Forecasting and Manipulating the Forecasts of Others

F orecasting and Manipulating the F orecasts of Others Sam Babic henk o ∗ Abstract When actions reshap e opp onen ts’ signals, each agen t’s optimal resp onse de- p ends on an inﬁnite hierarch y of b eliefs ab out b eliefs (T o wnsend, 1983) that has resisted exact analysis for four decades. W e pro vide the ﬁrst exact equilib- rium characterization of ﬁnite-play er contin uous-time linear–quadratic games with endogenous signals. Estimating primitive sho c ks rather than the state collapses the b elief hierarc h y on to deterministic impulse-resp onse maps, reduc- ing Nash equilibrium to a deterministic ﬁxed p oin t with no truncation and no large-p opulation limit. The characterization yields an information wedge pric- ing the marginal v alue of shifting opp onen ts’ p osteriors; it v anishes precisely when signals are exogenous. In a t w o-play er benchmark, nearly all w elfare gains from po oling information come from eliminating bilateral b elief manipulation rather than from improving state estimation. This strategic c hannel also gov- erns optimal precision allo cation: a planner should concen trate information on the eﬃcien t play er, since arming the ineﬃcien t pla y er maximizes the arms race while starving them of information collapses it. 1 In tro duction Dynamic games with disp ersed priv ate information perv ade economics: ﬁrms in- fer competitors’ pricing from mark et outcomes, traders learn from the order ﬂow ∗ Departmen t of Statistics and Applied Probabilit y , South Hall, Univ ersity of California, Santa Barbara, CA 93106, USA. E-mail: sam@sb abichenko.c om . I am grateful to my advisor T omo yuki Ic hiba for guidance and supp ort throughout this pro ject. I thank Javier Birc henall for v aluable discussions on the economic applications. Thiha Aung, Olivier Mulkin, Y an Lashchev, and Daniel Na ylor provided encouragemen t and companionship throughout m y do ctoral studies. Replication co de is a v ailable at https://github.com/sbabichenko/Noise- State- Games . 1 their own trades generate, central banks know their announcements reshap e priv ate agen ts’ signals. In eac h setting, actions feed back in to the information environ- men t, generating an inﬁnite hierarc h y of b eliefs that T o wnsend [ 32 ] iden tiﬁed and Sargen t [ 29 ] show ed resists ﬁnite-dimensional analysis even in linear–Gaussian en- vironmen ts. Bergemann and Morris [ 5 ] identify dynamic decen tralized information design as a central op en problem. F our decades after T o wnsend, this pap er resolves the hierarch y: estimating primitiv e sho cks rather than the state collapses the in- ﬁnite regress on to deterministic impulse-resp onse maps, reducing Nash equilibrium to a deterministic ﬁxed p oint. No truncation or large-p opulation limit is needed: a researc her who sp eciﬁes pa yoﬀs, a state equation, and a signal structure gets bac k equilibrium impulse-resp onse functions; the b elief hierarch y is handled internally . The conceptual mo ve parallels Harsan yi’s [ 12 ] resolution of the b elief hierarch y in incomplete-information games: just as a common prior o ver a type space closes the static regress, the common probabilit y space of primitiv e sho c ks closes the dynamic one, with all higher-order conditional exp ectations follo wing by linear projection onto deterministic impulse-resp onse maps. The diﬀerence is that Harsanyi’s types are dra wn once while noise unfolds contin uously and actions reshap e signals, so the closure requires a ﬁxed p oint: the maps go v erning b elief up dating must b e consistent with the strategy proﬁle they induce (Section 4 ). The resolution pro duces a new ob ject—the information we dge V i t —that prices the marginal v alue of shifting opp onen ts’ p osteriors. If play er i increases eﬀort by a small amoun t, the state drifts, opp onen t k ’s signal c hanges, k revises their p osterior and adjusts their action, and the wedge is the presen t v alue of that en tire chain. It v anishes precisely when signals do not dep end on actions, delineating the b oundary where strategic b elief manipulation matters. The wedge organizes four ﬁndings, ordered b y their economic con tent. W elfare decomp osition. In a t wo-pla y er game with opp osing targets, the simplest setting exhibiting the full bilateral externality , nearly all w elfare gains from p o oling information come from eliminating the c hannel b y which agents manipulate each other’s p osteriors, not from improving state estimation (Section 2 ). The estimation b eneﬁt of p o oling is identical whether agents co op erate or comp ete; the order-of- magnitude diﬀerence isolates the strategic externalit y . 2 Information starv ation. Allo wing asymmetric eﬀort costs, optimal precision al- lo cation under comp etition is sharply asymmetric and in the opp osite direction from what strategic in tuition suggests. One might exp ect a planner to balance precision to limit manipulation—concentrating information arms the eﬃcient pla yer, so strate- gic restraint should push to w ard equalit y . Instead, concentrating information on the eﬃcien t pla y er c ol lapses the arms race, while shifting it tow ard the ineﬃcient pla yer maximizes bilateral w aste (Section 2.5 ). Separation failure. Certain ty equiv alence fails: with multiple agents and endoge- nous signals, impro ving one pla yer’s information c hanges their p olicy rule itself, not merely their state estimate, b ecause the optimal action dep ends on what opp onen ts ha v e and hav e not learned. In any p olicy ev aluation exercise—central bank trans- parency , disclosure regulation, market structure reform—the information structure cannot b e treated as a parameter that shifts b eliefs while leaving decision rules ﬁxed; the w edge couples the t w o, and the equilibrium m ust b e recomputed. The obstruction arises wherever actions en ter the state and signals are not fully revealing—the regime Grossman and Stiglitz [ 10 ] show ed is generic when information acquisition is costly . Exogenous-signal reduction. Any mo del that treats signals as exogenous, in- cluding mean-ﬁeld limits, frequency-domain solutions with ﬁxed sp ectral densities, and team-theoretic b enc hmarks, is implicitly assuming aw a y the strategic c hannel that drives the ﬁrst three ﬁndings. The wedge mak es this precise: when con trols do not enter the state dynamics, V i t ≡ 0 for all i and t , and the equilibrium reduces to n indep enden t single-agen t LQG problems against a common exogenous state (Corol- lary 4.3 ). Section 2 previews the economic consequences in a tw o-pla y er game; Section 4 dev elops the general theory . 1.1 Related Literature The join t determination of ﬁltering and equilibrium has been approac hed from sev eral directions; eac h ac hiev es tractability by restricting or remo ving the informational externalit y that the presen t pap er resolv es exactly . 3 Higher-order exp ectations and the social v alue of information. The inﬁnite regress of “forecasting the forecasts of others” w as iden tiﬁed by T o wnsend [ 32 ] and sho wn b y Sargent [ 29 ] to resist ﬁnite-dimensional recursive analysis even in linear– Gaussian en vironmen ts. Morris and Shin [ 23 ] demonstrated that the so cial v alue of public information can b e negativ e when agents ov erweigh t common signals; Angele- tos and P av an [ 1 ] characterized the eﬃcient use of information in static co ordination games, sho wing that w elfare losses dep end on the gap b et w een equilibrium and so- cially optimal degrees of co ordination. W o o dford [ 34 ] sho wed that imperfect common kno wledge generates p ersisten t real eﬀects of monetary p olicy through higher-order exp ectations, with the inﬁnite hierarch y truncated by assuming exogenous priv ate signals. Section 2 pro vides a dynamic coun terpart: the information w edge V i t prices the strategic channel through which actions reshap e beliefs, and nearly all welfare gains from p o oling come from eliminating this c hannel rather than from impro ving state estimation. F requency-domain metho ds and exogenous-signal mo dels. A substan tial lit- erature solves disp ersed-information mo dels in the frequency domain [ 11 , 16 , 27 ], culminating in Huo and T aka y ama [ 14 ], who obtain ﬁnite-state represen tations for equilibrium aggregates under exogenous ARMA signals. These metho ds succeed b e- cause the ﬁltering environmen t is inv arian t to changes in control la ws—the sp ectral factorization can be solv ed indep enden tly of the p olicy . When con trols en ter the state dynamics, the signal pro cess dep ends on the p olicy , the endogenous-information case that Huo and T akay ama [ 14 ] show cannot admit a ﬁnite-state represen tation in gen- eral. The noise-state technique resolves this case exactly: the b elief hierarc hy collapses on to a deterministic ﬁxed p oint with no ﬁnite-state appro ximation required. Common-information approac hes. The common-information approach condi- tions on a shared information base to obtain a recursive reduction [ 24 ]. The approach handles asymmetric information but resolv es it through a co ordinator’s b eliefs; the c hannel b y which play er i ’s action shifts play er k ’s p osterior, which shifts k ’s action, whic h shifts i ’s signal, is not captured. T eam theory . The decen tralized decision problem with shared ob jectives originates with Marschak and Radner [ 22 ]; Radner [ 26 ] established certaint y equiv alence and 4 linear optimal p olicies for Linear–Quadratic–Gaussian (LQG) teams. T eam-theoretic solutions b ypass the b elief hierarch y b ecause a so cial planner with aligned ob jectives solv es a single-agent con trol problem—the ﬁxed-p oint structure never arises. The information w edge V i t displa y ed in Section 2 is an equilibrium ob ject: it prices the v alue of shifting an opp onent’s p osterior, which requires distinct ob jectives to b e w ell-deﬁned. Mean-ﬁeld LQG games. Mean-ﬁeld games take large-p opulation limits in which no single agen t’s action measurably aﬀects any other agen t’s signal [ 7 , 13 ]. Car- mona and Delarue [ 8 ] provide a comprehensive t w o-v olume treatment including LQG sp eciﬁcations with common noise. The bilateral informational externality is absent b y construction in the mean-ﬁeld limit; the information wedge V i t of Section 2 is a ﬁnite- n ob ject that v anishes in an y limit where individual actions b ecome negligible. T runcation of the b elief hierarch y . Nimark [ 25 ] approximates the higher-order b elief hierarch y b y truncation at ﬁnite order—the closest existing metho d in spirit to the presen t pap er. The deterministic impulse-resp onse system deriv ed here provides an exact b enc hmark against which truncation error can b e assessed. Applied contin uous-time ﬁltering games. Kyle [ 18 ], Bac k [ 3 ], and F oster and Visw anathan [ 9 ] solve strategic trading problems with priv ate information, each ex- ploiting sp ecial structure (a single informed trader, correlated-signal symmetry) to close the b elief hierarch y through mo del-sp eciﬁc conjectures. Sannik o v [ 28 ] solv es a con tin uous-time principal–agent problem by sho wing that the agent’s con tin uation v alue serv es as a suﬃcien t statistic for the principal’s problem, yielding a recursiv e c haracterization despite the agent’s priv ate action. The technique relies on the bilat- eral structure: the principal observes output directly , so the agen t’s deviation c hanges her pa yoﬀ but not her p osterior, and the w edge V i t is absen t. With multiple agents exerting unobserv able eﬀort on a shared outcome, eac h agen t infers others’ eﬀort from the shared output, the noise-state path replaces the scalar contin uation v alue, and the b elief hierarch y reapp ears. The present pap er provides a general framework that nests these as special cases: in the Kyle–Bac k em b edding (Section 5 ), the information w edge corresp onds to the strategic comp onen t of price impact, and the characteriza- tion extends to m ulti-trader settings where guess-and-v erify breaks do wn. 5 Rational inattention. Sims [ 31 ] in tro duced rational inatten tion; Ma ´ ck o wiak and Wiederholt [ 20 ] sho w ed that ﬁrms optimally attend to idiosyncratic ov er aggregate conditions, generating sluggish price adjustmen t (see Ma ´ ck ow iak, Ma ´ tejka, and Wieder- holt [ 21 ] for a survey). These mo dels take the strategic environmen t as giv en when solving the atten tion problem; the ﬁrst-order conditions derived here (Section D.3 ) extend the framework to settings where the marginal v alue of precision dep ends on the equilibrium through the information w edge. Information design. Kamenica and Gentzk ow [ 15 ] established Bay esian p ersua- sion; Bergemann and Morris [ 4 , 5 ] in tro duced Bay es correlated equilibrium and sur- v ey the ﬁeld, identifying dynamic decentralized settings as a cen tral op en problem. Viv es [ 33 ] develops the in terpla y b et w een information acquisition and comp etition in ﬁnancial markets that the Kyle–Back em b edding of Section 5 op erationalizes dynam- ically . Bergemann, Heumann, and Morris [ 6 ] pro vide precision comparativ e statics in static co ordination games where the eﬀect of signal precision on pay oﬀs admits direct analysis; dynamically , the information wedge couples the v alue of precision to the equilibrium, so that ﬁltering and control no longer separate. The equilibrium c haracterization derived here (Section 4 ) pro vides the mapping from signal structures to equilibrium outcomes that dynamic information design requires as an input. 2 Bilateral b elief manipulation in a t w o-pla y er game Before dev eloping the general framew ork, we display its consequences in the sim- plest game exhibiting the full bilateral externality: t wo symmetric pla y ers with scalar state, opp osing targets, and equal signal precision. Neither pla yer observ es the state directly; eac h receives a priv ate diﬀusion signal whose precision p > 0 is common kno wledge. Eac h pla yer’s con trol en ters the state dynamics and reshap es the op- p onen t’s signal, generating the bilateral informational externality of T o wnsend [ 32 ]. The general theory (Sections 3 – 4 ) collapses this hierarch y onto deterministic impulse- resp onse maps and pro duces an ob ject, the information we dge V i t , that prices the marginal v alue of shifting the opp onen t’s p osterior. 6 Primitiv e sho c ks W 0 (fundamen tal), W 1 , W 2 (signal noise) Priv ate signal Y 1 t (precision p ) State / fundamental X t Priv ate signal Y 2 t (precision p ) Filter / b elief c W 1 t ( · ) A ctions D i t = ¯ D i t + R t 0 D i t ( u ) d u c W i t ( u ) Filter / b elief c W 2 t ( · ) Figure 1: Information-to-action feedback in the w orked example. Actions mov e X t and shift opp onen ts’ b eliefs, so equilibrium mean actions dep end on precision p . 2.1 Primitiv es and ob jectiv es Fix a horizon [0 , T ]. The state X t ∈ R satisﬁes dX t =  D 1 t + D 2 t  dt + dW 0 t , X 0 = x 0 , (2.1) where W 0 is a standard Bro wnian motion. Play er i ∈ { 1 , 2 } do es not observe X directly; instead they observ e the priv ate diﬀusion channel d Y 1 t = √ p X t dt + dW 1 t , d Y 2 t = √ p X t dt + dW 2 t , (2.2) with ( W 0 , W 1 , W 2 ) mutually indep enden t. Play er i c ho oses D i adapted to their priv ate observ ation history F i t = σ ( Y i s : s ≤ t ). R unning costs are symmetric tracking losses with quadratic eﬀort p enalty r > 0: J 1 ( D 1 ; D 2 ) = E Z T 0  ( X t − 1) 2 + r ( D 1 t ) 2  dt, J 2 ( D 2 ; D 1 ) = E Z T 0  ( X t +1) 2 + r ( D 2 t ) 2  dt. (2.3) Pla y er 1 wan ts the state near +1; pla yer 2 w an ts it near − 1. There is no terminal cost in this work ed example. The mo del is in v ariant under the sign/lab el symmetry ( X , Y 1 , Y 2 , D 1 , D 2 ) 7→ ( − X, Y 2 , Y 1 , − D 2 , − D 1 ). 2.2 What the general theory deliv ers The equilibrium c haracterization dev elop ed in Sections 3 – 4 pro duces three ob jects that organize the analysis of this game; w e describ e them informally here and derive them in full generalit y b elow. 7 Noise-state and linear strategies. All randomness originates in the three prim- itiv e Brownian motions W = ( W 0 , W 1 , W 2 ). Pla y er i ’s noise-state c W i t ( u ) := E [ W u | F i t ] (0 ≤ u ≤ t ) is the running conditional mean of the en tire primitive noise path. In equilibrium, eac h pla y er’s con trol tak es the form D i t = ¯ D i t + Z t 0 D i t ( u ) d u c W i t ( u ) , (2.4) where ¯ D i t is a deterministic mean action and D i t ( u ) is a deterministic impulse-response k ernel gov erning how play er i ’s action at time t resp onds to news ab out the u -th sho c k. The key structural result (Corollary 4.5 ) is that the b est resp onse to an y such strategy proﬁle, o ver all admissible controls, is itself of this form—so equilibrium reduces to a deterministic ﬁxed p oin t in the maps ( ¯ D i , D i ). Unresolv ed impulse resp onse. The state’s impulse resp onse to the u -th sho ck decomp oses in to what play er i has learned and what remains hidden: X t ( u ) = c X i t ( u ) + f X i t ( u ). The unresolved comp onen t f X i t ( u ) is the p ortion of the u -th sho c k’s eﬀect on the state that play er i has not yet resolv ed—large when the sho ck is recen t or precision lo w, shrinking as observ ations accum ulate. It serves as the Kalman gain in the noise- state up date (Theorem 4.1 ). Information w edge. In the single-agen t problem, the backw ard equation for the costate (the shadow v alue of the state) is standard. With tw o agents and endogenous signals, a new term app ears: the information we dge ¯ V 1 t = P 2 ( t ) Z t 0 f X 2 t ( z ) · ¯ H 2 t ( z ) dz , (2.5) where f X 2 t ( z ) is the unresolv ed comp onent of the state’s resp onse to the z -th sho c k from pla y er 2’s p erspective and ¯ H 2 t ( z ) is the mean b elief adjoint —the shadow v alue to play er 1 of ha ving shifted play er 2’s estimate of the z -th sho ck (Theorem 4.2 ). The w edge enters the mean costate equation as d dt ¯ H X t = − ( ¯ X t − 1) − ¯ V 1 t , distorting mean actions relative to the single-agent b enc hmark. 8 2.3 Computation Equilibrium is computed by iterating the b est-resp onse op erator on the deterministic impulse-resp onse system (Section 4 ), discretized on a uniform grid of N = 40 p oin ts; no Mon te Carlo simulation is needed. Parameters: T = 1, x 0 = 0, r = 0 . 1, no terminal cost. The Picard residual reaches 10 − 5 in 19 iterations ( ≈ 10 ms in C++), deca ying geometrically . 1 A t longer horizons the binding constraint is numerical resolution (the ﬁltering kernel dev elops near-singularities) rather than economic. 2.4 Mean actions, separation failure, and the w elfare cost of b elief manipulation Since the sto chastic integral in ( 2.4 ) has zero unconditional mean, E [ D i t ] = ¯ D i t iso- lates the deterministic tug-of-war ov er the target ± 1, while the kernel term captures impulse resp onses to p osterior sho c ks. Separation failure. Under p erfect information, the mean p olicy is indep enden t of signal precision. Decentralized information breaks this: precision c hanges the sp eed of opp onents’ up dating, creating an incen tive to act on p osteriors rather than on the state. The mechanism is the information w edge ( 2.5 ): it couples the costate to the unresolv ed kernel f X 2 t ( z ) through the b elief adjoin t, introducing a dep endence of the equilibrium p olicy on precision that is absen t in the single-agent problem. More precisely , in the single-agen t case the p olicy kernel factors as D i t ( u ) = S ( t ) · X t ( u ) for a scalar Riccati gain S ( t ) indep endent of signal precision. With t w o agen ts, the w edge prev en ts this factorization: c hanging p changes the unresolv ed k ernel, which c hanges the b elief adjoint, which c hanges the p olicy kernel indep enden tly of the state estimate (Prop osition 4.6 ). The w edge across precision asymmetries. Figures 3 – 4 display the information w edge and its consequences when play er 1’s precision is ﬁxed at p 1 = 3 and play er 2’s precision p 2 v aries. Information p o oling as a Pigouvian in terv en tion. Figure 5 compares equilib- rium costs under priv ate signals with those under a p o oled signal of precision P 1 + P 2 , 1 An interactiv e bro wser visualization is av ailable at https://sbabichenko.com/lqg . 9 0.0 0.2 0.4 0.6 0.8 1.0 t 0 2 4 6 8 ¯ D 1 ( t ) P l a y e r 1 m e a n c o n t r o l p a t h ¯ D 1 ( t ) v s p e r f e c t i n f o ( c o l o r = p ) p = 1 p = 2 p = 3 p = 5 p = 1 0 P erfect info Figure 2: Equilibrium mean con trol ¯ D 1 t as a function of signal precision p , with the p erfect-information b enc hmark (dashed). Low precision makes opp onen ts’ p osteriors sluggish, amplifying the incen tiv e to manipulate b eliefs; as p → ∞ the mean p olicy con v erges to the p erfect-information limit. 0.0 0.2 0.4 0.6 0.8 1.0 t 6 4 2 0 2 4 6 8 ¯ D i ( t ) Mean contr ols ¯ D 1 ( s o l i d ) ¯ D 2 ( d a s h e d ) 0.0 0.2 0.4 0.6 0.8 1.0 t 0.2 0.1 0.0 0.1 0.2 ¯ X ( t ) Mean state path p 2 = 1 p 2 = 2 p 2 = 3 p 2 = 5 p 2 = 1 0 p 2 = 2 0 0.0 0.2 0.4 0.6 0.8 1.0 t 0 2 4 6 8 10 12 14 | ¯ D 1 | + | ¯ D 2 | Aggr egate mean effort P erfect info 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 p 2 A s y m m e t r i c e q u i l i b r i u m : p 1 = 3 f i x e d , p 2 v a r i e s ( r = 0 . 1 , T = 1 ) Figure 3: Asymmetric equilibrium with p 1 = 3 ﬁxed and p 2 v arying. Left: mean con trols ¯ D 1 t (solid) and ¯ D 2 t (dashed) diverge as the precision gap widens. Center: the mean state path ¯ X t tilts to w ard the b etter-informed pla yer’s target. Right: aggregate mean eﬀort | ¯ D 1 | + | ¯ D 2 | exceeds the p erfect-information b enc hmark (gra y dashed), with the excess gro wing in the precision asymmetry . for b oth comp etitive (opp osing targets ± 1) and co op erativ e (common target 0) sp ec- iﬁcations. P o oling is P areto-impro ving in b oth cases, but the gains diﬀer by an order of magnitude: roughly 0 . 02–0 . 05 under co op eration v ersus 0 . 3–0 . 5 under comp etition. Since the estimation b eneﬁt of po oling enters iden tically regardless of the target struc- ture, the diﬀerence isolates the strategic externality channel priced by V i t : nearly all of the welfare gain from mandatory disclosure comes from eliminating bilateral b elief manipulation, not from impro ving state estimation. R emark 2.1 (Impulse resp onses do not dep end on targets) . The equilibrium impulse- 10 0.0 0.2 0.4 0.6 0.8 1.0 t 0.0 0.2 0.4 0.6 0.8 1.0 1.2 V 1 ( t ) P l a y e r 1 w e d g e V 1 ( t ) p 2 = 1 p 2 = 2 p 2 = 3 p 2 = 5 p 2 = 1 0 p 2 = 2 0 0.0 0.2 0.4 0.6 0.8 1.0 t 1.0 0.8 0.6 0.4 0.2 0.0 V 2 ( t ) P l a y e r 2 w e d g e V 2 ( t ) p 2 = 1 p 2 = 2 p 2 = 3 p 2 = 5 p 2 = 1 0 p 2 = 2 0 I n f o r m a t i o n w e d g e s ( p 1 = 3 f i x e d , p 2 v a r i e s ) Figure 4: Mean information wedges ¯ V 1 t and ¯ V 2 t as p 2 v aries ( p 1 = 3 ﬁxed). Both w edges are hump-shaped in t , conﬁrming that b elief manipulation is most v aluable at intermediate dates. Increasing the opp onen t’s precision ampliﬁes the wedge: a b etter-informed opp onen t reacts more sharply to drift, raising the marginal v alue of manipulating their p osterior. The wedges v anish at the terminal date, consisten t with the zero terminal condition on the b elief adjoin t. resp onse maps—how play ers resp ond to sho c ks—do not dep end on the targets ± 1. All ﬁltering ob jects, unresolved comp onents, and p olicy resp onse maps are identical whether pla yers co op erate or comp ete; only the deterministic tug-of-war o v er mean actions sees the strategic structure, through the information w edge. 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 p 2 2.6 2.8 3.0 3.2 3.4 3.6 3.8 Cost C o m p e t i t i v e ( θ = ± 1 ) J 1 p r i v a t e J 1 p o o l e d J 2 p r i v a t e J 2 p o o l e d 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 p 2 0.225 0.250 0.275 0.300 0.325 0.350 0.375 Cost C o o p e r a t i v e ( θ = 0 , 0 ) J 1 p r i v a t e J 1 p o o l e d J 2 p r i v a t e J 2 p o o l e d E q u i l i b r i u m c o s t s : p r i v a t e v s p o o l e d ( p 1 = 3 , r = 0 . 1 ) Figure 5: Equilibrium costs under priv ate signals (dashed) and p o oled signals (solid) with p 1 = 3 ﬁxed and p 2 v arying. Left: opp osing targets ( ± 1). Right: common target (0). P o oling is P areto-improving in b oth cases, but the gain is an order of magnitude larger under comp etition, where it eliminates the bilateral b elief-manipulation exter- nalit y priced by V i t . Blue: play er 1; red: play er 2. 11 2.5 Optimal precision allo cation Information p o oling is a blunt instrument: it merges all signals into a common c han- nel, eliminating some of the bilateral externality but forecloses targeted allocation. Giv en a ﬁxed precision budget, how should a planner distribute it b etw een the t w o pla y ers? Setup. Retain the state dynamics ( 2.1 ) and observ ation channels ( 2.2 ) with opp os- ing targets ( ± 1), but allow asymmetric precisions ( p 1 , p 2 ) sub ject to p 2 1 + p 2 2 = ¯ P and asymmetric eﬀort costs ( r 1 , r 2 ) with r 1 < r 2 , so that play er 1 is the more eﬃcient mo v er of the state. The quadratic budget reﬂects diminishing returns to concen tra- tion. A planner c ho oses the split to minimize aggregate equilibrium cost; for each candidate allo cation we solv e the full equilibrium via Picard iteration. P arameters: ¯ P = 20, T = 1, with the fundamen tal volatilit y in ( 2.1 ) reduced to σ = 0 . 5. What one migh t exp ect. Under co op eration the answer is straigh tforw ard: giv e precision to the eﬃcient play er, p erhaps sharing some to ease the ineﬃcient play er’s burden. Under comp etition, concen trating precision arms the eﬃcien t play er to pull the state tow ard their target, so one migh t exp ect the planner to comp ensate by bal- ancing precision to limit manipulation. The presumed tradeoﬀ is pro ductiv e eﬃciency v ersus strategic restraint. Information starv ation. The tradeoﬀ is real but the strategic-restrain t side is far weak er than exp ected (Figure 6 , top row, r 1 = 0 . 05, r 2 = 0 . 2). Concen trating all precision on the eﬃcient play er do es shift the mean state further from zero, but the eﬀect on total eﬀort dominates: destructiv e eﬀort drops by nearly half. The pro ductiv e-eﬃciency and strategic-restraint c hannels p oint in the same direction: the presumed tradeoﬀ do es not exist. The mec hanism is information starvation : denying the ineﬃcient pla y er signal access collapses their abilit y to exert mean control (top left panel: pla y er 2’s eﬀort falls to near zero). With play er 2 largely neutralized, play er 1 has less reason to manipulate as w ell, and b oth pla y ers’ eﬀort falls. Conv ersely , giving all precision to the ineﬃcient pla yer pro duces the w orst outcome: the cheap mo v er pushes hard against a well-informed opp onent who retaliates sharply , maximizing bilateral w aste. The planner’s optimal resp onse is not to balance information but to create an information monop oly for the eﬃcien t play er. The cost of one-sided 12 0 1 2 3 4 5 6 Equilibrium cost A s y m m e t r i c c o s t s ( r 1 = 0 . 0 5 , r 2 = 0 . 2 ) Individual costs 0 1 2 3 4 5 T otal destructive effort T otal destructive effort 4 2 0 2 4 Mean contr ol / state Mean contr ols and state 0.0 0.2 0.4 0.6 0.8 F raction of pr ecision to player 1 0 1 2 3 4 5 6 Equilibrium cost S y m m e t r i c c o s t s ( r 1 = r 2 = 0 . 1 ) Individual costs 0.0 0.2 0.4 0.6 0.8 F raction of pr ecision to player 1 0 1 2 3 4 5 T otal destructive effort T otal destructive effort 0.0 0.2 0.4 0.6 0.8 F raction of pr ecision to player 1 4 2 0 2 4 Mean contr ol / state Mean contr ols and state J 1 ( e q u i l i b r i u m ) J 2 ( e q u i l i b r i u m ) J 1 ( f u l l i n f o ) J 2 ( f u l l i n f o ) Equilibrium F ull info ¯ D 1 ( e q ) ¯ D 2 ( e q ) ¯ D 1 ( C E ) ¯ D 2 ( C E ) ¯ X ( e q ) ¯ X ( C E ) Figure 6: Equilibrium quantities across the precision split, opp osing targets ( ± 1), p 2 1 + p 2 2 = ¯ P = 20, σ = 0 . 5, T = 1. T op: asymmetric eﬀort costs ( r 1 = 0 . 05, r 2 = 0 . 2); b ottom: symmetric ( r 1 = r 2 = 0 . 1). Left: individual mean con trol energies; center: total destructiv e eﬀort (equilibrium solid, full-information dashed); righ t: mean con trols and mean state (equilibrium solid, certaint y-equiv alen t dotted). dominance is a displacemen t in the mean state; the b eneﬁt is the collapse of bilateral w aste. The collapse of bilateral w aste dominates. R emark 2.2 (Commitment through incapacity) . The mec hanism parallels Sche lling’s [ 30 ] observ ation that in a game of c hick en, a driv er who visibly remo v es the steering wheel forces the opp onen t to sw erve. Here precision is common knowledge and enters the equilibrium through deterministic k ernels, so denying the ineﬃcien t play er signal ac- cess is a credible and observ able disarmament: b oth play ers b eneﬁt not from c hosen restrain t but from one side’s visible inabilit y to ﬁgh t. Separation failure in a picture. The righ t panels display separation failure di- rectly: certain ty-equiv alent mean con trols (dotted) are inv ariant to the precision split, while equilibrium mean controls (solid) slop e with the allo cation through the b elief adjoin ts. Under symmetric eﬀort costs (bottom row) the pattern is symmetric around 13 the equal split, conﬁrming that the asymmetry ab o v e is driven by cost diﬀerences in- teracting with the information channel. 3 The Decen tralized LQG Game Informal description of the game. The games w e study hav e three features: (i) m ultiple agents share a common ev olving state but observe it through priv ate noisy c hannels; (ii) eac h agent’s action feeds back into the state dynamics and thereb y in to every other agent’s signal pro cess; and (iii) agents minimize individual quadratic ob jectiv es o ver a ﬁnite horizon. F eatures (i)–(ii) generate the inﬁnite b elief hierar- c h y iden tiﬁed by T ownsend; feature (iii) pro vides the linear–Gaussian structure that mak es it tractable. 3.1 State dynamics and ob jectiv es Consider n play ers in teracting o v er a ﬁnite time horizon [ 0 , T ] on a ﬁltered probabilit y space (Ω , F , F , P ). State dynamics. The state X t ∈ R d ev olv es as dX t =  A ( t ) X t + n X i =1 D i t  dt + Σ( t ) dW 0 t , X 0 = x 0 , (3.1) where A ( t ) ∈ R d × d is deterministic and b ounded, W 0 is a d -dimensional Brownian motion, Σ( t ) ∈ R d × d is deterministic and b ounded, and D i t ∈ R d is play er i ’s control. R emark 3.1 (Con trol matrices and dimensions) . All results extend to D i t = B i ( t ) U i t with control U i t b y appropriately mo difying the cost function and ﬁrst order condi- tions. Ob jectiv es. The linear state-cost co eﬃcien t G X,i t = ¯ G X,i t + R t 0 G X,i t ( r ) dW r is a sto c hastic pro cess with deterministic mean ¯ G X,i t ∈ R d and deterministic k ernel G X,i t ( r ) ∈ R d × ( n +1) d ; deterministic co eﬃcients are the sp ecial case G X,i t ( r ) ≡ 0. Each pla y er i 14 incurs the random cost C i ( D i ; D − i ) := Z T 0  X ⊤ t G X X ,i ( t ) X t + 2( G X,i t ) ⊤ X t + G i ( t ) + ( D i t ) ⊤ G DD ,i ( t ) D i t  dt + X ⊤ T G X X ,i ( T ) X T + 2( G X,i T ) ⊤ X T , (3.2) where G X X ,i ( t ) and G DD ,i ( t ) are symmetric p ositive semideﬁnite matrices (with G DD ,i ( t ) p ositiv e deﬁnite), and D − i := ( D j ) j  = i denotes opp onents’ con trols. Each play er i seeks to minimize J i ( D i ; D − i ) := E h C i ( D i ; D − i ) i . (3.3) Opp onen ts’ con trols en ter play er i ’s cost only through X t , whic h they mov e via ( 3.1 ) and which pla yer i observes through ( 3.4 ). 3.2 Information structure Observ ations. Pla yer i do es not observe X t directly . Eac h play er monitors the ev olving state through priv ate information sources: analyst rep orts, proprietary sen- sors, researc h output, market data feeds. Some sources arrive without cost (publicly p osted prices, mandatory disclosures, direct observ ation of one’s o wn action) while others m ust b e actively acquired or pro cessed. The sources may b e discrete (a ﬁnite set of news feeds) or con tinuous (a ﬂow of independent measuremen ts whose aggregate is naturally mo deled by a Bro wnian sheet [ 17 ]). In this pap er we take each play er’s information-pro cessing capacit y as exogenous: the men u of sources and the attention allocated to each are ﬁxed and common kno wl- edge. The observ ation pro cess summarizing the play er’s information ab out the state then takes the form d Y i t = Γ i ( t ) X t dt + E i dW t , Y i 0 = 0 , (3.4) where Γ i ( t ) ∈ R d × d is a deterministic observ ation gain and E i ∈ R d × ( n +1) d is the blo c k selector extracting play er i ’s noise c hannel ( W i t := E i W t ). All Bro wnian mo- tions { W 0 , W 1 , . . . , W n } are mutually indep enden t. The precision matrix P i ( t ) := Γ i ( t ) ⊤ Γ i ( t ) measures the Fisher information rate ab out the state p er unit time; the gain Γ i is the primitiv e, and P i is derived. The blo c k-selector form is without 15 loss: replacing E i b y a general inv ertible noise loading Σ i ( t ) amoun ts to substituting Γ i, ⊤ (Σ i Σ i, ⊤ ) − 1 for Γ i, ⊤ throughout. Notation for primitiv e sho cks. W e collect all primitiv e noises into a single v ector W t := ( W 0 t , . . . , W n t ) ⊤ ∈ R ( n +1) d . The block pro jector Π i := E i, ⊤ E i is the iden tity on the i -th blo ck and zero elsewhere: Π i v = (0 , . . . , 0 , v i , 0 , . . . , 0). The blo ck-selector structure ensures that eac h play er’s observ ation noise is an indep enden t comp onen t of W , giving the noise-state its direct/indirect decomp osition (Theorem 4.1 ): pla y er i (almost) directly observ es Π i W and infers ( I − Π i ) W from drift-based learning. Pla y er i ’s information at time t is their observ ation history , F i t := σ ( Y i s : s ≤ t ). An admissible control is F i t -adapted with E [ R T 0 ∥ D i t ∥ 2 dt ] < ∞ , where ∥ · ∥ denotes the Euclidean norm on R d . Unresolv ed impulse resp onse. F or each pla yer i , deﬁne f X i t ( u ) := X t ( u ) − c X i t ( u ) , the comp onent of the state’s impulse resp onse to the u -th sho ck that play er i has not y et resolved at time t —large when the sho ck is recen t or the observ ation c hannel weak, and shrinking as observ ations accum ulate. F ormally , f X i t ( u ) = Cov( X t , dW u | F i t ) /du ; it serv es as the ﬁltering gain in the noise-state update (Theorem 4.1 ), with its explicit form ula in ( A.12 ). Throughout Sections 4 – 5 the gain Γ i ( · ) is ﬁxed; Section 6 discusses the extension to endogenous precision. 3.3 Nash Equilibrium Because each play er’s drift dep ends on opp onents’ estimates c X k t := E [ X t | F k t ], opti- mal actions generate an inﬁnite hierarc hy of conditional exp ectations. The following equilibrium concept ﬁxes opponents’ strategy maps rather than their realized actions, whic h is essential for the closure results in Section 4 . Deﬁnition 3.1 (Nash equilibrium in priv ate-signal strategies) . Throughout, a pla y er’s c hoice is a strategy map: for each t ∈ [ 0 , T ], the action D i t is a (progressively measur- able) functional of the priv ate observ ation history Y i ·∧ t . W e write D i t = D i t [ Y i ], where 16 the brack et notation D i t [ · ] denotes the map from the observ ation path ( Y i s ) s ≤ t to the action at time t . W e require E R T 0 ∥ D i t ∥ 2 dt < ∞ . A proﬁle D ∗ = ( D ∗ , 1 , . . . , D ∗ ,n ) is a Nash equilibrium if for ev ery pla yer i and ev ery admissible alternative strategy map f D i [ · ], J i ( D ∗ ,i ; D ∗ , − i ) ≤ J i ( f D i ; D ∗ , − i ) , where the inequality compares the costs induced by the corresp onding strategy pro- ﬁles. Equiv alen tly: in a unilateral deviation by i , opp onen ts’ maps are held ﬁxed and are applied to the deviated signal paths. In a unilateral deviation by i , opp onen ts’ strategy maps Y k ·∧ t 7→ D k t [ Y k ] are held ﬁxed and applied to the p erturb ed observ ation histories; in the noise-state linear class these maps ha v e deterministic co eﬃcients that do not c hange when the state path c hanges. Holding opp onents’ strategy maps ﬁxed under deviations (rather than their real- ized actions or b eliefs) is substantiv e here b ecause signals are endogenous: changing one play er’s control law changes the state dynamics, whic h c hanges ev ery opp onent’s signal, ﬁltering problem, and realized action, ev en though their decision rules hav e not changed. This pap er aims to establish that the class of noise-state linear strategies is closed under best resp onses and to deriv e a deterministic impulse-resp onse system whose ﬁxed p oints corresp ond to Nash equilibria in that class. R emark 3.2 (Sub jective equilibrium and p olicy ev aluation) . The best-resp onse closure (Corollary 4.5 ) do es not require pla yer i to kno w opp onen ts’ maps correctly . A policy- mak er can let each agen t optimize under its o wn b eliefs, then ev aluate welfare through the realized state dynamics ( 3.1 ); common kno wledge of rationalit y is suﬃcien t for Nash equilibrium but not required to apply the characterization as a b est-resp onse op erator. Noise-state. F or eac h play er i and eac h t ∈ [0 , T ], the noise-state encountered informally in Section 2.2 is deﬁned as the conditional mean path of the aggregated primitiv e noise: c W i t ( u ) := E [ W u | F i t ] , 0 ≤ u ≤ t. (3.5) 17 In linear–Gaussian settings, the conditional la w of primitive sho cks is Gaussian with deterministic conditional cov ariance, so c W i t ( · ) is a complete (path-v alued) summary of play er i ’s priv ate information. Deﬁnition 3.2 ((Primitive-noise) Linear Pro cess) . An L 2 sto c hastic pro cess L = ( L t ) t ∈ [0 ,T ] is a (primitiv e-noise) Linear pro cess if there exist deterministic functions ¯ L : [0 , T ] → R m , L : [0 , T ] 2 → R m × ( n +1) d , suc h that, for ev ery t ∈ [0 , T ], L t = ¯ L ( t ) + Z t 0 L ( t, s ) dW s , Z t 0 ∥ L ( t, s ) ∥ 2 ds < ∞ . When conv enien t, we extend L ( t, s ) by 0 for s > t . R emark 3.3 (Impulse-resp onse notation) . W e write X t ( s ) for the deterministic impulse resp onse, not the random v ariable X t ev aluated at an argumen t; the sho c k index s distinguishes the t w o. The same subscript- t , parenthetical-shock conv en tion applies to all t wo-time resp onse maps. R emark 3.4 (Prop erties of primitive-noise linear pro cesses) . By construction, primitiv e- noise linear pro cesses are F -adapted Gaussian pro cesses. Moreo v er, by Itˆ o isometry the square-in tegrabilit y condition ab o v e is equiv alent to L t ∈ L 2 for eac h t . By the martingale representation theorem, an y L 2 F t -measurable random v ariable has a sto c hastic integral represen tation against W ; the deﬁning restriction of the linear class is that the in tegrand L ( t, · ) is deterministic. Equiv alen tly , Deﬁnition 3.2 is the con tin uous-time analogue of an MA( ∞ ) represen tation: L ( t, s ) pla ys the role of the mo ving-a verage coeﬃcient at lag t − s , with the Itˆ o in tegral replacing the inﬁnite sum against i.i.d. inno v ations. Deﬁnition 3.3 (Noise-state Linear con trol / strategy) . Fix a pla yer i and recall their noise-state path c W i t ( · ) ( 3.5 ). An admissible control D i = ( D i t ) t ∈ [0 ,T ] is a noise-state linear con trol if it is aﬃne in the noise state. That is, if there exist deterministic functions ¯ D i : [0 , T ] → R d , D i : [0 , T ] 2 → R d × ( n +1) d , 18 (i) Candidate Opp onen ts use Noise-State Linear controls (ii) Closure State & ﬁlters ha ve det. resp onse maps (iii) Deviations Fixed maps ⇒ det. propagation (iv) Best resp onse F OC ⇒ control noise-state linear (v) Fixed p oin t BR map closes ⇐ ⇒ Nash eq. Figure 7: The Noise-State Linear ﬁxed-p oin t lo op. (i) Assume opp onen ts use Noise- state Linear con trols. (ii) The state and ﬁltering equations ha v e deterministic impulse resp onses. (iii) Under ﬁxed strategy maps, unobserved deviations propagate deter- ministically . (iv) The maximum principle pro duces a b est resp onse that is itself noise-state linear, closing the lo op. (v) A ﬁxed p oint of this b est-resp onse map is a Nash equilibrium. suc h that, for ev ery t ∈ [0 , T ], D i t = ¯ D i t + Z t 0 D i t ( u ) d u c W i t ( u ) , Z t 0 ∥ D i t ( u ) ∥ 2 du < ∞ , and we extend D i t ( u ) by 0 for u > t . Equiv alently , this deﬁnes the strategy map Y i ·∧ t 7− → D i t [ Y i ] := ¯ D i t + Z t 0 D i t ( u ) d u c W i t [ Y i ]( u ) . 4 Main results This section deriv es the ob jects preview ed in Section 2 in the general n -pla yer mo del of Section 3 . The strategy is analogous to Harsanyi’s [ 12 ]: just as a common prior o v er a t yp e space closes the b elief hierarch y in static incomplete-information games, the common probabilit y space of primitiv e sho cks W = ( W 0 , . . . , W n ) closes it here, with all higher-order conditional exp ectations following by linear projection onto deterministic impulse-resp onse maps. The diﬀerence is that Harsanyi’s types are drawn once while noise unfolds contin uously and actions reshap e signals, so the closure requires a ﬁxed p oin t: the maps go v erning b elief updating m ust be consisten t with the strategy proﬁle they induce (Figure 7 ). 19 Filtering closure. Recall the noise-state c W i t ( · ) ( 3.5 ). W e record the deterministic- k ernel represen tation of the estimated-noise increment and its inno v ations gain in noise-state co ordinates. T wo-time kernels throughout live on the causal triangle ∆ T := { ( t, s ) ∈ [0 , T ] 2 : 0 ≤ s ≤ t ≤ T } , extended b y 0 oﬀ ∆ T . W e write d u for incremen ts in the path index u at ﬁxed estimation time t , and d t for up dates in t at ﬁxed u . The state is assumed to admit the linear form X t = ¯ X t + R t 0 X t ( s ) dW s , where X t ( s ) ∈ R d × ( n +1) d is a deterministic impulse-resp onse k ernel mapping primitiv e sho c ks into the ph ysical state. Theorem 4.1 (Linear Filtering Closure) . Fix a player i . A ssume the state admits the primitive-noise line ar form X t = ¯ X t + Z t 0 X t ( s ) dW s , and player i observes d Y i t = Γ i ( t ) X t dt + E i dW t , with deterministic P i ( t ) ⪰ 0 and blo ck sele ctor E i . L et Π i := E i, ⊤ E i and c X i t := E [ X t | F i t ] , and deﬁne the innovation dI i t := d Y i t − Γ i ( t ) c X i t dt . (i) The estimate d-noise p ath incr ements admit the de c omp osition d u c W i t ( u ) = Π i dW u +  Z t 0 F i t ( u, s ) dW s  du, u < t, (4.1) wher e the deterministic kernel F i is given explicitly by F i t ( u, s ) = f X i s ( u ) ⊤ Γ i ( s ) E i + E i, ⊤ Γ i ( u ) f X i u ( s ) + Z t max( u,s ) f X i r ( u ) ⊤ P i ( r ) f X i r ( s ) dr . (4.2) (ii) The mixe d ( t, u ) -up date satisﬁes d t  d u c W i t ( u )  = f X i t ( u ) ⊤ Γ i ( t ) dI i t du. The unr esolve d kernel f X i is determine d algebr aic al ly fr om F i by ( A.12 ) ; the ﬁlter- ing kernel F i is the unique solution of the forwar d evolution system ( A.14 ) – ( A.15 ) (Deﬁnition A.1 ). 20 R emark 4.1 (Regression interpretation and scale separation) . Discretizing time, each observ ation increment is aﬃne in the noise incremen ts ∆ W 1: k , so by join t Gaussianit y the conditional mean E [∆ W 1: k | ∆ Y i 1: k ] is a deterministic linear pro jection; each lay er of “b eliefs ab out b eliefs” is a further deterministic comp osition of pro jections, whic h is why the regress closes. The decomp osition in part (i) reﬂects a scale separation: the direct comp onen t Π i dW u is O (1) (play er i observes their o wn noise contemporane- ously), while the indirect comp onen t captures drift-based inference ab out unobserved sho c ks at scale O ( dt ). Best-resp onse closure and information w edges. When opp onen ts use noise- state linear strategies, a play er’s b est resp onse ov er the full admissible L 2 class remains noise-state linear; the maximum principle yields a closed backw ard system of deter- ministic adjoint kernels including the information w edge V i ( t ) that prices bilateral b elief manipulation. Theorem 4.2 (Multiplay er b elief-adjoin t kernels and information w edges) . Fix a player i and supp ose e ach opp onent k  = i uses a noise-state line ar str ate gy with deterministic impulse-r esp onse maps, which pins down a deterministic forwar d en- vir onment (state impulse r esp onse X ( · , · ) , ﬁltering obje cts ( P k , f X k , F k ) , and p olicy kernels D k ). Structur e. A l l obje cts b elow ar e line ar pr o c esses (Deﬁnition 3.2 ): deterministic me an plus a sto chastic inte gr al against W with deterministic kernel. The single- agent c ostate ac quir es an additional c oupling term: player i ’s action shifts the state, which shifts opp onent k ’s signal, p osterior, and action, fe e ding b ack into the state. This channel intr o duc es b elief-adjoin t pr o c esses H k,i t ( u ) that tr ack the shadow value of p erturbing opp onent k ’s noise-state estimate. Backwar d system. The physic al c ostate H X,i t and b elief-adjoints H k,i t ( u ) ar e line ar pr o c esses satisfying d dt H X,i t = −  G X X ,i ( t ) X t + G X,i t  − A ( t ) ⊤ H X,i t − X k  = i P k ( t ) Z t 0 f X k t ( z ) H k,i t ( z ) dz | {z } =: V i t , information wedge , (4.3) d dt H k,i t ( u ) = −  D k t ( u )  ⊤ H X,i t + X t ( u ) ⊤ P k ( t ) Z t 0 f X k t ( z ) H k,i t ( z ) dz , k  = i, (4.4) 21 with terminal c onditions H X,i T = G X X ,i ( T ) X T + G X,i T and H k,i T ( · ) = 0 . A l l c o eﬃcients— P k ( t ) , f X k t ( z ) , D k t ( u ) , and X t ( u ) —ar e determine d by opp onents’ fr ozen str ate gy maps and do not change when player i deviates. The ﬁrst two terms in ( 4.3 ) ar e the standar d single-agent c ostate e quation; the thir d term V i t is the information we dge, absent in the single-agent pr oblem. In the b elief-adjoint e quation ( 4.4 ) , the for cing − D k t ( u ) ⊤ H X,i t tr ansmits the physic al c ostate into the b elief channel thr ough opp onent k ’s p olicy kernel: the lar ger D k t ( u ) , the mor e opp onent k ’s action at time t r esp onds to the u -th sho ck, and the mor e valuable it is for player i to have shifte d k ’s estimate of that sho ck. The inte gr al term is a self- c oupling thr ough which b elief p erturb ations pr op agate forwar d via the ﬁltering gain f X k ; the same inner pr o duct R t 0 f X k t ( z ) H k,i t ( z ) dz that deﬁnes the we dge in ( 4.3 ) drives the self-c oupling in ( 4.4 ) , so the b elief-adjoint fe e ds b ack into itself thr ough the channel it pric es. Sinc e al l c o eﬃcients in ( 4.3 ) – ( 4.4 ) ar e deterministic, taking exp e ctations r e c overs a close d system for the me an c o eﬃcients ( ¯ H X,i , ¯ H k,i ) ; matching the c o eﬃcient of dW r r e c overs a close d system for the r esp onse-map c o eﬃcients ( H X,i ( · , r ) , H k,i ( · , · , r )) . The two subsystems de c ouple and ar e solve d sep ar ately in the numeric al scheme of Se ction 2 . The me an we dge ¯ V i t is the gener al- n c ounterp art of the obje ct displaye d in Figur es 4 – 5 ; the kernel we dge V i t ( r ) distorts e quilibrium impulse r esp onses. Both vanish when signals ar e exo genous. Corollary 4.3 (Exogenous-signal reduction) . If c ontr ols do not enter the state dy- namics, then V i t ≡ 0 for al l i and t , the b elief-adjoint e quations ( 4.4 ) de c ouple, and the e quilibrium r e duc es to n indep endent single-agent LQG pr oblems against a c ommon exo genous state. R emark 4.2 (Mean-ﬁeld approximation error) . Mean-ﬁeld mo dels zero b oth w edge comp onen ts by construction: in the contin uum limit no single agen t’s action mea- surably aﬀects an y other agent’s signal. Eliminating b oth comp onen ts in a ﬁnite- n mo del requires either exogenous signals (Corollary 4.3 ) or n → ∞ with v anishing individual impact—the ﬁrst is ruled out when information is costly [ 10 ], and concen- tration of informed trading pushes real mark ets to w ard ﬁnite n . In the Kyle–Back em b edding (Section 5 ), the sign symmetry W 7→ − W zeros the mean wedge while the k ernel wedge remains active; this symmetry breaks in any market with asymmetric p ositions. 22 Lemma 4.4 (Strict conv exity of the b est-resp onse problem) . Fix opp onents’ str ate gy maps D − i in the deterministic impulse-r esp onse class (henc e ﬁxe d line ar op er ators fr om signal histories to actions). Then player i ’s induc e d obje ctive D i 7→ J i ( D i ; D − i ) is a strictly c onvex functional on the admissible L 2 c ontr ol sp ac e. Conse quently, the spike-variation stationarity c ondition ( B.8 ) is not only ne c essary but suﬃcient: any admissible c ontr ol satisfying it is the unique glob al b est r esp onse. Pr o of sketch. With opp onents’ maps ﬁxed and linear, the closed-lo op drift of X is aﬃne in D i , so ( X , D − i ) dep end aﬃnely on D i . The cost ( 3.3 ) is quadratic with G DD ,i ( t ) ≻ 0, hence strictly conv ex in D i . Corollary 4.5 (Noise-state linear b est resp onse (global closure)) . Under the c on- ditions of The or em 4.2 , ﬁx an opp onents’ pr oﬁle D − i of noise-state line ar str ate gies (Deﬁnition 3.3 ). If D i is a b est r esp onse to D − i over the ful l admissible L 2 c ontr ol class, then for a.e. t ∈ [0 , T ] , G DD ,i ( t ) D i t + E [ H X,i t | F i t ] = 0 , i.e., D i t = − ( G DD ,i ( t )) − 1 E [ H X,i t | F i t ] . Sinc e H X,i t is a line ar pr o c ess and E [ H X,i t | F i t ] = ¯ H X,i t + R t 0 H X,i t ( u ) d c W i t ( u ) , every b est r esp onse over the ful l admissible class is itself a noise-state line ar c ontr ol. Pr o of. With opponents’ maps ﬁxed and linear, the inno v ation dI i t := d Y i t − Γ i ( t ) c X i t dt is a standard ( F i , P )-Bro wnian motion whose law do es not dep end on D i (Liptser– Shiry aev [ 19 ], Theorem 7.12). Since c W i t ( · ) is a deterministic linear functional of I i [0 ,t ] (Theorem 4.1 in inno v ations co ordinates), play er i ’s con trol is a deterministic aﬃne functional of the exogenous pro cess I i . The induced ob jectiv e is strictly conv ex (Lemma 4.4 ), so the stationarit y condition ( B.8 ) is necessary and suﬃcient, yielding the stated formula. The exogeneit y of I i is a prop erty of the b est-resp onse problem: the resp onse-maps mediating b et w een I i and the noise-state dep end on the full strat- egy proﬁle through the forw ard closure, so the equilibrium remains a gen uine ﬁxed p oin t (Theorem D.5 ). R emark 4.3 . Since every b est resp onse ov er the full admissible L 2 class is noise- state linear (Corollary 4.5 ), any Nash equilibrium in noise-state linear con trols is automatically a Nash equilibrium of the unrestricted game; whether equilibria outside this class exist is discussed in Section 6 . 23 R emark 4.4 (Single-pla y er reduction) . When n = 1, V i t ≡ 0 and the ansatz H X,i t ( r ) = S ( t ) X ( t, r ) reduces the backw ard system to the standard matrix Riccati equation for S , reco vering classical separated LQG control. Prop osition 4.6 (F ailure of separation) . F or n ≥ 2 with endo genous signals, the e quilibrium p olicy kernel D i t ( u ) do es not factor as S ( t ) · X t ( u ) for any matrix-value d function S , unless the information we dge vanishes (Cor ol lary 4.3 ). A change in signal pr e cision alters the p olicy kernel itself, not mer ely the estimate to which it is applie d. Pr o of. With n ≥ 2 the information w edge couples the costate to b elief adjoints H k,i t ( u ), whic h dep end on f X k t ( u )—the p ortion of the state resp onse opp onent k has not yet learned. This in tro duces ( t, u )-dep endence that cannot factor through X t ( u ). P erturbing P i c hanges f X k through the equilibrium, c hanging H k,i and hence the p ol- icy k ernel indep enden tly of the state estimate. The factorization holds if and only if V i t ≡ 0. R emark 4.5 (Risk-sensitiv e extension) . The framew ork extends to the en tropic risk measure J i θ i := θ − 1 i log E [ e θ i C i ]: the b est-resp onse FOC is identical to Corollary 4.5 with c W i t replaced by a risk-adjusted noise-state that tilts conditional means and rescales conditional precision through the quadratic cost k ernel (App endix B.6 , The- orem B.1 ). R emark 4.6 (Inﬁnite horizon and the frequency domain) . Under time-homogeneous primitiv es, resp onse maps b ecome time-in v ariant K ( t, u ) = k ( t − u ) and the ﬁxed p oint b ecomes one in transfer functions, extending [ 14 , 16 , 27 ] to endogenous signals, where the transfer function of the equilibrium ﬁlter is itself an equilibrium ob ject. Rational transfer functions corresp ond to ﬁnite-dimensional Marko v suﬃcien t statistics, so the p ole structure of equilibrium k ernels diagnoses whether an exact state-space reduction exists. Reference: impulse-resp onse ob jects. F or conv enience, w e collect the kernel ob jects introduced ab o v e. 24 W = ( W 0 , . . . , W n ) primitive noise E i , Π i selector / pro jector c W i t ( u ) noise-state X t ( s ) state impulse resp onse F i t ( u, s ) ﬁltering impulse resp onse e X i t ( u ) unresolv ed state resp onse D i t ( u ) p olicy impulse resp onse D i t ( s ) primitiv e-noise con trol H X,i t ph ysical costate (linear pro cess) H k,i t ( u ) b elief adjoints (linear pro cess) V i t information wedge (linear pro cess) 5 T rading under asymmetric informedness The noise-state tec hnique extends to strategic trading with asymmetric priv ate learn- ing, a setting where guess-and-verify breaks do wn once traders diﬀer in precision or learn contin uously from priv ate sources. The price eac h trader faces decomp oses into an exogenous comp onen t and a history-linear function of their o wn demand, with Kyle’s lambda pla ying the role of G DD ,i ; the quadratic structure is distributed across time, so the ﬁrst-order condition yields an in tegral equation whose time deriv ative pins down the optimal trading rate. 5.1 Mo del and em b edding There are n informed traders and a comp etitive market maker. A fundamen tal V t := σ V W V t is driven by a Bro wnian motion W V indep enden t of ( W 0 , . . . , W n ), with selector E V . All play ers observ e aggregate order ﬂow d Z t = P j D j t dt + σ 0 dW 0 t ; trader i additionally observ es a priv ate signal d Y i t = Γ i ( t ) V t dt + dW i t . The mark et mak er observ es only Z and p osts the comp etitiv e price P t := E [ V T | F 0 t ]. T rader i maximizes exp ected proﬁt π i := E [ R T 0 D i t ( V T − P t ) dt ]. By the symmetry W 7→ − W , all mean terms v anish. Stac king trader i ’s t w o observ ation channels into a single equation reco vers the ob- serv ation mo del of Section 3 (App endix E ), so the ﬁltering machinery of App endix A applies verbatim. Price decomp osition. Fix opp onen ts’ noise-state linear strategy maps. The mar- k et mak er’s ﬁltering gains are deterministic, so the price decomp oses as P t = P − i t + Z t 0 ℓ ( t, s ) D i s ds, (5.1) 25 where P − i t is a linear pro cess indep enden t of D i oﬀ equilibrium (the price that would obtain if trader i submitted zero demand) and ℓ ( t, s ) is a deterministic kernel enco ding ho w eac h unit of past demand at time s has mo ved the current price by time t . The proﬁt b ecomes π i = E Z T 0 D i t ( V T − P − i t ) | {z } exogenous mispricing dt − E Z T 0 D i t Z t 0 ℓ ( t, s ) D i s ds | {z } endogenous price impact dt. The ﬁrst term is linear in D i ; the second is a quadratic form in the history of D i through the deterministic k ernel ℓ : − ( V T − P − i t ) pla ys the role of G X,i , and the in tegrated price-impact term plays the role of D i, ⊤ G DD ,i D i , distributed across time rather than concen trated at each instan t. Kyle’s lam b da and strict concavit y . Deﬁne the aggregate primitive-noise de- mand k ernel D Σ ,t ( s ) := P j D j ( t, s ). Since E V E 0 , ⊤ = 0, all price information comes through drift-based inference: dP t = λ ( t ) dI 0 t . (5.2) Assumption 5.1 (P ersisten t price informativeness) . λ ( t ) > 0 for al l t ∈ [0 , T ] . This is the economically relev an t case: if aggregate order ﬂow w ere temp orarily unin- formativ e ( λ = 0), traders could trade at zero price impact—an arbitrage opp ortunity that w ould restore informativ eness. The condition plays the role of G DD ,i ( t ) ≻ 0 in the LQG framework (Lemma 4.4 ), ensuring strict conca vity of the proﬁt in D i and hence necessit y and suﬃciency of the ﬁrst-order condition, just through the temp o- rally distributed quadratic form rather than p oint wise at eac h t . 5.2 Best-resp onse c haracterization Fix pla yer i and supp ose ev ery opp onent k  = i uses a noise-state linear strategy as in Section 4 . Because the quadratic structure is cross-temp oral, the ﬁrst-order condition do es not directly yield D i t ( u ); instead, the optimality condition equates the conditional mispricing to the integrated future impact cost (App endix E ): b V i t − P t | {z } H i, v al t ( u ): information rent density = E " Z T t l ( τ , t ) D i τ ( u ) dτ |F i t # | {z } H i, imp t : impact cost density , (5.3) 26 where δ P τ is the price p erturbation propagated by the deviation system ( E.1 )–( E.3 ). The left side is trader i ’s informational adv antage ov er the market maker; the righ t side is the presen t v alue of price distortion from a marginal unit of demand. Since the integral runs ov er [ t, T ], it do es not isolate D i t ( u )—only the en tire future trading plan weigh ted by price impact. Diﬀeren tiating b oth sides in t , the Leibniz b oundary term pro duces − λ ( t ) D i t ( u )— where λ ( t ) en ters as the endogenous control cost and D i t ( u ) ﬁrst app ears explicitly . Solving: λ ( t ) D i t = − d dt H i, v al t + E " Z T t ∂ l ∂ t ( τ , t ) D i τ dτ |F i t # , (5.4) with terminal condition D i T ( u ) = λ ( T ) − 1 H i, v al T ( u ). The deriv ative d dt H i, v al and the deviation system determining δ P τ are given in App endix E . 6 Conclusion Conditioning on primitiv e Brownian sho cks collapses the T o wnsend b elief hierarc h y on to deterministic impulse-response maps, yielding the ﬁrst exact equilibrium c harac- terization of ﬁnite-pla y er contin uous-time LQG games with endogenous signals. The information wedge V i t prices bilateral b elief manipulation and v anishes when signals are exogenous. The tw o-pla y er example shows that nearly all w elfare gains from disclosure come from eliminating this c hannel; the Kyle–Back embedding sho ws the tec hnique extends b ey ond quadratic costs to strategic trading with endogenous price impact, asymmetric precisions, and contin uous priv ate learning. F urther directions. The main open question is whether the deterministic impulse- resp onse class is without loss of generality . A natural program pro ceeds in three steps: (i) a Blac kw ell reduction showing that under quadratic costs the b est resp onse de- p ends on the p osterior π i t := L ( W [0 ,t ] | F i t ) only through its mean and cov ariance; (ii) a b elief-of-b elief collapse in which play er i ’s estimate of π k t factors through a deterministic op erator—the p osterior analogue of the kernel maps ( F i , f X i ); (iii) a con traction argumen t on the v ariance functional V ( D ) := P i E RR ∥ D i ( t, s, π i t ) − E [ D i ( t, s, π i t )] ∥ 2 ds dt , whic h v anishes exactly in the deterministic impulse-resp onse class. The precision optimization (App endix C.2 ) rev eals that multi-pla yer rational inatten tion couples the ﬁltering and con trol problems that separate in single-agen t 27 settings: the marginal v alue of precision dep ends on the equilibrium through the state kernel. Extending the framework to rational inatten tion requires care: v ary- ing the gain Γ i directly changes the ﬁltration in a non-monotone w ay , breaking the en v elop e argumen t. The diﬃcult y disapp ears when the control v ariable is the p ort- folio of indep endent sources monitored (a Brownian sheet formulation), since eac h additional source strictly enlarges the ﬁltration. The total precision decomp oses as P i = P i 0 + τ i , the marginal v alue of a source is the v alue of optimally exploiting its indep endent incremen t, and the env elop e theorem applies cleanly (App endix C ). Ov erlapping source p ortfolios endogenize cross-play er noise correlations, amplifying the b elief-manipulation c hannel priced by V i t . An alternative to the mean-ﬁeld limit is games on vertex-transitiv e graphs, where graph symmetry p ermits a representativ e-agent reduction while preserving the in- formation w edge. The natural conjecture is that equilibrium impulse-resp onse maps inherit the graph symmetry , reducing the ﬁxed p oin t to a single no de coupled to its neigh b orho o d—in terp olating b et w een the mean-ﬁeld limit (complete graph, v anishing edge weigh ts, ¯ V i t = 0) and the ﬁnite- n case. Sto chastic co eﬃcien ts (Remark D.2 ) and global uniqueness are natural next steps; the dissipative structure of Lemma D.3 sug- gests suﬃcien t regularity to preclude m ultiplicity , but a pro of requires monotonicity estimates b eyond the Gr¨ on wall b ounds used here. A Filtering Roadmap. W e iden tify the k ernel equations for F i via formal calculations, then de- ﬁne F i as their unique solution (uniqueness follows from L 2 -pro jection). The pip eline pro ceeds in three steps. Subsection A.1 derives the dynamics of t 7→ c W i t ( u ) from a characterizing identit y for conditional exp ectations. Subsection A.2 sp ecializes the abstract dynamics to the physical observ ation mo del, iden tifying f X i t ( u ) as the ﬁltering gain and establishing the algebraic identit y f X i = X ( I − Π i ) − X ∗ F i . Subsection A.3 deﬁnes F i as the primitiv e ﬁltering ob ject through its forw ard ev olution equation, deriv es the explicit three-term formula, and veriﬁes uniqueness. 28 A.1 Dynamics of the noise-state Fix a play er i . Recall the noise-state c W i t ( u ) := E [ W u | F i t ], 0 ≤ u ≤ t , and the observ ation c hannel d Y i t =  ¯ C ( t ) + Z t 0 C ( t, s ) dW s  dt + E i dW t , Y i 0 = 0 , where C ( t, s ) is deterministic. Since ¯ C is deterministic, we may assume without loss that ¯ C ≡ 0 by replacing Y i t with Y i t − R t 0 ¯ C ( r ) dr . Prop osition A.1 (Linear closure for c W ) . The map u 7→ c W i t ( u ) is a line ar pr o c ess: ther e exists a deterministic indir e ct-r esp onse map F i t ( · , · ) such that c W i t ( u ) = Π i W u + Z t 0 Z u 0 F i t ( z , s ) dz dW s , 0 ≤ u ≤ t. Prop osition A.2 (Dynamics of c W i t ( u )) . Ther e exist deterministic matrix-value d functions B Y ( u, t ) and B ( u ; t, s ) such that, for e ach ﬁxe d u , c W i t ( u ) − c W i t 0 ( u ) = Z t t 0 Z r 0 B ( u ; r, s ) d c W i r ( s ) dr + Z t t 0 B Y ( u, r ) d Y i r , t ≥ t 0 ≥ u, (A.1) or e quivalently in diﬀer ential form, d t c W i t ( u ) =  Z t 0 B ( u ; t, s ) d c W i t ( s )  dt + B Y ( u, t ) d Y i t , (A.2) with B Y ( u, t ) = Z t 0 ∂ s C i t ( u, s ) C ( t, s ) ⊤ ds + E i, ⊤ 1 { u ≥ t } , (A.3) B ( u ; t, s ) = − B Y ( u, t ) C ( t, s ) , (A.4) wher e C i t ( u, s ) := Cov( W u , W s | F i t ) is the deterministic c onditional err or c ovarianc e kernel. Pr o of. W e use the characterizing identit y for conditional exp ectations via exp onen tial test martingales. Fix a deterministic v ector function  r ( · ) and deﬁne φ r t b y φ r 0 = 1 and dφ r t = i φ r t ( d Y i t ) ⊤  r ( t ). A necessary and suﬃcien t condition characterizing c W i t ( u ) 29 is E h c W i t ( u ) φ r t i = E h W u φ r t i for all  r ( · ) . (A.5) W e compute the time deriv ativ e of each side indep endently , then matc h. L eft-hand side. Apply Itˆ o’s pro duct rule to c W i t ( u ) φ r t , using the ansatz ( A.2 ) and dφ r t = i φ r t ( d Y i t ) ⊤  r ( t ): d  c W i t ( u ) φ r t  = i c W i t ( u ) φ r t ( d Y i t ) ⊤  r ( t ) + φ r t d t c W i t ( u ) + d ⟨ c W i ( u ) , φ r ⟩ t . The quadratic co v ariation con tributes d ⟨ c W i ( u ) , φ r ⟩ t = i φ r t B Y ( u, t )  r ( t ) dt . T aking exp ectations and collecting the dt terms gives d dt E h c W i t ( u ) φ r t i = i E " φ r t  c W i t ( u ) Z t 0 C ( t, s ) d c W i t ( s ) ⊤ + B Y ( u, t )  #  r ( t ) + E " φ r t Z t 0  B ( u ; t, s ) + B Y ( u, t ) C ( t, s )  d c W i t ( s ) # . (A.6) R ight-hand side. Expand E [ W u φ r t ] using the SDE for φ r and apply the to wer prop ert y in the form E [ φ r s L ] = E [ φ r s E [ L | F i s ]]. Deﬁne the conditional second-momen t k ernel M i s ( u, z ) := E [ W u W ⊤ z | F i s ] = C i s ( u, z ) + c W i s ( u ) c W i s ( z ) ⊤ . Then E  W u Z s 0 C ( s, z ) dW ⊤ z     F i s  = Z s 0 ∂ z M i s ( u, z ) C ( s, z ) ⊤ dz . Splitting M i s = C i s + c W i s c W i, ⊤ s separates this in to a cov ariance piece and a pro duct- of-means piece: Z s 0 ∂ z M i s ( u, z ) C ( s, z ) ⊤ dz = Z s 0 ∂ z C i s ( u, z ) C ( s, z ) ⊤ dz + c W i s ( u ) Z s 0 C ( s, z ) d c W i s ( z ) ⊤ . (A.7) Diﬀeren tiating E [ W u φ r t ] and collecting dt terms then gives d dt E h W u φ r t i = i E " φ r t  Z t 0 ∂ s C i t ( u, s ) C ( t, s ) ⊤ ds + c W i t ( u ) Z t 0 C ( t, s ) d c W i t ( s ) ⊤ + E i, ⊤ 1 { u ≥ t }  #  r ( t ) . (A.8) Co eﬃcient matching. Equating ( A.6 ) and ( A.8 ) for all  r ( · ) and all t yields tw o condi- 30 tions. Matching the terms m ultiplying  r ( t ) (after canceling the common c W i t ( u ) R C d c W i, ⊤ terms that app ear on b oth sides via ( A.7 )) gives ( A.3 ). Requiring the residual drift in ( A.6 ) to v anish giv es Z t 0  B ( u ; t, s ) + B Y ( u, t ) C ( t, s )  d c W i t ( s ) = 0 , hence B ( u ; t, s ) = − B Y ( u, t ) C ( t, s ), whic h is ( A.4 ). A.2 Sp ecialization to the ph ysical observ ation mo del W e no w sp ecialize to the observ ation equation of Section 3 , d Y i t = Γ i ( t ) X t dt + E i dW t , whic h corresp onds to setting C ( t, s ) = Γ i ( t ) X t ( s ) in the general framework of Prop o- sition A.2 . Inno v ation pro cess. Deﬁne the inno v ation dI i t := d Y i t − Γ i ( t ) c X i t dt, where c X i t := E [ X t | F i t ]. By the inno v ations theorem for Gaussian linear ﬁltering (Liptser–Shiry aev [ 19 ], Theorem 7.12), I i is a standard ( F i , P )-Bro wnian motion. Inno v ation form of the dynamics. Substituting C ( t, s ) = Γ i ( t ) X t ( s ) in to Prop o- sition A.2 and using dI i t = d Y i t − Γ i ( t ) c X i t dt , the dynamics ( A.2 ) b ecome d t c W i t ( u ) =  Z t 0 ∂ s C i t ( u, s ) X t ( s ) ⊤ ds  Γ i ( t ) dI i t + E i, ⊤ 1 { u ≥ t } dI i t . (A.9) Iden tifying f X i as the ﬁltering gain. F or u < t , the indicator term in ( A.9 ) is lo cally constant in u , so the mixed ( t, u )-up date takes the form d t  d u c W i t ( u )  = f X i t ( u ) ⊤ Γ i ( t ) dI i t du, 0 ≤ u < t ≤ T . (A.10) T o verify the gain co eﬃcien t, recall that X t − c X i t = R t 0 f X i t ( s ) dW s and that the innov a- tion ( A.17 ) has martingale comp onen t E i dW t and absolutely con tin uous comp onen t 31 Γ i ( t ) R t 0 f X i t ( s ) dW s dt . The gain in ( A.10 ) equals the conditional cross-co v ariance den- sit y Cov( dW u , Γ i ( t )( X t − c X i t ) dt | F i t ) /dt ; b y Itˆ o isometry applied to R t 0 f X i t ( s ) dW s , this picks out f X i t ( u ). Th us f X i t ( u ) ⊤ Γ i ( t ) is the Kalman gain for updating the u -th noise incremen t: it maps observ ation innov ations at time t in to revisions of the noise estimate at the earlier time u . R emark A.1 (Conditional cross-cov ariance interpretation) . The iden tiﬁcation ab o v e can b e restated as f X i t ( u ) = Co v  X t , dW u | F i t  du . In tegrating in u reco v ers the full conditional cross-co v ariance Co v  X t , W u | F i t  = Z u 0 f X i t ( s ) ds = Z t 0 X t ( s ) ∂ s C i t ( s, u ) ds, (A.11) where the second equality follo ws from X t − c X i t = R t 0 X t ( s ) d f W i t ( s ) and Itˆ o isometry against f W i t ( u ). Algebraic iden tit y for f X i in terms of F i . Recall the induced estimated-state k ernel c X i t ( s ) := X t ( s )Π i + R t 0 X t ( z ) F i t ( z , s ) dz . Since f X i t ( u ) := X t ( u ) − c X i t ( u ), f X i t ( u ) = X t ( u )( I − Π i ) − Z t 0 X t ( z ) F i t ( z , u ) dz . (A.12) This identit y is algebraic: once F i is determined (Section A.3 ), f X i follo ws with no further ﬁxed-p oint step. R emark A.2 (State uncertain t y as a contraction) . The conditional state co v ariance satisﬁes Σ X X ,i ( t ) = Z t 0 X t ( s ) f X i t ( s ) ⊤ ds, (A.13) whic h follo ws from Itˆ o isometry applied to X t − c X i t = R t 0 f X i t ( s ) dW s . R emark A.3 (Kalman-ﬁlter interpretation) . Iden tit y ( A.13 ) sa ys that the d × d state uncertain t y Σ X X ,i ( t ) is obtained b y “pro jecting” the unresolv ed kernel f X i t ( s ) back through the state impulse resp onse X t ( s ). In the single-pla y er case with no endoge- nous drift, this reduces to the classical relation b etw een the Kalman gain and the error cov ariance. In the m ulti-pla y er setting the identit y p ersists b ecause opp onen ts’ strategy maps ha ve deterministic co eﬃcients. 32 A.3 The ﬁltering k ernel F i W e deﬁne F i as the primitive ﬁltering object through its evolution equation. The unresolv ed k ernel f X i is then a deriv ed quantit y via the algebraic iden tity ( A.12 ). Deﬁnition A.1 (Filtering kernel system) . Fix play er i . A deterministic causal k ernel F i on ∆ T solv es the ﬁltering kernel system if, with f X i deﬁned b y ( A.12 ), it satisﬁes F i 0 = 0 and ∂ t F i t ( u, s ) = f X i t ( u ) ⊤ P i ( t ) f X i t ( s ) , u, s < t, (A.14) F i t ( u, t ) = f X i t ( u ) ⊤ Γ i ( t ) E i , F i t ( t, u ) = E i, ⊤ Γ i ( t ) f X i t ( u ) . (A.15) Since ( A.12 ) expresses f X i as a deterministic linear functional of F i , the sys- tem ( A.14 )–( A.15 ) is a closed forward ODE for F i (quadratic through the substi- tution). Prop osition A.3 (Explicit formula) . Inte gr ating ( A.14 ) fr om max ( u, s ) to t and adding the b oundary c ontributions gives F i t ( u, s ) = f X i s ( u ) ⊤ Γ i ( s ) E i + E i, ⊤ Γ i ( u ) f X i u ( s ) + Z t max( u,s ) f X i r ( u ) ⊤ P i ( r ) f X i r ( s ) dr . (A.16) The thr e e terms ar e: (i) the unr esolve d r esp onse at observation time s , pr oje cte d thr ough the observation channel; (ii) the drift c omp onent of the innovation at time u , r eve aling information ab out e arlier incr ements at s ≤ u ; (iii) ac cumulate d indir e ct infer enc e fr om observations at times r ∈ [max ( u, s ) , t ] . R emark A.4 (Self-adjoin tness) . Under ( u, s ) 7→ ( s, u ) with transpose, the ﬁrst t w o terms in ( A.16 ) exchange and the integral is in v ariant (since P i ( r ) is symmetric). Hence F i t ( u, s ) = F i t ( s, u ) ⊤ , conﬁrming that the induced map is an orthogonal pro- jection. R emark A.5 (Uniqueness) . If F i and ˜ F i b oth yield the density representation of Theorem 4.1 (i), Itˆ o isometry giv es F i t ( u, s ) = ˜ F i t ( u, s ) a.e. R emark A.6 (Computation) . In the numerical scheme of Section 2 , f X i is computed from F i via ( A.12 ) at eac h time step, and the integral in ( A.16 ) is accum ulated forw ard alongside the F i up date. 33 W e now con ve rt the innov ations-based c haracterization of Subsections A.1 – A.2 in to primitive-noise co ordinates. The goal is to construct the deterministic k ernel F i suc h that for ev ery t ∈ [0 , T ] and u < t , d u c W i t ( u ) = Π i dW u +   Z t 0 F i t ( u, s ) dW s   du. A.3.1 Deriv ation Recall the inno v ation form of the estimated-noise dynamics (equation ( A.9 )): for u ≤ t , d u c W i t ( u ) = E i, ⊤ dI i u +   Z t u f X i s ( u ) ⊤ Γ i ( s ) dI i s   du, where the ﬁrst term is the direct observ ation at time u and the second is the accu- m ulated drift-based revision ov er ( u, t ]. T o pass from inno v ations to primitive noise, w e expand each dI i s in terms of dW . Expanding the inno v ation. The ph ysical-measure decomp osition of the inno v a- tion is dI i t = E i dW t + Γ i ( t ) ( X t − c X i t ) dt. The ﬁrst piece is the martingale comp onent; the second is absolutely contin uous. Substituting the linear forms and using f X i t ( r ) := X t ( r ) − c X i t ( r ) (the deﬁnition of f X i and ( A.13 )): dI i t = E i dW t + Γ i ( t ) Z t 0 f X i t ( r ) dW r dt. (A.17) Expanding the direct term. Substituting ( A.17 ) at t = u in to E i, ⊤ dI i u : E i, ⊤ dI i u = E i, ⊤ E i dW u | {z } = Π i dW u + E i, ⊤ Γ i ( u ) Z u 0 f X i u ( r ) dW r du. (A.18) The ﬁrst piece con tributes the Π i dW u in the density represen tation. The second is absolutely con tin uous in u with primitive-noise density E i, ⊤ Γ i ( u ) f X i u ( s ) at source- noise index s ≤ u . 34 Expanding the in tegral term. Substituting ( A.17 ) into R t u f X i s ( u ) ⊤ Γ i ( s ) dI i s : Z t u f X i s ( u ) ⊤ Γ i ( s ) dI i s = Z t u f X i s ( u ) ⊤ Γ i ( s ) E i dW s | {z } (a) martingale + Z t u f X i s ( u ) ⊤ P i ( s ) Z s 0 f X i s ( r ) dW r ds | {z } (b) abs. contin uous . T erm (a) con tributes f X i s ( u ) ⊤ Γ i ( s ) E i at source-noise index s ∈ ( u, t ]. Applying sto c hastic F ubini to term (b) and reading oﬀ the co eﬃcient of dW r : (b) = Z t 0   Z t max( u,r ) f X i s ( u ) ⊤ P i ( s ) f X i s ( r ) ds   dW r . Assem bling the densit y . Collecting the co eﬃcien t of dW s du from all three con- tributions and noting that causality ( f X i s ( u ) = 0 for u > s ) automatically zero es the irrelev an t indicators: F i t ( u, s ) = f X i s ( u ) ⊤ Γ i ( s ) E i + E i, ⊤ Γ i ( u ) f X i u ( s ) + Z t max( u,s ) f X i r ( u ) ⊤ P i ( r ) f X i r ( s ) dr . (A.19) The three terms ha v e the same in terpretation as in Prop osition A.3 . R emark A.7 (Boundary condition and ev olution) . Ev aluating ( A.19 ) at s = t (using f X i u ( t ) = 0 for u < t ): F i t ( u, t ) = f X i t ( u ) ⊤ Γ i ( t ) E i , F i t ( t, u ) = E i, ⊤ Γ i ( t ) f X i t ( u ) , whic h are eac h other’s transp oses. Diﬀerentiating ( A.19 ) in t at ﬁxed u, s < t gives the evolution ∂ t F i t ( u, s ) = f X i t ( u ) ⊤ P i ( t ) f X i t ( s ) , whic h is manifestly self-adjoin t under ( u, s ) 7→ ( s, u ) ⊤ . A.3.2 F ormal c haracterization Deﬁnition A.2 (Filtering kernel system (deﬁnition of F i )) . Fix play er i . The inputs are the state kernel X ( · , · ) and the unresolv ed k ernel f X i from Subsection A.2 . The 35 output is the primitiv e-noise density k ernel F i of Theorem 4.1 (i). A deterministic kernel F i is said to b e admissible if it is causal ( F i t ( u, s ) = 0 when max { u, s } > t ) and, together with the induced kernel F i t ( u, s ) = f X i s ( u ) ⊤ Γ i ( s ) E i + E i, ⊤ Γ i ( u ) f X i u ( s ) + Z t max( u,s ) f X i r ( u ) ⊤ P i ( r ) f X i r ( s ) dr , u, s ≤ t. (A.20) W e deﬁne F i to b e the unique causal solution of ( A.12 )–( A.20 ). Prop osition A.4 (V eriﬁcation: primitiv e-noise density of the noise-state) . L et F i b e the kernel deﬁne d by Deﬁnition A.1 . Then for every t ∈ [0 , T ] and u < t , d u c W i t ( u ) = Π i dW u +   Z t 0 F i t ( u, s ) dW s   du. Mor e over, F i is unique among deterministic c ausal kernels with this pr op erty. Pr o of of uniqueness. If F i and ˜ F i b oth satisfy the density represen tation, then R t 0 ( F i t ( u, s ) − ˜ F i t ( u, s )) dW s = 0 a.s. for each ( t, u ). By Itˆ o isometry , R t 0 ∥ F i t ( u, s ) − ˜ F i t ( u, s ) ∥ 2 ds = 0, hence F i t ( u, s ) = ˜ F i t ( u, s ) a.e. B Con trol First Order Conditions This app endix derives the stationarity condition ( B.8 ) and the closed backw ard sys- tem for the adjoint kernels ( ¯ H X , H X , { ¯ H k , H k } k  = i ). By the inno v ations exogeneity established in Corollary 4.5 , the b est-resp onse problem for pla yer i (opp onen ts’ maps ﬁxed) is an optimization ov er deterministic impulse-resp onse maps w eigh ting the ex- ogenous innov ation I i . 2 B.1 The b est-resp onse problem in inno v ations co ordinates Fix a baseline noise-state linear proﬁle and a deviating pla y er i . By the inno v ations exogeneit y established in Corollary 4.5 , c W i t ( · ) is a deterministic linear functional of 2 T o av oid circularity: opp onents’ deterministic k ernels mak e I i a standard Brownian motion indep enden t of D i [ 19 , Theorem 7.12]; Lemma 4.4 then gives strict conv exit y ov er the full L 2 class; the F OC derived here is therefore necessary and suﬃcient, yielding the deterministic-k ernel b est resp onse of Corollary 4.5 . 36 I i [0 ,t ] , and I i is a standard ( F i , P )-Bro wnian motion inv arian t to D i . Pla y er i ’s control D i t = ¯ D i t + Z t 0 D i t ( u ) d u c W i t ( u ) is a deterministic aﬃne functional of I i , and the optimization is o v er ( ¯ D i , D i ) with no self-referen tial signal dep endence. Since the state dynamics are linear in D i and the cost is quadratic with G DD ,i ( t ) ≻ 0, the induced ob jectiv e is strictly conv ex: the stationarit y condition derived b elo w is the unique global optim um. B.2 Spik e v ariation and the ﬁrst-v ariation system W e use spike v ariations to exp ose the adjoint structure. Fix t ∈ [0 , T ) and v ∈ L 2 (Ω , F i t ; R d ). The spike p erturbation D i,ρ,ε s := D i s + ρ v 1 [ t,t + ε ] ( s ) induces the nor- malized ﬁrst v ariation δ X s := lim ε ↓ 0 1 ε ∂ ∂ ρ X ρ,ε s    ρ =0 with δ X t = v . Because opp onen ts apply ﬁxed linear maps with deterministic kernels, the map v 7→ ( δ X s , { δ w k s ( · ) } k  = i ) is exactly linear in v with deterministic co eﬃcien ts. The deviator’s own innov ation is unaﬀected ( δ I i s = 0), since δ X s ∈ F i t implies δ c X i s = δ X s . Opp onen ts’ ﬁlter resp onse. F or each k  = i , δ ( d u c W k s ( u )) = δ w k s ( u ) du with δ w k t ( · ) ≡ 0 and ∂ s δ w k s ( u ) = f X k s ( u ) ⊤ P k ( s )  δ X s − Z s 0 X s ( r ) δ w k s ( r ) dr  , 0 ≤ u ≤ s. (B.1) The opp onen t con trol v ariation is δ D k s = R s 0 D k s ( u ) δ w k s ( u ) du , giving the coupled state system d ds δ X s = A ( s ) δ X s + X k  = i Z s 0 D k s ( u ) δ w k s ( u ) du, δ X t = v . (B.2) B.3 T ransition k ernels The coupled system ( B.1 )–( B.2 ) has b ounded deterministic co eﬃcien ts and admits a unique deterministic t wo-parameter ev olution family . Deﬁnition B.1 (Blo c k transition kernels) . F or s ≥ t , deﬁne Φ X X ( s, t ) ∈ R d × d and 37 Φ X k ( s, t, u ) ( k  = i , u ∈ [0 , t ]) b y δ X s = Φ X X ( s, t ) v + X k  = i Z t 0 Φ X k ( s, t, u ) η k ( u ) du, for initial data δ X t = v , δ w k t ( · ) = η k ( · ). In the spik e deviation η k ≡ 0, so δ X s = Φ X X ( s, t ) v . These blo cks satisfy ∂ ∂ t Φ X X ( s, t ) = − Φ X X ( s, t ) A ( t ) − X k  = i Z t 0 Φ X k ( s, t, z ) f X k t ( z ) ⊤ P k ( t ) dz , (B.3) ∂ ∂ t Φ X k ( s, t, u ) = − Φ X X ( s, t ) D k t ( u ) + Z t 0 Φ X k ( s, t, z ) f X k t ( z ) ⊤ P k ( t ) X t ( u ) dz , (B.4) with Φ X X ( t, t ) = I and Φ X k ( t, t, u ) = 0. R emark B.1 (Fixed co eﬃcien ts oﬀ equilibrium) . Ev ery co eﬃcien t in ( B.3 )–( B.4 )— D k t ( u ), f X k t ( z ) ⊤ P k ( t ), and X t ( u )—is determined by opp onen ts’ frozen strategy maps. A unilateral deviation b y play er i changes the true state kernel X through δ D i , but opp onen ts con tinue to ﬁlter and act as though the original state kernel obtained. The transition kernels therefore ha ve ﬁxed co eﬃcien ts; the deviation in the true state en ters only through the forcing G X X ,i ( t ) X t ( r ) in the backw ard system. B.4 First v ariation of costs and the adjoin t k ernels Fix play er i and suppress play er sup erscripts on w eigh t matrices. Using δ X s = Φ X X ( s, t ) v and the linear form of X s , the normalized cost v ariation is δ J i t ( v ) = 2 E " v ⊤  G DD ,i ( t ) D i t + ¯ H X t + Z T 0 H X t ( u ) d c W i t ( u )       F i t # , (B.5) where ¯ H X t := Z T t Φ X X ( s, t ) ⊤ h G X X ( s ) ¯ X s + ¯ G X,i s i ds + Φ X X ( T , t ) ⊤ h G X X ,i ( T ) ¯ X T + ¯ G X,i T i , (B.6) H X t ( u ) := Z T t Φ X X ( s, t ) ⊤ h G X X ( s ) X s ( u ) + G X,i s ( u ) i ds + Φ X X ( T , t ) ⊤ h G X X ,i ( T ) X T ( u ) + G X,i T ( u ) i . (B.7) 38 Stationarit y condition (globally necessary and suﬃcien t). Setting δ J i t ( v ) = 0 for all F i t -measurable v gives G DD ,i ( t ) D i t + ¯ H X t + Z T 0 H X t ( u ) d c W i t ( u ) = 0 a.s., (B.8) with equilibrium co eﬃcien ts ¯ D i t = − G DD ,i ( t ) − 1 ¯ H X t and D i t ( u ) = − G DD ,i ( t ) − 1 H X t ( u ). Equiv alen tly , D i t = − ( G DD ,i ( t )) − 1 E [ ¯ H X t + R T 0 H X t ( u ) dW u | F i t ]: the con trol is the conditional exp ectation of the full-information action (a certain t y-equiv alence prop ert y). B.5 Closed bac kw ard system for the adjoint k ernels Deﬁne the b elief-adjoin t ob jects ¯ H k t ( u ) := Z T t Φ X k ( s, t, u ) ⊤ h G X X ( s ) ¯ X s + ¯ G X,i s i ds + Φ X k ( T , t, u ) ⊤ h G X X ,i ( T ) ¯ X T + ¯ G X,i T i , (B.9) H k t ( u, r ) := Z T t Φ X k ( s, t, u ) ⊤ h G X X ( s ) X s ( r ) + G X,i s ( r ) i ds + Φ X k ( T , t, u ) ⊤ h G X X ,i ( T ) X T ( r ) + G X,i T ( r ) i . (B.10) Then ( ¯ H X , { ¯ H k } , H X , { H k } ) satisfy d dt ¯ H X t = − h G X X ,i ( t ) ¯ X t + ¯ G X,i t i − A ( t ) ⊤ ¯ H X t − X k  = i Z t 0 P k ( t ) f X k t ( z ) ¯ H k t ( z ) dz , (B.11) d dt ¯ H k t ( u ) = − D k t ( u ) ⊤ ¯ H X t + Z t 0 X t ( u ) ⊤ P k ( t ) f X k t ( z ) ¯ H k t ( z ) dz , (B.12) d dt H X t ( r ) = − h G X X ,i ( t ) X t ( r ) + G X,i t ( r ) i − A ( t ) ⊤ H X t ( r ) − X k  = i Z t 0 P k ( t ) f X k t ( z ) H k t ( z , r ) dz , (B.13) d dt H k t ( u, r ) = − D k t ( u ) ⊤ H X t ( r ) + Z t 0 X t ( u ) ⊤ P k ( t ) f X k t ( z ) H k t ( z , r ) dz , (B.14) with terminal conditions ¯ H X T = G X X ,i ( T ) ¯ X T + ¯ G X,i T , ¯ H k T ( · ) = 0, H X T ( r ) = G X X ,i ( T ) X T ( r )+ G X,i T ( r ), H k T ( · , r ) = 0. 39 B.6 Risk-sensitiv e extension: exp onen tial utilit y F OC Theorem B.1 (Risk-sensitiv e extension) . R eplac e ( 3.3 ) with the entr opic risk me a- sur e J i θ i := θ − 1 i log E h e θ i C i i , θ i > 0 , (B.15) wher e C i is the r andom c ost ( 3.2 ) . A T aylor exp ansion gives J i θ i = E [ C i ] + θ i 2 V ar( C i ) + O ( θ 2 i ) , so θ i governs the de gr e e of risk aversion (the risk-neutr al c ase θ i → 0 r e c ov- ers ( 3.3 ) ). Under a line ar pr oﬁle, the b est-r esp onse F OC is identic al to Cor ol lary 4.5 with c W i t r eplac e d by the risk-adjuste d noise-state c W i,θ t :=  I − θ i C t K t  − 1  c W i t + θ i C t k t  , wher e C t := C i t ( u, s ) = Co v( W u , W s | F i t ) is the deterministic c onditional c ovarianc e kernel of W given F i t (deterministic by Gaussianity of the c onditional law under the line ar pr oﬁle), K t is the symmetric quadr atic kernel of C i in the Wiener-chaos r epr esentation ( B.17 ) , and k t is the c orr esp onding line ar kernel ( B.16 ) . The formula is valid when I − θ i C 1 / 2 t K t C 1 / 2 t ≻ 0 , which holds for θ i suﬃciently smal l or T suﬃciently short. The ﬁltering structur e is unchange d; c ertainty e quivalenc e fails b e c ause the quadr atic c ost kernel r esc ales c onditional pr e cision via ( I − θ i C t K t ) − 1 while k t shifts the c onditional me an. W e now derive this result. Replace ( 3.3 ) with J i θ i := θ − 1 i log E [ e θ i C i ], where C i is the random cost ( 3.2 ). The spik e v ariation of Section B.2 carries ov er; diﬀerentiating J i θ i in the spik e amplitude gives δ J i θ i ,t ( v ) = E h Z i δ C i t ( v ) i , Z i := e θ i C i E [ e θ i C i ] , with δ C i t ( v ) identical to ( B.5 ). Quadratic structure of the cost. Expanding C i via the linear forms X s = ¯ X s + R s 0 X s ( u ) dW u and D i s = ¯ D i s + R s 0 D i s ( u ) dW u and applying sto c hastic F ubini yields the Wiener-c haos represen tation C i = c 0 + Z T 0 ℓ ( r ) ⊤ dW r + 1 2 Z T 0 Z T 0 dW ⊤ r K ( r, s ) dW s , 40 where c 0 absorbs deterministic and trace terms, and the linear and symmetric quadratic k ernels are ℓ ( r ) := Z T r  X t ( r ) ⊤  G X X ,i ( t ) ¯ X t + ¯ G X,i t  + G X,i t ( r ) ⊤ ¯ X t + D i t ( r ) ⊤ G DD ,i ( t ) ¯ D i t  dt + X T ( r ) ⊤  G X X ,i ( T ) ¯ X T + ¯ G X,i T  + G X,i T ( r ) ⊤ ¯ X T , (B.16) K ( r, s ) := Z T max( r,s )  X t ( r ) ⊤ G X X ,i ( t ) X t ( s ) + X t ( r ) ⊤ G X,i t ( s ) + G X,i t ( r ) ⊤ X t ( s ) + D i t ( r ) ⊤ G DD ,i ( t ) D i t ( s )  dt + X T ( r ) ⊤ G X X ,i ( T ) X T ( s ) + X T ( r ) ⊤ G X,i T ( s ) + G X,i T ( r ) ⊤ X T ( s ) . (B.17) Tilted conditional FOC. Since C i is quadratic in W , the w eigh t Z i tilts a Gaussian measure b y a quadratic form, preserving Gaussianity . Setting δ J i θ i ,t ( v ) = 0 for all F i t - measurable v and applying the tow er prop ert y giv es G DD ,i ( t ) D i t + E Q i  M i t     F i t  = 0 , E Q i [ · | F i t ] := E [ Z i ( · ) | F i t ] E [ Z i | F i t ] , whic h has the structure of ( B.8 ) under the tilted measure. Risk-adjusted noise-state. Since M i t is linear in W with deterministic k ernels, the FOC reduces to computing the tilted conditional mean c W i,θ t := E Q i [ W r | F i t ]. Under the deterministic-k ernel proﬁle, L ( W | F i t ) is Gaussian with mean m t := c W i t and deterministic co v ariance C t := C i t . The quadratic tilt shifts this to C − 1 t,θ = C − 1 t − θ i K t , m t,θ = C t,θ ( C − 1 t m t + θ i k t ) , pro vided I − θ i C 1 / 2 t K t C 1 / 2 t ≻ 0 (satisﬁed for | θ i | small or T short). Here C t and K t act as integral op erators on L 2 ([0 , t ]; R ( n +1) d ), so ( I − θ i C t K t ) − 1 is the resolven t (Neumann series) ( I − θ i C t K t ) − 1 f = f + θ i C t K t f + θ 2 i ( C t K t ) 2 f + · · · , con v ergen t precisely when ∥ θ i C 1 / 2 t K t C 1 / 2 t ∥ op < 1. Rearranging giv es c W i,θ t = ( I − θ i C t K t ) − 1 ( c W i t + θ i C t k t ) , 41 and substituting in to the tilted F OC yields G DD ,i ( t ) D i t + ¯ H X t + Z T 0 H X t ( r ) d r c W i,θ t ( r ) = 0 , (B.18) where c W i,θ t ( r ) := h ( I − θ i C t K t ) − 1  c W i t + θ i C t k t i ( r ) . (B.19) The risk-adjusted noise-state c W i,θ t diﬀers from the risk-neutral noise-state c W i t in tw o w a ys: the linear kernel k t ( B.16 ) shifts the conditional mean of W in the direction that reduces exp ected cost, while the quadratic kernel K t ( B.17 ) rescales conditional precision through ( I − θ i C t K t ) − 1 , amplifying the w eight on directions of W -space along whic h cost v ariance is large. As θ i → 0, c W i,θ t → c W i t and ( B.18 ) recov ers ( B.8 ). C Endogenous Signal Precision This app endix identiﬁes the obstruction to precision optimization in the multi-pla y er setting. C.1 Wh y precision v ariation is hard In single-agent LQG, the separation principle makes precision optimization straigh t- forw ard: the p olicy gain S ( t ) do es not dep end on signal precision, so the v alue of information can b e computed indep enden tly of the control problem. With m ultiple agen ts and endogenous signals, separation fails (Prop osition 4.6 ): the p olicy k ernel D i t ( u ) dep ends on precision through the information wedge, so the marginal v alue of precision dep ends on the equilibrium. A deep er diﬃcult y concerns the observ ation mo del itself. The observ ation equa- tion ( 3.4 ), d Y i t = Γ i ( t ) X t dt + E i dW t , treats each pla y er’s information channel as exogenously given. This is adequate when the analyst asks: “holding the information structure ﬁxed, what is the equilibrium?” It do es not extend to a play er choosing whic h sources to monitor, b ecause rescaling the gain Γ i → Γ i + δ Γ i c hanges the ﬁltration in a w ay that is not monotone. Concretely , the ﬁltrations generated by Γ i X dt + dW i and (Γ i + δ Γ i ) X dt + dW i 42 are generically incomparable: neither contains the other. A play er op erating at the p erturb ed precision cannot reconstruct the noise-state path c W i t ( · ) they w ould ha v e main tained under the original gain—they ha v e diﬀerent information, not more infor- mation. This breaks the env elop e argument that makes single-agent rational inatten- tion tractable: the play er at P i + δ P i cannot implement the control they w ould hav e c hosen at P i . The incomparabilit y disapp ears when precision is increased b y adding an indepen- den t source (the Brownian-sheet form ulation of Section 3.2 ). Monitoring an additional c hannel strictly enlarges the ﬁltration, and the marginal v alue of the source is the v alue of optimally exploiting its indep enden t increment. The full rational-inattention game—in which source p ortfolios are chosen endogenously—requires this formulation and is left to future w ork. C.2 T w o c hannels from a precision p erturbation Despite the incomparability , one can compute the ﬁrst-order eﬀect of a gain p er- turbation on equilibrium ob jects by linearizing the forw ard–bac kward system. The calculation reveals t wo c hannels through which δ P i aﬀects cost. Fix an equilibrium proﬁle and spike-perturb pla y er i ’s gain at time t 0 . By the en v elop e theorem, δ D i has no ﬁrst-order cost eﬀect (the FOC is satisﬁed). But the primitiv e-noise control D i τ ( r ) = D i τ ( r )Π i + R τ 0 D i τ ( u ) F i τ ( u, r ) du resp onds at ﬁrst order through δ F i : the ﬁltering kernel shifts, changing whic h part of the ﬁxed p olicy kernel is resolved b y the play er’s own information. The v ariation of the unresolved adjoin t decomp oses as δ f H X,i t ( s ) = − Z t s H X,i t ( u ) δ F i t ( u, s ) du | {z } (a) ﬁltering channel + Z T t 0 G i ( t, τ ) g δ D i t ( τ , s ) dτ | {z } (b) equilibrium channel , (C.1) where G i is the cost propagator and g δ D i is the unresolved comp onen t of the con trol v ariation induced by δ F i . Channel (a) resolves more of the ﬁxed adjoint—the standard v alue-of-information eﬀect present in single-agen t problems. Channel (b) arises because the same δ F i shifts the pro jection b oundary , pushing part of D i τ outside the play er’s curren t resolution. This is the equilibrium channel: it op erates through the information wedge, and 43 v anishes precisely when V i t = 0 (exogenous signals, Corollary 4.3 ). Multi-pla y er rational inattention therefore couples the ﬁltering and control problems that separate in single-agen t settings: the marginal v alue of precision dep ends on the equilibrium through the wedge, and optimizing attention requires solving for the equilibrium ﬁrst. D W ell-p osedness of the deterministic impulse-resp onse ﬁxed p oin t This app endix form ulates the equilibrium as a dynamic programming problem on the impulse-resp onse state space, records the a priori b ounds that follow from the pro jection structure of the ﬁltering map, and establishes w ell-p osedness: con traction for short horizons, existence for arbitrary horizons. Notation. C denotes a generic constan t dep ending on ( n, d, L, λ ); C ( M ) when it also dep ends on a p olicy-norm b ound M . D.1 Standing assumptions Assumption D.1. Ther e exist L > 0 and λ > 0 such that, for al l t ∈ [0 , T ] and al l i , ∥ A ( t ) ∥ + ∥ Σ( t ) ∥ + ∥ P i ( t ) ∥ + ∥ G X X ,i ( t ) ∥ + ∥ ¯ G X,i t ∥ + ∥ G X,i ( t, · ) ∥ L ∞ + ∥ G X X ,i ( T ) ∥ + ∥ ¯ G X,i T ∥ + ∥ G X,i ( T , · ) ∥ L ∞ ≤ L and G DD ,i ( t ) ⪰ λ I d . D.2 The k ernel state space and its dynamics A t time t the kernel state is Z t :=  ¯ X t , X t ( · ) , { F j t } n j =1  ∈ R d × L 2 ([0 , t ]; R d × ( n +1) d ) × n Y j =1 L 2 sym ([0 , t ] 2 ; R ( n +1) d × ( n +1) d ) . (D.1) The unresolv ed kernel f X j is not a state v ariable; it is determined algebraically from ( Z t ) by f X j t ( u ) = X t ( u )( I − Π j ) − Z t 0 X t ( z ) F j t ( z , u ) dz . (D.2) 44 Under a noise-state linear proﬁle D = { ( ¯ D j , D j ) } n j =1 , the k ernel state evolv es by ˙ ¯ X ( t ) = A ( t ) ¯ X t + P k ¯ D k t , (D.3) ˙ X t ( s ) = A ( t ) X t ( s ) + P k D k t ( s ) , X ( s, s ) = Σ( s ) , (D.4) ∂ t F j t ( u, s ) = f X j t ( u ) ⊤ P j ( t ) f X j t ( s ) , F j 0 = 0 , (D.5) with b oundary F j t ( u, t ) = f X j t ( u ) ⊤ Γ j ( t ) E j , and primitive-noise con trol D k t ( s ) := D k t ( s )Π k + R t 0 D k t ( u ) F k t ( u, s ) du . The p olicy D en ters ( D.3 )–( D.4 ) but do es not enter ( D.5 ) directly: F j dep ends on D only through X via ( D.2 ). D.3 The Hamilton–Jacobi–Bellman equation Since the k ernel-state dynamics ( D.3 )–( D.5 ) are deterministic, the equilibrium con- tin uation cost V i ( t, Z ) satisﬁes a ﬁrst-order terminal-v alue PDE (no diﬀusion term): ∂ t V i + D ¯ X V i · ˙ ¯ X + ⟨ D X V i , ˙ X ⟩ + n X j =1 ⟨ D F j V i , ˙ F j ⟩ + ℓ i ( t, Z , D ) = 0 , V i ( T , · ) = g i ( · ) , (D.6) where ⟨· , ·⟩ denotes the L 2 pairing. Play er i ’s con trol en ters ˙ ¯ X and ˙ X but not ˙ F j , so the Hamiltonian minimization yields ¯ D i t = − ( G DD ,i ( t )) − 1 D ¯ X V i , D i t ( u ) = − ( G DD ,i ( t )) − 1 D X V i ( u ) , (D.7) reco v ering Corollary 4.5 along the equilibrium path. The remaining P j ⟨ D F j V i , ˙ F j ⟩ term captures the indirect cost of opp onen ts’ ﬁlter ev olution; diﬀerentiating through D i → X → f X k → F k iden tiﬁes the mean information wedge as a shadow price: ¯ V i t = X k  = i Z t 0 P k ( t ) f X k t ( z ) ¯ H k,i t ( z ) dz = X k  = i ⟨ D F k V i , ∂ X ˙ F k ⟩ . (D.8) The kernel w edge V i t ( r ) admits the same in terpretation with ¯ H k,i replaced by H k,i ( · , · , r ). 45 D.4 A priori b ounds from pro jection structure The ﬁltering k ernel F j t parametrizes a p ortion of the conditional pro jection ( P j t f )( u ) := Π j f ( u )+ R t 0 F j t ( u, s ) f ( s ) ds , an orthogonal pro jection on L 2 ([0 , t ]; R ( n +1) d ) (Remark A.4 ). Lemma D.2 (Pro jection b ounds) . F or e ach j and t ≥ 0 : (i) ∥P j t ∥ op = 1 ; (ii) b oth P j t and I − P j t ar e L 2 -c ontr actions; (iii) ∥F j t ∥ op ≤ 2 wher e F j t := P j t − Π j . Pr o of. Standard prop erties of orthogonal pro jections; (iii) is the triangle inequalit y . Lemma D.3 (Dissipation) . {P j t } t ≥ 0 is non-de cr e asing in the p ositive semideﬁnite or der, and ∥ f X j ( t, · ) ∥ L 2 ([0 ,t ]) ≤ ∥ X ( t, · ) ∥ L 2 ([0 ,t ]) for al l t . Pr o of. Equation ( D.5 ) adds a p ositiv e semideﬁnite increment to F j t , so P j t is non- decreasing. Since f X j ( t, · ) = ( I − P j t ) X ( t, · ) by ( D.2 ), Lemma D.2 (ii) gives the b ound. These b ounds hold uniformly in the precision paths, the p olicy proﬁle, and the horizon T . The quadratic right-hand side of ( D.5 ) is self-limiting: as P j t gro ws to w ard I , the complementary pro jection contracts, driving f X j → 0 and decelerating F j . This prev en ts ﬁnite-time blowup despite the quadratic nonlinearit y—the noise- state analogue of Kalman v ariance collapse. D.5 W ell-posedness Deﬁnition D.1. B M T := n D = { ( ¯ D i , D i ) } n i =1 : ¯ D i ∈ C ([0 , T ]; R d ) , D i ∈ L ∞ (∆ T ) , ∥ D ∥ ≤ M o , where ∥ D ∥ := max i ( ∥ ¯ D i ∥ C + ∥ D i ∥ L ∞ ). Deﬁnition D.2 (Best-resp onse op erator) . T ( D ) := n ( − ( G DD ,i ) − 1 ¯ H X,i, D , − ( G DD ,i ) − 1 H X,i, D ) o n i =1 , where the adjoints solv e the backw ard system of Theorem 4.2 at the forward en viron- men t induced by D . Fixed p oin ts are Nash equilibria (Corollary 4.5 , Remark 4.3 ). The following estimates are pro ved b y Gr¨ on w all arguments on the forward sys- tem ( D.3 )–( D.5 ) (using the a priori b ounds of Lemmas D.2 – D.3 to con trol the quadratic nonlinearit y in ( D.5 )) and b y bac kw ard Gr¨ on wall on the linear adjoin t system ( B.11 )– ( B.14 ) (using Cauc hy–Sc hw arz on the b elief-adjoin t coupling integrals R t 0 P k f X k ¯ H k,i dz ). 46 Prop osition D.4 (Bounds and Lipschitz contin uity) . Under A ssumption D.1 , for D ∈ B M T : (i) The forwar d system has a unique glob al solution with ∥ ¯ X ∥ C + ∥ X ∥ L ∞ ≤ K fwd , ∥ f X j ( t, · ) ∥ L 2 ≤ K fwd √ T , ∥P j t ∥ op = 1 . (ii) The b ackwar d system has a unique solution with ∥ ¯ H X,i ∥ C + ∥ H X,i ∥ L ∞ ≤ K bwd , and time derivatives b ounde d by C ( M ) K bwd . (iii) Both maps D 7→ ( ¯ X , X , f X j ) and D 7→ ( ¯ H X,i , H X,i ) ar e Lipschitz with c onstants C fwd ( T , M ) and C bwd ( T , M ) r esp e ctively, b oth b ounde d by C ( M ) T e C ( M ) T . Her e K fwd , K bwd ≤ C ( M ) e C ( M ) T . Pr o of sketch. F or the forw ard system: ( ¯ X , X ) solve linear equations with b ounded co eﬃcien ts once D k is con trolled. Since D k in v olves F k through a linear integral, and ∥F k t ∥ op ≤ 2 (Lemma D.2 ), the coupled system for ϕ ( t ) := ∥ X ∥ L ∞ (∆ t ) closes as ϕ ( t ) ≤ C ( M ) + C ( M ) R t 0 ϕ ( r ) dr . F or the Lipschitz b ound: the v ariation δ D k decomp oses in to a direct term b ounded b y C ( M ) ∥ δ D ∥ and an indirect term through δ F k , whic h couples to δ f X k and back to δ X via ( D.2 ). A Gr¨ on wall argumen t on the aggregate Ψ( t ) := ∥ δ X ∥ L ∞ (∆ t ) + max j sup r ≤ t ∥ δ f X j ( r , · ) ∥ L 2 giv es Ψ( T ) ≤ C ( M ) T e C ( M ) T ∥ δ D ∥ ; the factor of T arises b ecause the p olicy enters through integrals of length ≤ T . The backw ard estimates follo w from linearity of the adjoin t system, Cauc hy– Sc h warz on coupling terms, and backw ard Gr¨ on w all. Theorem D.5 (Short-horizon equilibrium) . Under A ssumption D.1 , ther e exists T ∗ > 0 dep ending only on ( n, d, L, λ ) such that for T ∈ (0 , T ∗ ] , T is a c ontr action on B M 0 T for a suitable M 0 . In p articular, ther e exists a unique Nash e quilibrium in noise- state line ar str ate gies with b ounde d kernels, Pic ar d iter ates c onver ge ge ometric al ly, and the e quilibrium is Nash over the ful l admissible L 2 class. Pr o of. By Prop osition D.4 (iii), Lip( T ) ≤ λ − 1 C ( M 0 ) T e C ( M 0 ) T . Self-mapping ( T : B M 0 T → B M 0 T ) holds b ecause at M = 0 the b elief-adjoin t decouples ( D k ≡ 0 kills the forcing in ( B.12 )), giving ﬁnite K bwd (0 , T ); the intermediate v alue theorem yields M 0 . Since M 0 is b ounded as T → 0, c ho osing T ∗ small enough makes Lip( T ) ≤ 1 2 . Ba- nac h’s theorem gives existence, uniqueness, and geometric conv ergence. Corollary 4.5 extends to the full L 2 class. 47 Theorem D.6 (Global existence) . Under A ssumption D.1 , for every T > 0 ther e exists a Nash e quilibrium over the ful l admissible L 2 class. Pr o of sketch. Apply the Schauder ﬁxed-p oin t theorem to T on B M 0 T , w orking in the top ology τ := ( C ([0 , T ]) × L 2 (∆ T )) n . Self-mapping holds by the same argument as ab o v e. τ -closedness of B M 0 T : w eak- ∗ low er semicontin uit y of the L ∞ norm pre- serv es the b ound under L 2 limits. τ -contin uit y of T : on the b ounded set B M 0 T , L 2 -con v ergence of p olicy kernels propagates through the forw ard linear equations (via Cauc hy–Sc hw arz) and the linear backw ard system. τ -precompactness of the im- age: the uniform time-deriv ativ e b ound on ¯ H X,i (Prop osition D.4 (ii)) giv es equicon- tin uit y of ¯ D i, new in C ([0 , T ]); the same b ound gives p oint wise-in- u equicontin uit y of D i, new ( t, u ) in t , whic h with the uniform L ∞ b ound yields L 2 -equicon tin uity of t 7→ D i, new ( t, · ) and hence sequential compactness in L 2 (∆ T ) b y the vector-v alued Arzel` a–Ascoli theorem ([ 2 , Theorem 1.3.2]). R emark D.1 (Uniqueness b ey ond the contrac tion regime) . Theorem D.6 gives exis- tence but not uniqueness for T > T ∗ . The dissipation of Lemma D.3 suggests enough structure to preclude multiplicit y , but a proof requires monotonicit y estimates b ey ond the Gr¨ on w all b ounds used here. In Section 2 , Picard iteration conv erges to the same limit from m ultiple initial conditions for all parameter v alues tested. R emark D.2 (Sto c hastic co eﬃcien ts) . If primitiv es dep end on a public factor pro- cess ξ t (e.g. a ﬁnite-state Mark o v c hain representing regime switc hes), the k ernel state augmen ts to ( Z t , ξ t ). The pro jection b ounds of Lemmas D.2 – D.3 are algebraic and hold pathwise, so forward well-posedness extends betw een jumps of ξ . Eac h jump triggers a regime transition in which ﬁltering gains, strategies, and the information w edge shift discontin uously , while Z is contin uous across jumps. The backw ard ODE b ecomes a BSDE driven by jumps of ξ ; the Gr¨ on wall argumen ts extend to path wise b ounds conditional on the regime path. E Kyle–Bac k deriv ations This app endix derives the price-impact kernel l ( τ , t ), the ﬁrst-order condition ( 5.3 ), and the bac kw ard ODE ( 5.4 ), with all opp onen ts’ noise-state linear strategy maps held ﬁxed. 48 E.1 The price-impact k ernel and Kyle’s lam b da A unit spike p erturbation of trader i ’s demand at time t triggers a coupled forw ard resp onse: the mark et maker revises the price, each opp onent revises their estimate, adjusts their trading, and the adjustmen ts feed bac k in to prices. Let δ D Σ ,τ denote the aggregate demand p erturbation and δ w j τ ( u ) the noise-state density p erturbation for observer j . The deviation system is: δ D Σ ,τ = X k  = i Z τ 0 D k τ ( u ) δ w k τ ( u ) du, (E.1) ∂ τ δ w 0 τ ( u ) = e C 0 ( τ , u ) ⊤ σ − 1 0 δ D Σ ,τ − Z τ 0 C 0 ( τ , r ) δ w 0 τ ( r ) dr ! , (E.2) ∂ τ δ w k τ ( u ) = e C k ( τ , u ) ⊤   σ − 1 0 δ D Σ ,τ 0   − Z τ 0 C k ( τ , r ) δ w k τ ( r ) dr ! , k  = i, (E.3) with initial conditions δ w j t ≡ 0, δ D Σ ,t = 1, and δ I i τ = 0 (the deviator’s o wn inno v ation is unaﬀected). Equation ( E.1 ) closes the lo op: each opp onent’s noise-state shift propagates through their p olicy k ernel bac k into aggregate demand, whic h in turn shifts the mark et mak er’s and other opp onen ts’ noise-states. Price-impact kernel. Deﬁne the price-impact kernel as the price resp onse to a unit demand impulse at time t : l ( τ , t ) := σ V Z τ 0 E V δ w 0 τ ( u ) du, τ ≥ t. (E.4) This is the object ℓ ( t, s ) of the price decomposition ( 5.1 ): the market mak er’s ﬁltering gains are deterministic, so l ( τ , t ) is deterministic. Kyle’s lam b da. Since E V E 0 , ⊤ = 0, the mark et maker learns ab out the fundamen- tal only through drift-based inference, and the instan taneous price impact is l ( t, t ) = λ ( t ) := σ V Z t 0 E V e C 0 ( t, u ) ⊤ du. (E.5) 49 E.2 Leibniz diﬀeren tiation and the bac kw ard ODE Diﬀeren tiating ( 5.3 ) in t , Leibniz on the impact side pro duces − λ ( t ) D i t plus E [ R T t ∂ l ∂ t ( τ , t ) D i τ dτ | F i t ]. The v alue side requires more care. V alue side. The conditional mispricing H i, v al t = b V i t − P t = σ V E V  c W i t ( t ) − c W 0 t ( t )  can b e written as σ V E V Z t 0 " Z t 0  F i t ( u, s ) − F 0 t ( u, s )  du # dW s . The integrand k ernel is deterministic and evolv es through ∂ t F j t ( v , u ) = e C j ( t, v ) ⊤ e C j ( t, u ). Diﬀeren tiating b y Leibniz: dH i, v al t = σ V E V Z t 0 " Z t 0  e C i ( t, u ) ⊤ e C i ( t, s ) − e C 0 ( t, u ) ⊤ e C 0 ( t, s )  du # dW s dt + dM t , (E.6) where dM t collects the martingale increment from the b oundary at s = t . 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