We develop a geometric theory of incentives that unifies games, nudges, language, market structure, and collective cascades within a single framework. Canonical strategic environments—such as the Prisoner’s Dilemma, Stag Hunt, Chicken, and Harmony—are shown to correspond to open regions in payoff space, separated by low-codimension indifference boundaries across which equilibrium structure changes discontinuously. We formalize nudging as payoff engineering: minimal deterministic or stochastic perturbations designed to move a system across these boundaries. This perspective accommodates benevolent nudging, adversarial manipulation, linguistic reframing, and institutional design as instances of the same underlying operation. Extending the analysis to N-player settings, we show how threshold effects and strategic complementarities give rise to phase transitions and, dynamically, to self-organized criticality near coordination boundaries. Risk, noise, and variance are shown to matter only insofar as they deform incentive geometry, clarifying when stochastic interventions can and cannot alter strategic type. The framework yields concrete design principles for robustness and defensive engineering, emphasizing distance from critical boundaries, control of incentive gradients, and resistance to adversarial nudging. Incentive geometry thus provides a unifying lens for understanding robustness, escalation, and coordination across economic, institutional, and social systems.