Forecasting and Manipulating the Forecasts of Others

TL;DR

This paper develops a recursive representation for finite-player dynamic games with private information, revealing how beliefs and policies evolve and interact, and analyzing strategic information manipulation.

Contribution

It introduces a recursive framework that explicitly characterizes beliefs, value gradients, and policies as deterministic functions, providing new insights into strategic information manipulation.

Findings

Beliefs and policies are deterministic impulse-response functions.

The information wedge explains strategic pooling and information allocation effects.

Signal precision influences policy rules and can cause separation failure.

Abstract

Finite-player dynamic games with dispersed private information are difficult because actions both move payoffs and reshape what opponents learn, generating hierarchies of beliefs about beliefs. This paper provides a recursive representation for this problem. The noise state records agents' beliefs about the underlying shocks that generate histories, so higher-order beliefs are generated by composition rather than tracked as separate state variables. In the canonical continuous-time LQG benchmark, the representation becomes explicit: beliefs, value gradients, and policy rules are deterministic impulse-response functions, and equilibrium is a deterministic fixed point in those functions. Any fixed point in the noise-state linear class is a Nash equilibrium against arbitrary admissible \(L^2\) deviations. The first-order system contains an information wedge, the shadow price of changing opponents' posteriors. In a two-player benchmark, the wedge explains why pooling gains are mostly strategic, why optimal precision allocation can starve an inefficient player of information, and why signal precision changes policy rules themselves, so separation fails.