As a fundamental tool for interval-valued intuitionistic fuzzy sets (IVIFSs), the score function (SF) plays a pivotal role in quantifying interval-valued intuitionistic fuzzy values (IVIFVs) and facilitating their comparative analysis. However, a notable limitation of existing SFs is their potential to assign identical scores to distinct IVIFVs, thereby compromising the discrimination capability. To address this challenge, this study introduces IVIFS-SF, a novel score function grounded in prospect theory, and proposes two innovative assessment methodologies. First, we employed prospect theory to develop an interval-valued evaluation method (IVEM), which converts the interval into a crisp number. Second, using IVEM, we developed the new score function IVIFS-SF and present its properties. Third, we put forward pass rate and variance as metrics to analyze and compare SFs. Rigorous comparative analysis demonstrated that IVIFS-SF achieves superior performance in both pass rate and variance metrics when benchmarked against existing state-of-the-art SFs. Furthermore, sensitivity analysis confirmed the robustness of IVIFS-SF across the parameter spectrum of prospect theory. Empirical case studies revealed that while IVIFS-SF identifies the same optimal alternative as competing SFs, it exhibits the highest variance among them, suggesting enhanced discriminative power.