorem C1 identifies conditions under which worst-equilibrium financial safety admits an exact convex rational-polyhedral representation, allowing minimum-cost protective interventions to be characterized rigorously. Theorem M1, the central theoretical contribution, reveals a more intricate mechanism. When committed capital support changes creditors' strategic incentives through a funded priority-payment structure, worst-equilibrium losses can become nonmonotonic, and the set of safe interventions can become nonconvex. The analysis establishes these phenomena over an open interval of model parameters and demonstrates an exact discrepancy between policies assessed using only pure Nash equilibria and those protecting against all relevant equilibria, including mixed strategies. Theorem N1 translates this framework into an exact computational procedure for directed interbank networks, combining financial clearing, complete two-player Nash-equilibrium enumeration, and globally optimal capital allocation over specified finite rational policy lattices. The computational investigation evaluates 41 synthetic financial-network configurations and 46,404 candidate capital allocations, identifying seven strict improvements over defined benchmark policies and 34 ties. These experiments provide reproducible, model-specific evidence of the framework's computational operation, rather than causal estimates from observed banking systems. The study's principal contribution is to establish a mathematically explicit distinction between financial intervention under fixed strategic incentives and intervention that endogenously changes equilibrium behavior. Its central message is consequential: financial stability cannot always be assessed by measuring the resources injected into a system; it may also require understanding how those resources change the strategic behavior of its participants. By identifying the conditions under which convex optimization remains valid—and demonstrating precisely how those conditions can fail—the research contributes a rigorous theoretical and computational foundation for equilibrium-aware financial risk analysis, capital-support design, and future investigations of systemic crisis prevention.