N-firm oligopolies with pollution control and random profits

An n-firm oligopoly is introduced in which, in addition to production levels, the pollution emissions are also included. A regulator cannot monitor individual emission volumes of firms, so uniform incentives are introduced to firms to reduce pollution concentrations. The regulator cannot observe the exact concentrations, so the incentives are also uncertain. Therefore, each firm considers random profit with expectations that it is maximized by minimizing variances or standard deviations. This idea leads to a multi-objective optimization problem for each firm, so two different concepts are applied as a solution. The unique positive Nash equilibrium is proven in all cases examined, and the effects of the environmental tax rate on industry output, prices, and pollution emission levels are analyzed.