Strategic Gaussian Signaling under Linear Sensitivity Mismatch

Strategic Gaussian Signaling under Linear Sensitivit y Mismatc h ⋆ Hassan Mohamad ∗ , ∗∗ , Vineeth S. V arma ∗ , Samson Lasaulce ∗ , ∗ Universit ´ e de L orr aine, CNRS, CRAN, F-54000 Nancy, F r anc e. ∗∗ Universit ´ e Internationale de R ab at, TICL ab, R ab at, Mor o c c o. Abstract: This paper analyzes Stac kelberg Gaussian signaling games under linear sensitivit y mismatc h, generalizing standard additiv e and constant-bias mo dels. W e c haracterize the Stac kelberg equilibrium structure for b oth noiseless and noisy signaling regimes. In the noiseless case, we show that the enco der selectiv ely rev eals information along specific eigenspaces of a cost-mismatc h matrix. W e then extend the analysis to the noisy regime and deriv e analytical thresholds for the existence of informative equilibria, demonstrating a sharp phase transition where comm unication collapses into silence if the sensitivity mismatch is sufficiently high, in con trast with the fully revealing equilibria often found in constan t-bias mo dels. Keywor ds: Signaling games, Cheap talk, Bay esian p ersuasion, Linear bias. 1. INTR ODUCTION In decen tralized con trol and cyb er-physical systems, in- formation exchange frequen tly inv olves agen ts with mis- aligned incentiv es. Unlik e classical comm unication, whic h prioritizes reliable data reconstruction, Str ate gic Informa- tion T r ansmission (SIT) arises when an informed sender comm unicates with a receiver whose actions impact b oth parties’ costs. In this setting, divergen t ob jectiv es create a fundamen tal tension b et ween information revelation and strategic manipulation. Understanding the limits of com- m unication under such misalignment is crucial for design- ing robust decen tralized systems, with applications rang- ing from smart grids Larrousse et al. (2014) and federated learning (Sun et al., 2024; Hassan et al., 2024) to economic and p olitical interactions Kamenica (2019). In the foundational cheap-talk mo del of Crawford and Sob el (1982), costless but non-verifiable communication b et ween misaligned agents leads to coarse, quan tized infor- mation transmission. The Ba yesian p ersuasion framework of Kamenica and Gen tzko w (2011) considers a complemen- tary setting where the sender commits to an information disclosure p olicy , yielding a Stack elb erg game structure that often enables ric her communication. Ov er the past decade, these ideas ha ve been extensiv ely de- v elop ed in control, signal pro cessing, and information the- ory , where the sender is t ypically a sensor or enco der, the receiv er is an estimator or controller, and both agen ts ha v e quadratic ob jectives with Gaussian information structures Aky ol et al. (2017); Sarıta¸ s et al. (2017); F arokhi et al. (2017). This has pro duced a rich bo dy of results on signal- ing games, including conditions for linear equilibria (a fun- damen tal issue dating back to Witsenhausen (1968)), dy- namic extensions,Sarıta¸ s et al. (2020); Sayin et al. (2019), priv acy-signaling games Akyol et al. (2015); Kazikli et al. (2022), and multidimensional geometric c haracterizations ⋆ This work was funded b y the CNRS MITI pro ject BLESS. E-mail corresponding author: hassan.mohamad@univ-lorraine.fr Kazıklı et al. (2023). In parallel, the information-design lit- erature has dev elop ed semidefinite programming metho ds and sp ectral characterizations for quadratic Gaussian p er- suasion T am ura (2018); Sa yin and Ba¸ sar (2022); V elic heti et al. (2025), rev ealing a geometric principle: the sender rev eals state-space directions where information b enefits her ob jectiv e and suppresses directions where it is harmful. A common thread in the signaling literature is that pref- erence misalignment is typically modeled through additive bias , where the enco der prefers the deco der’s action to trac k a shifted target rather than the true state. Under this assumption, a key result in Ba y esian p ersuasion (or Stack- elb erg c heap talk) is that equilibria are fully revealing: the enco der, an ticipating the deco der’s optimal resp onse, gains nothing from withholding information Sarıta¸ s et al. (2017, 2020). How ev er, in man y c yber-physical and human– mac hine systems, misalignmen t arises not from a shift in lo cation but from a difference in sensitivity . Tw o agents ma y both care about a v ector-v alued state but w eigh its comp onen ts differently . A strategic sensor, for instance, ma y be o ver-sensitiv e to certain state dimensions due to priv acy constrain ts Aky ol et al. (2015) or safety considera- tions, effectively preferring the deco der’s action to track a sc ale d or r otate d version of the state rather than a shifted one. This paper inv estigates how such line ar sensitivity mis- match fundamentally alters strategic Gaussian signaling. W e consider a sender who observ es a Gaussian source and communicates o ver either a noiseless (c heap-talk) or noisy c hannel to a receiver. The receiver minimizes stan- dard mean-squared estimation error (MMSE), while the sender’s ob jectiv e is cen tered at a linear transformation of the state. The sender announces her p olicy first, and the receiv er b est responds, yielding a Stac k elb erg game. Our main con tributions are as follows: (1) W e in tro duce a Gaussian signaling game where mis- alignmen t is captured b y a general matrix A that trans- forms the state. This formulation generalizes the standard additiv e bias models, showing that equilibrium b eha vior is gov erned by the eigenv alues of the sensitivit y mismatch rather than the magnitude of a bias v ector. (2) F or multidimensional c heap talk, w e sp ecialize quadratic Gaussian p ersuasion results (T amura, 2018; Sa yin and Ba¸ sar, 2022) to the linear sensitivit y mismatch mo del. W e sho w that equilibrium disclosure is go verned b y the sp ectral decomp osition of a transformed mismatch matrix, and w e provide an explicit linear enco der that realizes the optimal p osterior cov ariance. (3) Distinct from the persuasion literature, we analyze the noisy channel setting with transmission costs. W e derive necessary and sufficient conditions for the existence of informativ e equilibria in the scalar case and a necessary condition in the vector case with isotropic sensitivit y . W e demonstrate a phase transition where communication collapses if the transmission cost exceeds a threshold determined by the channel capacit y and the degree of sensitivit y mismatch. Notation. W e denote vectors with b old low er-case letters (e.g., x ) and matrices with regular upp ercase letters (e.g., A, Σ). Scalars are denoted by lo wer-case non-bold letters (e.g., ρ, a ). Unless specified otherwise, v ectors represen t random v ariables. Realizations are denoted b y the same sym b ols, distinguished by context. F or a given matrix A and a vector y , A ⊤ and y ⊤ denote the transp ose. The identit y and zero matrices are denoted by I and O , resp ectiv ely . F or t wo symmetric matrices A and B , A ≻ B ( A ⪰ B ) indicates that A − B is p ositive (semi)definite. N ( 0 , Σ) denotes a Gaussian distribution with mean 0 and co v ariance Σ. E [ · ] denotes the exp ectation op erator, ∥ · ∥ denotes the standard Euclidean norm, and T r( · ) denotes the matrix trace. 2. PR OBLEM FORMULA TION 2.1 System Mo del W e consider a strategic comm unication system consisting of tw o decision makers: an enco der (sender) and a deco der (receiv er). The system op erates as follows (see Fig. 1): 1. Sour c e: A random v ector m ∈ R n is drawn from a zero-mean Gaussian distribution with co v ariance Σ m ≻ O , i.e., m ∼ N ( 0 , Σ m ). 2. Enc o ding: The enco der observes the realization m of the source and transmits a signal x = γ e ( m ) ∈ R n . W e define the set of admissible enco der policies, Γ e , as the set of all deterministic (Borel-measurable) functions γ e : R n → R n . 3. Channel: The signal x passes through an additiv e Gaussian noise c hannel y = x + w , where w ∼ N ( 0 , Σ w ) is indep endent of m . The noiseless (cheap-talk) setting corresp onds to Σ w = O , yielding y = x . 4. De c o ding: The deco der observes the realization y and pro duces an estimate u = γ d ( y ) ∈ R n . Similarly , we define the set of admissible deco der p olicies, Γ d , as the set of all deterministic (Borel-measurable) functions γ d : R n → R n . 2.2 Obje ctive F unctions with Line ar Bias The enco der and deco der aim to minimize their resp ective exp ected costs. These ob jectives are non-aligne d , making the problem a game rather than a joint optimization m Enco der γ e + w ∼ N ( 0 , Σ w ) Deco der γ d u x y Fig. 1. System mo del. The enco der observ es the source m , transmits x ov er a noisy channel, and the deco der pro duces an estimate u of m . problem. W e define the instantane ous c ost functions for the deco der and enco der as follows: c d ( m , u ) = ∥ m − u ∥ 2 , (1) c e ( m , x , u ) = ∥ A m − b − u ∥ 2 + ρ ∥ x ∥ 2 . (2) Here, c d represen ts the standard squared error distortion. F or the encoder, c e incorp orates a line ar bias through a matrix A ∈ R n × n , an additive bias b ∈ R n , and a soft transmission p o wer constraint weigh ted b y ρ ≥ 0. Decision makers seek to minimize the exp ected v alue of these instantaneous costs ov er the join t distribution induced by the source, the channel noise, and their chosen p olicies. The deco der’s exp ected cost is J d ( γ e , γ d ) = E  c d ( m , u )  . (3) The enco der’s exp ected cost is J e ( γ e , γ d ) = E [ c e ( m , x , u )] . (4) R emark 1. The matrix A captures the sensitivity mis- match b et ween the encoder and decoder. If A = I , this setup reco vers the classical model of Sarıta ¸ s et al. (2017) with additiv e bias b . R emark 2. (Cheap talk vs. Signaling). When ρ = 0 and the channel is noiseless, the transmitted signal x has no direct cost and serves purely as a message. This setting is referred to as Stac kelberg che ap talk (Crawford and Sobel, 1982) (with commitment), effectively making the problem one of Bayesian Persuasion (Kamenica and Gentzk o w, 2011). When ρ > 0 or the c hannel is noisy , the problem b ecomes a signaling game , where the enco der faces a trade- off b et ween communication fidelity and transmission cost. 2.3 Stackelb er g Equilibrium W e study the Stackelb er g game where the enco der acts as the le ader and the decoder as the fol lower . This models scenarios where the enco der’s p olicy is designed and com- mitted to b efore the deco der optimizes its resp onse. Definition 3. (Stack elb erg Equilibrium). A pair of policies constitutes a Stack elb erg Equilibrium (SE) with the en- co der as the leader if the enco der commits to a p olicy γ e and the decoder pla ys the optimal best resp onse. The p olicy γ ∗ ,e satisfies J e ( γ ∗ ,e , γ ∗ ,d ( γ ∗ ,e )) ≤ J e ( γ e , γ ∗ ,d ( γ e )) ∀ γ e ∈ Γ e , where γ ∗ ,d ( γ e ) is the deco der’s b est resp onse to γ e : γ ∗ ,d ( γ e ) = arg min γ d ∈ Γ d J d ( γ e , γ d ) . Pr op osition 4. (Deco der’s b est resp onse). F or an y fixed en- co der policy γ e , the unique optimal decoder strategy is the conditional exp ectation of the source given the observ a- tion: γ ∗ ,d ( y ) = E [ m | y ] . (5) Pro of. Let c m ( y ) = E [ m | y ] and let u ( y ) b e any arbitrary estimator. W e decomp ose the mean squared error as E [ ∥ m − u ∥ 2 ] = E [ ∥ ( m − c m ) + ( c m − u ) ∥ 2 ] = E [ ∥ m − c m ∥ 2 ] + E [ ∥ c m − u ∥ 2 ] + 2 E [( c m − u ) ⊤ ( m − c m )] . By the la w of iterated exp ectations, we hav e E  ( c m − u ) ⊤ ( m − c m )  = E h E [( c m − u ) ⊤ ( m − c m ) | y ] i = 0 . Th us, the cost is minimized if and only if E [ ∥ c m − u ∥ 2 ] = 0, whic h implies u ( y ) = c m ( y ) almost surely . 2 W e define the following equilibrium outcomes. Definition 5. W e say that an SE is informative if the enco der reveals information related to the source, i.e., the source m and the message x are not indep enden t random v ariables. An SE is non-informative if the signal pro vided by the encoder do es not alter the decoder’s b elief relativ e to the prior mean, resulting in u = E [ m ] almost surely . F urthermore, an SE is ful ly r eve aling if the signal x allo ws the deco der to perfectly reconstruct the source (i.e., u = m almost surely). 3. NOISELESS CASE: BA YESIAN PERSUASION 3.1 Sc alar Case W e begin by analyzing the scalar instance ( n = 1) under the c heap-talk assumption with no noise ( w = 0 , ρ = 0). In this setting, the signal is y = x . By Prop osition 4, for an y enco ding p olicy γ e , the deco der’s unique optimal resp onse is u = E [ m | x ] . (6) The enco der, anticipating this resp onse, seeks a p olicy γ e that minimizes E [( am − b − u ) 2 ]. The or em 6. (Stack elb erg informative threshold). The na- ture of the SE is determined solely b y the m ultiplicative bias parameter a : 1. If a > 1 / 2, ev ery SE is fully revealing. 2. If a < 1 / 2, ev ery SE is non-informative (babbling). 3. If a = 1 / 2, the enco der is indifferen t and every p olicy is an SE. Pro of. The game pro ceeds sequen tially . Given the de- co der’s optimal response u = E [ m | x ] (Prop. 4), the enco der selects γ e to minimize J e = E  ( am − b − u ) 2  . Expanding the quadratic term, J e = E  (( am − u ) − b ) 2  = E  ( am − u ) 2  − 2 b E [ am − u ] + b 2 . W e analyze the linear and quadratic terms separately . By the la w of iterated exp ectations, E [ u ] = E [ E [ m | x ]] = E [ m ]. Th us, the linear term b ecomes − 2 b ( a E [ m ] − E [ u ]) = − 2 b ( a − 1) E [ m ] . This term dep ends only on the prior statistics of the source and the constants a and b ; it is indep endent of the enco der’s p olicy γ e . No w consider the quadratic term E  ( am − u ) 2  . W e can decomp ose the source m into the estimate u and the estimation error m − u . Then am − u = ( a − 1) u + a ( m − u ), and w e obtain E h  ( a − 1) u + a ( m − u )  2 i = ( a − 1) 2 E [ u 2 ] + a 2 E [( m − u ) 2 ] + 2 a ( a − 1) E [ u ( m − u )] . By the orthogonality principle of conditional exp ectation, the error m − u is orthogonal to an y function of the observ ation x , and specifically to the estimator u . Thus, E [ u ( m − u )] = 0. F urthermore, applying orthogonality to the v ariance of m , we hav e E [ m 2 ] = E [ u 2 ] + E [( m − u ) 2 ], whic h implies E [( m − u ) 2 ] = E [ m 2 ] − E [ u 2 ]. Substituting this bac k gives ( a − 1) 2 E [ u 2 ] + a 2 ( E [ m 2 ] − E [ u 2 ]) =  ( a − 1) 2 − a 2  E [ u 2 ] + a 2 E [ m 2 ] = (1 − 2 a ) E [ u 2 ] + a 2 E [ m 2 ] . The total cost for the enco der is therefore J e = (1 − 2 a ) E [ u 2 ] + cte , where cte = a 2 E [ m 2 ] − 2 b ( a − 1) E [ m ] + b 2 is a constant indep enden t of γ e . The minimization problem reduces to minimizing (1 − 2 a ) E [ u 2 ]. T o minimize the cost, the enco der m ust choose E [ u 2 ] sub ject to ( E [ m ]) 2 ≤ E [ u 2 ] ≤ E [ m 2 ]. W e hav e the follo wing cases: 1. If a > 1 2 , the co efficient (1 − 2 a ) is negative. The enco der minimizes cost b y maximizing E [ u 2 ]. The upp er b ound E [ u 2 ] = E [ m 2 ] is achiev able by any injectiv e policy , such as the iden tit y map γ e ( m ) = m . In this case, u = m almost surely , resulting in a fully revealing equilibrium. 2. If a < 1 2 , the co efficient (1 − 2 a ) is p ositiv e. The enco der minimizes cost b y minimizing E [ u 2 ]. The low er b ound E [ u 2 ] = ( E [ m ]) 2 is achiev able b y any constan t p olicy , suc h as γ e ( m ) = 0. In this case, the signal carries no information, u = E [ m ], and the equilibrium is non- informativ e. 3. If a = 1 2 , the coefficient is z ero. The cost is indep endent of the p olicy γ e , meaning any admissible policy constitutes a Stac kelberg equilibrium. 2 3.2 Multidimensional Case W e now consider the multidimensional c heap-talk game describ ed in Section 2, where the source is m ∼ N ( 0 , Σ m ) with Σ m ≻ O , and the enco der has a bias matrix A . In this cheap-talk setting, w e ha v e no noise (Σ w = O ), and signaling is cost-free ( ρ = 0). F or any fixed enco der p olicy γ e , the deco der observes x = γ e ( m ). The deco der’s ob jectiv e is to minimize the mean squared error; the unique optimal b est resp onse (by Prop. 4) is the conditional mean: γ ∗ ,d ( x ) = E [ m | x = γ e ( m )] . Let u = γ ∗ ,d ( x ) denote the deco der’s estimate. By the prop erties of conditional exp ectation, w e can decomp ose the source in to the estimate and the error m − u . The error is orthogonal to the estimate, i.e., E [ u ( m − u ) ⊤ ] = O . Consequen tly , the cov ariance of the source decomp oses as Σ m = Σ u + Σ e , where Σ u = E [ uu ⊤ ] and Σ e = E [( m − u )( m − u ) ⊤ ]. Since co v ariance matrices are positive semidefinite, any achiev able p osterior m ean cov ariance Σ u m ust satisfy the constraint Σ m ⪰ Σ u ⪰ O . Expanding the norm and noting that E [ m ] = E [ u ] (b y the la w of iterated expectations), the linear terms inv olving b boil do wn to constants indep enden t of the p olicy . The enco der’s problem reduces to minimizing the quadratic term: E [ ∥ A m − u ∥ 2 ] = E [ ∥ ( A − I ) u + A ( m − u ) ∥ 2 ] . Using the orthogonality E [ u ( m − u ) ⊤ ] = O and introduc- ing the trace op erator (recall that E [ ∥ z ∥ 2 ] = T r( E [ z z ⊤ ])), w e rewrite the cost as T r  ( A − I ) ⊤ ( A − I )Σ u  + T r  A ⊤ A Σ e  = T r  ( A − I ) ⊤ ( A − I )Σ u  + T r  A ⊤ A (Σ m − Σ u )  . Grouping terms dep endent on Σ u , the enco der’s ob jectiv e b ecomes J e (Σ u ) = T r( V Σ u ) + const , (7) where const = T r  A ⊤ A Σ m  , and V : = ( A − I ) ⊤ ( A − I ) − A ⊤ A = I − ( A + A ⊤ ) is the cost k ernel. The optimization is o ver the set of ac hiev able p osterior mean cov ariances Σ u , which must satisfy Σ m ⪰ Σ u ⪰ O . Directly optimizing ov er Σ u is difficult due to the gen- eralized inequality constraint Σ m ⪰ Σ u . T o solv e this, w e utilize a transformation to standardize the constraint, adapted from V elicheti et al. (2023). L emma 7. (Equiv alent SDP formulation). Let Σ m ≻ O . The optimization problem min Σ u T r( V Σ u ) s.t. Σ m ⪰ Σ u ⪰ O (8) can b e equiv alently written as min Π ∈ S n T r( B Π) s.t. I ⪰ Π ⪰ O , (9) where B = Σ 1 2 m V Σ 1 2 m , S n denotes the set of p ositive semi- definite matrices of dimension n × n , and the optimization v ariable is transformed via Σ u = Σ 1 2 m ΠΣ 1 2 m . Pro of. Since Σ m ≻ O , the matrix Σ 1 2 m exists and is in vertible. W e introduce the c hange of v ariable Π = Σ − 1 2 m Σ u Σ − 1 2 m . First, we transform the ob jective function using the definition Σ u = Σ 1 2 m ΠΣ 1 2 m and the cyclic property of the trace op erator 1 : T r( V Σ u ) = T r( V Σ 1 2 m ΠΣ 1 2 m ) = T r(Σ 1 2 m V Σ 1 2 m Π) = T r( B Π) . Second, we transform the constraints. Recall that for any in vertible matrix M , A ⪰ B if and only if M AM ⊤ ⪰ M B M ⊤ . W e apply this with M = Σ − 1 2 m : Σ m ⪰ Σ u ⪰ O ⇐ ⇒ Σ − 1 2 m Σ m Σ − 1 2 m ⪰ Σ − 1 2 m Σ u Σ − 1 2 m ⪰ Σ − 1 2 m O Σ − 1 2 m ⇐ ⇒ I ⪰ Π ⪰ O . Th us, the problem is equiv alen t to minimizing T r( B Π) sub ject to I ⪰ Π ⪰ O . 2 R emark 8. (Relation to quadratic p ersuasion). The sender’s problem in (7) coincides with the quadratic persuasion form ulations in (T amura, 2018; Sayin and Ba ¸ sar, 2022), where the sender’s cost is linear in Σ u and the feasible set 1 i.e., T r( X Y Z ) = T r( Z X Y ) is { 0 ⪯ Σ u ⪯ Σ m } . In our setting, the linear functional V is induced b y the sensitivity mismatch matrix A , and Prop osition 10 b elow sho ws how to construct an explicit linear enco der achieving the optimal Σ u , thereb y instanti- ating the p ersuasion solution in a Gaussian SIT context. Applying Lemma 7 with the sender’s cost kernel V = I − ( A + A ⊤ ), we define the weigh t matrix B = Σ 1 2 m V Σ 1 2 m . The solution to the transformed problem (9) allows us to c haracterize the SE as follows. The or em 9. (Equilibrium information structure). Let β 1 ≤ β 2 ≤ · · · ≤ β n b e the ordered eigenv alues of B = Σ 1 2 m V Σ 1 2 m , and let q 1 , . . . , q n b e the corresponding orthonormal eigen- v ectors. Let k ∈ N denote the n um b er of strictly negative eigen v alues (i.e., β k < 0 and β k +1 ≥ 0). Among the set of optimal p olicies, let us select the solution minimizing the rank of Σ u 2 . The resulting unique SE p osterior mean cov ariance is given by Σ ∗ u = Σ 1 2 m Π ∗ Σ 1 2 m , (10) where Π ∗ = P k i =1 q i q ⊤ i is the pro jection on to the subspace spanned by the eigen vectors associated with the strictly negativ e eigenv alues of B . In particular: 1. If k = 0 (all β i ≥ 0), the equilibrium is non- informativ e (Σ ∗ u = O ). 2. If k = n (all β i < 0), the equilibrium is fully rev ealing (Σ ∗ u = Σ m ). 3. If 0 < k < n , the equilibrium is partially revealing. Pro of. F ollo wing Lemma 7, we minimize T r( B Π) sub ject to O ⪯ Π ⪯ I . W e perform the sp ectral decomp osition B = Q Λ Q ⊤ , where Λ = diag( β 1 , . . . , β n ) and Q = [ q 1 . . . q n ]. Let F = Q ⊤ Π Q . Using the cyclic property of the trace, w e obtain T r( B Π) = T r( Q Λ Q ⊤ Π) = T r(Λ Q ⊤ Π Q ) = T r(Λ F ) . The constrain t O ⪯ Π ⪯ I is equiv alen t to O ⪯ F ⪯ I since Q is orthogonal. Th us, we are solving min F ∈ S n T r(Λ F ) s.t. O ⪯ F ⪯ I . This is the same SDP (up to a sign change in the cost matrix) as in (T am ura, 2018, Thm 1), where it is sho wn that there exists an optimal solution to the SDP that is an orthogonal pro jection matrix in the eigen basis of the cost matrix. Th us, without loss of generality we ma y restrict attention to F that are diagonal in the eigenbasis of B , with diagonal en tries in { 0 , 1 } . Hence w e can write F = diag( f 11 , . . . , f nn ) with f ii ∈ { 0 , 1 } , and the ob jectiv e b ecomes the following sum T r(Λ F ) = n X i =1 β i f ii . T o minimize it, each f ii is c hosen based on the sign of β i : f ∗ ii =  1 if β i < 0 , 0 if β i ≥ 0 . (F or β i = 0, the choice do es not affect the cost; w e set f ii = 0 to obtain the minim um-rank, least-informative solution.) 2 If any eigenv alues are exactly zero, the equilibrium is not unique. Similar to V elic heti et al. (2023); Sayin and Ba¸ sar (2022), we effec- tively select the solution with the minimum rank (least informative) among the set of optimal policies. Since the eigenv alues are sorted suc h that the first k are negativ e, the optimal matrix F ∗ is diagonal with the first k en tries equal to 1 and the rest 0. T ransforming bac k yields Π ∗ = QF ∗ Q ⊤ = Q  I k O O O  Q ⊤ = Q k Q ⊤ k , where Q k ∈ R n × k is the matrix containing the first k columns of Q (the eigenv ectors corresp onding to negative eigen v alues). 2 The follo wing proposition, adapted from (T am ura, 2018, Thm 2), characterizes an encoding policy that achiev es the equilibrium describ ed in Theorem 9. Pr op osition 10. (Equilibrium signaling p olicy). Let k b e the num b er of strictly negativ e eigen v alues of B , and let Q k ∈ R n × k b e the matrix of the corresp onding eigen vectors (the first k columns of Q ). The equilibrium cov ariance Σ ∗ u c haracterized in Theo- rem 9 is ac hieved by a deterministic linear enco der p olicy γ ∗ ,e ( m ) = L m , where L ∈ R n × n is constructed as L =  Q ⊤ k Σ − 1 2 m O ( n − k ) × n  , (11) where O ( n − k ) × n is a zero matrix padding the remaining dimensions. Pro of. The decoder’s b est resp onse to a linear Gaussian map x = L m is the linear MMSE estimator, explicitly giv en by u = Σ mx Σ † xx x , where ( · ) † denotes the pseudoin- v erse of a matrix Σ † xx whic h naturally handles the rank deficiency of the noiseless signal cov ariance, ensuring the estimator is w ell-defined on the signal subspace. First, w e compute the cov ariance of the signal x : Σ xx = L Σ m L ⊤ =  Q ⊤ k Σ − 1 2 m O  Σ m h Σ − 1 2 m Q k O ⊤ i =  Q ⊤ k Q k O O O  =  I k O O O n − k  , where w e used the orthonormality condition Q ⊤ k Q k = I k . Next, the cross-co v ariance Σ mx is Σ mx = Σ m L ⊤ = Σ m h Σ − 1 2 m Q k O ⊤ i = h Σ 1 2 m Q k O i . The posterior estimate is u = Σ mx Σ † xx x . Noting that Σ † xx = Σ xx (as it is a diagonal pro jection matrix), the co v ariance of the estimate is Σ u = Σ mx Σ † xx Σ ⊤ mx = h Σ 1 2 m Q k O i  I k O O O  " Q ⊤ k Σ 1 2 m O ⊤ # = Σ 1 2 m Q k I k Q ⊤ k Σ 1 2 m = Σ 1 2 m ( Q k Q ⊤ k )Σ 1 2 m . Since Π ∗ = Q k Q ⊤ k (from the pro of of Theorem 9), w e hav e Σ u = Σ 1 2 m Π ∗ Σ 1 2 m = Σ ∗ u . Thus, the policy achiev es the SE. 2 4. NOISY CASE: SIGNALING GAME W e now turn to the general setting where the c hannel is noisy and the encoder faces a transmission cost ( ρ > 0). Unlik e the cheap-talk setting, the signal is corrupted by noise, preven ting the deco der from p erfectly inv erting the enco der’s map even if it is injective. 4.1 Sc alar Case W e first analyze the scalar case ( n = 1) where m ∼ N (0 , σ 2 m ) and w ∼ N (0 , σ 2 w ) with σ 2 w > 0. The enco der c ho oses a policy γ e : R → R , resulting in x = γ e ( m ) and y = x + w . The or em 11. (Stack elb erg signaling thresholds). The exis- tence and structure of SE are determined b y the bias parameter a : 1. If a ≤ 1 2 , the unique SE is non-informativ e ( P ∗ = 0). 2. If a > 1 2 , an informativ e SE exists if and only if the transmission cost satisfies 0 < ρ < σ 2 m σ 2 w (2 a − 1) . (12) where the optimal transmission p o wer is given by P ∗ = σ w s (2 a − 1) σ 2 m ρ − σ 2 w . (13) Otherwise, the SE is non-informativ e ( P ∗ = 0). Pro of. W e analyze the game where m ∼ N (0 , σ 2 m ) and w ∼ N (0 , σ 2 w ). Let D : = E [( m − u ) 2 ] and P : = E [ x 2 ] denote the scalar mean squared estimation error and a verage transmission p o wer, resp ectively . Step 1: Enc o der’s c ost. By Prop osition 4, the deco der’s b est resp onse is u = E [ m | y ]. F ollo wing the same orthogo- nal decomp osition logic used in Section 3.1, we substitute E [ u 2 ] = σ 2 m − D in to the enco der’s cost function (2). The exp ected cost simplifies to J e = (2 a − 1) D + ρP + ( a − 1) 2 σ 2 m + b 2 . (14) The enco der’s problem is to minimize (14) sub ject to the ph ysical constraints imp osed by the channel. Step 2: Information-the or etic lower b ound. First, we relate the mutual information to the distortion using differential en tropy: I ( m ; y ) = h ( m ) − h ( m | y ) = h ( m ) − h ( m − E [ m | y ] | y ) ≥ h ( m ) − h ( m − E [ m | y ]) ( a ) ≥ 1 2 log 2 (2 π eσ 2 m ) − 1 2 log 2 (2 π eD ) = 1 2 log 2  σ 2 m D  . In verting this relation yields D ≥ σ 2 m 2 − 2 I ( m ; y ) . Using the data-pro cessing inequalit y and the definition of c hannel capacit y C ( P ) = sup p ( x ): E [ x 2 ] ≤ P I ( x ; y ), we obtain: D ( b ) ≥ σ 2 m 2 − 2 I ( x ; y ) ≥ σ 2 m 2 − 2 C ( P ) ( c ) = σ 2 m 2 − 2 [ 1 2 log 2 ( 1+ P /σ 2 w )] ⇒ D = E [( m − u ) 2 ] ≥ σ 2 m 1 + P /σ 2 w . (15) Here, (a) holds since h ( m ) = 1 2 log 2 (2 π eσ 2 m ) for a Gaussian source, (b) follows from the data-pro cessing inequality , and (c) substitutes the Gaussian c hannel capacity . Step 3: Optimization. The encoder seeks to minimize the cost J e = (2 a − 1) D + ρP + const. W e analyze the tw o regimes for a separately . 1. Case a ≤ 1 2 . Then (2 a − 1) ≤ 0. T o minimize (2 a − 1) D , the enco der must maximize the distortion D . The maxim um MSE (error v ariance) is the prior v ariance σ 2 m , implying D ≤ σ 2 m . Sim ultaneously , the enco der seeks to minimize the p o wer cost ρP (since ρ > 0). Both terms (2 a − 1) D and ρP are minimized when the enco der transmits no information ( x = 0), resulting in P = 0 and maximal distortion D = σ 2 m . Thus, the unique global minim um is at P ∗ = 0. 2. Case a > 1 2 . The co efficient (2 a − 1) is p ositiv e. In this regime, the enco der faces a trade-off b et ween reducing distortion and sa ving p ow er. Substituting the lo wer b ound (15) into (14) yields J e ≥ ( a − 1) 2 σ 2 m + b 2 + (2 a − 1) σ 2 m σ 2 w σ 2 w + P + ρP | {z } : = f ( P ) . (16) W e minimize f ( P ) o ver P ≥ 0. Since f ′′ ( P ) > 0, the function is strictly conv ex. Setting the deriv ative f ′ ( P ) = ρ − (2 a − 1) σ 2 m σ 2 w ( σ 2 w + P ) 2 to zero yields the unique unconstrained minimizer P ∗ giv en in (13). Th us, an informative equilibrium exists if and only if P ∗ > 0. Imp osing this inequalit y yields q (2 a − 1) σ 2 m ρ > σ w , which simplifies to condition (12). If this condition is not met, f ′ (0) ≥ 0 and the minim um o ccurs at the b oundary P ∗ = 0 (non- informativ e). Step 4: A chievability. The b ound (15) is tight for Gaus- sian sources o v er A WGN channels using linear enco ding. Sp ecifically , a linear policy γ e ( m ) = αm with α 2 = P ∗ /σ 2 m results in the low er b ound MMSE D = σ 2 m 1+ P ∗ /σ 2 w . Thus, the linear p olicy ac hieves the global lo wer b ound of the cost function J e . Consequen tly: 1. If a ≤ 1 2 , P ∗ = 0, leading to x = 0 (non-informative). 2. If condition (12) holds, the optimal p olicy is the linear map corresp onding to P ∗ > 0 (informativ e). 2 4.2 Multidimensional Signaling W e now extend the analysis to the m ultidimensional signaling game with noisy c hannel and transmission cost. The source is m ∼ N ( 0 , Σ m ) with Σ m ≻ O , and the enco der has a general bias matrix A ∈ R n × n . By the same decomp osition as in Section 3.2, the encoder’s exp ected cost can b e written as J e = T r( V Σ u ) + ρP + const , (17) where V = I − ( A + A ⊤ ) is the cost k ernel from the c heap-talk analysis, Σ u = E [ uu ⊤ ] is the posterior mean co v ariance, P = E [ ∥ x ∥ 2 ] is the transmission pow er, and const = T r( A ⊤ A Σ m ) + ∥ b ∥ 2 is indep endent of the enco der p olicy . The key difference from the c heap-talk setting is that the ac hiev able pairs (Σ u , P ) are no w constrained b y the c hannel capacity . First, following the approac h of Sarıta¸ s et al. (2017), w e deriv e b ounds on the achiev able p osterior cov ariance. L emma 12. (Determinant b ound). F or an y enco der p olicy with total transmission pow er P = E [ ∥ x ∥ 2 ], the error co v ariance Σ e = Σ m − Σ u satisfies | Σ e | ≥ | Σ m | 2 − 2 C tot ( P ) , (18) where C tot ( P ) is the total capacit y of the n -dimensional additiv e Gaussian noise channel with p ow er constraint P . Pro of. By the data-pro cessing inequalit y and the defini- tion of c hannel capacity , I ( m ; y ) = h ( m ) − h ( m | y ) = h ( m ) − h ( m − E [ m | y ] | y ) ≥ h ( m ) − h ( m − E [ m | y ]) ≥ 1 2 log 2 ((2 π e ) n | Σ m | ) − 1 2 log 2 ((2 π e ) n | Σ e | ) = 1 2 log 2  | Σ m | | Σ e |  . Since I ( m ; y ) ≤ I ( x ; y ) ≤ C tot ( P ), we obtain | Σ e | ≥ | Σ m | 2 − 2 C tot ( P ) . 2 F or the colored Gaussian noise channel with co v ariance Σ w , w e use the water-filling capacity expression and the arithmetic–geometric mean inequalit y to b ound the capac- it y term. L emma 13. (Capacity b ound). The capacity C tot ( P ) of the additive Gaussian noise c hannel with cov ariance Σ w ≻ O satisfies 2 − 2 C tot ( P ) /n ≥ | Σ w | 1 /n P /n + 1 n T r(Σ w ) . (19) Pro of. The capacity of the additive colored Gaussian noise channel is ac hieved b y the water-filling pow er allo ca- tion. The total capacity is giv en by (Co ver and Thomas, 1999, Eq. (9.166)): C tot ( P ) = n X i =1 1 2 log 2  1 + max( ν − λ i , 0) λ i  , where λ 1 , . . . , λ n are the eigen v alues of Σ w , and ν is the w ater lev el c hosen such that the total p ow er constraint is satisfied: P n i =1 max( ν − λ i , 0) = P . W e examine the term 2 − 2 C tot ( P ) /n : 2 − 2 C tot ( P ) /n = 2 − 2 n P n i =1 1 2 log 2  1+ max( ν − λ i , 0) λ i  = n Y i =1  1 + max( ν − λ i , 0) λ i  − 1 /n = n Y i =1  λ i max( ν, λ i )  1 /n = ( Q n i =1 λ i ) 1 /n ( Q n i =1 max( ν, λ i )) 1 /n . The n umerator is the geometric mean of the eigen v alues, whic h is | Σ w | 1 /n . F or the denominator, we apply the inequalit y of arithmetic and geometric means (AM–GM): 2 − 2 C tot ( P ) /n ( a ) ≥ | Σ w | 1 /n 1 n P n i =1 max( ν, λ i ) = | Σ w | 1 /n 1 n P n i =1 (max( ν − λ i , 0) + λ i ) = | Σ w | 1 /n 1 n ( P + T r(Σ w )) . Here, (a) follo ws as the geometric mean is less than or equal to the arithmetic mean. 2 Com bining Lemmas 12 and 13 with the AM–GM inequal- it y yields b ounds on the achiev able p osterior cov ariance. Pr op osition 14. (Posterior cov ariance b ounds). The trace of the p osterior mean cov ariance Σ u is upp er-bounded by T r(Σ u ) ≤ T r(Σ m ) − n 2 | Σ m | 1 n | Σ w | 1 n ( P + T r(Σ w )) − 1 . (20) Pro of. By the AM–GM inequalit y applied to the diagonal en tries of Σ e , T r(Σ e ) = n X i =1 Σ e ( i, i ) ≥ n n Y i =1 Σ e ( i, i ) ! 1 n ( ∗ ) ≥ n | Σ e | 1 n . Where (*) follows from the Hadamard inequality (as Σ e is p ositiv e se mi-definite). F rom Lemma 12, w e hav e | Σ e | 1 n ≥ | Σ m | 1 n 2 − 2 C tot ( P ) n . Substituting the b ound from Lemma 13 giv es T r(Σ e ) ≥ n | Σ m | 1 n n | Σ w | 1 n P + T r(Σ w ) ! = n 2 | Σ m | 1 n | Σ w | 1 n P + T r(Σ w ) . The result follo ws from T r(Σ u ) = T r(Σ m ) − T r(Σ e ). 2 W e now analyze the p otential for informative equilibria. T o maintain tractability , w e consider the case of isotropic sensitivit y A = aI . The or em 15. (Signaling p otential). Consider the case where A = aI for scalar a > 0. The cost kernel is V = (1 − 2 a ) I . 1. If a ≤ 1 2 , the unique equilibrium is non-informative for an y ρ > 0. 2. If a > 1 2 , the information-theoretic low er b ound on the enco der’s cost, derived from Prop osition 14, is minimized b y a strictly positive pow er P > 0 if and only if ρ < (2 a − 1) n 2 | Σ m | 1 n | Σ w | 1 n (T r(Σ w )) 2 . (21) This condition c haracterizes the regime where the fundamen tal information-theoretic constraints allow for a cost reduction via signaling. Pro of. With V = (1 − 2 a ) I , the enco der minimizes J e = (1 − 2 a ) T r(Σ u ) + ρP . 1. If a ≤ 1 2 . The co efficien t (1 − 2 a ) is non-negative. Minimizing the cost requires minimizing T r(Σ u ) and P . The global minim um is attained at Σ u = O and P = 0, which is achiev able b y transmitting no signal. Th us, the equilibrium is non-informative. 2. If a > 1 2 . The co efficient (1 − 2 a ) is negativ e. Substituting the b ound from Prop osition 14 into the ob jective yields the low er b ound function f ( P ): J e ( P ) ≥ f ( P ) : = (1 − 2 a )  T r(Σ m ) − κ P + τ  + ρP , where κ = n 2 | Σ m | 1 n | Σ w | 1 n and τ = T r(Σ w ). W e min- imize f ( P ) ov er P ≥ 0. Computing the deriv atives: f ′ ( P ) = − (2 a − 1) κ ( P + τ ) 2 + ρ, f ′′ ( P ) = 2(2 a − 1) κ ( P + τ ) 3 . Since a > 1 / 2, f ′′ ( P ) > 0 for all P ≥ 0, so f ( P ) is strictly conv ex. The unique global minim um o ccurs at P ∗ > 0 if and only if f ′ (0) < 0, which is equiv alent to ρ < (2 a − 1) κ τ 2 = (2 a − 1) n 2 | Σ m | 1 n | Σ w | 1 n (T r(Σ w )) 2 . 0 . 0 0 . 5 1 . 0 1 . 5 Bias parameter a 10 − 2 10 − 1 10 0 10 1 T ransmission cost ρ a = 0 . 5 ( P ∗ = 0) 0.5 2.0 4.0 6.0 8.0 ρ = σ 2 m σ 2 w (2 a − 1) 2 4 6 8 Optimal Pow er P ∗ Fig. 2. Phase diagram for the scalar signaling game. The solid b oundary separates the region where commu- nication is beneficial for the enco der (informative) from the region where the optimal strategy is silence (non-informativ e). The color in tensity represen ts the optimal p o wer P ∗ . Th us, condition (21) is necessary and sufficient for the low er b ound f ( P ) to b e minimized at a non-zero p o wer. 2 Cor ol lary 16. (The i.i.d. case). Consider the case where Σ m = σ 2 m I and Σ w = σ 2 w I . In this setting, an informativ e affine equilibrium exists if and only if ρ < (2 a − 1) σ 2 m σ 2 w . (22) reco vering the scalar threshold. Pro of. In the i.i.d. case, T r(Σ m ) = nσ 2 m and | Σ m | 1 /n = σ 2 m , so T r(Σ m ) = n | Σ m | 1 /n . Similarly , T r(Σ w ) = n | Σ w | 1 /n . Consequen tly , the AM–GM inequalities used in Lem- mas 13 and Prop osition 14 hold with equalit y at P = 0 and for uniform p ow er allo cations. Sp ecifically , f (0) = J e (0), and the lo wer bound f ( P ) is ac hiev able by scalar linear p olicies x = α m . Therefore, the condition (21), whic h simplifies to (22) in this case, b ecomes necessary and sufficien t for the existence of an informativ e equilibrium. 2 5. NUMERICAL ILLUSTRA TIONS W e illustrate our theoretical findings through n umerical examples cov ering b oth the scalar signaling game and the m ultidimensional cheap-talk setting. Scalar Case. Figure 2 sho ws the phase transition betw een informativ e and non-informative equilibria in the ( a, ρ ) parameter space. W e set σ 2 m = 1 and σ 2 w = 0 . 5. The solid black curve represents the theoretical boundary ρ = σ 2 m σ 2 w (2 a − 1). F or a ≤ 1 / 2 (left of the dashed line) or when the cost is to o high (hatc hed region), the equilibrium is strictly non-informative, resulting in zero transmission p o wer ( P ∗ = 0). F or a > 1 / 2, an informative equilibrium exists b elo w the b oundary . The color gradient indicates the magnitude of the optimal transmission p ow er P ∗ , showing that communication intensit y increases as the sensitivit y mismatc h decreases (higher a ) or the cost ρ decreases. − 2 0 2 Source realization m i − 2 0 2 Estimate comp onent u i Corr( m 1 , u 1 ) = 1 . 00 Corr( m 2 , u 2 ) = 0 . 00 (a) Independent Source ([Σ m ] 12 = 0 . 0) F ully-informative Non-informative i = 1: a 1 = 0 . 8 i = 2: a 2 = 0 . 2 − 2 0 2 Source realization m i Corr( m 1 , u 1 ) = 0 . 99 Corr( m 2 , u 2 ) = 0 . 35 (b) Correlated Source ([Σ m ] 12 = 0 . 3) Fig. 3. Illustration of equilibrium b eha vior for 2D c heap talk with A = diag(0 . 8 , 0 . 2). (a) Indep endent Source: The problem decouples; Comp onen t 1 is rev ealed, while Comp onent 2 is suppressed ( u 2 = 0). (b) Correlated Source: The enco der still do esn’t rev eal m 2 , but the decoder infers partial information ab out m 2 via its correlation with the rev ealed m 1 , illustrating the in teraction b etw een mismatch geome- try and source statistics. Multidimensional Case. T o visualize the sp ectral c har- acterization of information revelation Theorem 9, we con- sider a 2-dimensional c heap-talk game (noiseless, ρ = 0). W e set the sensitivit y matrix A = diag (0 . 8 , 0 . 2) and as- sume a zero-mean source with marginal v ariances Σ 11 = 1 and Σ 22 = 1 . 5. Figure 3 compares the equilibrium out- comes for an indep enden t source (Σ 12 = 0) versus a correlated source (Σ 12 = 0 . 3). In panel (a) (independent source), the encoder fully rev eals m 1 but completely sup- presses m 2 , confirming that the enco der filters out direc- tions with high mismatc h. In panel (b) (correlated source), the enco der pro jects onto the subspace spanned b y the eigenv ector of the trans- formed cost matrix B corresp onding to the negativ e eigen- v alue. Due to the source correlation, this optimal signaling direction is rotated relativ e to the canonical axes (deviat- ing from the first dimension). Consequently , the decoder infers information about m 2 through this rotated pro jec- tion, resulting in the estimate u 2 b eing correlated with m 1 (orange p oints) in a manner that optimally balances information rev elation against the sensitivity mismatch. 6. CONCLUSION AND PERSPECTIVES This pap er inv estigated a Gaussian signaling game where ob jective misalignment arises from a linear sensitivit y mismatc h. By mo deling this interaction as a Stac kelberg game, we derived analytical conditions for informative equilibria, rev ealing that linear bias introduces a sharp phase transition in information disclosure. 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