Differential quality game with fuzzy information for assessing financial resources for air quality monitoring in cities

Abstract

Rapid urbanization intensifies pressure on city air quality, making costeffective monitoring a governance priority. Existing sensor-placement approaches optimize coverage but often ignore strategic behavior of polluters and budget uncertainty, leading to fragile deployments. We propose a decision model that allocates monitoring funds via a bilinear differential game with fuzzy information between an environmental defender and a polluter. Unlike linear differential games solvable via the Cauchy formula, bilinear dynamics and non-measurable adversary strategies require a novel discreteapproximation method within a positional game scheme. The model captures dynamic financial interactions through membership functions and yields analytical characterizations of the defender's preference set and optimal pure strategies. Computational experiments on realistic scenarios illustrate stable funding regimes and support actionable guidance for urban planners: how much to invest, when, and where to expand monitoring stations to achieve resilient oversight under uncertainty. The framework can be embedded in intelligent decision-support tools for smart-city environmental management.