Technological change in water use: A mean-field game approach to optimal investment timing

The need for clean water is expected to substantially increase while further reductions of water availability in sufficient quantity and quality are projected owing to climate change and anthropogenic activities. Accordingly, the debate on water security has recently intensified and reached the intergovernmental arena. Industry is, in particular, one of the largest (non-consumptive) water users, accountable for massive toxic wastewater discharges and facing stringent and costly environmental oversight. However, the management of reservoirs is intricate and operational research must be further expanded to design tools that enhance water security while improving operators’ profitability.

We therefore consider a game-theoretic framework to study the strategies adopted by a large group of similar producers sharing a water reservoir for their manufacturing activities. Each operator faces random demand for its outputs and chooses the optimal time to invest in a technology that ends its reliance on the reservoir. This technology introduces cost saving opportunities for the operator and benefits for the environment. Each producer therefore solves a so-called optimal stopping problem, and all problems are coupled through the reservoir level. We formulate the problem of finding a Nash equilibrium as a mean-field game (MFG) of optimal stopping. We then apply the model to the paper milling industry, an extensive water user facing a tightening of environmental regulations. This paper provides fresh insights into how to rethink the problem of technological change and water management, by offering an innovative application of operational research that builds on recent mathematical developments made in MFG theory.