Perturbed Utility Functionals: A Functional-Analytic Framework for Adaptive Decision-Making

We propose a functional-analytic framework for modeling bounded rationality by introducing perturbed utility functionals. By extending classical expected utility with a structured perturbation term, we capture behavioral deviations such as loss aversion and risk sensitivity while maintaining analytical tractability. We establish fundamental existence results and demonstrate the consistency of our framework by proving convergence to classical models in the limit of vanishing bias. Through numerical simulations, we illustrate two key findings: (i) in portfolio optimization, our framework captures non-linear "regime shifts" in risk appetite, and (ii) in sequential decision-making, it generates "emergent cautiousness, " allowing AI agents to navigate safely around high-risk states. This framework unifies descriptive behavioral insights with prescriptive optimization, offering a scalable pathway for integrating human-like heuristics into AI safety and control systems.