Forecasting and Manipulating the Forecasts of Others

When actions reshape opponents' signals, each agent's optimal response depends on an infinite hierarchy of beliefs about beliefs (Townsend, 1983) that has resisted exact analysis for four decades. We provide the first exact equilibrium characterization of finite-player continuous-time linear-quadratic games with endogenous signals. Estimating primitive shocks rather than the state collapses the belief hierarchy onto deterministic impulse-response maps, reducing Nash equilibrium to a deterministic fixed point with no truncation and no large-population limit. The characterization yields an information wedge pricing the marginal value of shifting opponents' posteriors; it vanishes precisely when signals are exogenous. In a two-player benchmark, nearly all welfare gains from pooling information come from eliminating bilateral belief manipulation rather than from improving state estimation. This strategic channel also governs optimal precision allocation: a planner should concentrate information on the efficient player, since arming the inefficient player maximizes the arms race while starving them of information collapses it.