A Minimal-Assumption Approach to Deriving a Unique Utility Function: Measuring the Psychological Utility of COVID-19 Vaccinations Worldwide

“Psychic utility or satisfaction could scarcely be defined, let alone be measured”, a Nobel Prize laureate in Economic Sciences once said. Utility or objective functions are important for optimisation and strategic analysis across different fields, but it is challenging to identify the functions. Here, we present a pure mathematical method to solve an unknown utility function from a partial differential equation (PDE) β ² U _xx = U _yy under minimal assumptions, where the solution is proved unique, stable, reliable, valid, and generalisable worldwide. A discount function measuring both pure time preference and shadow price can be derived from utility to infer vaccine hesitancy. A probability measure and a decision weight function are formulated by utility to compute the likelihood of vaccination behaviours. We prove the uniqueness and stability of the utility function in theory. For empirical evaluation using global data from 2020 to 2024, the psychometric measurement demonstrates strong internal-consistency reliability (|ρ|≥0.957), and strong convergent (ρ>0.83), discriminant (ρ=±0.2673) and predictive (ρ=0.8816, R ² =0.984, RMSE=6.63×10 ^-5 ) validity. Our model addresses anomalies such as bounded rationality and inconsistent preferences. From first principles, we use the PDE to prove the properties of prospect theory's value function and the Friedman-Savage hypothesis. Despite its simplicity, this novel approach enables rich analysis and effectively handles complexity. Our ultimate goal is to derive a general equation to establish a theory of everything in economic science, under minimal assumptions and the least prerequisite data for modeling.