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Economics of Age of Information (AoI) Management:Competition and Pricing
Presenter: Lingjie Duan
Engineering Systems and Design PillarSingapore University of Technology and Design (SUTD)
May 4, 2019
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 1 / 61
Acknowledgement
This is a joint work withShugang Hao (PhD student)
Xuehe Wang (Postdoc fellow)
Parts of results here will appear in ACM MobiHoc 2019 and IEEEISIT 2019 Symposia.
S. Hao and L. Duan, “Economics of Age of Information Management under
Network Externalities,” to appear in the Twentieth International Symposiumon Mobile Ad Hoc Networking and Computing (ACM MobiHoc), 2019.[Online]. Available: https://arxiv.org/pdf/1904.01841.pdf
X. Wang and L. Duan, “Dynamic pricing for controlling age of information,”
to appear in IEEE International Symposium on Information Theory (ISIT),
2019. [Online]. Available: https://arxiv.org/pdf/1904.01185.pdf
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 3 / 61
Background: Who care about AoI?
Age of Information (AoI): duration from the moment that the latest
content was generated to current reception time.
Today many customers do not want to lose any breaking news or
useful information in smartphone even if in minute.
Online content platforms (such as navigation and shopping
applications) aim to keep their information update fresh.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 4 / 61
Background: Who care about AoI?
Age of Information (AoI): duration from the moment that the latest
content was generated to current reception time.
Today many customers do not want to lose any breaking news or
useful information in smartphone even if in minute.
Online content platforms (such as navigation and shopping
applications) aim to keep their information update fresh.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 4 / 61
Background: Who care about AoI?
Age of Information (AoI): duration from the moment that the latest
content was generated to current reception time.
Today many customers do not want to lose any breaking news or
useful information in smartphone even if in minute.
Online content platforms (such as navigation and shopping
applications) aim to keep their information update fresh.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 4 / 61
Background: Crowdsourcing for reducing AoI
Crowdsourcing: To keep high sampling rate, platforms invite and pay
crowd to collect information updates, introducing large sampling cost.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 5 / 61
Background: Crowdsourcing for reducing AoI
Crowdsourcing: To keep high sampling rate, platforms invite and pay
crowd to collect information updates, introducing large sampling cost.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 5 / 61
Research Questions
Economic Issues on AoI was largely overlooked in the literature.
Information supply: Platform crowdsourcing incur large sampling cost.
Tradeo↵ between AoI reduction & sampling cost hasn’t been studied.
Information delivery: More than one platform selfishly shares the
content delivery network
Updates of platforms may preempt or jam each other.
Negative network externalities between platforms.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 6 / 61
Research Questions
Economic Issues on AoI was largely overlooked in the literature.
Information supply: Platform crowdsourcing incur large sampling cost.
Tradeo↵ between AoI reduction & sampling cost hasn’t been studied.
Information delivery: More than one platform selfishly shares the
content delivery network
Updates of platforms may preempt or jam each other.
Negative network externalities between platforms.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 6 / 61
Research Questions
Economic Issues on AoI was largely overlooked in the literature.
Information supply: Platform crowdsourcing incur large sampling cost.
Tradeo↵ between AoI reduction & sampling cost hasn’t been studied.
Information delivery: More than one platform selfishly shares the
content delivery network
Updates of platforms may preempt or jam each other.
Negative network externalities between platforms.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 6 / 61
Research Questions
Economic Issues on AoI was largely overlooked in the literature.
Information supply: Platform crowdsourcing incur large sampling cost.
Tradeo↵ between AoI reduction & sampling cost hasn’t been studied.
Information delivery: More than one platform selfishly shares the
content delivery network.
Updates of platforms may preempt or jam each other.
Negative network externalities between platforms.
How to best tradeo↵ between AoI reduction and sampling cost?
How bad is platform competition and how to enforce e�cient cooperation
between selfish platforms?
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 7 / 61
Related Works on AoI
Queueing analysis on average AoI estimation
Single link: Costa et al. (2016), Huang et al. (2015), Kaul et al.
(2012) and Sun et al. (2017).
Multi-source LCFS queue with preemption: Kaul et al. (2012).
Such work do not consider sampling cost or the tradeo↵ between AoI
reduction and sampling cost.
Scheduling broadcast channel among multiple sources for AoI
Hsu et al. (2017). Bedewy et al. (2017).
Such work assumes sources/platforms will follow recommendation,
and do not consider selfish sources’ update competition over the
content delivery network.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 8 / 61
Related Works on AoI
Queueing analysis on average AoI estimation
Single link: Costa et al. (2016), Huang et al. (2015), Kaul et al.
(2012) and Sun et al. (2017).
Multi-source LCFS queue with preemption: Kaul et al. (2012).
Such work do not consider sampling cost or the tradeo↵ between AoI
reduction and sampling cost.
Scheduling broadcast channel among multiple sources for AoI
Hsu et al. (2017). Bedewy et al. (2017).
Such work assumes sources/platforms will follow recommendation,
and do not consider selfish sources’ update competition over the
content delivery network.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 8 / 61
Related Work on Platform Competition
Selfish sharing under negative network externalities
Duopoly competition in network sharing: Gibbens et al. (2000),
Roughgarden et al. (2002), Duan et al. (2015).
Mechanism design to mitigate competition:
Direct pricing as penalty: Courcoubetis (2003), Varian (2004), though
di�cult to enforce additional penalty on platforms here.
Repeated games approach with indirect punishment: Treust et al.
(2010), Xiao et al. (2012), Lanctot et al. (2017), which requires
complete information and su�ciently large discount factor to work.
We will investigate how to regulate platform competition under
incomplete information, and the proposed non-monetary approach
works for any discount factor.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 9 / 61
Related Work on Platform Competition
Selfish sharing under negative network externalities
Duopoly competition in network sharing: Gibbens et al. (2000),
Roughgarden et al. (2002), Duan et al. (2015).
Mechanism design to mitigate competition:
Direct pricing as penalty: Courcoubetis (2003), Varian (2004), though
di�cult to enforce additional penalty on platforms here.
Repeated games approach with indirect punishment: Treust et al.
(2010), Xiao et al. (2012), Lanctot et al. (2017), which requires
complete information and su�ciently large discount factor to work.
We will investigate how to regulate platform competition under
incomplete information, and the proposed non-monetary approach
works for any discount factor.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 9 / 61
Related Work on Platform Competition
Selfish sharing under negative network externalities
Duopoly competition in network sharing: Gibbens et al. (2000),
Roughgarden et al. (2002), Duan et al. (2015).
Mechanism design to mitigate competition:
Direct pricing as penalty: Courcoubetis (2003), Varian (2004), though
di�cult to enforce additional penalty on platforms here.
Repeated games approach with indirect punishment: Treust et al.
(2010), Xiao et al. (2012), Lanctot et al. (2017), which requires
complete information and su�ciently large discount factor to work.
We will investigate how to regulate platform competition under
incomplete information, and the proposed non-monetary approach
works for any discount factor.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 9 / 61
Related Work on Platform Competition
Selfish sharing under negative network externalities
Duopoly competition in network sharing: Gibbens et al. (2000),
Roughgarden et al. (2002), Duan et al. (2015).
Mechanism design to mitigate competition:
Direct pricing as penalty: Courcoubetis (2003), Varian (2004), though
di�cult to enforce additional penalty on platforms here.
Repeated games approach with indirect punishment: Treust et al.
(2010), Xiao et al. (2012), Lanctot et al. (2017), which requires
complete information and su�ciently large discount factor to work.
We will investigate how to regulate platform competition under
incomplete information, and the proposed non-monetary approach
works for any discount factor.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 9 / 61
Part I: Competition regulation for AoI-driven platforms in long run
Part II: Dynamic pricing to control AoI in short-term
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 10 / 61
System Model on Platforms
Two platforms (Groupon and Waze) need to decide how many samples to
buy from their own crowdsourcing pools with sampling rates �1
and �2
,
and then update through the delivery network of bandwidth µ.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 11 / 61
System Model on AoI
Consider each platform:
We consider LCFS M/M/1 queue with preemption (Kaul et al.
(2012)).
Preemption happen within and between platform(s).
Status update age = completion time - generation time.
Average AoI for a single platform in long run is
� =
1
�+
1
µ.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 12 / 61
System Model on AoI
Consider each platform:
We consider LCFS M/M/1 queue with preemption (Kaul et al.
(2012)).
Preemption happen within and between platform(s).
Status update age = completion time - generation time.
Average AoI for a single platform in long run is
� =
1
�+
1
µ.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 12 / 61
System Model on AoI
Consider each platform:
We consider LCFS M/M/1 queue with preemption (Kaul et al.
(2012)).
Preemption happen within and between platform(s).
Status update age = completion time - generation time.
Average AoI for a single platform in long run is
� =
1
�+
1
µ.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 12 / 61
Average AoI for Duopoly Platforms
Average AoI of platform 1 and platform 2 under network sharing:
�
1
=
1
�1
+
�1
+ �2
�1
µ,
�
2
=
1
�2
+
�1
+ �2
�2
µ.
�
1
decreases with sampling rate �1
and bandwidth µ, and increases
with the other platform’s sampling rate �2
.
Negative network externalities due to competition on µ.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 13 / 61
Average AoI for Duopoly Platforms
Average AoI of platform 1 and platform 2 under network sharing:
�
1
=
1
�1
+
�1
+ �2
�1
µ,
�
2
=
1
�2
+
�1
+ �2
�2
µ.
�
1
decreases with sampling rate �1
and bandwidth µ, and increases
with the other platform’s sampling rate �2
.
Negative network externalities due to competition on µ.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 13 / 61
Average AoI for Duopoly Platforms
Average AoI of platform 1 and platform 2 under network sharing:
�
1
=
1
�1
+
�1
+ �2
�1
µ,
�
2
=
1
�2
+
�1
+ �2
�2
µ.
�
1
decreases with sampling rate �1
and bandwidth µ, and increases
with the other platform’s sampling rate �2
.
Negative network externalities due to competition on µ.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 13 / 61
Complete Information Scenario
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 14 / 61
System Model under Complete Information
Model ci as unit cost per sampling rate. Sampling cost is �ici wheninviting sensors of density �i to contribute.
Both platforms have full information on their sampling costs.
Platform 1’s cost function:
⇡1
(�1
,�2
) = �
1
(�1
,�2
) + c1
�1
.
implying the tradeo↵ between AoI and sampling cost by deciding �1
.
Platform 2’s cost function:
⇡2
(�1
,�2
) = �
2
(�1
,�2
) + c2
�2
,Social cost function:
⇡(�1
,�2
) = ⇡1
(�1
,�2
) + ⇡2
(�1
,�2
).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 15 / 61
System Model under Complete Information
Model ci as unit cost per sampling rate. Sampling cost is �ici wheninviting sensors of density �i to contribute.
Both platforms have full information on their sampling costs.
Platform 1’s cost function:
⇡1
(�1
,�2
) = �
1
(�1
,�2
) + c1
�1
.
implying the tradeo↵ between AoI and sampling cost by deciding �1
.
Platform 2’s cost function:
⇡2
(�1
,�2
) = �
2
(�1
,�2
) + c2
�2
,Social cost function:
⇡(�1
,�2
) = ⇡1
(�1
,�2
) + ⇡2
(�1
,�2
).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 15 / 61
System Model under Complete Information
Model ci as unit cost per sampling rate. Sampling cost is �ici wheninviting sensors of density �i to contribute.
Both platforms have full information on their sampling costs.
Platform 1’s cost function:
⇡1
(�1
,�2
) = �
1
(�1
,�2
) + c1
�1
.
implying the tradeo↵ between AoI and sampling cost by deciding �1
.
Platform 2’s cost function:
⇡2
(�1
,�2
) = �
2
(�1
,�2
) + c2
�2
,Social cost function:
⇡(�1
,�2
) = ⇡1
(�1
,�2
) + ⇡2
(�1
,�2
).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 15 / 61
System Model under Complete Information
Model ci as unit cost per sampling rate. Sampling cost is �ici wheninviting sensors of density �i to contribute.
Both platforms have full information on their sampling costs.
Platform 1’s cost function:
⇡1
(�1
,�2
) = �
1
(�1
,�2
) + c1
�1
.
implying the tradeo↵ between AoI and sampling cost by deciding �1
.
Platform 2’s cost function:
⇡2
(�1
,�2
) = �
2
(�1
,�2
) + c2
�2
,
Social cost function:
⇡(�1
,�2
) = ⇡1
(�1
,�2
) + ⇡2
(�1
,�2
).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 15 / 61
System Model under Complete Information
Model ci as unit cost per sampling rate. Sampling cost is �ici wheninviting sensors of density �i to contribute.
Both platforms have full information on their sampling costs.
Platform 1’s cost function:
⇡1
(�1
,�2
) = �
1
(�1
,�2
) + c1
�1
.
implying the tradeo↵ between AoI and sampling cost by deciding �1
.
Platform 2’s cost function:
⇡2
(�1
,�2
) = �
2
(�1
,�2
) + c2
�2
,Social cost function:
⇡(�1
,�2
) = ⇡1
(�1
,�2
) + ⇡2
(�1
,�2
).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 15 / 61
Non-cooperative Static Game under Complete Information
Non-cooperative game with equilibrium (�⇤1
,�⇤2
):
min
�1
>0
⇡1
(�1
,�2
)
min
�2
>0
⇡2
(�1
,�2
)
Min-social-cost problem: centrally follow social optimizers (�⇤⇤1
,�⇤⇤2
):
min
�1
,�2
>0
⇡(�1
,�2
)
Question: equilibrium (�⇤1
,�⇤2
) versus optimal (�⇤⇤1
,�⇤⇤2
)?
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 16 / 61
Non-cooperative Static Game under Complete Information
Non-cooperative game with equilibrium (�⇤1
,�⇤2
):
min
�1
>0
⇡1
(�1
,�2
)
min
�2
>0
⇡2
(�1
,�2
)
Min-social-cost problem: centrally follow social optimizers (�⇤⇤1
,�⇤⇤2
):
min
�1
,�2
>0
⇡(�1
,�2
)
Question: equilibrium (�⇤1
,�⇤2
) versus optimal (�⇤⇤1
,�⇤⇤2
)?
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 16 / 61
Non-cooperative Static Game under Complete Information
Non-cooperative game with equilibrium (�⇤1
,�⇤2
):
min
�1
>0
⇡1
(�1
,�2
)
min
�2
>0
⇡2
(�1
,�2
)
Min-social-cost problem: centrally follow social optimizers (�⇤⇤1
,�⇤⇤2
):
min
�1
,�2
>0
⇡(�1
,�2
)
Question: equilibrium (�⇤1
,�⇤2
) versus optimal (�⇤⇤1
,�⇤⇤2
)?
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 16 / 61
Competition Equilibrium and Social Optimizers
Proposition 1 (Equilibrium vs Social Optimizers under complete information)
Under complete information, the competition equilibrium (�⇤1
,�⇤2
) are the uniquesolutions to
� 1�2
1
(1 +�2
µ) + c
1
= 0,
� 1�2
2
(1 +�1
µ) + c
2
= 0. (1)
Di↵erently, the social optimizers (�⇤⇤1
,�⇤⇤2
), are the unique solutions to
� 1�2
1
(1 +�2
µ) + c
1
+1
�2
µ= 0,
� 1�2
2
(1 +�1
µ) + c
2
+1
�1
µ= 0. (2)
By comparing (1) and (2), we conclude competition leads over-sampling (�⇤i � �⇤⇤
i fori = 1, 2) at the equilibrium.
Direct pricing approach: charge price
1
�⇤⇤j µ per sampling rate from platform i to
reach (�⇤⇤1
,�⇤⇤2
).Yet our focus is non-monetary approach!
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 17 / 61
Competition Equilibrium and Social Optimizers
Proposition 1 (Equilibrium vs Social Optimizers under complete information)
Under complete information, the competition equilibrium (�⇤1
,�⇤2
) are the uniquesolutions to
� 1�2
1
(1 +�2
µ) + c
1
= 0,
� 1�2
2
(1 +�1
µ) + c
2
= 0. (1)
Di↵erently, the social optimizers (�⇤⇤1
,�⇤⇤2
), are the unique solutions to
� 1�2
1
(1 +�2
µ) + c
1
+1
�2
µ= 0,
� 1�2
2
(1 +�1
µ) + c
2
+1
�1
µ= 0. (2)
By comparing (1) and (2), we conclude competition leads over-sampling (�⇤i � �⇤⇤
i fori = 1, 2) at the equilibrium.
Direct pricing approach: charge price
1
�⇤⇤j µ per sampling rate from platform i to
reach (�⇤⇤1
,�⇤⇤2
).
Yet our focus is non-monetary approach!
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 17 / 61
Competition Equilibrium and Social Optimizers
Proposition 1 (Equilibrium vs Social Optimizers under complete information)
Under complete information, the competition equilibrium (�⇤1
,�⇤2
) are the uniquesolutions to
� 1�2
1
(1 +�2
µ) + c
1
= 0,
� 1�2
2
(1 +�1
µ) + c
2
= 0. (1)
Di↵erently, the social optimizers (�⇤⇤1
,�⇤⇤2
), are the unique solutions to
� 1�2
1
(1 +�2
µ) + c
1
+1
�2
µ= 0,
� 1�2
2
(1 +�1
µ) + c
2
+1
�1
µ= 0. (2)
By comparing (1) and (2), we conclude competition leads over-sampling (�⇤i � �⇤⇤
i fori = 1, 2) at the equilibrium.
Direct pricing approach: charge price
1
�⇤⇤j µ per sampling rate from platform i to
reach (�⇤⇤1
,�⇤⇤2
).Yet our focus is non-monetary approach!
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 17 / 61
Equilibrium under Complete Informatiion
Corollary 1
Equilibrium �⇤1
increases with �⇤2
, and decreases with c1
, c2
, µ, respectively.
�⇤1
increases with �⇤2
: competition to preempt and occupy µ.
�⇤1
decreases with µ: less competition given ample bandwidth.
�⇤1
decreases with c1
: avoid high samping cost.
�⇤1
decreases with c2
: �⇤2
decreases with c2
.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 18 / 61
Equilibrium under Complete Informatiion
Corollary 1
Equilibrium �⇤1
increases with �⇤2
, and decreases with c1
, c2
, µ, respectively.
�⇤1
increases with �⇤2
: competition to preempt and occupy µ.
�⇤1
decreases with µ: less competition given ample bandwidth.
�⇤1
decreases with c1
: avoid high samping cost.
�⇤1
decreases with c2
: �⇤2
decreases with c2
.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 18 / 61
Equilibrium under Complete Informatiion
Corollary 1
Equilibrium �⇤1
increases with �⇤2
, and decreases with c1
, c2
, µ, respectively.
�⇤1
increases with �⇤2
: competition to preempt and occupy µ.
�⇤1
decreases with µ: less competition given ample bandwidth.
�⇤1
decreases with c1
: avoid high samping cost.
�⇤1
decreases with c2
: �⇤2
decreases with c2
.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 18 / 61
Equilibrium under Complete Informatiion
Corollary 1
Equilibrium �⇤1
increases with �⇤2
, and decreases with c1
, c2
, µ, respectively.
�⇤1
increases with �⇤2
: competition to preempt and occupy µ.
�⇤1
decreases with µ: less competition given ample bandwidth.
�⇤1
decreases with c1
: avoid high samping cost.
�⇤1
decreases with c2
: �⇤2
decreases with c2
.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 18 / 61
Equilibrium under Complete Informatiion
Corollary 1
Equilibrium �⇤1
increases with �⇤2
, and decreases with c1
, c2
, µ, respectively.
�⇤1
increases with �⇤2
: competition to preempt and occupy µ.
�⇤1
decreases with µ: less competition given ample bandwidth.
�⇤1
decreases with c1
: avoid high samping cost.
�⇤1
decreases with c2
: �⇤2
decreases with c2
.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 18 / 61
E�ciency loss with Competition: versus optimum
Price of Anarchy (PoA):
PoA = max
c1
,c2
,µ>0
⇡(�⇤1
,�⇤2
)
⇡(�⇤⇤1
,�⇤⇤2
)
� 1.
We use PoA � 1 to tell e�ciency loss due to competition in the worst case.
Proposition 2 (Huge e�ciency loss under complete information)
Price of anarchy under complete information is PoA = 1, which is
achieved when platform 1’s sampling cost c1
is infinitesimal.
When c1
! 0,
equilibrium �⇤1
! 1, �⇤2
! 1.
optimal �⇤⇤1
< 1, �⇤⇤2
< 1.
Need non-monetary mechanism to remedy the huge e�ciency loss!
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 19 / 61
E�ciency loss with Competition: versus optimum
Price of Anarchy (PoA):
PoA = max
c1
,c2
,µ>0
⇡(�⇤1
,�⇤2
)
⇡(�⇤⇤1
,�⇤⇤2
)
� 1.
We use PoA � 1 to tell e�ciency loss due to competition in the worst case.
Proposition 2 (Huge e�ciency loss under complete information)
Price of anarchy under complete information is PoA = 1, which is
achieved when platform 1’s sampling cost c1
is infinitesimal.
When c1
! 0,
equilibrium �⇤1
! 1, �⇤2
! 1.
optimal �⇤⇤1
< 1, �⇤⇤2
< 1.
Need non-monetary mechanism to remedy the huge e�ciency loss!
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 19 / 61
E�ciency loss with Competition: versus optimum
Price of Anarchy (PoA):
PoA = max
c1
,c2
,µ>0
⇡(�⇤1
,�⇤2
)
⇡(�⇤⇤1
,�⇤⇤2
)
� 1.
We use PoA � 1 to tell e�ciency loss due to competition in the worst case.
Proposition 2 (Huge e�ciency loss under complete information)
Price of anarchy under complete information is PoA = 1, which is
achieved when platform 1’s sampling cost c1
is infinitesimal.
When c1
! 0,
equilibrium �⇤1
! 1, �⇤2
! 1.
optimal �⇤⇤1
< 1, �⇤⇤2
< 1.
Need non-monetary mechanism to remedy the huge e�ciency loss!
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 19 / 61
E�ciency loss with Competition: versus optimum
Price of Anarchy (PoA):
PoA = max
c1
,c2
,µ>0
⇡(�⇤1
,�⇤2
)
⇡(�⇤⇤1
,�⇤⇤2
)
� 1.
We use PoA � 1 to tell e�ciency loss due to competition in the worst case.
Proposition 2 (Huge e�ciency loss under complete information)
Price of anarchy under complete information is PoA = 1, which is
achieved when platform 1’s sampling cost c1
is infinitesimal.
When c1
! 0,
equilibrium �⇤1
! 1, �⇤2
! 1.
optimal �⇤⇤1
< 1, �⇤⇤2
< 1.
Need non-monetary mechanism to remedy the huge e�ciency loss!
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 19 / 61
E�ciency loss with Competition: versus optimum
Price of Anarchy (PoA):
PoA = max
c1
,c2
,µ>0
⇡(�⇤1
,�⇤2
)
⇡(�⇤⇤1
,�⇤⇤2
)
� 1.
We use PoA � 1 to tell e�ciency loss due to competition in the worst case.
Proposition 2 (Huge e�ciency loss under complete information)
Price of anarchy under complete information is PoA = 1, which is
achieved when platform 1’s sampling cost c1
is infinitesimal.
When c1
! 0,
equilibrium �⇤1
! 1, �⇤2
! 1.
optimal �⇤⇤1
< 1, �⇤⇤2
< 1.
Need non-monetary mechanism to remedy the huge e�ciency loss!
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 19 / 61
Trigger-and-punishment Mechanism under Complete Information
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 20 / 61
Our Repeated Game Approach
We consider how to regulate competition in long run in repeated
games with discount factor ⇢ < 1.
Each time slot in the repeated games is long enough for each
platform i ’s AoI statistic to converge to its average value �i .
Definition 1 (Non-forgiving trigger mechanism of punishment undercomplete information)
In each round, recommend cooperation profile (
˜�1
(⇢), ˜�2
(⇢)) tofollow, if neither was found to deviate in the past.
Once a deviation was found in the past, the two platforms will keepplaying the punishment/equilibrium profile (�⇤
1
,�⇤2
) forever.
How to detect deviation and trigger punishment?
Platform 1 can identify the other platform’s sampling rate �2
from its
own average AoI experience �
1
(�1
,�2
) and �1
.
The key is how to design (
˜�1
(⇢), ˜�2
(⇢)) to never trigger punishment.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 21 / 61
Our Repeated Game Approach
We consider how to regulate competition in long run in repeated
games with discount factor ⇢ < 1.
Each time slot in the repeated games is long enough for each
platform i ’s AoI statistic to converge to its average value �i .
Definition 1 (Non-forgiving trigger mechanism of punishment undercomplete information)
In each round, recommend cooperation profile (
˜�1
(⇢), ˜�2
(⇢)) tofollow, if neither was found to deviate in the past.
Once a deviation was found in the past, the two platforms will keepplaying the punishment/equilibrium profile (�⇤
1
,�⇤2
) forever.
How to detect deviation and trigger punishment?
Platform 1 can identify the other platform’s sampling rate �2
from its
own average AoI experience �
1
(�1
,�2
) and �1
.
The key is how to design (
˜�1
(⇢), ˜�2
(⇢)) to never trigger punishment.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 21 / 61
Our Repeated Game Approach
We consider how to regulate competition in long run in repeated
games with discount factor ⇢ < 1.
Each time slot in the repeated games is long enough for each
platform i ’s AoI statistic to converge to its average value �i .
Definition 1 (Non-forgiving trigger mechanism of punishment undercomplete information)
In each round, recommend cooperation profile (
˜�1
(⇢), ˜�2
(⇢)) tofollow, if neither was found to deviate in the past.
Once a deviation was found in the past, the two platforms will keepplaying the punishment/equilibrium profile (�⇤
1
,�⇤2
) forever.
How to detect deviation and trigger punishment?
Platform 1 can identify the other platform’s sampling rate �2
from its
own average AoI experience �
1
(�1
,�2
) and �1
.
The key is how to design (
˜�1
(⇢), ˜�2
(⇢)) to never trigger punishment.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 21 / 61
Our Repeated Game Approach
We consider how to regulate competition in long run in repeated
games with discount factor ⇢ < 1.
Each time slot in the repeated games is long enough for each
platform i ’s AoI statistic to converge to its average value �i .
Definition 1 (Non-forgiving trigger mechanism of punishment undercomplete information)
In each round, recommend cooperation profile (
˜�1
(⇢), ˜�2
(⇢)) tofollow, if neither was found to deviate in the past.
Once a deviation was found in the past, the two platforms will keepplaying the punishment/equilibrium profile (�⇤
1
,�⇤2
) forever.
How to detect deviation and trigger punishment?
Platform 1 can identify the other platform’s sampling rate �2
from its
own average AoI experience �
1
(�1
,�2
) and �1
.
The key is how to design (
˜�1
(⇢), ˜�2
(⇢)) to never trigger punishment.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 21 / 61
Our Repeated Game Approach
We consider how to regulate competition in long run in repeated
games with discount factor ⇢ < 1.
Each time slot in the repeated games is long enough for each
platform i ’s AoI statistic to converge to its average value �i .
Definition 1 (Non-forgiving trigger mechanism of punishment undercomplete information)
In each round, recommend cooperation profile (
˜�1
(⇢), ˜�2
(⇢)) tofollow, if neither was found to deviate in the past.
Once a deviation was found in the past, the two platforms will keepplaying the punishment/equilibrium profile (�⇤
1
,�⇤2
) forever.
How to detect deviation and trigger punishment?
Platform 1 can identify the other platform’s sampling rate �2
from its
own average AoI experience �
1
(�1
,�2
) and �1
.
The key is how to design (
˜�1
(⇢), ˜�2
(⇢)) to never trigger punishment.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 21 / 61
Condition for No Deviation for Both Platforms
Ideally, we want to ensure no deviation of each platform from
(
˜�1
(⇢), ˜�2
(⇢)) = (�⇤⇤1
,�⇤⇤2
).
Platform 1’s long-term cost over all time stages without deviation:
⇧
1
= ⇡1
(�⇤⇤1
,�⇤⇤2
) + ⇢⇡1
(�⇤⇤1
,�⇤⇤2
) + ⇢2⇡1
(�⇤⇤1
,�⇤⇤2
) + · · · ,
=
1
1� ⇢⇡1
(�⇤⇤1
,�⇤⇤2
).
Platform 1’s long-term cost over all time stages by deviating in the
first round with best response �1
=
q1+�⇤⇤
2
/µc1
:
ˆ
⇧
1
= ⇡1
✓s1 + �⇤⇤
2
/µ
c1
,�⇤⇤2
◆+ ⇢⇡
1
(�⇤1
,�⇤2
) + ⇢2⇡1
(�⇤1
,�⇤2
) + · · · ,
= ⇡1
✓s1 + �⇤⇤
2
/µ
c1
,�⇤⇤2
◆+
⇢
1� ⇢⇡1
(�⇤1
,�⇤2
).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 22 / 61
Condition for No Deviation for Both Platforms
Ideally, we want to ensure no deviation of each platform from
(
˜�1
(⇢), ˜�2
(⇢)) = (�⇤⇤1
,�⇤⇤2
).
Platform 1’s long-term cost over all time stages without deviation:
⇧
1
= ⇡1
(�⇤⇤1
,�⇤⇤2
) + ⇢⇡1
(�⇤⇤1
,�⇤⇤2
) + ⇢2⇡1
(�⇤⇤1
,�⇤⇤2
) + · · · ,
=
1
1� ⇢⇡1
(�⇤⇤1
,�⇤⇤2
).
Platform 1’s long-term cost over all time stages by deviating in the
first round with best response �1
=
q1+�⇤⇤
2
/µc1
:
ˆ
⇧
1
= ⇡1
✓s1 + �⇤⇤
2
/µ
c1
,�⇤⇤2
◆+ ⇢⇡
1
(�⇤1
,�⇤2
) + ⇢2⇡1
(�⇤1
,�⇤2
) + · · · ,
= ⇡1
✓s1 + �⇤⇤
2
/µ
c1
,�⇤⇤2
◆+
⇢
1� ⇢⇡1
(�⇤1
,�⇤2
).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 22 / 61
Condition for No Deviation for Both Platforms
Ideally, we want to ensure no deviation of each platform from
(
˜�1
(⇢), ˜�2
(⇢)) = (�⇤⇤1
,�⇤⇤2
).
Platform 1’s long-term cost over all time stages without deviation:
⇧
1
= ⇡1
(�⇤⇤1
,�⇤⇤2
) + ⇢⇡1
(�⇤⇤1
,�⇤⇤2
) + ⇢2⇡1
(�⇤⇤1
,�⇤⇤2
) + · · · ,
=
1
1� ⇢⇡1
(�⇤⇤1
,�⇤⇤2
).
Platform 1’s long-term cost over all time stages by deviating in the
first round with best response �1
=
q1+�⇤⇤
2
/µc1
:
ˆ
⇧
1
= ⇡1
✓s1 + �⇤⇤
2
/µ
c1
,�⇤⇤2
◆+ ⇢⇡
1
(�⇤1
,�⇤2
) + ⇢2⇡1
(�⇤1
,�⇤2
) + · · · ,
= ⇡1
✓s1 + �⇤⇤
2
/µ
c1
,�⇤⇤2
◆+
⇢
1� ⇢⇡1
(�⇤1
,�⇤2
).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 22 / 61
Condition for No Deviation for Both Platforms
Ideally, we want to ensure no deviation of each platform from
(
˜�1
(⇢), ˜�2
(⇢)) = (�⇤⇤1
,�⇤⇤2
).
Platform 1’s long-term cost over all time stages without deviation:
⇧
1
= ⇡1
(�⇤⇤1
,�⇤⇤2
) + ⇢⇡1
(�⇤⇤1
,�⇤⇤2
) + ⇢2⇡1
(�⇤⇤1
,�⇤⇤2
) + · · · ,
=
1
1� ⇢⇡1
(�⇤⇤1
,�⇤⇤2
).
Platform 1’s long-term cost over all time stages by deviating in the
first round with best response �1
=
q1+�⇤⇤
2
/µc1
:
ˆ
⇧
1
= ⇡1
✓s1 + �⇤⇤
2
/µ
c1
,�⇤⇤2
◆+ ⇢⇡
1
(�⇤1
,�⇤2
) + ⇢2⇡1
(�⇤1
,�⇤2
) + · · · ,
= ⇡1
✓s1 + �⇤⇤
2
/µ
c1
,�⇤⇤2
◆+
⇢
1� ⇢⇡1
(�⇤1
,�⇤2
).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 22 / 61
Condition for No Deviation for Both Platforms
Ideally, we want to ensure no deviation of each platform from
(
˜�1
(⇢), ˜�2
(⇢)) = (�⇤⇤1
,�⇤⇤2
).
Platform 1’s long-term cost over all time stages without deviation:
⇧
1
= ⇡1
(�⇤⇤1
,�⇤⇤2
) + ⇢⇡1
(�⇤⇤1
,�⇤⇤2
) + ⇢2⇡1
(�⇤⇤1
,�⇤⇤2
) + · · · ,
=
1
1� ⇢⇡1
(�⇤⇤1
,�⇤⇤2
).
Platform 1’s long-term cost over all time stages by deviating in the
first round with best response �1
=
q1+�⇤⇤
2
/µc1
:
ˆ
⇧
1
= ⇡1
✓s1 + �⇤⇤
2
/µ
c1
,�⇤⇤2
◆+ ⇢⇡
1
(�⇤1
,�⇤2
) + ⇢2⇡1
(�⇤1
,�⇤2
) + · · · ,
= ⇡1
✓s1 + �⇤⇤
2
/µ
c1
,�⇤⇤2
◆+
⇢
1� ⇢⇡1
(�⇤1
,�⇤2
).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 22 / 61
Condition for No Deviation for Both Platforms (Cont.)
No deviation for platform 1: ⇧
1
ˆ
⇧
1
is equivalent to
discount factor ⇢ � ⇢th1
:=
✓r1+�⇤⇤
2
/µ
c1
+
1
�⇤⇤2
µ
�q
1+�⇤⇤2
/µc1
◆2
2�⇤⇤1
✓�⇤1
�q
1+�⇤⇤2
/µc1
◆ .
Similarly, no deviation for platform 2:
discount factor ⇢ � ⇢th2
:=
✓r1+�⇤⇤
1
/µ
c2
+
1
�⇤⇤1
µ
�q
1+�⇤⇤1
/µc2
◆2
2�⇤⇤2
✓�⇤2
�q
1+�⇤⇤1
/µc2
◆ .
We assume c1
c2
. Which platform is more likely to deviate?
Platform 1 is more likely to deviate with ⇢th1
� ⇢th2
.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 23 / 61
Condition for No Deviation for Both Platforms (Cont.)
No deviation for platform 1: ⇧
1
ˆ
⇧
1
is equivalent to
discount factor ⇢ � ⇢th1
:=
✓r1+�⇤⇤
2
/µ
c1
+
1
�⇤⇤2
µ
�q
1+�⇤⇤2
/µc1
◆2
2�⇤⇤1
✓�⇤1
�q
1+�⇤⇤2
/µc1
◆ .
Similarly, no deviation for platform 2:
discount factor ⇢ � ⇢th2
:=
✓r1+�⇤⇤
1
/µ
c2
+
1
�⇤⇤1
µ
�q
1+�⇤⇤1
/µc2
◆2
2�⇤⇤2
✓�⇤2
�q
1+�⇤⇤1
/µc2
◆ .
We assume c1
c2
. Which platform is more likely to deviate?
Platform 1 is more likely to deviate with ⇢th1
� ⇢th2
.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 23 / 61
Condition for No Deviation for Both Platforms (Cont.)
No deviation for platform 1: ⇧
1
ˆ
⇧
1
is equivalent to
discount factor ⇢ � ⇢th1
:=
✓r1+�⇤⇤
2
/µ
c1
+
1
�⇤⇤2
µ
�q
1+�⇤⇤2
/µc1
◆2
2�⇤⇤1
✓�⇤1
�q
1+�⇤⇤2
/µc1
◆ .
Similarly, no deviation for platform 2:
discount factor ⇢ � ⇢th2
:=
✓r1+�⇤⇤
1
/µ
c2
+
1
�⇤⇤1
µ
�q
1+�⇤⇤1
/µc2
◆2
2�⇤⇤2
✓�⇤2
�q
1+�⇤⇤1
/µc2
◆ .
We assume c1
c2
. Which platform is more likely to deviate?
Platform 1 is more likely to deviate with ⇢th1
� ⇢th2
.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 23 / 61
Condition for No Deviation for Both Platforms (Cont.)
No deviation for platform 1: ⇧
1
ˆ
⇧
1
is equivalent to
discount factor ⇢ � ⇢th1
:=
✓r1+�⇤⇤
2
/µ
c1
+
1
�⇤⇤2
µ
�q
1+�⇤⇤2
/µc1
◆2
2�⇤⇤1
✓�⇤1
�q
1+�⇤⇤2
/µc1
◆ .
Similarly, no deviation for platform 2:
discount factor ⇢ � ⇢th2
:=
✓r1+�⇤⇤
1
/µ
c2
+
1
�⇤⇤1
µ
�q
1+�⇤⇤1
/µc2
◆2
2�⇤⇤2
✓�⇤2
�q
1+�⇤⇤1
/µc2
◆ .
We assume c1
c2
. Which platform is more likely to deviate?
Platform 1 is more likely to deviate with ⇢th1
� ⇢th2
.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 23 / 61
Cooperation Profile for Large ⇢ Regime
Large ⇢ Regime: ⇢ � max{⇢th1
, ⇢th2
} = ⇢th1
. Both will never deviate.
Proposition 3 (Large ⇢ Regime)
Under complete information, if ⇢ � ⇢th1
, both platforms will follow the
perfect cooperation profile (
˜�1
(⇢), ˜�2
(⇢)) = (�⇤⇤1
,�⇤⇤2
) all the time without
triggering the punishment profile (�⇤1
,�⇤2
).
Corollary 2 (symmetric ⇢th1
= ⇢th2
versus µ, c1
= c2
)
Given c1
= c2
under complete information, ⇢th1
= ⇢th2
decreases with µand c
1
if
pc1
µ 1/4 and then increases with µ and c1
if
pc1
µ > 1/4.
Small µ: scarce bandwidth causes intense competition, and as µincreases, competition is mitigated.
Large µ: ample bandwidth allures both platforms keep their AoI
small, and as µ increases, both sample more aggressively.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 24 / 61
Cooperation Profile for Large ⇢ Regime
Large ⇢ Regime: ⇢ � max{⇢th1
, ⇢th2
} = ⇢th1
. Both will never deviate.
Proposition 3 (Large ⇢ Regime)
Under complete information, if ⇢ � ⇢th1
, both platforms will follow the
perfect cooperation profile (
˜�1
(⇢), ˜�2
(⇢)) = (�⇤⇤1
,�⇤⇤2
) all the time without
triggering the punishment profile (�⇤1
,�⇤2
).
Corollary 2 (symmetric ⇢th1
= ⇢th2
versus µ, c1
= c2
)
Given c1
= c2
under complete information, ⇢th1
= ⇢th2
decreases with µand c
1
if
pc1
µ 1/4 and then increases with µ and c1
if
pc1
µ > 1/4.
Small µ: scarce bandwidth causes intense competition, and as µincreases, competition is mitigated.
Large µ: ample bandwidth allures both platforms keep their AoI
small, and as µ increases, both sample more aggressively.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 24 / 61
Cooperation Profile for Large ⇢ Regime
Large ⇢ Regime: ⇢ � max{⇢th1
, ⇢th2
} = ⇢th1
. Both will never deviate.
Proposition 3 (Large ⇢ Regime)
Under complete information, if ⇢ � ⇢th1
, both platforms will follow the
perfect cooperation profile (
˜�1
(⇢), ˜�2
(⇢)) = (�⇤⇤1
,�⇤⇤2
) all the time without
triggering the punishment profile (�⇤1
,�⇤2
).
Corollary 2 (symmetric ⇢th1
= ⇢th2
versus µ, c1
= c2
)
Given c1
= c2
under complete information, ⇢th1
= ⇢th2
decreases with µand c
1
if
pc1
µ 1/4 and then increases with µ and c1
if
pc1
µ > 1/4.
Small µ: scarce bandwidth causes intense competition, and as µincreases, competition is mitigated.
Large µ: ample bandwidth allures both platforms keep their AoI
small, and as µ increases, both sample more aggressively.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 24 / 61
Cooperation Profile for Large ⇢ Regime
Large ⇢ Regime: ⇢ � max{⇢th1
, ⇢th2
} = ⇢th1
. Both will never deviate.
Proposition 3 (Large ⇢ Regime)
Under complete information, if ⇢ � ⇢th1
, both platforms will follow the
perfect cooperation profile (
˜�1
(⇢), ˜�2
(⇢)) = (�⇤⇤1
,�⇤⇤2
) all the time without
triggering the punishment profile (�⇤1
,�⇤2
).
Corollary 2 (symmetric ⇢th1
= ⇢th2
versus µ, c1
= c2
)
Given c1
= c2
under complete information, ⇢th1
= ⇢th2
decreases with µand c
1
if
pc1
µ 1/4 and then increases with µ and c1
if
pc1
µ > 1/4.
Small µ: scarce bandwidth causes intense competition, and as µincreases, competition is mitigated.
Large µ: ample bandwidth allures both platforms keep their AoI
small, and as µ increases, both sample more aggressively.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 24 / 61
Cooperation Profile for Large ⇢ Regime
Large ⇢ Regime: ⇢ � max{⇢th1
, ⇢th2
} = ⇢th1
. Both will never deviate.
Proposition 3 (Large ⇢ Regime)
Under complete information, if ⇢ � ⇢th1
, both platforms will follow the
perfect cooperation profile (
˜�1
(⇢), ˜�2
(⇢)) = (�⇤⇤1
,�⇤⇤2
) all the time without
triggering the punishment profile (�⇤1
,�⇤2
).
Corollary 2 (symmetric ⇢th1
= ⇢th2
versus µ, c1
= c2
)
Given c1
= c2
under complete information, ⇢th1
= ⇢th2
decreases with µand c
1
if
pc1
µ 1/4 and then increases with µ and c1
if
pc1
µ > 1/4.
Small µ: scarce bandwidth causes intense competition, and as µincreases, competition is mitigated.
Large µ: ample bandwidth allures both platforms keep their AoI
small, and as µ increases, both sample more aggressively.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 24 / 61
Cooperation Profile Design for Medium ⇢ Regime
⇢ < max{⇢th1
, ⇢th2
}, we cannot use (
˜�1
(⇢), ˜�2
(⇢)) = (�⇤⇤1
,�⇤⇤2
) as
cooperation profile.
⇢th2
⇢ < ⇢th1
Platform 2 will still follow social optimizer �⇤⇤2
.
Platform 1 will deviate and we should redesign
˜�1
(⇢) to satisfy
⇧
1
(
˜�1
(⇢),�⇤⇤2
) =
ˆ
⇧
1
(�⇤1
,�⇤2
) or simply,
⇢ =
⇡1
(
˜�1
(⇢),�⇤⇤2
)� ⇡1
✓q1+�⇤⇤
2
/µc1
,�⇤⇤2
◆
⇡1
(�⇤1
,�⇤2
)� ⇡1
✓q1+�⇤⇤
2
/µc1
,�⇤⇤2
◆ .
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 25 / 61
Cooperation Profile Design for Medium ⇢ Regime
⇢ < max{⇢th1
, ⇢th2
}, we cannot use (
˜�1
(⇢), ˜�2
(⇢)) = (�⇤⇤1
,�⇤⇤2
) as
cooperation profile.
⇢th2
⇢ < ⇢th1
Platform 2 will still follow social optimizer �⇤⇤2
.
Platform 1 will deviate and we should redesign
˜�1
(⇢) to satisfy
⇧
1
(
˜�1
(⇢),�⇤⇤2
) =
ˆ
⇧
1
(�⇤1
,�⇤2
) or simply,
⇢ =
⇡1
(
˜�1
(⇢),�⇤⇤2
)� ⇡1
✓q1+�⇤⇤
2
/µc1
,�⇤⇤2
◆
⇡1
(�⇤1
,�⇤2
)� ⇡1
✓q1+�⇤⇤
2
/µc1
,�⇤⇤2
◆ .
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 25 / 61
Cooperation Profile Design for Medium ⇢ Regime
⇢ < max{⇢th1
, ⇢th2
}, we cannot use (
˜�1
(⇢), ˜�2
(⇢)) = (�⇤⇤1
,�⇤⇤2
) as
cooperation profile.
⇢th2
⇢ < ⇢th1
Platform 2 will still follow social optimizer �⇤⇤2
.
Platform 1 will deviate and we should redesign
˜�1
(⇢) to satisfy
⇧
1
(
˜�1
(⇢),�⇤⇤2
) =
ˆ
⇧
1
(�⇤1
,�⇤2
) or simply,
⇢ =
⇡1
(
˜�1
(⇢),�⇤⇤2
)� ⇡1
✓q1+�⇤⇤
2
/µc1
,�⇤⇤2
◆
⇡1
(�⇤1
,�⇤2
)� ⇡1
✓q1+�⇤⇤
2
/µc1
,�⇤⇤2
◆ .
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 25 / 61
Cooperation Profile Design for Medium ⇢ Regime
⇢ < max{⇢th1
, ⇢th2
}, we cannot use (
˜�1
(⇢), ˜�2
(⇢)) = (�⇤⇤1
,�⇤⇤2
) as
cooperation profile.
⇢th2
⇢ < ⇢th1
Platform 2 will still follow social optimizer �⇤⇤2
.
Platform 1 will deviate and we should redesign
˜�1
(⇢) to satisfy
⇧
1
(
˜�1
(⇢),�⇤⇤2
) =
ˆ
⇧
1
(�⇤1
,�⇤2
) or simply,
⇢ =
⇡1
(
˜�1
(⇢),�⇤⇤2
)� ⇡1
✓q1+�⇤⇤
2
/µc1
,�⇤⇤2
◆
⇡1
(�⇤1
,�⇤2
)� ⇡1
✓q1+�⇤⇤
2
/µc1
,�⇤⇤2
◆ .
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 25 / 61
Cooperation Profile Design for Medium ⇢ Regime
⇢ < max{⇢th1
, ⇢th2
}, we cannot use (
˜�1
(⇢), ˜�2
(⇢)) = (�⇤⇤1
,�⇤⇤2
) as
cooperation profile.
⇢th2
⇢ < ⇢th1
Platform 2 will still follow social optimizer �⇤⇤2
.
Platform 1 will deviate and we should redesign
˜�1
(⇢) to satisfy
⇧
1
(
˜�1
(⇢),�⇤⇤2
) =
ˆ
⇧
1
(�⇤1
,�⇤2
) or simply,
⇢ =
⇡1
(
˜�1
(⇢),�⇤⇤2
)� ⇡1
✓q1+�⇤⇤
2
/µc1
,�⇤⇤2
◆
⇡1
(�⇤1
,�⇤2
)� ⇡1
✓q1+�⇤⇤
2
/µc1
,�⇤⇤2
◆ .
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 25 / 61
Cooperation Profile Design for Medium ⇢ Regime (Cont.)
Proposition 4 (Medium ⇢ Regime)
In the repeated game under complete information, if ⇢th2
⇢ < ⇢th1
, both
platforms will always follow the cooperation profile below without
deviating to trigger punishment (�⇤1
,�⇤2
):
(
˜�1
(⇢), ˜�2
(⇢)) =
✓⇢�⇤
1
+ (1� ⇢)
s1 +
�⇤⇤2
µ
c1
�
vuut✓⇢�⇤
1
+ (1� ⇢)
s1 +
�⇤⇤2
µ
c1
◆2
�1 +
�⇤⇤2
µ
c1
,�⇤⇤2
◆.
�⇤⇤1
< ˜�1
(⇢) < �⇤1
.
˜�1
(⇢) decreases with ⇢ 2 [⇢th2
, ⇢th1
) and eventually
˜�1
(⇢) ! �⇤⇤1
.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 26 / 61
Cooperation Profile Design for Medium ⇢ Regime (Cont.)
Proposition 4 (Medium ⇢ Regime)
In the repeated game under complete information, if ⇢th2
⇢ < ⇢th1
, both
platforms will always follow the cooperation profile below without
deviating to trigger punishment (�⇤1
,�⇤2
):
(
˜�1
(⇢), ˜�2
(⇢)) =
✓⇢�⇤
1
+ (1� ⇢)
s1 +
�⇤⇤2
µ
c1
�
vuut✓⇢�⇤
1
+ (1� ⇢)
s1 +
�⇤⇤2
µ
c1
◆2
�1 +
�⇤⇤2
µ
c1
,�⇤⇤2
◆.
�⇤⇤1
< ˜�1
(⇢) < �⇤1
.
˜�1
(⇢) decreases with ⇢ 2 [⇢th2
, ⇢th1
) and eventually
˜�1
(⇢) ! �⇤⇤1
.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 26 / 61
Cooperation Profile Design for Small ⇢ Regime
What if ⇢ < min{⇢th1
, ⇢th2
} = ⇢th2
?
Neither platforms will follow the social optimizers (�⇤⇤1
,�⇤⇤2
).
We redesign (
˜�1
(⇢), ˜�1
(⇢)) jointly such that
⇧
1
(
˜�1
(⇢), ˜�2
(⇢)) = ˆ
⇧
1
(�⇤1
,�⇤2
) for platform 1 and
⇧
2
(
˜�1
(⇢), ˜�2
(⇢)) = ˆ
⇧
2
(�⇤1
,�⇤2
) for platform 2.
We jointly solve (
˜�1
(⇢), ˜�1
(⇢)) according to
⇢ =
⇡1
(
˜�1
(⇢), ˜�2
(⇢))� ⇡1
✓q1+
˜�2
(⇢)/µc1
, ˜�2
(⇢)
◆
⇡1
(�⇤1
,�⇤2
)� ⇡1
✓q1+
˜�2
(⇢)/µc1
, ˜�2
(⇢)
◆ ,
⇢ =
⇡2
(
˜�1
(⇢), ˜�2
(⇢))� ⇡2
✓˜�1
(⇢),q
1+
˜�1
(⇢)/µc2
◆
⇡2
(�⇤1
,�⇤2
)� ⇡2
✓˜�1
(⇢),q
1+
˜�1
(⇢)/µc2
◆ .
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 27 / 61
Cooperation Profile Design for Small ⇢ Regime
What if ⇢ < min{⇢th1
, ⇢th2
} = ⇢th2
?
Neither platforms will follow the social optimizers (�⇤⇤1
,�⇤⇤2
).
We redesign (
˜�1
(⇢), ˜�1
(⇢)) jointly such that
⇧
1
(
˜�1
(⇢), ˜�2
(⇢)) = ˆ
⇧
1
(�⇤1
,�⇤2
) for platform 1 and
⇧
2
(
˜�1
(⇢), ˜�2
(⇢)) = ˆ
⇧
2
(�⇤1
,�⇤2
) for platform 2.
We jointly solve (
˜�1
(⇢), ˜�1
(⇢)) according to
⇢ =
⇡1
(
˜�1
(⇢), ˜�2
(⇢))� ⇡1
✓q1+
˜�2
(⇢)/µc1
, ˜�2
(⇢)
◆
⇡1
(�⇤1
,�⇤2
)� ⇡1
✓q1+
˜�2
(⇢)/µc1
, ˜�2
(⇢)
◆ ,
⇢ =
⇡2
(
˜�1
(⇢), ˜�2
(⇢))� ⇡2
✓˜�1
(⇢),q
1+
˜�1
(⇢)/µc2
◆
⇡2
(�⇤1
,�⇤2
)� ⇡2
✓˜�1
(⇢),q
1+
˜�1
(⇢)/µc2
◆ .
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 27 / 61
Cooperation Profile Design for Small ⇢ Regime
What if ⇢ < min{⇢th1
, ⇢th2
} = ⇢th2
?
Neither platforms will follow the social optimizers (�⇤⇤1
,�⇤⇤2
).
We redesign (
˜�1
(⇢), ˜�1
(⇢)) jointly such that
⇧
1
(
˜�1
(⇢), ˜�2
(⇢)) = ˆ
⇧
1
(�⇤1
,�⇤2
) for platform 1 and
⇧
2
(
˜�1
(⇢), ˜�2
(⇢)) = ˆ
⇧
2
(�⇤1
,�⇤2
) for platform 2.
We jointly solve (
˜�1
(⇢), ˜�1
(⇢)) according to
⇢ =
⇡1
(
˜�1
(⇢), ˜�2
(⇢))� ⇡1
✓q1+
˜�2
(⇢)/µc1
, ˜�2
(⇢)
◆
⇡1
(�⇤1
,�⇤2
)� ⇡1
✓q1+
˜�2
(⇢)/µc1
, ˜�2
(⇢)
◆ ,
⇢ =
⇡2
(
˜�1
(⇢), ˜�2
(⇢))� ⇡2
✓˜�1
(⇢),q
1+
˜�1
(⇢)/µc2
◆
⇡2
(�⇤1
,�⇤2
)� ⇡2
✓˜�1
(⇢),q
1+
˜�1
(⇢)/µc2
◆ .
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 27 / 61
Cooperation Profile Design for Small ⇢ Regime
What if ⇢ < min{⇢th1
, ⇢th2
} = ⇢th2
?
Neither platforms will follow the social optimizers (�⇤⇤1
,�⇤⇤2
).
We redesign (
˜�1
(⇢), ˜�1
(⇢)) jointly such that
⇧
1
(
˜�1
(⇢), ˜�2
(⇢)) = ˆ
⇧
1
(�⇤1
,�⇤2
) for platform 1 and
⇧
2
(
˜�1
(⇢), ˜�2
(⇢)) = ˆ
⇧
2
(�⇤1
,�⇤2
) for platform 2.
We jointly solve (
˜�1
(⇢), ˜�1
(⇢)) according to
⇢ =
⇡1
(
˜�1
(⇢), ˜�2
(⇢))� ⇡1
✓q1+
˜�2
(⇢)/µc1
, ˜�2
(⇢)
◆
⇡1
(�⇤1
,�⇤2
)� ⇡1
✓q1+
˜�2
(⇢)/µc1
, ˜�2
(⇢)
◆ ,
⇢ =
⇡2
(
˜�1
(⇢), ˜�2
(⇢))� ⇡2
✓˜�1
(⇢),q
1+
˜�1
(⇢)/µc2
◆
⇡2
(�⇤1
,�⇤2
)� ⇡2
✓˜�1
(⇢),q
1+
˜�1
(⇢)/µc2
◆ .
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 27 / 61
Cooperation Profile Design for Small ⇢ Regime (Cont.)
Proposition 5 (Small ⇢ Regime)In the repeated game under complete information, if ⇢ < ⇢th
2
, the two platforms will always
follow the optimal cooperation profile (
˜�1
(⇢), ˜�2
(⇢)) below as unique solutions to
˜�1
(⇢)� ⇢�⇤1
� (1� ⇢)
vuut1 +
˜�2
(⇢)µ
c1
+
vuuut✓⇢�⇤
1
+ (1� ⇢)
vuut1 +
˜�2
(⇢)µ
c1
◆2
�1 +
˜�2
(⇢)µ
c1
= 0,
˜�2
(⇢)� ⇢�⇤2
� (1� ⇢)
vuut1 +
˜�1
(⇢)µ
c2
+
vuuut✓⇢�⇤
2
+ (1� ⇢)
vuut1 +
˜�1
(⇢)µ
c2
◆2
�1 +
˜�1
(⇢)µ
c2
= 0,
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 28 / 61
Cooperation Profile Design for Small ⇢ Regime (Cont.)
Proved properties:
�⇤⇤1
< ˜�1
(⇢) �⇤1
and �⇤⇤2
< ˜�2
(⇢) �⇤2
.
As ⇢ ! 0, (
˜�1
(⇢), ˜�2
(⇢)) approach competition equilibrium (�⇤1
,�⇤2
),
and the repeated game degenerates to one-shot static game.
As ⇢ increases in the low regime (0, ⇢th2
), both
˜�1
(⇢) and ˜�2
(⇢)decrease and the duopoly competition mitigates.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 29 / 61
Cooperation Profile Design for Small ⇢ Regime (Cont.)
Proved properties:
�⇤⇤1
< ˜�1
(⇢) �⇤1
and �⇤⇤2
< ˜�2
(⇢) �⇤2
.
As ⇢ ! 0, (
˜�1
(⇢), ˜�2
(⇢)) approach competition equilibrium (�⇤1
,�⇤2
),
and the repeated game degenerates to one-shot static game.
As ⇢ increases in the low regime (0, ⇢th2
), both
˜�1
(⇢) and ˜�2
(⇢)decrease and the duopoly competition mitigates.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 29 / 61
Cooperation Profile Design for Small ⇢ Regime (Cont.)
Proved properties:
�⇤⇤1
< ˜�1
(⇢) �⇤1
and �⇤⇤2
< ˜�2
(⇢) �⇤2
.
As ⇢ ! 0, (
˜�1
(⇢), ˜�2
(⇢)) approach competition equilibrium (�⇤1
,�⇤2
),
and the repeated game degenerates to one-shot static game.
As ⇢ increases in the low regime (0, ⇢th2
), both
˜�1
(⇢) and ˜�2
(⇢)decrease and the duopoly competition mitigates.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 29 / 61
Numerical Results
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Discount Factor
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
Co
op
era
tio
n P
rofile
fo
r 2
pla
tfo
rms
Low ⇢ regime: 0 - 0.3, Medium ⇢ regime: 0.3 - 0.7, High ⇢ regime: 0.7 - 1.
Cooperation profile (
˜�1
(⇢), ˜�2
(⇢)) decrease with ⇢ and converge to
social optimizers (�⇤⇤1
,�⇤⇤2
).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 30 / 61
Numerical Results (Cont.)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Discount Factor
1
1.1
1.2
1.3
1.4
1.5
Socia
l C
ost R
atio
*/
**
Fix c1
= 1, c2
= 1.5, and alter µ and ⇢.Compare the social costs under recommendation (
˜�1
(⇢), ˜�2
(⇢)) versussocial optimum (�⇤⇤
1
,�⇤⇤2
).
Social cost under our trigger-and-punishment mechanism
(
˜�1
(⇢), ˜�2
(⇢)) decreases with ⇢ and converges to social optimum.
Social cost ratio under (
˜�1
(⇢), ˜�2
(⇢)) decreases to 1 as µ increases.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 31 / 61
Numerical Results (Cont.)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Discount Factor
1
1.1
1.2
1.3
1.4
1.5
Socia
l C
ost R
atio
*/
**
Fix c1
= 1, c2
= 1.5, and alter µ and ⇢.Compare the social costs under recommendation (
˜�1
(⇢), ˜�2
(⇢)) versussocial optimum (�⇤⇤
1
,�⇤⇤2
).
Social cost under our trigger-and-punishment mechanism
(
˜�1
(⇢), ˜�2
(⇢)) decreases with ⇢ and converges to social optimum.
Social cost ratio under (
˜�1
(⇢), ˜�2
(⇢)) decreases to 1 as µ increases.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 31 / 61
Numerical Results (Cont.)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Discount Factor
1
1.1
1.2
1.3
1.4
1.5
Socia
l C
ost R
atio
*/
**
Fix c1
= 1, c2
= 1.5, and alter µ and ⇢.Compare the social costs under recommendation (
˜�1
(⇢), ˜�2
(⇢)) versussocial optimum (�⇤⇤
1
,�⇤⇤2
).
Social cost under our trigger-and-punishment mechanism
(
˜�1
(⇢), ˜�2
(⇢)) decreases with ⇢ and converges to social optimum.
Social cost ratio under (
˜�1
(⇢), ˜�2
(⇢)) decreases to 1 as µ increases.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 31 / 61
Incomplete Information Scenario
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 32 / 61
System model under one-sided incomplete information
Platform 1’s cost function when c1
= cH :
⇡1
(�1
(cH),�2
) =
�1
(cH) + �2
�1
(cH)
✓1
�1
(cH) + �2
+
1
µ
◆+ cH�1
(cH).
Platform 1’s cost function when c1
= cL:
⇡1
(�1
(cL),�2
) =
�1
(cL) + �2
�1
(cL)
✓1
�1
(cL) + �2
+
1
µ
◆+ cL�1
(cL).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 33 / 61
System model under one-sided incomplete information
Platform 1’s cost function when c1
= cH :
⇡1
(�1
(cH),�2
) =
�1
(cH) + �2
�1
(cH)
✓1
�1
(cH) + �2
+
1
µ
◆+ cH�1
(cH).
Platform 1’s cost function when c1
= cL:
⇡1
(�1
(cL),�2
) =
�1
(cL) + �2
�1
(cL)
✓1
�1
(cL) + �2
+
1
µ
◆+ cL�1
(cL).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 33 / 61
System model under one-sided incomplete information
Platform 1’s cost function when c1
= cH :
⇡1
(�1
(cH),�2
) =
�1
(cH) + �2
�1
(cH)
✓1
�1
(cH) + �2
+
1
µ
◆+ cH�1
(cH).
Platform 1’s cost function when c1
= cL:
⇡1
(�1
(cL),�2
) =
�1
(cL) + �2
�1
(cL)
✓1
�1
(cL) + �2
+
1
µ
◆+ cL�1
(cL).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 33 / 61
System Model under Incomplete Information
Indi↵erent between cH and cL instances, platform 2’s cost function:
⇡2
((�1
(cH),�1
(cL)),�2
) = pH ·✓�1
(cH) + �2
�2
✓1
�1
(cH) + �2
+
1
µ
◆◆
+ (1� pH) ·✓�1
(cL) + �2
�2
✓1
�1
(cL) + �2
+
1
µ
◆◆+ c
2
�2
.
Social cost function in average sense:
⇡((�1
(cH),�1
(cL)),�2
) =pH⇡1
(�1
(cH),�2
) + (1� pH)⇡1
(�1
(cL),�2
)
+ ⇡2
((�1
(cH),�1
(cL)),�2
).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 34 / 61
System Model under Incomplete Information
Indi↵erent between cH and cL instances, platform 2’s cost function:
⇡2
((�1
(cH),�1
(cL)),�2
) = pH ·✓�1
(cH) + �2
�2
✓1
�1
(cH) + �2
+
1
µ
◆◆
+ (1� pH) ·✓�1
(cL) + �2
�2
✓1
�1
(cL) + �2
+
1
µ
◆◆+ c
2
�2
.
Social cost function in average sense:
⇡((�1
(cH),�1
(cL)),�2
) =pH⇡1
(�1
(cH),�2
) + (1� pH)⇡1
(�1
(cL),�2
)
+ ⇡2
((�1
(cH),�1
(cL)),�2
).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 34 / 61
Bayesian Game under Incomplete Information
Non-cooperative Bayesian game with equilibrium
((�⇤1
(cH),�⇤1
(cL)),�⇤2
):
min
�1
(cH)>0
⇡1
(�1
(cH),�2
)
min
�1
(cL)>0
⇡1
(�1
(cL),�2
)
min
�2
>0
⇡2
((�1
(cH),�1
(cL)),�2
)
Min-social-cost problem with social optimizers
((�⇤⇤1
(cH),�⇤⇤1
(cL)),�⇤⇤2
):
min
�1
(cH),�1
(cL),�2
>0
⇡((�1
(cH),�1
(cL)),�2
)
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 35 / 61
Bayesian Game under Incomplete Information
Non-cooperative Bayesian game with equilibrium
((�⇤1
(cH),�⇤1
(cL)),�⇤2
):
min
�1
(cH)>0
⇡1
(�1
(cH),�2
)
min
�1
(cL)>0
⇡1
(�1
(cL),�2
)
min
�2
>0
⇡2
((�1
(cH),�1
(cL)),�2
)
Min-social-cost problem with social optimizers
((�⇤⇤1
(cH),�⇤⇤1
(cL)),�⇤⇤2
):
min
�1
(cH),�1
(cL),�2
>0
⇡((�1
(cH),�1
(cL)),�2
)
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 35 / 61
Competition Equilibrium and Social Optimizers
Proposition 6 (Equilibrium vs social optimizers under incomplete information)
The competition equilibrium ((�⇤1
(cH),�⇤1
(cL)),�⇤2
) are the unique solutions to
�1
�2
1
(cH )
✓1 +
�2
µ
◆+ cH = 0,
�1
�2
1
(cL)
✓1 +
�2
µ
◆+ cL = 0,
�pH
�2
2
✓1 +
�1
(cH )
µ
◆�
1 � pH
�2
2
✓1 +
�1
(cL)
µ
◆+ c
2
= 0.
Social optimizers ((�⇤⇤1
(cH),�⇤⇤1
(cL)),�⇤⇤2
) are the unique solutions to
�1
�2
1
(cH )
✓1 +
�2
µ
◆+ cH +
1
�2
µ= 0,
�1
�2
1
(cL)
✓1 +
�2
µ
◆+ cL +
1
�2
µ= 0,
pH
✓�
1
�2
2
✓1 +
�1
(cH )
µ
◆+ c
2
+
1
�1
(cH )µ
◆+ (1 � pH )
✓�
1
�2
2
✓1 +
�1
(cL)
µ
◆+ c
2
+
1
�1
(cL)µ
◆= 0.
Both platforms will over-sample at equilibrium, i.e., �⇤1
(cH) � �⇤⇤1
(cH),
�⇤1
(cL) � �⇤⇤1
(cL) and �⇤2
� �⇤⇤2
. Additionally, �⇤1
(cH)/�⇤1
(cL) =pcL/cH .
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 36 / 61
Ine�ciency of Competition Equilibrium
Price of Anarchy (PoA):
PoA = max
cL,cH ,c2,µ,pH
⇡((�⇤1
(cH),�⇤1
(cL)),�⇤2
)
⇡((�⇤⇤1
(cH),�⇤⇤1
(cL)),�⇤⇤2
)
� 1.
Proposition 7
Price of anarchy under incomplete information is PoA = 1, which is
achieved when platform 1’s smaller sampling cost cL is infinitesimal.
cL ! 0, �⇤1
(cL) ! 1, �⇤2
! 1.
((�⇤⇤1
(cH),�⇤⇤1
(cL)),�⇤⇤2
) < 1.
Need non-monetary mechanism to remedy huge e�ciency loss!
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 37 / 61
Ine�ciency of Competition Equilibrium
Price of Anarchy (PoA):
PoA = max
cL,cH ,c2,µ,pH
⇡((�⇤1
(cH),�⇤1
(cL)),�⇤2
)
⇡((�⇤⇤1
(cH),�⇤⇤1
(cL)),�⇤⇤2
)
� 1.
Proposition 7
Price of anarchy under incomplete information is PoA = 1, which is
achieved when platform 1’s smaller sampling cost cL is infinitesimal.
cL ! 0, �⇤1
(cL) ! 1, �⇤2
! 1.
((�⇤⇤1
(cH),�⇤⇤1
(cL)),�⇤⇤2
) < 1.
Need non-monetary mechanism to remedy huge e�ciency loss!
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 37 / 61
Ine�ciency of Competition Equilibrium
Price of Anarchy (PoA):
PoA = max
cL,cH ,c2,µ,pH
⇡((�⇤1
(cH),�⇤1
(cL)),�⇤2
)
⇡((�⇤⇤1
(cH),�⇤⇤1
(cL)),�⇤⇤2
)
� 1.
Proposition 7
Price of anarchy under incomplete information is PoA = 1, which is
achieved when platform 1’s smaller sampling cost cL is infinitesimal.
cL ! 0, �⇤1
(cL) ! 1, �⇤2
! 1.
((�⇤⇤1
(cH),�⇤⇤1
(cL)),�⇤⇤2
) < 1.
Need non-monetary mechanism to remedy huge e�ciency loss!
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 37 / 61
Ine�ciency of Competition Equilibrium
Price of Anarchy (PoA):
PoA = max
cL,cH ,c2,µ,pH
⇡((�⇤1
(cH),�⇤1
(cL)),�⇤2
)
⇡((�⇤⇤1
(cH),�⇤⇤1
(cL)),�⇤⇤2
)
� 1.
Proposition 7
Price of anarchy under incomplete information is PoA = 1, which is
achieved when platform 1’s smaller sampling cost cL is infinitesimal.
cL ! 0, �⇤1
(cL) ! 1, �⇤2
! 1.
((�⇤⇤1
(cH),�⇤⇤1
(cL)),�⇤⇤2
) < 1.
Need non-monetary mechanism to remedy huge e�ciency loss!
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 37 / 61
Ine�ciency of Competition Equilibrium
Price of Anarchy (PoA):
PoA = max
cL,cH ,c2,µ,pH
⇡((�⇤1
(cH),�⇤1
(cL)),�⇤2
)
⇡((�⇤⇤1
(cH),�⇤⇤1
(cL)),�⇤⇤2
)
� 1.
Proposition 7
Price of anarchy under incomplete information is PoA = 1, which is
achieved when platform 1’s smaller sampling cost cL is infinitesimal.
cL ! 0, �⇤1
(cL) ! 1, �⇤2
! 1.
((�⇤⇤1
(cH),�⇤⇤1
(cL)),�⇤⇤2
) < 1.
Need non-monetary mechanism to remedy huge e�ciency loss!
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 37 / 61
Hurt with More Information for Platform 1
Question: Does platform 1 take advantage from knowing more information
about the sampling costs of both platforms?
Answer: Not exactly even in average sense!
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 38 / 61
Hurt with More Information for Platform 1
Question: Does platform 1 take advantage from knowing more information
about the sampling costs of both platforms?
Answer: Not exactly even in average sense!
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 38 / 61
Hurt with More Information for Platform 1 (Cont.)
Proposition 8
Under incomplete information, the cost objective of platform 1 under each
c1
= cH realization is greater than that under complete information, and
becomes smaller under each c1
= cL realization.
Perhaps surprisingly, its average cost
pH⇡1(�⇤1
(cH),�⇤2
) + (1� pH)⇡1(�⇤1
(cL),�⇤2
) becomes greater once
pH �pcL(
p1 +
¯�2
(cL)/µ�p1 + �⇤
2
/µ)pcL(
p1 +
¯�2
(cL)/µ�p
1 + �⇤2
/µ) +pcH(
p1 + �⇤
2
/µ�p
1 +
¯�2
(cH)/µ)
Platform 2 cannot identify c1
= cH or c1
= cL, and its over-sampling
when c1
= cH forces platform 1 to over-sample.
When pH is large, this happens more often and platform 1 loses in
average sense.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 39 / 61
Hurt with More Information for Platform 1 (Cont.)
Proposition 8
Under incomplete information, the cost objective of platform 1 under each
c1
= cH realization is greater than that under complete information, and
becomes smaller under each c1
= cL realization.
Perhaps surprisingly, its average cost
pH⇡1(�⇤1
(cH),�⇤2
) + (1� pH)⇡1(�⇤1
(cL),�⇤2
) becomes greater once
pH �pcL(
p1 +
¯�2
(cL)/µ�p1 + �⇤
2
/µ)pcL(
p1 +
¯�2
(cL)/µ�p
1 + �⇤2
/µ) +pcH(
p1 + �⇤
2
/µ�p
1 +
¯�2
(cH)/µ)
Platform 2 cannot identify c1
= cH or c1
= cL, and its over-sampling
when c1
= cH forces platform 1 to over-sample.
When pH is large, this happens more often and platform 1 loses in
average sense.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 39 / 61
Hurt with More Information for Platform 1 (Cont.)
Proposition 8
Under incomplete information, the cost objective of platform 1 under each
c1
= cH realization is greater than that under complete information, and
becomes smaller under each c1
= cL realization.
Perhaps surprisingly, its average cost
pH⇡1(�⇤1
(cH),�⇤2
) + (1� pH)⇡1(�⇤1
(cL),�⇤2
) becomes greater once
pH �pcL(
p1 +
¯�2
(cL)/µ�p1 + �⇤
2
/µ)pcL(
p1 +
¯�2
(cL)/µ�p
1 + �⇤2
/µ) +pcH(
p1 + �⇤
2
/µ�p
1 +
¯�2
(cH)/µ)
Platform 2 cannot identify c1
= cH or c1
= cL, and its over-sampling
when c1
= cH forces platform 1 to over-sample.
When pH is large, this happens more often and platform 1 loses in
average sense.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 39 / 61
Hurt with More Information for Platform 1 (Cont.)
Proposition 8
Under incomplete information, the cost objective of platform 1 under each
c1
= cH realization is greater than that under complete information, and
becomes smaller under each c1
= cL realization.
Perhaps surprisingly, its average cost
pH⇡1(�⇤1
(cH),�⇤2
) + (1� pH)⇡1(�⇤1
(cL),�⇤2
) becomes greater once
pH �pcL(
p1 +
¯�2
(cL)/µ�p1 + �⇤
2
/µ)pcL(
p1 +
¯�2
(cL)/µ�p
1 + �⇤2
/µ) +pcH(
p1 + �⇤
2
/µ�p
1 +
¯�2
(cH)/µ)
Platform 2 cannot identify c1
= cH or c1
= cL, and its over-sampling
when c1
= cH forces platform 1 to over-sample.
When pH is large, this happens more often and platform 1 loses in
average sense.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 39 / 61
Approximate Mechanism under Incomplete Information
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 40 / 61
New Challenge for Mechanism Design under IncompleteInformation
Even if ⇢ is large enough, can we still use social optimizers
((�⇤⇤1
(cL),�⇤⇤1
(cH)),�⇤⇤2
) as in complete information scenario?
Lemma 5.1.
Given the cooperation profile (�⇤⇤1
(cL),�⇤⇤1
(cH)) for platform 1 under
su�ciently large ⇢, platform 1 will not deviate from �⇤⇤1
(cL) when c1
= cLbut may deviate from �⇤⇤
1
(cH) when c1
= cH .
Platform 2 over-samples when c1
= cH compared to complete
information scenario.
Platform 1 benefits from choosing �⇤⇤1
(cL) and sampling more than
�⇤⇤1
(cH).
When sampling according to larger �⇤⇤1
(cL), he will not be caught.
Need to design new ((
˜�1
(cH , ⇢), ˜�1
(cL, ⇢)), ˜�2
(⇢)) even for large ⇢ regime!
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 41 / 61
New Challenge for Mechanism Design under IncompleteInformation
Even if ⇢ is large enough, can we still use social optimizers
((�⇤⇤1
(cL),�⇤⇤1
(cH)),�⇤⇤2
) as in complete information scenario?
Lemma 5.1.
Given the cooperation profile (�⇤⇤1
(cL),�⇤⇤1
(cH)) for platform 1 under
su�ciently large ⇢, platform 1 will not deviate from �⇤⇤1
(cL) when c1
= cLbut may deviate from �⇤⇤
1
(cH) when c1
= cH .
Platform 2 over-samples when c1
= cH compared to complete
information scenario.
Platform 1 benefits from choosing �⇤⇤1
(cL) and sampling more than
�⇤⇤1
(cH).
When sampling according to larger �⇤⇤1
(cL), he will not be caught.
Need to design new ((
˜�1
(cH , ⇢), ˜�1
(cL, ⇢)), ˜�2
(⇢)) even for large ⇢ regime!
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 41 / 61
New Challenge for Mechanism Design under IncompleteInformation
Even if ⇢ is large enough, can we still use social optimizers
((�⇤⇤1
(cL),�⇤⇤1
(cH)),�⇤⇤2
) as in complete information scenario?
Lemma 5.1.
Given the cooperation profile (�⇤⇤1
(cL),�⇤⇤1
(cH)) for platform 1 under
su�ciently large ⇢, platform 1 will not deviate from �⇤⇤1
(cL) when c1
= cLbut may deviate from �⇤⇤
1
(cH) when c1
= cH .
Platform 2 over-samples when c1
= cH compared to complete
information scenario.
Platform 1 benefits from choosing �⇤⇤1
(cL) and sampling more than
�⇤⇤1
(cH).
When sampling according to larger �⇤⇤1
(cL), he will not be caught.
Need to design new ((
˜�1
(cH , ⇢), ˜�1
(cL, ⇢)), ˜�2
(⇢)) even for large ⇢ regime!
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 41 / 61
New Challenge for Mechanism Design under IncompleteInformation
Even if ⇢ is large enough, can we still use social optimizers
((�⇤⇤1
(cL),�⇤⇤1
(cH)),�⇤⇤2
) as in complete information scenario?
Lemma 5.1.
Given the cooperation profile (�⇤⇤1
(cL),�⇤⇤1
(cH)) for platform 1 under
su�ciently large ⇢, platform 1 will not deviate from �⇤⇤1
(cL) when c1
= cLbut may deviate from �⇤⇤
1
(cH) when c1
= cH .
Platform 2 over-samples when c1
= cH compared to complete
information scenario.
Platform 1 benefits from choosing �⇤⇤1
(cL) and sampling more than
�⇤⇤1
(cH).
When sampling according to larger �⇤⇤1
(cL), he will not be caught.
Need to design new ((
˜�1
(cH , ⇢), ˜�1
(cL, ⇢)), ˜�2
(⇢)) even for large ⇢ regime!
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 41 / 61
New Challenge for Mechanism Design under IncompleteInformation
Even if ⇢ is large enough, can we still use social optimizers
((�⇤⇤1
(cL),�⇤⇤1
(cH)),�⇤⇤2
) as in complete information scenario?
Lemma 5.1.
Given the cooperation profile (�⇤⇤1
(cL),�⇤⇤1
(cH)) for platform 1 under
su�ciently large ⇢, platform 1 will not deviate from �⇤⇤1
(cL) when c1
= cLbut may deviate from �⇤⇤
1
(cH) when c1
= cH .
Platform 2 over-samples when c1
= cH compared to complete
information scenario.
Platform 1 benefits from choosing �⇤⇤1
(cL) and sampling more than
�⇤⇤1
(cH).
When sampling according to larger �⇤⇤1
(cL), he will not be caught.
Need to design new ((
˜�1
(cH , ⇢), ˜�1
(cL, ⇢)), ˜�2
(⇢)) even for large ⇢ regime!
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 41 / 61
Approximate Profile Design under Incomplete Information
New idea: recommend platform 1 to behave indi↵erently no matter
c1
= cH or c1
= cL. That is, ˜�1
(cH , ⇢) = ˜�1
(cL, ⇢) = ˜�1
(⇢).
Definition 2 (Approximate trigger mechanism of punishment underincomplete information)
In each round, two platforms follow approximate cooperation profile(
˜�1
(⇢), ˜�2
(⇢)) if neither was detected to deviate from tits profile inthe past.
Once a deviation was found in the past, the two platforms will keepplaying the equilibrium punishment profile ((�⇤
1
(cH), �⇤1
(cL)), �⇤2
)
forever.
What is the best design for (
˜�1
(⇢), ˜�2
(⇢))?
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 42 / 61
Approximate Profile Design under Incomplete Information
New idea: recommend platform 1 to behave indi↵erently no matter
c1
= cH or c1
= cL. That is, ˜�1
(cH , ⇢) = ˜�1
(cL, ⇢) = ˜�1
(⇢).
Definition 2 (Approximate trigger mechanism of punishment underincomplete information)
In each round, two platforms follow approximate cooperation profile(
˜�1
(⇢), ˜�2
(⇢)) if neither was detected to deviate from tits profile inthe past.
Once a deviation was found in the past, the two platforms will keepplaying the equilibrium punishment profile ((�⇤
1
(cH), �⇤1
(cL)), �⇤2
)
forever.
What is the best design for (
˜�1
(⇢), ˜�2
(⇢))?
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 42 / 61
Approximate Profile Design under Incomplete Information
New idea: recommend platform 1 to behave indi↵erently no matter
c1
= cH or c1
= cL. That is, ˜�1
(cH , ⇢) = ˜�1
(cL, ⇢) = ˜�1
(⇢).
Definition 2 (Approximate trigger mechanism of punishment underincomplete information)
In each round, two platforms follow approximate cooperation profile(
˜�1
(⇢), ˜�2
(⇢)) if neither was detected to deviate from tits profile inthe past.
Once a deviation was found in the past, the two platforms will keepplaying the equilibrium punishment profile ((�⇤
1
(cH), �⇤1
(cL)), �⇤2
)
forever.
What is the best design for (
˜�1
(⇢), ˜�2
(⇢))?
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 42 / 61
Approximate Cooperation Profile
New min-social cost problem with optimal profile (
˜�1
(⇢), ˜�2
(⇢)):
min
�1
(cH ,⇢),�1
(cL,⇢),�2
(⇢)>0
⇡((�1
(cH , ⇢),�1
(cL, ⇢)),�2
(⇢))
s.t. �1
(cH , ⇢) = �1
(cL, ⇢) := ˜�1
(⇢)
Optimal approximate profile:
˜�1
(⇢) =
vuut 1 +
˜�2
(⇢)/µ
pHcH + (1� pH)cL +1
˜�2
(⇢)µ
,
˜�2
(⇢) =
vuut1 +
˜�1
(⇢)/µ
c2
+
1
˜�1
(⇢)µ
,
which proves to provide at most 2-approximation of minimum social cost.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 43 / 61
Approximate Cooperation Profile
New min-social cost problem with optimal profile (
˜�1
(⇢), ˜�2
(⇢)):
min
�1
(cH ,⇢),�1
(cL,⇢),�2
(⇢)>0
⇡((�1
(cH , ⇢),�1
(cL, ⇢)),�2
(⇢))
s.t. �1
(cH , ⇢) = �1
(cL, ⇢) := ˜�1
(⇢)
Optimal approximate profile:
˜�1
(⇢) =
vuut 1 +
˜�2
(⇢)/µ
pHcH + (1� pH)cL +1
˜�2
(⇢)µ
,
˜�2
(⇢) =
vuut1 +
˜�1
(⇢)/µ
c2
+
1
˜�1
(⇢)µ
,
which proves to provide at most 2-approximation of minimum social cost.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 43 / 61
Approximate Mechanism Design under IncompleteInformation
Derive ⇢th1
and ⇢th2
similarly as under complete information.
Divide profile design into three di↵erent ⇢ regimes (low, medium and
high):
High ⇢ regime (⇢ � ⇢th1): both platforms follow optimal
recommendation.
Medium ⇢ regime (⇢th2 ⇢ < ⇢th1): only one platform follows optimal
recommendation.
Low ⇢ regime (⇢ < ⇢th2): neither follows optimal recommendation.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 44 / 61
Approximate Mechanism Design under IncompleteInformation
Derive ⇢th1
and ⇢th2
similarly as under complete information.
Divide profile design into three di↵erent ⇢ regimes (low, medium and
high):
High ⇢ regime (⇢ � ⇢th1): both platforms follow optimal
recommendation.
Medium ⇢ regime (⇢th2 ⇢ < ⇢th1): only one platform follows optimal
recommendation.
Low ⇢ regime (⇢ < ⇢th2): neither follows optimal recommendation.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 44 / 61
Numerical Results
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Discount Factor
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Co
op
era
tio
n P
rofile
fo
r 2
pla
tfo
rms
Low ⇢ regime: 0 - 0.3, Medium ⇢ regime: 0.3 - 0.7, High ⇢ regime: 0.7 - 1
Cooperation profile (
˜�1
(⇢), ˜�2
(⇢)) decrease with ⇢ and converge to
optimal recommended profile.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 45 / 61
Numerical Results
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Discount Factor
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Co
op
era
tio
n P
rofile
fo
r 2
pla
tfo
rms
Low ⇢ regime: 0 - 0.3, Medium ⇢ regime: 0.3 - 0.7, High ⇢ regime: 0.7 - 1
Cooperation profile (
˜�1
(⇢), ˜�2
(⇢)) decrease with ⇢ and converge to
optimal recommended profile.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 45 / 61
Conclusion
The first work to analyze tradeo↵ between AoI reduction and
sampling cost for online content platforms in the long run.
Competition between platforms when co-using content delivery
network can lead to huge e�ciency loss (PoA ! 1) under both
complete and incomplete info.
Under complete information, propose repeated games mechanism
with the threat of future punishment to enforce e�cient cooperation
under any discount factor.
Under incomplete information, propose approximate mechanism to
negate the platform with information advantage.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 46 / 61
Conclusion
The first work to analyze tradeo↵ between AoI reduction and
sampling cost for online content platforms in the long run.
Competition between platforms when co-using content delivery
network can lead to huge e�ciency loss (PoA ! 1) under both
complete and incomplete info.
Under complete information, propose repeated games mechanism
with the threat of future punishment to enforce e�cient cooperation
under any discount factor.
Under incomplete information, propose approximate mechanism to
negate the platform with information advantage.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 46 / 61
Conclusion
The first work to analyze tradeo↵ between AoI reduction and
sampling cost for online content platforms in the long run.
Competition between platforms when co-using content delivery
network can lead to huge e�ciency loss (PoA ! 1) under both
complete and incomplete info.
Under complete information, propose repeated games mechanism
with the threat of future punishment to enforce e�cient cooperation
under any discount factor.
Under incomplete information, propose approximate mechanism to
negate the platform with information advantage.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 46 / 61
Conclusion
The first work to analyze tradeo↵ between AoI reduction and
sampling cost for online content platforms in the long run.
Competition between platforms when co-using content delivery
network can lead to huge e�ciency loss (PoA ! 1) under both
complete and incomplete info.
Under complete information, propose repeated games mechanism
with the threat of future punishment to enforce e�cient cooperation
under any discount factor.
Under incomplete information, propose approximate mechanism to
negate the platform with information advantage.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 46 / 61
Part I: Competition regulation for AoI-driven platforms in long run
Long-term tradeo↵ between average AoI and sampling cost:
min�1
⇡1
(�1
,�2
) = �1
(�1
,�2
) + c1
�1
.
Part II: Dynamic pricing to control AoI in short-term
Require to examine AoI evolution over time t
Use age-dependent pricing p(t) instead of fixed c1
to motivatetime-varying sampling
Complex even for a single platform’s problem
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 47 / 61
Part I: Competition regulation for AoI-driven platforms in long run
Long-term tradeo↵ between average AoI and sampling cost:
min�1
⇡1
(�1
,�2
) = �1
(�1
,�2
) + c1
�1
.
Part II: Dynamic pricing to control AoI in short-term
Require to examine AoI evolution over time t
Use age-dependent pricing p(t) instead of fixed c1
to motivatetime-varying sampling
Complex even for a single platform’s problem
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 47 / 61
System model: dynamic pricing to control AoI
A price p(t) is announced at time t to motivate a user arrival to
sample and contribute information in a discrete time horizon.
A user may arrive randomly (s(t) = 1) with probability ↵ in this time
slot and (if so) he further decides to sample or not based on the price
p(t) and its private sampling cost ⇡ ⇠ F (·).The sensor data (e.g., about tra�c and road condition) is transmitted
with fixed delay A0
to finally post in the platform.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 48 / 61
System model: dynamic pricing to control AoI
A price p(t) is announced at time t to motivate a user arrival to
sample and contribute information in a discrete time horizon.
A user may arrive randomly (s(t) = 1) with probability ↵ in this time
slot and (if so) he further decides to sample or not based on the price
p(t) and its private sampling cost ⇡ ⇠ F (·).
The sensor data (e.g., about tra�c and road condition) is transmitted
with fixed delay A0
to finally post in the platform.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 48 / 61
System model: dynamic pricing to control AoI
A price p(t) is announced at time t to motivate a user arrival to
sample and contribute information in a discrete time horizon.
A user may arrive randomly (s(t) = 1) with probability ↵ in this time
slot and (if so) he further decides to sample or not based on the price
p(t) and its private sampling cost ⇡ ⇠ F (·).The sensor data (e.g., about tra�c and road condition) is transmitted
with fixed delay A0
to finally post in the platform.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 48 / 61
System model: AoI dynamics over time
The probability that a user will appear and accept the price is
Pr(s(t) = 1)Pr(⇡ < p(t)) = ↵F (p(t)) to reduce A(t + 1) to A0
,
otherwise, age increases by 1 unit of time.
E.g., for uniform cost distribution ⇡ ⇠ [0, b], update probability is
↵F (p(t)) = ↵p(t)b and average payment is
↵p2(t)b .
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 49 / 61
System model: AoI dynamics over time
The probability that a user will appear and accept the price is
Pr(s(t) = 1)Pr(⇡ < p(t)) = ↵F (p(t)) to reduce A(t + 1) to A0
,
otherwise, age increases by 1 unit of time.
E.g., for uniform cost distribution ⇡ ⇠ [0, b], update probability is
↵F (p(t)) = ↵p(t)b and average payment is
↵p2(t)b .
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 49 / 61
Dynamic pricing problem formulation
Dynamic of expected age at time t + 1 is
The platform’s problem formulation to tradeo↵ between AoI and
sample cost in T time slots is
where ⇢ is the discount factor.
This is a constrained nonlinear dynamic program, challenging to solve
analytically due to curse of dimensionality. Complexity is O((
b✏ )
T).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 50 / 61
Dynamic pricing problem formulation
Dynamic of expected age at time t + 1 is
The platform’s problem formulation to tradeo↵ between AoI and
sample cost in T time slots is
where ⇢ is the discount factor.
This is a constrained nonlinear dynamic program, challenging to solve
analytically due to curse of dimensionality. Complexity is O((
b✏ )
T).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 50 / 61
Dynamic pricing problem formulation
Dynamic of expected age at time t + 1 is
The platform’s problem formulation to tradeo↵ between AoI and
sample cost in T time slots is
where ⇢ is the discount factor.
This is a constrained nonlinear dynamic program, challenging to solve
analytically due to curse of dimensionality. Complexity is O((
b✏ )
T).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 50 / 61
Approximate Dynamic pricing
We reformulate the nonlinear age dynamic into linear term:
where � is a time-average term to approximate the dynamic age
reduction A(t)� A0
over time.
We can show the best estimator of � is
� =
1� ⇢
1� ⇢T
T�1X
t=0
⇢t(A(t)� A0
),
a↵ected by all A(t) over any time t, which will in turn a↵ect A(t).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 51 / 61
Approximate Dynamic pricing
We reformulate the nonlinear age dynamic into linear term:
where � is a time-average term to approximate the dynamic age
reduction A(t)� A0
over time.
We can show the best estimator of � is
� =
1� ⇢
1� ⇢T
T�1X
t=0
⇢t(A(t)� A0
),
a↵ected by all A(t) over any time t, which will in turn a↵ect A(t).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 51 / 61
Approximate Dynamic pricing
We reformulate the nonlinear age dynamic into linear term:
where � is a time-average term to approximate the dynamic age
reduction A(t)� A0
over time.
We can show the best estimator of � is
� =
1� ⇢
1� ⇢T
T�1X
t=0
⇢t(A(t)� A0
),
a↵ected by all A(t) over any time t, which will in turn a↵ect A(t).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 51 / 61
Approximate Dynamic pricing: Analysis of pricing
We want to obtain the structure of value function V (A(t), t) w.r.t. A(t):
Backward induction from time T : by minimizing A2
(T ) +
↵b p
2
(T ),
optimal price p(T ) = 0 without considering future age.
Back to time T � 1:
Iteratively, we can prove that optimal p(t) is a linear function of A(t)!Value function is in the quadratic structure of A(t):
V (A(t), t) = QtA2
(t) +MtA(t) + St , QT = 1,MT = 0, St = 0.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 52 / 61
Approximate Dynamic pricing: Analysis of pricing
We want to obtain the structure of value function V (A(t), t) w.r.t. A(t):
Backward induction from time T : by minimizing A2
(T ) +
↵b p
2
(T ),
optimal price p(T ) = 0 without considering future age.
Back to time T � 1:
Iteratively, we can prove that optimal p(t) is a linear function of A(t)!Value function is in the quadratic structure of A(t):
V (A(t), t) = QtA2
(t) +MtA(t) + St , QT = 1,MT = 0, St = 0.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 52 / 61
Approximate Dynamic pricing: Analysis of pricing
We want to obtain the structure of value function V (A(t), t) w.r.t. A(t):
Backward induction from time T : by minimizing A2
(T ) +
↵b p
2
(T ),
optimal price p(T ) = 0 without considering future age.
Back to time T � 1:
Iteratively, we can prove that optimal p(t) is a linear function of A(t)!Value function is in the quadratic structure of A(t):
V (A(t), t) = QtA2
(t) +MtA(t) + St , QT = 1,MT = 0, St = 0.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 52 / 61
Approximate Dynamic pricing: Analysis of pricing
We want to obtain the structure of value function V (A(t), t) w.r.t. A(t):
Backward induction from time T : by minimizing A2
(T ) +
↵b p
2
(T ),
optimal price p(T ) = 0 without considering future age.
Back to time T � 1:
Iteratively, we can prove that optimal p(t) is a linear function of A(t)!
Value function is in the quadratic structure of A(t):
V (A(t), t) = QtA2
(t) +MtA(t) + St , QT = 1,MT = 0, St = 0.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 52 / 61
Approximate Dynamic pricing: Analysis of pricing
We want to obtain the structure of value function V (A(t), t) w.r.t. A(t):
Backward induction from time T : by minimizing A2
(T ) +
↵b p
2
(T ),
optimal price p(T ) = 0 without considering future age.
Back to time T � 1:
Iteratively, we can prove that optimal p(t) is a linear function of A(t)!Value function is in the quadratic structure of A(t):
V (A(t), t) = QtA2
(t) +MtA(t) + St , QT = 1,MT = 0, St = 0.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 52 / 61
Approximate Dynamic pricing: Analysis of pricing
Optimal dynamic pricing p(t) is increasing in A(t), given by
where
The resulting expected age A(t) at time t is
which increases in initial age A(0).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 53 / 61
Approximate Dynamic pricing: Analysis of pricing
Optimal dynamic pricing p(t) is increasing in A(t), given by
where
The resulting expected age A(t) at time t is
which increases in initial age A(0).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 53 / 61
Approximate Dynamic pricing: Analysis of pricing
Optimal dynamic pricing p(t) is increasing in A(t), given by
where
The resulting expected age A(t) at time t is
which increases in initial age A(0).
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 53 / 61
Approximate Dynamic pricing: Simulations
Dynamic pricing p(t) first increases with A(t) until both reach
steady-states.
But when close to the end of time horizon T , price decreases to 0.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 54 / 61
Approximate Dynamic pricing in the long-run
As T ! 1, without recursive computing we further simplify p(t) to
where
How good is p1(t) as compared to optimal p(t) for finite time
horizon T?
Theorem: 8T > 0, there exists an ✏T > 0 such that
Uopt(T ) U1
(T ) =
TX
t=0
⇢t(A2
(t) +↵(p1(t))2
b) Uopt
(T ) + ✏T ,
and limT!1 ✏T = 0, i.e., p1(t) is asymptotically optimal.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 55 / 61
Approximate Dynamic pricing in the long-run
As T ! 1, without recursive computing we further simplify p(t) to
where
How good is p1(t) as compared to optimal p(t) for finite time
horizon T?
Theorem: 8T > 0, there exists an ✏T > 0 such that
Uopt(T ) U1
(T ) =
TX
t=0
⇢t(A2
(t) +↵(p1(t))2
b) Uopt
(T ) + ✏T ,
and limT!1 ✏T = 0, i.e., p1(t) is asymptotically optimal.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 55 / 61
Approximate Dynamic pricing in the long-run
As T ! 1, without recursive computing we further simplify p(t) to
where
How good is p1(t) as compared to optimal p(t) for finite time
horizon T?
Theorem: 8T > 0, there exists an ✏T > 0 such that
Uopt(T ) U1
(T ) =
TX
t=0
⇢t(A2
(t) +↵(p1(t))2
b) Uopt
(T ) + ✏T ,
and limT!1 ✏T = 0, i.e., p1(t) is asymptotically optimal.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 55 / 61
Approximate Dynamic pricing in the long-run
Theorem: 8T > 0, there exists an ✏T > 0 such that
Uopt(T ) U1
(T ) =
TX
t=0
⇢t(A2
(t) +↵(p1(t))2
b) Uopt
(T ) + ✏T ,
and limT!1 ✏T = 0, i.e., p1(t) achieves asymptotically optimal.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 56 / 61
Approximate Dynamic pricing: Analysis of Estimator �
Note that in Stage I, A(t) is a function of estimator �:
Our estimator � is a fixed point to equation:
� =
1� ⇢
1� ⇢T
T�1X
t=0
⇢t(A(t)� A0
),
The fixed point � exists according to Brouwer’s fixed-point theorem:
Every continuous function from a closed disk [A0
,A(0) + 1]⇥ · · ·⇥[A
0
,A(0) + (T � 1)] to itself has at least one fixed point.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 57 / 61
Approximate Dynamic pricing: Analysis of Estimator �
Note that in Stage I, A(t) is a function of estimator �:
Our estimator � is a fixed point to equation:
� =
1� ⇢
1� ⇢T
T�1X
t=0
⇢t(A(t)� A0
),
The fixed point � exists according to Brouwer’s fixed-point theorem:
Every continuous function from a closed disk [A0
,A(0) + 1]⇥ · · ·⇥[A
0
,A(0) + (T � 1)] to itself has at least one fixed point.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 57 / 61
Approximate Dynamic pricing: Analysis of Estimator �
Note that in Stage I, A(t) is a function of estimator �:
Our estimator � is a fixed point to equation:
� =
1� ⇢
1� ⇢T
T�1X
t=0
⇢t(A(t)� A0
),
The fixed point � exists according to Brouwer’s fixed-point theorem:
Every continuous function from a closed disk [A0
,A(0) + 1]⇥ · · ·⇥[A
0
,A(0) + (T � 1)] to itself has at least one fixed point.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 57 / 61
Conclusion of Part II: Dynamic pricing for controlling AoI
First work to tradeo↵ dynamic age A(t) and sampling cost
↵p2(t)b over
time.
Linearize the AoI evolution to simplify the constrained nonlinear
dynamic programming problem by using time-average estimator �.
Dynamic pricing under incomplete information: the content provider
should o↵er a high price p(t) if current age A(t) is high.
In the long run, the approximate dynamic pricing can be further
simplified to an ✏T -optimal pricing version without recursive
computing over time.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 58 / 61
Conclusion of Part II: Dynamic pricing for controlling AoI
First work to tradeo↵ dynamic age A(t) and sampling cost
↵p2(t)b over
time.
Linearize the AoI evolution to simplify the constrained nonlinear
dynamic programming problem by using time-average estimator �.
Dynamic pricing under incomplete information: the content provider
should o↵er a high price p(t) if current age A(t) is high.
In the long run, the approximate dynamic pricing can be further
simplified to an ✏T -optimal pricing version without recursive
computing over time.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 58 / 61
Conclusion of Part II: Dynamic pricing for controlling AoI
First work to tradeo↵ dynamic age A(t) and sampling cost
↵p2(t)b over
time.
Linearize the AoI evolution to simplify the constrained nonlinear
dynamic programming problem by using time-average estimator �.
Dynamic pricing under incomplete information: the content provider
should o↵er a high price p(t) if current age A(t) is high.
In the long run, the approximate dynamic pricing can be further
simplified to an ✏T -optimal pricing version without recursive
computing over time.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 58 / 61
Conclusion of Part II: Dynamic pricing for controlling AoI
First work to tradeo↵ dynamic age A(t) and sampling cost
↵p2(t)b over
time.
Linearize the AoI evolution to simplify the constrained nonlinear
dynamic programming problem by using time-average estimator �.
Dynamic pricing under incomplete information: the content provider
should o↵er a high price p(t) if current age A(t) is high.
In the long run, the approximate dynamic pricing can be further
simplified to an ✏T -optimal pricing version without recursive
computing over time.
Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 58 / 61
References
Maice Costa, Marian Codreanu, and Anthony Ephremides, “On the age ofinformation in status update systems with packet management.” IEEE Transactionson Information Theory 62, no. 4 (2016): 1897-1910.
Yu-Pin Hsu, Eytan Modiano, and Lingjie Duan, “Age of information: Design andanalysis of optimal scheduling algorithms.” In 2017 IEEE International Symposiumon Information Theory (ISIT), pp. 561-565. IEEE, 2017.
Longbo Huang, and Eytan Modiano, “Optimizing age-of-information in amulti-class queueing system.” In 2015 IEEE International Symposium onInformation Theory (ISIT), pp. 1681-1685. IEEE, 2015.
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Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 60 / 61
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Presenter: Lingjie Duan (SUTD) eco-AoI May 4, 2019 61 / 61
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