Appendix AAdditional Finite-Horizon Extensions

A.1Endogenous signal precision

This section identifies the obstruction to optimizing precision in the multi-player setting.

A.1.1Why choosing precision is hard

In single-agent LQG, the separation principle makes precision easy to optimize: the policy gain $S (t)$ does not depend on signal precision, so the value of information can be computed independently of the control problem. With multiple agents and endogenous signals, separation fails (Remark 1.13), because the policy kernel $D_{t}^{i} (u)$ depends on precision through the information wedge, the shadow price of moving another player’s noise-state.

The filtrations generated by $B_{X}^{Y, i} X d t + d W^{i}$ and $(B_{X}^{Y, i} + δ B_{X}^{Y, i}) X d t + d W^{i}$ are generically incomparable: neither contains the other. This breaks the envelope argument that makes single-agent rational inattention tractable, because the player at $P^{i} + δ P^{i}$ cannot implement the control it would have chosen at $P^{i}$ .

There is also a more basic problem with calling $δ B_{X}^{Y, i}$ a change in precision. The $d W^{i}$ in the reduced observation equation need not be the error from one literal measurement. It can already be the noise left after several measurements have been collapsed to a sufficient statistic. For example, if $d Y^{m} = b_{m} X d t + d V^{m}, m \in S,$ with independent unit-variance measurement noises, then the informative part of these measurements can be written $d \overset{ˉ}{Y}^{S} = P_{S} X d t + d \overset{ˉ}{V}^{S}, P_{S} = m \in S \sum b_{m}^{2}, d \overset{ˉ}{V}^{S} = \frac{1}{P _{S}} m \in S \sum b_{m} d V^{m} .$ Adding another measurement increases $P_{S}$ , but it also changes $d \overset{ˉ}{V}^{S}$ . At the level of the actual measurements, more precision therefore does not usually mean increasing the gain while holding the same measurement noise fixed. The latter is a perfectly good perturbation of the reduced-form filtering problem, including for the calculations below, but it needs some care before it is interpreted as acquiring more information.

Correlation makes this more important rather than less. If the measurement errors have covariance $R$ , the corresponding precision is $b^{⊤} R^{- 1} b$ and the sufficient statistic weights the measurements by $R^{- 1} b$ . Adding a source can then change the weights on measurements the player already had, and correlations with other information the player observes have to be carried along as well. None of this makes the filtering mathematics difficult; it means that an economically meaningful deviation in precision should come from a specified change in the underlying measurements, with the new reduced-form gain and noise derived from that change.

The incomparability disappears when an independent source adds precision (the Brownian-sheet formulation sketched in Section 7.1.3). Monitoring an additional channel strictly enlarges the filtration, and the marginal value of the source is the value of optimally exploiting its independent increment. The full rational-inattention game in which players choose source portfolios requires this formulation and is left to later work. Whether opponents are privy or naive to the precision change, in the sense of Chapter 6, depends on whether they see it. If they do not see it, they filter its effects as primitive shock. This is coherent: the change reaches them through the deviating player’s control, so it moves the drift of their observations and not the volatility, and treating a deviation as primitive shock breaks down only when the deviation changes the volatility. The channels described in Section A.1.2 are what a naive opponent filters as noise. If they see it, through a data purchase or disclosed research, they are privy and respond through the gains of Chapter 6. In the naive case the best response to deterministic precisions is deterministic, because fixed gains make the conditional variances deterministic, so the precision game can be solved over deterministic gain paths. Unfortunately my intuition does not stretch far enough for players with both privy and naive opponents, although when all opponents are privy it should still apply.

A.1.2Two channels from a precision perturbation

Despite the incomparability, one can compute the first-order effect of a gain perturbation on equilibrium objects by linearizing the forward–backward system.

Fix an equilibrium profile and spike-perturb player $i$ ’s gain at time $t_{0}$ . By the envelope theorem, $δ D^{i}$ has no first-order cost effect, because the FOC is satisfied. But the primitive-shock control $D_{W, τ}^{i} (r) = D_{τ}^{i} (r) Π^{i} + \int_{0}^{τ} D_{τ}^{i} (u) F_{τ}^{i} (u, r) d u$ responds at first order through $δ F^{i}$ , changing which part of the fixed policy kernel the player’s own information resolves.

The variation of the unresolved adjoint decomposes as $δ H_{t}^{X, i} (s) = (a) filtering channel - \int_{s}^{t} H_{t}^{X, i} (u) δ F_{t}^{i} (u, s) d u \int_{t_{0}}^{T} + (b) equilibrium channel \int_{t_{0}}^{T} Φ^{HD, i} (t, τ) δD_{W, t}^{i} (τ, s) d τ, (A.1.1)$ where $Φ^{HD, i}$ is the deterministic kernel through which a control variation at time $τ$ enters the time- $t$ adjoint, and $δD_{W}^{i}$ is the unresolved component of the control variation induced by $δ F^{i}$ .

Through channel (a) the player resolves more of the fixed adjoint from the perturbed signal, the standard value-of-information effect present in single-agent problems. Channel (b) arises because the same $δ F^{i}$ changes the projection, leaving part of $D_{W, τ}^{i}$ unresolved by the player’s current information. The equilibrium channel operates through the information wedge and vanishes when $V_{t}^{i} = 0$ (exogenous signals, Corollary 1.9). Multi-player rational inattention couples the filtering and control problems that separate in single-agent settings: the marginal value of precision depends on the equilibrium through the wedge. Optimizing attention requires solving for the equilibrium first.

Local profile-space notation.

In the remainder of this appendix, bold symbols denote stacked policy profiles, and calligraphic and sans-serif symbols denote maps on the profile space. These are local functional-analytic abbreviations and do not carry into the substantive chapters.

A.2Well-posedness of the deterministic impulse-response fixed point

This section records the a priori bounds that follow from the projection structure of the filtering map and separates three well-posedness questions. A unilateral best response against fixed linear opponent maps is globally well posed for every finite horizon, and the joint policy update is a contraction on short horizons. For arbitrary horizons, existence and uniqueness follow under a weighted monotonicity condition that one can also check numerically near a computed equilibrium.

Notation.

$C$ denotes a generic constant depending on $(n, d, L, λ)$ ; $C (M)$ when it also depends on a policy-norm bound $M$ .

A.2.1Standing assumptions

Assumption A.1. There exist $L > 0$ and $λ > 0$ such that, for all $t \in [0, T]$ and all $i$ , $∥ B_{X}^{X} (t) ∥ + ∥ B_{i}^{X} (t) ∥ + ∥ σ (t) ∥ + ∥ P^{i} (t) ∥ + ∥ G^{XX, i} (t) ∥ + ∥ \overset{ˉ}{G}_{t}^{X, i} ∥ + ∥ G^{X, i} (t, \cdot) ∥_{L^{\infty}} + ∥ G^{D X, i} (t) ∥ + ∥ G^{XX, i} (T) ∥ + ∥ \overset{ˉ}{G}_{T}^{X, i} ∥ + ∥ G^{X, i} (T, \cdot) ∥_{L^{\infty}} \leq L$ , the joint running Hessian is positive semidefinite, and the Schur condition (1.3.3) holds with constant $λ$ (which implies $G^{DD, i} (t) ⪰ λ I_{d}$ , since $S^{i} ⪯ G^{DD, i}$ ).

A.2.2The kernel state space and its dynamics

At time $t$ the kernel state is $Z_{t} := (\overset{ˉ}{X}_{t}, X_{t} (\cdot), {F_{t}^{j}}_{j = 1}^{n}) \in R^{d} \times L^{2} ([0, t]; R^{d \times (n + 1) d}) \times j = 1 \prod n L_{sym}^{2} ([0, t]^{2}; R^{(n + 1) d \times (n + 1) d}) . (A.2.1)$ The unresolved response $X^{j}$ is not a state variable, but follows algebraically from $(Z_{t})$ by $X_{t}^{j} (u) = X_{t} (u) (I - Π^{j}) - \int_{0}^{t} X_{t} (v) F_{t}^{j} (v, u) d v . (A.2.2)$ It is the part of $X_{t}$ that player $j$ has not resolved from its own observations.

Under a noise-state linear profile $D = {(\overset{ˉ}{D}^{j}, D^{j})}_{j = 1}^{n}$ , the kernel state evolves by $\dot{\overset{ˉ}{X}} (t) \dot{X}_{t} (s) \partial_{t} F_{t}^{j} (u, s) = B_{X}^{X} (t) \overset{ˉ}{X}_{t} + \sum_{k} B_{k}^{X} (t) \overset{ˉ}{D}_{t}^{k}, = B_{X}^{X} (t) X_{t} (s) + \sum_{k} B_{k}^{X} (t) D_{W, t}^{k} (s), X_{s} (s) = σ (s) E^{0}, = X_{t}^{j} (u)^{⊤} P^{j} (t) X_{t}^{j} (s), F_{0}^{j} = 0, (A.2.3) (A.2.4) (A.2.5)$ with $E^{0}$ the block selector for the state noise $W^{0}$ , boundary $F_{t}^{j} (u, t) = X_{t}^{j} (u)^{⊤} B_{X}^{Y, j} (t)^{⊤} E^{j}$ , and primitive-shock control $D_{W, t}^{k} (s) := D_{t}^{k} (s) Π^{k} + \int_{0}^{t} D_{t}^{k} (u) F_{t}^{k} (u, s) d u$ .

The policy $D$ enters (A.2.3)–(A.2.4) but does not enter (A.2.5) directly, since $F^{j}$ depends on $D$ only through $X$ by (A.2.2).

A.2.3Dynamic-programming interpretation of a fixed-profile best response

The deterministic kernels in (A.2.1) are useful for computing equilibrium, but they are not themselves a state observed by any player. Their domains also grow with time and create the birth terms at the current date derived in Appendix 1.B. For these reasons, the well-posedness results below do not rely on a Hamilton–Jacobi–Bellman equation on $(\overset{ˉ}{X}, X, F)$ .

Dynamic programming instead becomes available after fixing the opponents’ complete linear strategy maps. For player $i$ , let $S_{t}^{i}$ denote the augmented hidden state containing the physical state and the opponents’ noise-state coordinates that enter their future controls, lifted to a fixed Hilbert space so that its generator includes transport and coordinate births. The player-specific separated state is $m_{t}^{i} := E [S_{t}^{i} ∣ F_{t}^{i}] .$ Under the frozen linear profile the problem is linear Gaussian, the conditional covariance is deterministic, and the separated state has the formal innovation representation $d m_{t}^{i} = (A_{t}^{i} m_{t}^{i} + B_{t}^{i} u_{t}^{i} + b_{t}^{i}) d t + K_{t}^{i} d I_{t}^{i} .$ The control $u_{t}^{i}$ moves the drift, and the innovation $d I_{t}^{i}$ carries what the player learns. For the value function normalized as one half of the continuation cost, the corresponding formal HJB equation is $0 = \partial_{t} V^{i} (t, m) + u in f {\frac{1}{2} \overset{ˉ}{ℓ}^{i} (t, m, u) + ⟨ D V^{i} (t, m), A_{t}^{i} m + B_{t}^{i} u + b_{t}^{i} ⟩ + \frac{1}{2} Tr [K_{t}^{i} (K_{t}^{i})^{*} D^{2} V^{i} (t, m)]} . (A.2.6)$ The first-order condition is $u_{t}^{i, *} = - (G^{DD, i} (t))^{- 1} [G^{D X, i} (t) Π_{X} m_{t}^{i} + (B_{t}^{i})^{*} D V^{i} (t, m_{t}^{i})], (A.2.7)$ where $Π_{X} m_{t}^{i} = E [X_{t} ∣ F_{t}^{i}]$ is the physical component of the separated state. A quadratic operator ansatz for $V^{i}$ gives an operator Riccati system. Its off-diagonal blocks, coupling the physical component of $m^{i}$ to the opponents’ noise-state coordinates, are the dynamic-programming counterparts of the belief prices and the information wedge. A rigorous HJB treatment would require specifying the fixed-space lift and the unbounded transport and birth operators.

A.2.4A priori bounds from projection structure

The filter kernel $F_{t}^{j}$ parametrizes the integral part of player $j$ ’s blueprint, the conditional projection $(P_{t}^{j} f) (u) := Π^{j} f (u) + \int_{0}^{t} F_{t}^{j} (u, s) f (s) d s$ , which is an orthogonal projection on $L^{2} ([0, t]; R^{(n + 1) d})$ (Theorem 1.5 and the projection interpretation following it).

Lemma A.2 (Projection bounds). For each $j$ and $t \geq 0$ : (i) $∥ P_{t}^{j} ∥_{op} = 1$ ; (ii) both $P_{t}^{j}$ and $I - P_{t}^{j}$ are $L^{2}$ -contractions; (iii) $∥ P_{t}^{j} - Π^{j} ∥_{op} \leq 2$ .

Proof. Standard properties of orthogonal projections; (iii) is the triangle inequality. ◻

Lemma A.3 (Dissipation). ${P_{t}^{j}}_{t \geq 0}$ is non-decreasing in the positive semidefinite order, and $∥ X^{j} (t, \cdot) ∥_{L^{2} ([0, t])} \leq ∥ X (t, \cdot) ∥_{L^{2} ([0, t])}$ for all $t$ .

Proof. $P_{t}^{j}$ is the orthogonal projection onto the $F_{t}^{j}$ -measurable linear functionals of $W$ (Lemma 1.7); since $F_{s}^{j} \subseteq F_{t}^{j}$ for $s \leq t$ , these projections are nested and $P_{t}^{j}$ is non-decreasing. Since $X^{j} (t, \cdot) = (I - P_{t}^{j}) X (t, \cdot)$ by (A.2.2), Lemma A.2(ii) gives the bound. ◻

These bounds hold uniformly in the precision paths, the policy profile, and the horizon $T$ . As $P_{t}^{j}$ grows toward $I$ , the complementary projection contracts, driving $X^{j} \to 0$ and decelerating $F^{j}$ , so the quadratic right-hand side of (A.2.5) is self-limiting. This prevents finite-time blowup despite the quadratic nonlinearity, just as the Kalman filter covariance remains bounded.

A.2.5Well-posedness

Definition A.4. Define $B_{T}^{M} := {D = {(\overset{ˉ}{D}^{i}, D^{i})}_{i = 1}^{n} : \overset{ˉ}{D}^{i} \in C ([0, T]; R^{d}), D^{i} \in C (Δ_{T}), ∥ D ∥ \leq M}$ , where $∥ D ∥ := max_{i} (∥ \overset{ˉ}{D}^{i} ∥_{C} + ∥ D^{i} ∥_{L^{\infty}})$ . With continuous data the Volterra structure of (1.A.5)–(1.A.6) and of the adjoint equations propagates continuity to $F^{i}$ and to the belief prices, so the birth traces $F_{t}^{i} (u, t)$ , $F_{t}^{i} (t, u)$ and $H_{t}^{k, i} (t)$ are defined pointwise.

Definition A.5 (Forward–backward policy update). Define $T (D) := {(- (G^{DD, i})^{- 1} [G^{D X, i} \overset{ˉ}{X}^{D} + \overset{ˉ}{H}^{X, i, D}], - (G^{DD, i})^{- 1} [G^{D X, i} X^{D} + H^{X, i, D}])}_{i = 1}^{n},$ where the adjoints solve the backward system of Theorem 1.8 at the forward environment induced by $D$ . Fixed points are Nash equilibria (Corollary 1.11).

Proof of Proposition 1.12. Use the control-free reference filtration of Proposition 1.6 and let $U_{i} := L_{prog}^{2} (Ω \times [0, T], F^{i}; R^{d}) .$ The exact system for the difference between two controls makes the physical state and all induced opponent controls affine and continuous functions of the realized input $u_{i} \in U_{i}$ . The cost then has the form $J^{i} (u_{i}; D^{- i}) = \frac{1}{2} ⟨ u_{i}, A_{i} u_{i} ⟩_{U_{i}} + ⟨ b_{i}, u_{i} ⟩_{U_{i}} + c_{i},$ where $A_{i}$ is bounded (Proposition 1.6), deterministic, self-adjoint, and $⟨ u_{i}, A_{i} u_{i} ⟩ \geq λ ∥ u_{i} ∥^{2}$ by Lemma 1.10. There is then a unique minimizer.

It remains to show that the minimizer is noise-state linear, without assuming in advance that the optimum lies in that class. Split every square-integrable functional of the reference noise into its affine part, a constant plus a linear functional of the noise, and a remainder orthogonal to all affine functionals. The frozen linear causal response maps send affine functionals to affine functionals and remainders to remainders (they are built from deterministic kernels and from conditional expectations given filtrations generated by linear functionals of the noise), so $A_{i}$ does the same, $b_{i}$ is affine, and the cost has no cross term between the two parts. Any remainder in a control contributes only its nonnegative quadratic cost and cannot occur in the unique minimizer. Since $F_{t}^{i}$ is generated by linear functionals of the reference noise, conditioning on it also maps affine functionals to affine functionals and remainders to remainders, so the affine part of an adapted control is adapted and the minimization over $U_{i}$ may be carried out part by part. The minimizer is consequently affine Gaussian and adapted. By the Gaussian-span statement in Lemma 1.7, it has a deterministic noise-state representation. The corresponding deterministic kernels satisfy the stationarity system. ◻

Fixed coordinates for arbitrary-horizon equilibrium.

The monotonicity result below is stated on a fixed Hilbert space of complete linear causal strategy maps. Let $S_{T} := i = 1 \prod n S_{T}^{i}$ be such a realization, using a profile-independent observation or reference-innovation coordinate system. If profile-dependent noise-state coordinates are used computationally, the coordinate-change terms are included when the best-response derivative is formed. Proposition 1.12 defines the exact best-response map $B (s) := (B_{i} (s^{- i}))_{i = 1}^{n}, s \in S_{T},$ and the equilibrium residual $R (s) := s - B (s) . (A.2.8)$ A zero of $R$ is a fixed point of the exact linear-strategy best-response map, and so a Nash equilibrium (Corollary 1.11).

The forward estimates below follow by Grönwall arguments on the forward system (A.2.3)–(A.2.5), using the a priori bounds of Lemmas A.2–A.3 to control the quadratic nonlinearity in (A.2.5). The adjoint estimates follow by backward Grönwall on the linear adjoint system (1.B.19)–(1.B.20), using Cauchy–Schwarz on the belief-price coupling terms, including the birth term $B_{X}^{Y, k} (t)^{⊤} E^{k} \overset{ˉ}{H}_{t}^{k, i} (t)$ and the drift-inference integral $\int_{0}^{t} P^{k} (t) X_{t}^{k} (r) \overset{ˉ}{H}_{t}^{k, i} (r) d r$ .

Proposition A.6 (Bounds and Lipschitz continuity). Under Assumption A.1, for $D \in B_{T}^{M}$ :

(i) The forward system has a unique global solution with $∥ \overset{ˉ}{X} ∥_{C} + ∥ X ∥_{L^{\infty}} \leq K_{fwd}$ , $∥ X^{j} (t, \cdot) ∥_{L^{2}} \leq K_{fwd} T$ , $∥ P_{t}^{j} ∥_{op} = 1$ .

(ii) The backward system has a unique solution with $∥ \overset{ˉ}{H}^{X, i} ∥_{C} + ∥ H^{X, i} ∥_{L^{\infty}} \leq K_{bwd}$ , and time derivatives bounded by $C (M) K_{bwd}$ .

(iii) Both maps $D \mapsto (\overset{ˉ}{X}, X, X^{j})$ and $D \mapsto (\overset{ˉ}{H}^{X, i}, H^{X, i})$ are Lipschitz with constants $C_{fwd} (T, M)$ and $C_{bwd} (T, M)$ respectively, both bounded by $C (M) T e^{C (M) T}$ .

Here $K_{fwd}, K_{bwd} \leq C (M) e^{C (M) T}$ .

Proof sketch. For the forward system, $(\overset{ˉ}{X}, X)$ solve linear equations with bounded coefficients once $D_{W}^{k}$ is controlled. The operator bound $∥ P_{t}^{k} - Π^{k} ∥_{op} \leq 2$ (Lemma A.2(iii)) on the filter kernel closes the system in $L^{2}$ by Grönwall, ruling out finite-time blowup. The $L^{\infty}$ closure follows by bootstrapping. With $ϕ (t) := ∥ X ∥_{L^{\infty} (Δ_{t})} + max_{k} ∥ X^{k} ∥_{L^{\infty} (Δ_{t})}$ , the explicit formula (1.A.10) gives $∥ F_{t}^{k} (\cdot, s) ∥_{L^{2} ([0, t])} \leq C (M) (1 + ϕ (t))$ , so Cauchy–Schwarz on $D_{W, t}^{k} (s) = D_{t}^{k} (s) Π^{k} + \int_{0}^{t} D_{t}^{k} (u) F_{t}^{k} (u, s) d u$ gives $∣ D_{W, t}^{k} (s) ∣ \leq C (M) (1 + ϕ (t))$ , and $ϕ$ satisfies $ϕ (t) \leq C (M) + C (M) \int_{0}^{t} ϕ (r) d r$ .

For the Lipschitz bound, the variation $δ D_{W}^{k}$ decomposes into a direct term bounded by $C (M) ∥ δ D ∥$ and an indirect term through $δ F^{k}$ , which couples to $δ X^{k}$ and back to $δ X$ by (A.2.2). A Grönwall argument on the aggregate $Ψ (t) := ∥ δ X ∥_{L^{\infty} (Δ_{t})} + max_{j} sup_{r \leq t} ∥ δ X^{j} (r, \cdot) ∥_{L^{2}}$ gives $Ψ (T) \leq C (M) T e^{C (M) T} ∥ δ D ∥$ ; the factor of $T$ arises because the policy enters through integrals of length $\leq T$ .

The backward estimates follow from linearity of the adjoint system, Cauchy–Schwarz on coupling terms, and backward Grönwall. ◻

Theorem A.7 (Short-horizon equilibrium). Under Assumption A.1, there exists $T^{*} > 0$ depending only on $(n, d, L, λ)$ such that for $T \in (0, T^{*}]$ , $T$ is a contraction on $B_{T}^{M_{0}}$ for a suitable $M_{0}$ . There is then a unique noise-state linear equilibrium in $B_{T}^{M_{0}}$ , it is Nash over the full admissible $L^{2}$ class, and Picard iterates converge geometrically.

Proof. Write $T_{T}$ for the update on horizon $T$ . By Proposition A.6(ii), there are $T_{0} > 0$ and $C_{0} < \infty$ such that $0 < T \leq T_{0} sup ∥ T_{T} (0) ∥ \leq C_{0} .$ Set $M_{0} := 2 C_{0}$ . Proposition A.6(iii) gives $Lip (T_{T}; B_{T}^{M_{0}}) \leq λ^{- 1} C (M_{0}) T e^{C (M_{0}) T} .$ Choose $T^{*} \leq T_{0}$ so that the right-hand side is at most $1/2$ for $T \leq T^{*}$ . Then, for $D \in B_{T}^{M_{0}}$ , $∥ T_{T} (D) ∥ \leq ∥ T_{T} (0) ∥ + \frac{1}{2} ∥ D ∥ \leq C_{0} + \frac{1}{2} M_{0} = M_{0} .$ The map $T_{T}$ is then a self-map and a contraction on $B_{T}^{M_{0}}$ . Banach’s theorem gives existence, uniqueness, and geometric convergence in $B_{T}^{M_{0}}$ . ◻

Proposition A.8 (Weighted Jacobian criterion). Let $W : S_{T} \to S_{T}$ be bounded, self-adjoint, and uniformly positive, and write $⟨ h, k ⟩_{W} := ⟨ W h, k ⟩, ∥ h ∥_{W}^{2} := ⟨ h, h ⟩_{W} .$ Let $K \subset S_{T}$ be convex, and suppose that the exact residual $R$ is continuously Fréchet differentiable on $K$ . If there is $μ > 0$ such that $\frac{1}{2} (W D R (s) + D R (s)^{*} W) ⪰ μ W for every s \in K, (A.2.9)$ then $R$ is $μ$ -strongly monotone on $K$ : $⟨ R (s) - R (s), s - s ⟩_{W} \geq μ ∥ s - s ∥_{W}^{2} .$

Proof. Set $Δ = s - s$ . Convexity of $K$ and the fundamental theorem of calculus give $R (s) - R (s) = \int_{0}^{1} D R (s + θ Δ) Δ d θ .$ Taking the $W$ -inner product with $Δ$ and using (A.2.9) proves the claim. ◻

Theorem A.9 (Certified equilibrium on a strategy ball). Fix a reference profile $s_{0} \in S_{T}$ and $R > 0$ , and set $K_{R} (s_{0}) := {s \in S_{T} : ∥ s - s_{0} ∥_{W} \leq R} .$ Suppose that the exact best-response map is well defined on $K_{R} (s_{0})$ and that its residual $R = I - B$ is $μ$ -strongly monotone and $L$ -Lipschitz there. If $∥ R (s_{0}) ∥_{W} < μ R, (A.2.10)$ then $K_{R} (s_{0})$ contains a unique profile $s^{*}$ satisfying $R (s^{*}) = 0.$ The profile $s^{*}$ is a Nash equilibrium over the full admissible $L^{2}$ control class. For every $0 < η < \frac{2 μ}{L ^{2}},$ the projected residual iteration $s^{m + 1} = P_{K_{R} (s_{0})}^{W} [s^{m} - η R (s^{m})] (A.2.11)$ converges geometrically to $s^{*}$ .

Proof. For $s, s \in K_{R} (s_{0})$ , nonexpansiveness of metric projection gives $P_{K_{R} (s_{0})}^{W} (s - η R (s)) - P_{K_{R} (s_{0})}^{W} (s - η R (s))_{W}^{2} \leq (1 - 2 η μ + η^{2} L^{2}) ∥ s - s ∥_{W}^{2} .$ The stated range of $η$ makes the projected map a contraction. Banach’s fixed-point theorem supplies a unique fixed point $s^{*}$ of that map. The projection characterization gives $⟨ R (s^{*}), y - s^{*} ⟩_{W} \geq 0 for every y \in K_{R} (s_{0}) .$ If $∥ s^{*} - s_{0} ∥_{W} = R$ , choosing $y = s_{0}$ gives $⟨ R (s^{*}), s^{*} - s_{0} ⟩_{W} \leq 0.$ Strong monotonicity relative to $s_{0}$ instead gives $⟨ R (s^{*}), s^{*} - s_{0} ⟩_{W} \geq μ R^{2} + ⟨ R (s_{0}), s^{*} - s_{0} ⟩_{W} \geq μ R^{2} - ∥ R (s_{0}) ∥_{W} R > 0,$ a contradiction. The profile $s^{*}$ then lies in the interior of the ball, and testing the variational inequality in both directions gives $R (s^{*}) = 0$ . Strong monotonicity shows that two zeros in the ball must coincide. ◻

Remark 8.10 (A two-player check). For two players and scalar weights $r_{1}, r_{2} > 0$ , a sufficient form of (A.2.9) on a convex set $K$ is $s \in K sup \frac{1}{2} [\frac{r _{1}}{r _{2}} D_{2} B_{1} (s) + \frac{r _{2}}{r _{1}} D_{1} B_{2} (s)^{*}]_{op} < 1.$ Only the symmetrically reinforcing part of reciprocal best responses enters. Monotonicity can then hold even when ordinary Picard iteration is not a contraction because the dominant feedback is antisymmetric or rotational.

Corollary A.11 (Numerical certification near a computed profile). Let $s$ be a computed strategy profile for a finite-dimensional discretization, let $J_{R} \approx D R (s)$ be the computed Jacobian of the best-response residual at $s$ , with $∥ J_{R} - D R (s) ∥_{W} \leq ε_{J}$ in the weighted operator norm $∥ A ∥_{W} := ∥ W^{1/2} A W^{- 1/2} ∥_{op}$ , and define the local weighted margin $μ := λ_{m i n} [W^{- 1/2} \frac{W J _{R} + J _{R}^{T} W}{2} W^{- 1/2}] . (A.2.12)$ Suppose that, throughout $K_{R} (s)$ , $∥ D R (s) - D R (s) ∥_{W} \leq Λ R .$ Then $R$ is strongly monotone on that ball with constant $μ_{R} := μ - ε_{J} - Λ R .$ If $μ_{R} > 0, ∥ R (s) ∥_{W} < μ_{R} R, (A.2.13)$ then the ball contains a unique equilibrium.

Proof. The Jacobian-variation bound and the variational characterization of the smallest eigenvalue imply $\frac{1}{2} (W D R (s) + D R (s)^{*} W) ⪰ μ_{R} W$ throughout the ball. Proposition A.8 and Theorem A.9 then apply with center $s$ . ◻

Numerical implementation.

After discretizing the deterministic policy kernels, the margin (A.2.12) is the smallest generalized eigenvalue of $\frac{1}{2} (W J_{R} + J_{R}^{T} W) ψ = μ W ψ .$ Take the Jacobian for the exact best-response residual, computing it from tangent equations, automatic differentiation, complex-step differentiation, or centered finite differences. A positive $μ$ alone certifies local monotonicity and local uniqueness, but the variation and residual bounds in (A.2.13) turn it into a rigorous certificate on a whole ball. Report separately $μ, σ_{m i n} (J_{R}), ρ (D B (s)) .$ The first measures the monotonicity margin, the second local invertibility, and the third local convergence of undamped Picard iteration. Then $μ > 0$ together with $ρ (D B) > 1$ describes a locally unique equilibrium for which ordinary Picard iteration is unstable but sufficiently damped residual descent remains convergent.

Remark 8.12 (What remains model-specific). Theorem A.9 is an arbitrary-horizon result, but one must verify its hypotheses on the chosen strategy ball. In the noise-state system this requires differentiating the deterministic forward filter and backward belief-price equations with respect to the fixed linear strategy maps. The resulting tangent equations are linear Volterra–backward systems. Proposition A.6 supplies the zeroth-order bounds; a corresponding first-order estimate gives $D B$ , while a second-order estimate or validated numerical bound gives $Λ$ in Corollary A.11.

Remark 8.13 (Stochastic coefficients). If primitives depend on a public factor process $ξ_{t}$ , the kernel state augments to $(Z_{t}, ξ_{t})$ . The projection bounds of Lemmas A.2–A.3 are algebraic and hold pathwise, so forward well-posedness extends between jumps of $ξ$ . At each jump the filtering gains, strategies, and the information wedge shift discontinuously, while $Z$ is continuous across jumps.