Two players jointly control a scalar state $X_t$ over $[0,T]$. Each player privately observes one component of the driving Brownian noise and chooses a control to steer the state toward their own target.
The state evolves as
$$dX_t = \bigl(D_t^1 + D_t^2\bigr)\,dt + \sigma\,dW_t^0, \qquad X_0 = x_0,$$where $W^0$ is common noise observed by neither player, $\sigma$ is the state diffusion coefficient, and $x_0$ is the initial state.
Player $i$ continuously observes a private signal
$$dY_t^i = p_i\,X_t\,dt + dW_t^i, \qquad i=1,2,$$where $W^1, W^2$ are independent Brownian motions (also independent of $W^0$), and $p_i > 0$ is the signal root-precision (observation precision is $p_i^2$). Player 1 sees only $Y^1$; player 2 sees only $Y^2$. Neither player observes the state $X_t$ directly.
Player $i$ minimizes
$$J^i = \mathbb{E}\!\int_0^T \!\Bigl[\,(X_t - b_i)^2 + r_i\,(D_t^i)^2\,\Bigr]\,dt,$$where $b_i$ is player $i$'s bliss point (target state) and $r_i > 0$ is the control cost weight.
We look for a Nash equilibrium amongst all admissible strategies. In equilibrium, each player's control decomposes into a deterministic mean path and a feedback response to their innovation process:
$$D_t^i = \bar{D}^i(t) + \int_0^t D^i(t,s)\,d\widehat{W}^i_t(s),$$where $\bar{D}^i(t)$ is the mean control, $D^i(t,s)$ is the feedback kernel, and $\widehat{W}^i_t$ is player $i$'s noise state. The solver finds $\bar{D}^1, \bar{D}^2, D^1, D^2$ such that neither player can reduce their cost by deviating.
| $p_1, p_2$ | Signal root-precisions — observation precision is $p_i^2$ |
| $b_1, b_2$ | Bliss points — each player's preferred state level |
| $r_1, r_2$ | Control cost weights — how expensive it is to act |
| $T$ | Time horizon |
| $N$ | Grid points (numerical resolution) |
Decentralized LQG games normally require each player to form beliefs about the other's beliefs, and so on — the Townsend belief hierarchy. This tool displays the equilibrium objects that emerge once that hierarchy is resolved via a coordinate change into noise-state (primitive Brownian) coordinates.
Instead of tracking conditional distributions, work in the coordinates of the three underlying Brownian shocks $W^0, W^1, W^2$. Every adapted process admits a stochastic-integral representation governed by a kernel (a causal function of two time indices), and the equilibrium reduces to a system of coupled kernel equations with no infinite regress. The channels have distinct roles: $W^0$ is common noise observed by neither player, $W^1$ is player 1's observation noise, $W^2$ is player 2's.
Kernels tab.
$X(t,s)$ is the impulse response of the state: how much a unit shock at source time $s$ persists in $X_t$. The $W^0$ channel starts at 1 and decays as both players stabilize it. The $W^1, W^2$ channels start at zero (observation noise doesn't enter the state directly) but build up as each player's privately-adapted control feeds back through the drift.
Kernels tab.
$D^i(t,s)$ is player $i$'s feedback kernel in filtered coordinates (what the player knows). $\mathcal{D}^i(t,s)$ is the same control re-expressed in primitive coordinates (what drives the state), obtained by convolving $D^i$ with the filtration kernel $F^i$. $\mathcal{D}^i$ is the kernel that enters the state equation and determines equilibrium volatility.
Kernels tab.
$\widetilde{X}^i(t,s)$ is the component of the state's impulse response that player $i$ has not yet resolved at time $t$ — the filtering gain in the noise-state innovation update (Theorem 3.1). The $W^i$ channel is identically zero: player $i$'s own observation noise enters the state only through their own (measurable) actions.
Kernels tab.
$F^i(t,u,s)$ maps from primitive coordinates $(s)$ to filtered coordinates $(u)$ for player $i$ at time $t$ — the density kernel of the noise-state conditional mean. The 3×3 subplot grid shows each source channel (column) mapping into each target channel (row).
Equilibrium tab (top panel).
The deterministic components of the equilibrium. Dotted lines show the perfect-information Riccati benchmark. The gap is the mean-channel distortion from bilateral belief manipulation.
Mean $\bar{V}^i(t)$ and full kernel $\mathcal{V}^i(t,r)$ on the Equilibrium tab.
Player $i$'s optimality condition gives $D^i(t,s) = -(1/r_i)\,H^i_x(t,s)$, where $H^i_x$ is the kernel costate. The backward equation for $H^i_x$ is
$$H^i_x(t,r) = H^i_x(T,r) + \int_t^T \bigl[\, X(u,r) + \mathcal{V}^i(u,r) \,\bigr]\,du$$where $X(u,r)$ is the standard state-tracking term and $\mathcal{V}^i$ is the information wedge:
$$\mathcal{V}^i(t,r) \;=\; p_{-i}^2 \int_0^t \widetilde{X}^{-i}(t,z) \cdot H^{-i,i}(t,z,r)\,dz$$$\widetilde{X}^{-i}$ governs how much a drift perturbation shifts the opponent's posterior; $H^{-i,i}$ governs how that shifted posterior changes the opponent's future actions; $p_{-i}^2$ scales by opponent precision. Without the wedge, the backward equation collapses to the perfect-information Riccati solution. With it, player $i$ shades the drift to manipulate what the opponent learns. The entire strategic information friction enters through this single additive correction to the costate.
Statics tab.
Sweeps $p_2$ with all else fixed. Dashed lines show pooled-information costs (common filtration, no strategic friction). The gap reflects both the direct cost of having less information and the additional distortion from bilateral belief manipulation.
Allocation tab.
Sweeps the precision split under a quadratic budget $p_1^2 + p_2^2 = \bar{P}$. The paper's headline result (Section 2.5): under competition with asymmetric effort costs, a planner should concentrate all precision on the efficient player. The mechanism is information starvation: denying the inefficient player signal access collapses their ability to exert mean control, which in turn reduces the efficient player's incentive to manipulate, collapsing bilateral waste. The certainty-equivalent benchmark (dotted lines in the controls panel) is invariant to the precision split, directly visualizing separation failure (Proposition 4.6).
Simulation tab.
Generates pathwise realizations of the state, player estimates, and controls using the computed kernels. Each path is constructed as $X_j = \bar{X}_j + \sum_{k \le j}\sum_{\mathrm{ch}} X(j,k,\mathrm{ch})\,\Delta W^{\mathrm{ch}}_k$ where $\Delta W^{\mathrm{ch}}_k = \sqrt{\Delta t}\,Z$ with $Z\sim\mathcal{N}(0,1)$. The $\pm 2\sigma$ bands are computed analytically from kernel norms: $\mathrm{Var}(X_j) = \Delta t \sum_{k \le j}\sum_{\mathrm{ch}} X(j,k,\mathrm{ch})^2$. Player estimates use $\hat{X}^i = X - \tilde{X}^i$ (state minus unresolved component). The filter-error plot shows $X - \hat{X}^i = \tilde{X}^i$, the part of the state invisible to each player. Use the Resample button or change the seed to draw new paths.
Diagnostics tab.
Anderson-accelerated Picard iteration on the best-response map $D \mapsto G(D)$. The residual $\|G(D)-D\|/\|G(D)\|$ may be non-monotonic — Anderson mixing is not a descent method.
All kernel plots show three sub-panels for noise channels $W^0$ (system), $W^1$ (player 1 obs.), $W^2$ (player 2 obs.). Curves colored by evaluation time $t$ (purple $\to$ yellow = early $\to$ late).
SVD of the selected kernel per channel. Rapid decay indicates low-rank structure.