Gaussian Cheap Talk Game with Quadratic Cost Functions: When Herding between Strategic Senders Is a Virtue Farhad Farokhi ⋆ , Andr´ e Teixeira ⋆ , and C´ edric Langbort † ⋆ KTH Royal Institute of Technology, Sweden † University of Illinois Urbana-Champaign, USA American Control Conference Thursday June 5, 2014 C´ edric Langbort (UIUC) Gaussian Cheap Talk Game ... Thursday June 5, 2014 1 / 14
Crowd-Sourcing Estimation Crowd-sourcing estimation: - Indirect: Use their devices, e.g., Mobile Millennium; - Direct: Ask them to report, e.g., Waze. C´ edric Langbort (UIUC) Gaussian Cheap Talk Game ... Thursday June 5, 2014 2 / 14
Crowd-Sourcing Estimation What if I intentionally under-estimate? C´ edric Langbort (UIUC) Gaussian Cheap Talk Game ... Thursday June 5, 2014 2 / 14
Crowd-Sourcing Estimation What if I intentionally under-estimate? What if I intentionally over-estimate? C´ edric Langbort (UIUC) Gaussian Cheap Talk Game ... Thursday June 5, 2014 2 / 14
Cheap Talk Game Cheap Talk Game A game in which better informed senders are communicating with a receiver, who ultimately takes a decision regarding the social welfare (e.g., negotiations in organizations, voting in subcommittees in congress, etc). [Crawford & Sobel, 82] C´ edric Langbort (UIUC) Gaussian Cheap Talk Game ... Thursday June 5, 2014 3 / 14
Cheap Talk Game Cheap Talk Game A game in which better informed senders are communicating with a receiver, who ultimately takes a decision regarding the social welfare (e.g., negotiations in organizations, voting in subcommittees in congress, etc). [Crawford & Sobel, 82] In our example: • Better informed senders: Crowd; • Receiver: Traffic estimation application (e.g., Waze); • Decision regarding the social welfare: Traffic estimate. C´ edric Langbort (UIUC) Gaussian Cheap Talk Game ... Thursday June 5, 2014 3 / 14
Quadratic Gaussian Cheap Talk Game θ 1 S 1 y 1 θ 2 y 2 S 2 x x (( y i ) N ˆ i =1 ) R y n . . . θ N S N • x ∼ N (0 , V xx ) ; • Receiver cost: E {� x − ˆ x � 2 } . C´ edric Langbort (UIUC) Gaussian Cheap Talk Game ... Thursday June 5, 2014 4 / 14
Quadratic Gaussian Cheap Talk Game θ 1 S 1 y 1 θ 2 y 2 S 2 x x (( y i ) N ˆ i =1 ) R y n . . . θ N S N At the first step, we deploy N strategic sensors: • Sensor i cost: E {� ( x + θ i ) − ˆ x � 2 } ; • Sensor i has perfect measurements of x, θ i (nothing about others); • θ = ( θ i ) N i =1 ∼ N (0 , V θθ ) . C´ edric Langbort (UIUC) Gaussian Cheap Talk Game ... Thursday June 5, 2014 4 / 14
Quadratic Gaussian Cheap Talk Game θ 1 S 1 y 1 θ 2 y 2 S 2 x x (( y i ) N ˆ i =1 ) R y n . . . θ N S N At the second step, sensors transmit scalar signals: • y i = γ i ( x, θ i ) ∈ R where γ i ( x, θ i ) = a ⊤ i x + b ⊤ i θ i + v i ; • v i ∼ N (0 , V v i v i ) ; • The set of such mappings is Γ i (isomorph to R n x × R n x × R ≥ 0 ). C´ edric Langbort (UIUC) Gaussian Cheap Talk Game ... Thursday June 5, 2014 4 / 14
Quadratic Gaussian Cheap Talk Game θ 1 S 1 y 1 θ 2 y 2 S 2 x x (( y i ) N ˆ i =1 ) R y n . . . θ N S N At the third step, the receiver announces its estimate: • ˆ x = ˆ x ( y 1 , . . . , y n ) where ˆ x ∈ Ψ ; • Ψ is the set of all Lebesgue-measurable functions from R N to R n x . C´ edric Langbort (UIUC) Gaussian Cheap Talk Game ... Thursday June 5, 2014 4 / 14
Quadratic Gaussian Cheap Talk Game θ 1 S 1 y 1 θ 2 y 2 S 2 x x (( y i ) N ˆ i =1 ) R y n . . . θ N S N At the fourth step, the cost functions are realized: • Receiver: E {� x − ˆ x � 2 } ; • Sensor i : E {� ( x + θ i ) − ˆ x � 2 } . C´ edric Langbort (UIUC) Gaussian Cheap Talk Game ... Thursday June 5, 2014 4 / 14
Independent Senders Equilibrium x ∗ , ( γ ∗ i ) N A tuple (ˆ i =1 ) ∈ Ψ × Γ 1 × · · · × Γ N constitutes an equilibrium in affine strategies if x ∗ ∈ arg min x (( γ ∗ j ( x, θ j )) N ˆ E {� x − ˆ j =1 ) � 2 } , x ∈ Ψ ˆ γ ∗ x ( γ i ( x, θ i ) , ( γ ∗ i ∈ arg min E {� ( x + θ i ) − ˆ j ( x, θ j )) j � = i ) � 2 } , ∀ i. γ i ∈ Γ i As always, equilibrium is tuple of actions (i.e., policies) in which no one (i.e., the receiver and the sensors) can gain by unilaterally deviating from her action. C´ edric Langbort (UIUC) Gaussian Cheap Talk Game ... Thursday June 5, 2014 5 / 14
Do we have an equilibrium? Theorem Assume that V xθ = 0 , V θ i θ i = V θθ , and V θ i θ j = 0 for j � = i . There exists a symmetric equilibrium in affine strategies where the receiver follows x ∗ ( y ) = E { x | ( y 1 + · · · + y N ) /N } ˆ and sender S i , 1 ≤ i ≤ N , employs the affine policy γ ∗ ( x, θ i ) = a ∗⊤ x + b ∗⊤ θ i , where � � � NV − 1 / 2 � b ∗ � 1 0 = θθ ξ a ∗ V − 1 / 2 1 + ( N − 1) ξ ⊤ 1 ξ 1 0 xx � ⊤ is the normalized eigenvector ( i.e., � ξ � 2 = 1 ) � ξ ⊤ ξ ⊤ and ξ = 1 2 corresponding to the smallest eigenvalue of the matrix � � − V − 1 / 2 V − 1 / 2 0 xx θθ . − V − 1 / 2 V − 1 / 2 − V xx xx θθ C´ edric Langbort (UIUC) Gaussian Cheap Talk Game ... Thursday June 5, 2014 6 / 14
How good is the equilibrium? Corollary Assume that V xθ = 0 , V θ i θ i = V θθ , and V θ i θ j = 0 for j � = i . At the presented symmetric equilibrium 1 x ∗ (( γ ∗ ( x, θ j )) N E {� x − ˆ j =1 ) � 2 } = V xx − α + βN U, where α, β ∈ R ≥ 0 and U ∈ R n x × n x such that 0 < U ≤ ( α + β ) V xx . C´ edric Langbort (UIUC) Gaussian Cheap Talk Game ... Thursday June 5, 2014 7 / 14
How good is the equilibrium? Corollary Assume that V xθ = 0 , V θ i θ i = V θθ , and V θ i θ j = 0 for j � = i . At the presented symmetric equilibrium 1 x ∗ (( γ ∗ ( x, θ j )) N E {� x − ˆ j =1 ) � 2 } = V xx − α + βN U, where α, β ∈ R ≥ 0 and U ∈ R n x × n x such that 0 < U ≤ ( α + β ) V xx . • Sadly, this implies that x ∗ (( γ ∗ ( x, θ j )) N N →∞ E {� x − ˆ j =1 ) � 2 } = V xx . lim • This equilibrium is not good for anyone! It is even worse for the sensors in comparison to the receiver. x ∗ (( γ ∗ ( x, θ j )) N N →∞ E {� x + θ i − ˆ lim j =1 ) � 2 } = V xx + V θθ . C´ edric Langbort (UIUC) Gaussian Cheap Talk Game ... Thursday June 5, 2014 7 / 14
What went wrong? • All the sensors are strategic (benevolent users); • All the sensors measure x perfectly (looking vs. measuring); • There is no correlation between the private information (shopping street vs. residential area); C´ edric Langbort (UIUC) Gaussian Cheap Talk Game ... Thursday June 5, 2014 8 / 14
What went wrong? • All the sensors are strategic (benevolent users); • All the sensors measure x perfectly (looking vs. measuring); • There is no correlation between the private information (shopping street vs. residential area); It is not all doom and gloom! • We are dealing with Humans and not Econs (bounded rationality, intuition, etc); C´ edric Langbort (UIUC) Gaussian Cheap Talk Game ... Thursday June 5, 2014 8 / 14
Herding Senders Herding Equilibrium x ∗ , γ ∗ ) ∈ Ψ × Γ constitutes a herding equilibrium in affine A tuple (ˆ strategies if x ∗ ∈ arg min x (( γ ∗ ( x, θ j )) N ˆ E {� x − ˆ j =1 ) � 2 } , x ∈ Ψ ˆ γ ∗ ∈ arg min E {� ( x + θ i ) − ˆ x ( γ ( x, θ i ) , ( γ ( x, θ j )) j � = i ) � 2 } , ∀ i. γ ∈ Γ As opposed to before, the senders are constrained to imitate each other. C´ edric Langbort (UIUC) Gaussian Cheap Talk Game ... Thursday June 5, 2014 9 / 14
Do we have a herding equilibrium? Theorem Assume that n y i = 1 for all i , V xθ = 0 , and V θθ = 0 . There exists a herding equilibrium in affine strategies where the receiver follows x ∗ ( y ) = E { x | ( y 1 + · · · + y N ) /N } ˆ and sender S i , 1 ≤ i ≤ N , employs a linear policy γ ∗ ( x, θ i ) = a ∗⊤ x + b ∗⊤ θ i where � √ � NV − 1 / 2 � b ∗ � 0 θθ = ζ, a ∗ V − 1 / 2 0 xx and ζ is the normalized eigenvector (i.e., � ζ � 2 = 1 ) corresponding to the smallest eigenvalue of the matrix N V − 1 / 2 V − 1 / 2 � � − 1 0 √ xx θθ . N V − 1 / 2 V − 1 / 2 1 − − V xx √ xx θθ C´ edric Langbort (UIUC) Gaussian Cheap Talk Game ... Thursday June 5, 2014 10 / 14
When herding becomes a virtue Proposition Assume that V xθ = 0 , V θ i θ i = V θθ , and V θ i θ j = 0 for j � = i . At a herding equilibrium x ∗ (( γ ∗ ( x, θ j )) N N →∞ E {� x − ˆ lim j =1 ) � 2 } = 0 . C´ edric Langbort (UIUC) Gaussian Cheap Talk Game ... Thursday June 5, 2014 11 / 14
When herding becomes a virtue Proposition Assume that V xθ = 0 , V θ i θ i = V θθ , and V θ i θ j = 0 for j � = i . At a herding equilibrium x ∗ (( γ ∗ ( x, θ j )) N N →∞ E {� x − ˆ lim j =1 ) � 2 } = 0 . Actually, the rate of converge is faster than employing nonstrategic sensors with measurement noise! C´ edric Langbort (UIUC) Gaussian Cheap Talk Game ... Thursday June 5, 2014 11 / 14
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