Mean-field games of controls with reflected state processes

Establish a general theory of mean-field games of controls in the presence of reflected state processes, including existence and characterization results beyond the specific reflected Stackelberg pollution model considered here.

Background

Mean-field games of controls differ from standard mean-field games because the population distribution depends on the agents’ control processes rather than only on their state distributions. Introducing state reflection imposes a hard state constraint through a regulating or local-time process, creating additional analytical difficulties in the associated stochastic control and equilibrium problems. The introduction notes that prior work has addressed reflected diffusions and some mean-field models, but that the combination of control-based mean-field interaction and reflected state dynamics remains unresolved in general.

The paper develops a particular finite-horizon mean-field Stackelberg game for production and carbon-emission reduction with a reflected cumulative-emission state. It constructs an approximate Nash equilibrium for heterogeneous followers and an approximate Stackelberg equilibrium for the leader, thereby addressing a specialized instance of the broader unresolved problem rather than providing a general theory for all mean-field games of controls with reflected states.