Paper: arXiv 2603.12140

Authors: Sam Babichenko

Abstract

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper resolves a decades-old open problem in game theory and filtering by providing an exact equilibrium characterization for continuous-time linear-quadratic games with endogenous signals via a novel ’noise-state’ technique. While mathematically rich and highly novel conceptually, it is entirely theoretical with minimal empirical or market data validation. A GitHub repository with replication code is provided, enhancing its methodological reproducibility.

Research Flowchart

  flowchart TD
    A["<b>Research Goal</b><br>Solve infinite belief hierarchy in continuous-time LQ games with endogenous signals"] --> B["<b>Key Innovation</b><br>Estimate primitive shocks instead of state variables"]
    
    B --> C["<b>Computational Process</b><br>Collapse belief hierarchy onto deterministic impulse-response maps<br><i>(Nash Eq = Deterministic Fixed Point)</i>"]
    
    C --> D["<b>Theoretical Outcome: Information Wedge</b><br>Quantify marginal value of shifting opponents' posteriors<br><i>(Wedge = 0 when signals are exogenous)</i>"]
    
    C --> E["<b>Application 1: Information Pooling</b><br>Welfare gains driven by eliminating bilateral belief manipulation, not better state estimation"]
    
    C --> F["<b>Application 2: Precision Allocation</b><br>Planner concentrates info on efficient player to collapse strategic 'arms race'"]

Cited References (24)

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  2. Equilibrium with signal extraction from endogenous variables (1991)
  3. Information Design: A Unified Perspective (2019)
  4. Games with Incomplete Information Played by ``Bayesian’’ Players, I–III . Part I . The Basic Model (1967)
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  6. Forecasting the Forecasts of Others in the Frequency Domain (2000)
  7. Confounding dynamics (2021)
  8. Rational Expectations Models with Higher-Order Beliefs (2025)
and 16 more
  1. Dynamic Higher Order Expectations (2017)
  2. Social Value of Public Information (2002)
  3. Efficient Use of Information and Social Value of Information (2007)
  4. Imperfect Common Knowledge and the Effects of Monetary Policy (2003)
  5. Information and volatility (2015)
  6. Team Decision Problems (1962)
  7. Economic Theory of Teams (1972)
  8. Decentralized Stochastic Control with Partial History Sharing: A Common Information Approach (2013)
  9. Linear–Quadratic Gaussian Games with Asymmetric Information: Belief Corrections Using the Opponents Actions (2023)
  10. Statistics of Random Processes I : General Theory (2001)
  11. Large population stochastic dynamic games: Closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle (2006)
  12. Probabilistic Theory of Mean Field Games with Applications I : Mean Field FBSDEs , Control, and Games (2018)
  13. Bayesian Persuasion (2011)
  14. Bayes correlated equilibrium and the comparison of information structures in games (2016)
  15. Implications of Rational Inattention (2003)
  16. The Strategy of Conflict (1960)

0 of 24 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.