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Forecasting and Manipulating the Forecasts of Others

Sam Babichenko

Abstract

In strategic environments with private information, evaluating a change in policy requires predicting how the equilibrium responds -- but when actions reshape opponents' signals, each agent's optimal response depends on an infinite hierarchy of beliefs about beliefs that has resisted exact analysis for four decades. We provide the first exact equilibrium characterization of finite-player continuous-time LQG games with endogenous signals. Conditioning on primitive Brownian shocks rather than the physical state -- a dynamic analogue of Harsanyi's common-prior construction -- collapses the belief hierarchy onto deterministic two-time kernels, reducing Nash equilibrium to a deterministic fixed point with no truncation and no large-population limit. The characterization yields an explicit information wedge $\mathcal{V}^i_t$ -- a deterministic Volterra process -- that prices the marginal value of shifting opponents' posteriors. The wedge vanishes precisely when signals are exogenous to controls, formally delineating the boundary where strategic belief manipulation matters, and provides a closed-form mapping from information primitives to equilibrium outcomes.

Forecasting and Manipulating the Forecasts of Others

Abstract

In strategic environments with private information, evaluating a change in policy requires predicting how the equilibrium responds -- but when actions reshape opponents' signals, each agent's optimal response depends on an infinite hierarchy of beliefs about beliefs that has resisted exact analysis for four decades. We provide the first exact equilibrium characterization of finite-player continuous-time LQG games with endogenous signals. Conditioning on primitive Brownian shocks rather than the physical state -- a dynamic analogue of Harsanyi's common-prior construction -- collapses the belief hierarchy onto deterministic two-time kernels, reducing Nash equilibrium to a deterministic fixed point with no truncation and no large-population limit. The characterization yields an explicit information wedge -- a deterministic Volterra process -- that prices the marginal value of shifting opponents' posteriors. The wedge vanishes precisely when signals are exogenous to controls, formally delineating the boundary where strategic belief manipulation matters, and provides a closed-form mapping from information primitives to equilibrium outcomes.
Paper Structure (96 sections, 17 theorems, 111 equations, 7 figures)

This paper contains 96 sections, 17 theorems, 111 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Higher-order expectations and the social value of information.
  4. Frequency-domain methods and exogenous-signal models.
  5. Common-information approaches.
  6. Team theory.
  7. Mean-field LQG games.
  8. Truncation of the belief hierarchy.
  9. Applied continuous-time filtering games.
  10. Information design.
  11. Rational inattention.
  12. The Decentralized LQG Game
  13. Informal description of the game.
  14. Primitives
  15. State dynamics.
  16. ...and 81 more sections

Key Result

Theorem 3.1

Fix a player $i$. Assume the state admits the primitive-noise Volterra form and player $i$ observes with deterministic $P^i(t)\succeq 0$ and block selector $E^i$. Let $\Pi^i:=E^{i,\top}E^i$ and $\widehat{X}_t^i:=\mathbb{E}[X_t\mid\mathcal{F}_t^i]$, and define the innovation $dI_t^i:=dY_t^i-\sqrt{P^i(t)}\,\widehat{X}_t^i\,dt$. (i) The estimated-noise path increments admit the decomposition where

Figures (7)

  • Figure 1: The Volterra fixed-point loop. (i) Assume opponents use Volterra controls with deterministic kernels. (ii) The state and filtering equations close on deterministic two-time kernels. (iii) Under fixed strategy maps, unobserved deviations propagate deterministically. (iv) The maximum principle produces a best response that is itself Volterra, closing the loop. (v) A fixed point of this best-response map is a Nash equilibrium.
  • Figure 2: Information-to-action feedback in the worked example. Actions move $X_t$ and shift opponents' beliefs, so equilibrium mean actions depend on precision $p$.
  • Figure 3: Equilibrium kernel time-slices, shown channel-by-channel for $(W^0,W^1,W^2)$ with curves colored by evaluation time $t$. (a) Both players jointly stabilize older fundamental shocks (decay in the $W^0$ panel); observation-noise channels show smaller, opposed effects mediated by the belief loop. (b) The system-noise channel dominates; observation-noise channels exhibit curvature reflecting drift-based inference.
  • Figure 4: Equilibrium mean control $\bar{D}^1(t)$ as a function of signal precision $p$, with the perfect-information benchmark (dashed). Low precision makes opponents' posteriors sluggish, amplifying the incentive to manipulate beliefs; as $p\to\infty$ the mean policy converges to the perfect-information limit.
  • Figure 5: Asymmetric equilibrium with $p_1=3$ fixed and $p_2$ varying. Left: mean controls $\bar{D}^1(t)$ (solid) and $\bar{D}^2(t)$ (dashed) diverge as the precision gap widens. Center: the mean state path $\bar{X}(t)$ tilts toward the better-informed player's target. Right: aggregate mean effort $|\bar{D}^1|+|\bar{D}^2|$ exceeds the perfect-information benchmark (gray dashed), with the excess growing in the precision asymmetry.
  • ...and 2 more figures

Theorems & Definitions (53)

  • Remark 2.1: Control matrices and dimensions
  • Definition 2.1: Nash equilibrium in private-signal strategies
  • Definition 2.2: Primitive-noise Volterra process
  • Remark 2.2: Properties of primitive-noise Volterra processes
  • Definition 2.3: Noise-state Volterra control / strategy
  • Theorem 3.1: Volterra Filtering Closure
  • Remark 3.1: Regression interpretation and scale separation
  • Theorem 3.2: Multiplayer belief-adjoint kernels and information wedges
  • Corollary 3.3: Exogenous-signal reduction
  • Lemma 3.4: Why the maximum-principle stationarity condition is global here
  • ...and 43 more