A thousand renewable energy producers enter a market. Each decides independently how much capacity to install, when to produce, and how much to sell. Their decisions interact through a single variable: the market price, which drops when everyone produces and rises when no one does. The individual optimization is straightforward. The collective outcome is not.
Campi and Wu model this as a mean-field game — the limit of an N-player stochastic differential game as N approaches infinity. In this limit, each producer treats the market as a continuum rather than tracking individual competitors. The equilibrium is characterized by a forward-backward stochastic differential equation: forward in time for the state dynamics (capacity evolves under investment, depreciation, and noise), backward in time for the value function (each producer optimizes against the anticipated future price trajectory).
The social planner enters as a Stackelberg leader — a higher-level optimizer who sets subsidies before the producers choose their strategies. The planner observes the mean-field equilibrium and adjusts the subsidy to steer aggregate capacity toward a social optimum.
The key finding: optimal subsidy design depends on prevailing market conditions. When capacity is scarce, subsidies accelerate installation. When capacity is excessive — overproduction from a previous subsidy round — the optimal policy is to reduce or withdraw support. The same subsidy that solves undercapacity in one regime causes overproduction in another.
This is the generic problem with static incentive policies. A fixed subsidy is optimal at one market state and harmful at another. The mean-field framework makes the state-dependence explicit: the equilibrium shifts continuously with the subsidy level, and the optimal subsidy shifts continuously with the equilibrium. Policy and market co-evolve.
The right subsidy is a function, not a number.