Gaussian Cheap Talk Game with Quadratic Cost Functions: When - - PowerPoint PPT Presentation

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

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Crowd-Sourcing Estimation

Crowd-sourcing estimation:

• Indirect: Use their devices, e.g., Mobile Millennium;
• Direct: Ask them to report, e.g., Waze.

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Crowd-Sourcing Estimation

What if I intentionally under-estimate?

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Crowd-Sourcing Estimation

What if I intentionally under-estimate? What if I intentionally

• ver-estimate?

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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]

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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.

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Quadratic Gaussian Cheap Talk Game

R S1 S2

. . .

SN

x θ1 θ2 θN

ˆ x((yi)N

i=1)

y1 y2 yn

• x ∼ N(0, Vxx);
• Receiver cost: E{x − ˆ

x2}.

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Quadratic Gaussian Cheap Talk Game

R S1 S2

. . .

SN

x θ1 θ2 θN

ˆ x((yi)N

i=1)

y1 y2 yn

At the first step, we deploy N strategic sensors:

• Sensor i cost: E{(x + θi) − ˆ

x2};

• Sensor i has perfect measurements of x, θi (nothing about others);
• θ = (θi)N

i=1 ∼ N(0, Vθθ).

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Quadratic Gaussian Cheap Talk Game

R S1 S2

. . .

SN

x θ1 θ2 θN

ˆ x((yi)N

i=1)

y1 y2 yn

At the second step, sensors transmit scalar signals:

• yi = γi(x, θi) ∈ R where γi(x, θi) = a⊤

i x + b⊤ i θi + vi;

• vi ∼ N(0, Vvivi);
• The set of such mappings is Γi (isomorph to Rnx × Rnx × R≥0).

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Quadratic Gaussian Cheap Talk Game

R S1 S2

. . .

SN

x θ1 θ2 θN

ˆ x((yi)N

i=1)

y1 y2 yn

At the third step, the receiver announces its estimate:

• ˆ

x = ˆ x(y1, . . . , yn) where ˆ x ∈ Ψ;

• Ψ is the set of all Lebesgue-measurable functions from RN to Rnx.

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Quadratic Gaussian Cheap Talk Game

R S1 S2

. . .

SN

x θ1 θ2 θN

ˆ x((yi)N

i=1)

y1 y2 yn

At the fourth step, the cost functions are realized:

• Receiver: E{x − ˆ

x2};

• Sensor i: E{(x + θi) − ˆ

x2}.

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Independent Senders

Equilibrium

A tuple (ˆ x∗, (γ∗

i )N i=1) ∈ Ψ × Γ1 × · · · × ΓN constitutes an equilibrium in

affine strategies if ˆ x∗ ∈ arg min

ˆ x∈Ψ

E{x − ˆ x((γ∗

j (x, θj))N j=1)2},

γ∗

i ∈ arg min γi∈Γi

E{(x + θi) − ˆ x(γi(x, θi), (γ∗

j (x, θj))j=i)2}, ∀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.

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Do we have an equilibrium?

Theorem

Assume that Vxθ = 0, Vθiθi = Vθθ, and Vθiθj = 0 for j = i. There exists a symmetric equilibrium in affine strategies where the receiver follows ˆ x∗(y) = E{x|(y1 + · · · + yN)/N} and sender Si, 1 ≤ i ≤ N, employs the affine policy γ∗(x, θi) = a∗⊤x + b∗⊤θi, where

• b∗

a∗

• =
• 1

1 + (N − 1)ξ⊤

1 ξ1

• NV −1/2

θθ

V −1/2

xx

• ξ

and ξ =

• ξ⊤

1

ξ⊤

2

⊤ is the normalized eigenvector (i.e., ξ2 = 1) corresponding to the smallest eigenvalue of the matrix

• −V −1/2

θθ

V −1/2

xx

−V −1/2

xx

V −1/2

θθ

−Vxx

• .

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How good is the equilibrium?

Corollary

Assume that Vxθ = 0, Vθiθi = Vθθ, and Vθiθj = 0 for j = i. At the presented symmetric equilibrium E{x − ˆ x∗((γ∗(x, θj))N

j=1)2} = Vxx −

1 α + βN U, where α, β ∈ R≥0 and U ∈ Rnx×nx such that 0 < U ≤ (α + β)Vxx.

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How good is the equilibrium?

Corollary

Assume that Vxθ = 0, Vθiθi = Vθθ, and Vθiθj = 0 for j = i. At the presented symmetric equilibrium E{x − ˆ x∗((γ∗(x, θj))N

j=1)2} = Vxx −

1 α + βN U, where α, β ∈ R≥0 and U ∈ Rnx×nx such that 0 < U ≤ (α + β)Vxx.

• Sadly, this implies that

lim

N→∞ E{x − ˆ

x∗((γ∗(x, θj))N

j=1)2} = Vxx.

• This equilibrium is not good for anyone! It is even worse for the

sensors in comparison to the receiver. lim

N→∞ E{x + θi − ˆ

x∗((γ∗(x, θj))N

j=1)2} = Vxx + Vθθ.

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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);

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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);

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Herding Senders

Herding Equilibrium

A tuple (ˆ x∗, γ∗) ∈ Ψ × Γ constitutes a herding equilibrium in affine strategies if ˆ x∗ ∈ arg min

ˆ x∈Ψ

E{x − ˆ x((γ∗(x, θj))N

j=1)2},

γ∗ ∈ arg min

γ∈Γ

E{(x + θi) − ˆ x(γ(x, θi), (γ(x, θj))j=i)2}, ∀i. As opposed to before, the senders are constrained to imitate each other.

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Do we have a herding equilibrium?

Theorem

Assume that nyi = 1 for all i, Vxθ = 0, and Vθθ = 0. There exists a herding equilibrium in affine strategies where the receiver follows ˆ x∗(y) = E{x|(y1 + · · · + yN)/N} and sender Si, 1 ≤ i ≤ N, employs a linear policy γ∗(x, θi) = a∗⊤x + b∗⊤θi where

• b∗

a∗

• =

√ NV −1/2

θθ

V −1/2

xx

• ζ,

and ζ is the normalized eigenvector (i.e., ζ2 = 1) corresponding to the smallest eigenvalue of the matrix

• −

1 √ N V −1/2 θθ

V −1/2

xx

−

1 √ N V −1/2 xx

V −1/2

θθ

−Vxx

• .

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When herding becomes a virtue

Proposition

Assume that Vxθ = 0, Vθiθi = Vθθ, and Vθiθj = 0 for j = i. At a herding equilibrium lim

N→∞ E{x − ˆ

x∗((γ∗(x, θj))N

j=1)2} = 0.

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When herding becomes a virtue

Proposition

Assume that Vxθ = 0, Vθiθi = Vθθ, and Vθiθj = 0 for j = i. At a herding equilibrium lim

N→∞ E{x − ˆ

x∗((γ∗(x, θj))N

j=1)2} = 0.

Actually, the rate of converge is faster than employing nonstrategic sensors with measurement noise!

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Numerical Example

Let us fix nx = 1 and nyi = 1 for all i = 1, . . . , N and assume that Vxθ = 0, Vθθ = 1, and Vxx = 1.

• Independent Senders

Receiver’s cost: E{x − ˆ x(y)2

2} =

0.2763N 0.7236 + 0.2763N ,

• Herding Senders

Receiver’s cost: E{x − ˆ x(y)2

2} =

2 N2 + N(

• N(N + 4) + 2) + 2
• Nonstrategic Senders

Senders transmit yi = x + ui where ui are i.i.d. zero-mean Gaussian random variables so that E{u2

i } = σ.

Receiver’s cost: E{x − ˆ x(y)2

2} =

σ σ + N

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Numerical Example

10 10

1

10

2

10

3

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10 10

1

Estimation Error N

Independent Senders Herding Senders Nonstrategic Senders

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Conclusion and Future Works

Conclusions

• Including strategic sensors in networked estimation;
• Characterized an equilibrium and studied its efficiency;
• “Better to employ a handful of naively-strategic but accurate sensors

than many nonstrategic but noisy sensors”.

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Conclusion and Future Works

Conclusions

• Including strategic sensors in networked estimation;
• Characterized an equilibrium and studied its efficiency;
• “Better to employ a handful of naively-strategic but accurate sensors

than many nonstrategic but noisy sensors”.

Future Work

• Asynchronous communication;
• Arbitrary communication graphs;
• Dynamic or repeated cheap talk game.

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