Gaussian Cheap Talk Game with Quadratic Cost
Functions:
When Herding between Strategic Senders Is a Virtue
Farhad Farokhi? , André Teixeira? , and Cédric Langbort†
?
†
KTH Royal Institute of Technology, Sweden
University of Illinois Urbana-Champaign, USA
American Control Conference
Thursday June 5, 2014
Cédric Langbort (UIUC)
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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.
Cédric Langbort (UIUC)
Gaussian Cheap Talk Game ...
Thursday June 5, 2014
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Crowd-Sourcing Estimation
What if I intentionally
under-estimate?
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Thursday June 5, 2014
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Crowd-Sourcing Estimation
What if I intentionally
under-estimate?
What if I intentionally
over-estimate?
Cédric Langbort (UIUC)
Gaussian Cheap Talk Game ...
Thursday June 5, 2014
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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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Thursday June 5, 2014
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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.
Cédric Langbort (UIUC)
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Thursday June 5, 2014
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Quadratic Gaussian Cheap Talk Game
θ1
S1
θ2
S2
x
..
θN
y1
y2
yn
R
x̂((yi )N
i=1 )
SN
• x ∼ N (0, Vxx );
• Receiver cost: E{kx − x̂k2 }.
Cédric Langbort (UIUC)
Gaussian Cheap Talk Game ...
Thursday June 5, 2014
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Quadratic Gaussian Cheap Talk Game
θ1
S1
θ2
S2
x
..
θN
y1
y2
yn
R
x̂((yi )N
i=1 )
SN
At the first step, we deploy N strategic sensors:
• Sensor i cost: E{k(x + θi ) − x̂k2 };
• Sensor i has perfect measurements of x, θi (nothing about others);
• θ = (θi )N
i=1 ∼ N (0, Vθθ ).
Cédric Langbort (UIUC)
Gaussian Cheap Talk Game ...
Thursday June 5, 2014
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Quadratic Gaussian Cheap Talk Game
θ1
S1
θ2
S2
x
..
θN
y1
y2
yn
R
x̂((yi )N
i=1 )
SN
At the second step, sensors transmit scalar signals:
>
• yi = γi (x, θi ) ∈ R where γi (x, θi ) = a>
i x + bi θi + vi ;
• vi ∼ N (0, Vvi vi );
• The set of such mappings is Γi (isomorph to Rnx × Rnx × R≥0 ).
Cédric Langbort (UIUC)
Gaussian Cheap Talk Game ...
Thursday June 5, 2014
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Quadratic Gaussian Cheap Talk Game
θ1
S1
θ2
S2
x
..
θN
y1
y2
yn
R
x̂((yi )N
i=1 )
SN
At the third step, the receiver announces its estimate:
• x̂ = x̂(y1 , . . . , yn ) where x̂ ∈ Ψ;
• Ψ is the set of all Lebesgue-measurable functions from RN to Rnx .
Cédric Langbort (UIUC)
Gaussian Cheap Talk Game ...
Thursday June 5, 2014
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Quadratic Gaussian Cheap Talk Game
θ1
S1
θ2
S2
x
..
θN
y1
y2
yn
R
x̂((yi )N
i=1 )
SN
At the fourth step, the cost functions are realized:
• Receiver: E{kx − x̂k2 };
• Sensor i: E{k(x + θi ) − x̂k2 }.
Cédric Langbort (UIUC)
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Thursday June 5, 2014
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Independent Senders
Equilibrium
A tuple (x̂∗ , (γi∗ )N
i=1 ) ∈ Ψ × Γ1 × · · · × ΓN constitutes an equilibrium in
affine strategies if
x̂∗ ∈ arg min E{kx − x̂((γj∗ (x, θj ))N
j=1 )k2 },
x̂∈Ψ
γi∗
∈ arg min E{k(x + θi ) − x̂(γi (x, θi ), (γj∗ (x, θj ))j6=i )k2 }, ∀i.
γi ∈Γi
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.
Cédric Langbort (UIUC)
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Thursday June 5, 2014
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Do we have an equilibrium?
Theorem
Assume that Vxθ = 0, Vθi θi = Vθθ , and Vθi θj = 0 for j 6= i. There exists a
symmetric equilibrium in affine strategies where the receiver follows
x̂∗ (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∗
s
=
1
1 + (N − 1)ξ1> ξ1
"
−1/2
N Vθθ
0
0
−1/2
Vxx
#
ξ
>
and ξ = ξ1> ξ2>
is the normalized eigenvector (i.e., kξk2 = 1)
corresponding to the smallest eigenvalue of the matrix
"
#
−1/2 −1/2
0
−Vθθ Vxx
.
−1/2 −1/2
−Vxx Vθθ
−Vxx
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How good is the equilibrium?
Corollary
Assume that Vxθ = 0, Vθi θi = Vθθ , and Vθi θj = 0 for j 6= i. At the
presented symmetric equilibrium
E{kx − x̂∗ ((γ ∗ (x, θj ))N
j=1 )k2 } = Vxx −
1
U,
α + βN
where α, β ∈ R≥0 and U ∈ Rnx ×nx such that 0 < U ≤ (α + β)Vxx .
Cédric Langbort (UIUC)
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Thursday June 5, 2014
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How good is the equilibrium?
Corollary
Assume that Vxθ = 0, Vθi θi = Vθθ , and Vθi θj = 0 for j 6= i. At the
presented symmetric equilibrium
E{kx − x̂∗ ((γ ∗ (x, θj ))N
j=1 )k2 } = Vxx −
1
U,
α + βN
where α, β ∈ R≥0 and U ∈ Rnx ×nx such that 0 < U ≤ (α + β)Vxx .
• Sadly, this implies that
lim E{kx − x̂∗ ((γ ∗ (x, θj ))N
j=1 )k2 } = Vxx .
N →∞
• This equilibrium is not good for anyone! It is even worse for the
sensors in comparison to the receiver.
lim E{kx + θi − x̂∗ ((γ ∗ (x, θj ))N
j=1 )k2 } = Vxx + Vθθ .
N →∞
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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);
Cédric Langbort (UIUC)
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Thursday June 5, 2014
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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);
Cédric Langbort (UIUC)
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Thursday June 5, 2014
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Herding Senders
Herding Equilibrium
A tuple (x̂∗ , γ ∗ ) ∈ Ψ × Γ constitutes a herding equilibrium in affine
strategies if
x̂∗ ∈ arg min E{kx − x̂((γ ∗ (x, θj ))N
j=1 )k2 },
x̂∈Ψ
∗
γ ∈ arg min E{k(x + θi ) − x̂(γ(x, θi ), (γ(x, θj ))j6=i )k2 }, ∀i.
γ∈Γ
As opposed to before, the senders are constrained to imitate each other.
Cédric Langbort (UIUC)
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Thursday June 5, 2014
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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
x̂∗ (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∗
" √
=
−1/2
N Vθθ
0
0
−1/2
Vxx
#
ζ,
and ζ is the normalized eigenvector (i.e., kζk2 = 1) corresponding to the smallest
eigenvalue of the matrix
#
"
−1/2 −1/2
0
− √1N Vθθ Vxx
.
−1/2 −1/2
− √1N Vxx Vθθ
−Vxx
Cédric Langbort (UIUC)
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Thursday June 5, 2014
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When herding becomes a virtue
Proposition
Assume that Vxθ = 0, Vθi θi = Vθθ , and Vθi θj = 0 for j 6= i. At a herding
equilibrium
lim E{kx − x̂∗ ((γ ∗ (x, θj ))N
j=1 )k2 } = 0.
N →∞
Cédric Langbort (UIUC)
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Thursday June 5, 2014
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When herding becomes a virtue
Proposition
Assume that Vxθ = 0, Vθi θi = Vθθ , and Vθi θj = 0 for j 6= i. At a herding
equilibrium
lim E{kx − x̂∗ ((γ ∗ (x, θj ))N
j=1 )k2 } = 0.
N →∞
Actually, the rate of converge is faster than employing nonstrategic sensors
with measurement noise!
Cédric Langbort (UIUC)
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Thursday June 5, 2014
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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:
0.2763N
,
E{kx − x̂(y)k22 } =
0.7236 + 0.2763N
• Herding Senders
Receiver’s cost:
2
p
E{kx − x̂(y)k22 } =
2
N + 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{u2i } = σ.
Receiver’s cost:
σ
E{kx − x̂(y)k22 } =
σ+N
Cédric Langbort (UIUC)
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Thursday June 5, 2014
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Numerical Example
1
10
0
10
Estimation Error
−1
10
−2
10
−3
10
−4
10
−5
10
−6
Independent Senders
Herding Senders
Nonstrategic Senders
10
−7
10
0
10
1
2
10
10
3
10
N
Cédric Langbort (UIUC)
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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”.
Cédric Langbort (UIUC)
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Thursday June 5, 2014
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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.
Cédric Langbort (UIUC)
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Thursday June 5, 2014
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